Quantum Groups and Their Representations

Anatoli Klimyk Konrad Schmtidgen

Quantum Groups and Their Representations

Springer

This book starts with an introduction to quantum groups for the beginner and continues as a textbook for graduate students in physics and mathematics. It can also be used as a reference by more advanced readers. The authors cover a large but well-chosen variety of subjects from the theory of quantum groups (quantized universal enveloping algebras, quantized algebras of functions) and q-deformed algebras(q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. The book is written with potential applications in physics and mathematics in mind. The basic quantum groups and quantum algebras and their representations are given in detail and accompanied by explicit formulas. A number of topics and recent results from the more advanced general theory are developed and discussed.

Preface

The invention of quantum groups is one of the outstanding achievements of mathematical physics and mathematics in the late twentieth century. The birth of the new theory and its rapid development are results of a strong interrelation between mathematics and physics. Quantum groups arose in the work of L.D. Faddeev and the Leningrad school on the inverse scattering method in order to solve integrable models. The algebra Uq (512 ) appeared first in 1981 in a paper by P.P. Kulish and N.Yu. Reshetikhin on the study of integrable XYZ models with highest spin. Its Hopf algebra structure was discovered later by E.K. Sklyanin. A major event was the discovery by V.G. Drinfeld and M. Jimbo around 1985 of a class of Hopf algebras which can be considered as one-parameter deformations of universal enveloping algebras of semisimple complex Lie algebras. These Hopf algebras will be called Drinfeld—Jimbo algebras in this book. Almost simultaneously, S.L. Woronowicz invented the quantum group SUq (2) and developed his theory of compact quantum matrix groups. An algebraic approach to quantized coordinate algebras was given about this time by Yu.I. Manin. A striking feature of quantum group theory is the surprising connections with many, sometimes at first glance unrelated, branches of mathematics and physics. There are links with mathematical fields such as Lie groups, Lie algebras and their representations, special functions, knot theory, low-dimensional topology, operator algebras, noncommutative geometry, and combinatorics. On the physical side there are interrelations with the quantum inverse scattering method, the theory of integrable models, elementary particle physics, conformal and quantum field theories, and others. It is expected that quantum groups will lead to a deeper understanding of the concept of symmetry in physics. Currently there is no satisfactory general definition of a quantum group. It is commonly accepted that quantum groups are certain "nice" Hopf algebras and that the standard deformations of the enveloping Hopf algebras of semisimple Lie algebras and of coordinate Hopf algebras of the corresponding Lie groups are guiding eiamples. Instead of searching for a rigorous definition of a quantum group it seems to be more fruitful to look for classes of Hopf algebras that give rise to a rich theory with important applications and con-

VI

Preface

tain enough interesting examples. In this book at least three such classes are extensively studied: quasitriangular Hopf algebras, coquasitriangular Hopf algebras, and compact quantum group algebras. The aim of this book is to provide a treatment of the theory of quantum algebras (quantized universal enveloping algebras), quantum groups (quantized algebras of functions), their representations and corepresentations, and the noncommutative differential calculus on quantum groups. The exposition is organized such that different parts of the text can be read and used (almost) independently of others. Sections 1.2 and 1.3 contain the main general definitions and notions on Hopf algebras needed in the text. This book is divided into four parts. Part I serves (among others) as an introduction to the theory of Hopf algebras, to the quantum algebra Uq (s12), the quantum group SL q (2), the q-oscillator algebra, and to their representations. The reader can use the corresponding chapters as first steps in order to learn the theory of quantum groups. A beginner might try to become aquainted with the language of Hopf algebras by reading Sect. 1.1 and portions of Sects. 1.2 and 1.3 and then passing immediately to Chaps. 3 or 4 (or start with Chaps. 3 or 4 and read parallel to them the relevant parts of Chap. 1). The main parts of the material of Chaps. 1, 3, and 4 can also be taken as a basis for an introductory course on quantum groups. Parts II—IV cover some of the more advanced topics of the theory. In Part II (quantized universal enveloping algebras).. and Part III (quantized algebras of functions) both fundamental approaches to quantum groups are developed in detail and as independently as possible, so readers interested in only one of these parts can restrict themselves to the corresponding chapters. Nevertheless the connections between both approaches appear to be very fruitful and instructive (see Sects. 4.4, 4.5.5, 9.4, 11.2.3, 11.5, and 11.6.6). A reader who is interested in only noncommutative differential calculus should pass directly to Part IV of the book and begin with Chaps. 12 or 14 (of course, some knowledge about the corresponding quantum groups from Chap. 9 and the L-functionals from Subsect. 10.1.3 is still required there). Together with Sects. 1.2 and 1.3, Parts II—IV form an advanced text on quantum groups. Selected material from these parts can also be used for graduate courses or seminars on quantum groups. Moreover, a large number of explicit formulas and new material (for instance, in Sects. 8.5, 10.1.3, 10.3.1, 13.2, and 14.3-5) are provided throughout the text, so we hope the book may be useful for experts as well. Let us say a few words about the selected topics and the presentation in the book. Our objective in choosing the material was to cover important and useful tools and methods for (possible) applications in theoretical and mathematical physics (especially in representation theory and in noncommutative differential calculus). Of course, this depends on our personal view of the matter. We have tried to give a comprehensive treatment of the chosen

Preface

VII

topics at the price of not including some concepts (for instance, the quantum Weyl group). Although we develop a number of general concepts too, the emphasis in the book is always placed on the study of concrete quantum groups and quantum algebras and their representations. Most of the results are presented with complete (but sometimes concise) proofs. Often, missing proofs or gaps in the existing literature have been filled. For some rather technical proofs (in particular of advanced algebraic results) readers are referred to the original papers. In many cases we have omitted proofs that are similar to the classical case. Having the potential reader in mind, we have avoided abstract mathematical theories whenever it was possible. For instance, we do not use cohomology theory, category theory (apart from Subsect. 10.3.4), Poisson—Lie groups, deformation theory, and knot theory in the book. We assume, however, that the reader has some standard knowledge of Lie groups and Lie algebras and their representation theory. The book is organized as follows. Formulas, results, definitions, examples, and remarks are numbered and quoted consecutively within the chapters. When a reference to an item in another chapter is made, the number of the chapter is added. For instance, (30) means formula (30) in the same chapter and Propostion 9.7 refers to Proposition 7 in Chap. 9. The end of a proof is marked by 11 and of an example or a remark by A. The reader should also notice that often assumptions are fixed and kept in force throughout the whole chapter, section, or subsection. Bibliographical comments are usually gathered at the end of each chapter. There the sources of some results or notions are cited (as far as the authors are aware) and some related references are listed, but no attempt has been made to report the origins of all items. We want to express our gratitude to A. Schiiler and I. Heckenberger for their indispensible help and valuable suggestions in writing this book and to Mrs. K. Schmidt for typing parts of the manuscript. We also thank Yu. Bespalov, A. Gavrilik, and L. Vainerman for reading parts of the book. Kiev and Leipzig, March 1997

A. U. Klimyk, K. Schmiidg en

Table of Contents

Part L An Introduction to Quantum Groups 1.

2.

Hopf Algebras 1.1 Prolog: Examples of Hopf Algebras of Functions on Groups 1.2 Coalgebras, Bialgebras and Hopf Algebras 1.2.1 Algebras 1.2.2 Coalgebras 1.2.3 Bialgebras 1.2.4 Hopf Algebras 1.2.5* Dual Pairings of Hopf Algebras 1.2.6 Examples of Hopf Algebras 1.2.7 *-Structures 1.2.8* The Dual Hopf Algebra A° 1.2.9* Super Hopf Algebras 1.2.10*h-Adic Hopf Algebras 1.3 Modules and Comodules of Hopf Algebras 1.3.1 Modules and Representations 1.3.2 Comodules and Corepresentations 1.3.3 Comodule Algebras and Related Concepts 1.3.4* Adjoint Actions and Coactions of Hopf Algebras 1.3.5* Corepresentations and Representations of Dually Paired Coalgebras and Algebras 1.4 Notes

3 3 6 6 8 11 13 16 18 20 22 23 25 27 27 29 32 34

q Calculus 2.1 Main Notions on q-Calculus 2.1.1 q-Numbers and q-Factorials 2.1.2 q-Binomial Coefficients 2.1.3 Basic Hypergeometric Functions 2.1.4 The 'Function 1(P0( 4 ; q Z) 2.1.5 The Basic Hypergeometric Function Zoi 2.1.6 Transformation Formulas for 0 2 and 4 (10 3 2.1.7 q-Analog of the Binomial Theorem 2.2 q-Differentiation and q-Integration 2.2.1 q-Differentiation

37 37 37 39 40 41 42 43 44 44 44

-

35 36

Table of Contents

X

2.2.2 g-Integral 2.2.3 q-Analog of the Exponential Function 2.2.4 g-Analog of the Gamma Function 2.3 g-Orthogonal Polynomials 2.3.1 Jacobi Matrices and Orthogonal Polynomials 2.3.2 g-Hermite Polynomials 2.3.3 Little q-Jacobi Polynomials 2.3.4 Big g-Jacobi Polynomials 2.4 Notes

3.

The Quantum Algebra Uq (512 ) and Its Representations 3.1

3.2

3.3

3.4

3.5

3.6

The Quantum Algebras Uq (s12) and Uh(s12) 3.1.1 The Algebra Uq (s12 ) 3.1.2 The Hopf Algebra Uq (s12 ) 3.1.3 The Classical Limit of the Hopf Algebra Uq (512) 3.1.4 Real Forms of the Quantum Algebra Uq (512) 3.1.5 The h-Adic Hopf Algebra Uh(s12) Finite-Dimensional Representations of Uq (512 ) for g not a Root of Unity 3.2.1 The Representations Tw i 3.2.2 Weight Representations and Complete Reducibility 3.2.3 Finite-Dimensional Representations of ("q.( s 1 2 ) and Uh(s12) Representations of Uq (s12 ) for g a Root of Unity 3.3.1 The Center of Uq (512 ) 3.3.2 Representations of Uq (512) 3.3.3 Representations of Uqre8 (s12) Tensor Products of Representations. Clebsch—Gordan Coefficients 3.4.1 Tensor Products of Representations Ti 3.4.2 Clebsch—Gordan Coefficients 3.4.3 Other Expressions for Clebsch—Gordan Coefficients 3.4.4 Symmetries of Clebsch—Gordan Coefficients Racah Coefficients and 6j Symbols of Uq (su2) 3.5.1 Definition of the Racah Coefficients 3.5.2 Relations Between Racah and Clebsch—Gordan Coefficients 3.5.3 Symmetry Relations 3.5.4 Calculation of Racah Coefficients 3.5.5 The Biedenharn—Elliott Identity 3.5.6 The Hexagon Relation 3.5.7 Clebsch—Gordan Coefficients as Limits of Racah Coefficients Tensor Operators and the Wigner—Eckart Theorem 3.6.1 Tensor Operators for Compact Lie Groups

46 47 48 49 49 50 51 52 52 • 53 53 53 55 57 58 60 61 61 • 63 65 66 66 67 71 72 72 74 • 78 81 82 82 84 84 85 88 90 90 92 92

Table of Contents

Tensor Operators and the Wigner—Eckart Theorem for Ù (su 2) Applications 3.7.1 The Uq (512 ) Rotator Model of Deformed Nuclei 3.7.2 Electromagnetic Transitions in the (4012) Model Notes

XI

3.6.2 3.7

3.8 4.

93 94 94 95 96

The Quantum Group SL q (2) and Its Representations 4.1 The Hopf Algebra 0(SL q (2)) 4.1.1 The Bialgebra 0(Mq (2)) 4.1.2 The Hopf Algebra 0(SL q (2)) 4.1.3 A Geometric Approach to SL q (2) 4.1.4 Real Forms of 0(841, (2)) 4.1.5 The Diamond Lemma 4.2 Representations of the Quantum Group SL q (2) 4.2.1 Finite-Dimensional Corepresentations of 0(S.L q (2)):

97 97 97 99 101 102 103 104

Main Results 4.2.2 A Decomposition of 0(S L q (2)) 4.2.3 Finite-Dimensional Subcomodules of 0(SL q (2)) 4.2.4 Calculation of the Matrix Coefficients 4.2.5 The Peter—Weyl Decomposition of 0(S.L q (2)) 4.2.6 The Haar Functional of 0(SL q (2)) 4.3 The Compact Quantum Group SUq (2) and Its Representations 4.3.1 Unitary Representations of the Quantum Group SUq (2) 4.3.2 The Haar State and the Peter—Weyl Theorem for 0(SUq (2)) 4.3.3 The Fourier Transform on SUq (2) 4.3.4 *-Representations and the C4-Algebra of 0(SUg (2)) 4.4 Duality of the Hopf Algebras Uq (512 ) and O(SL q (2)) 4.4.1 Dual Pairing of the Hopf Algebras Uq (s12) and 0(SL q (2)) 4.4.2 Corepresentations of 0(S L q (2)) and Representations of Uq (512) 4.5 Quantum 2-Spheres 4.5.1 A Family of Quantum Spaces for SL q (2) 4.5.2 Decomposition of the Algebra 0(4) 4.5.3 Spherical Functions on 51 4.5.4 An Infinitesimal Characterization of 0(SL) 4.6 Notes \

104 105 106 108 110 111 113 113 114 117 117 119 119 123 124 124 126 129 129 132

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XII

5.

The q-Oscillator Algebras and Their Representations 5.1 The g-Oscillator Algebras Aqc and Aq 5.1.1 Definitions and Algebraic Properties 5.1.2 Other Forms of the g-Oscillator Algebra 5.1.3 The g-Oscillator Algebra and the Quantum Algebra Û q (s12) 5.1.4 The g-Oscillator Algebras and the Quantum Space Mq 2 (2) 5.2

5.3

5.4

133 133 133 136 137 140 140 140

Representations of g-Oscillator Algebras 5.2.1 N-Finite Representations 5.2.2 Irreducible Representations 141 with Highest (Lowest) Weights 5.2.3 Representations Without Highest and Lowest Weights 143 5.2.4 Irreducible Representations of Acq 145 for q a Root of Unity 147 5.2.5 Irreducible *-Representations of and A g 5.2.6 Irreducible *-Representations 148 of Another g-Oscillator Algebra 149 The Fock Representation of the g-Oscillator Algebra 149 5.3.1 The Fock Representation 150 5.3.2 The Bargmann-Fock Realization 152 5.3.3 Coherent States 5.3.4 Bargmann-Fock Space Realization 153 of Irreducible Representations of Oq (s12) 154 Notes

Part II. Quantized Universal Enveloping Algebras 6.

Drinfeld-Jimbo Algebras 6.1 Definitions of Drinfeld-Jimbo Algebras 6.1.1 Semisimple Lie Algebras 6.1.2 The Drinfeld-Jimbo Algebras Uq (g) 6.1.3 The h-Adic Drinfeld-Jimbo Algebras Uh(9) 6.1.4 Some Algebra Autornorphisms of Drinfeld-Jimbo Algebras 6.1.5 Triangular Decomposition of Uq (g) 6.1.6 Hopf Algebra Automorphisms of Uq (g) 6.1.7 Real Forms of Drinfeld-Jimbo Algebras 6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules 6.2.1 Braid Groups 6.2.2 Action of Braid Groups on Drinfeld-Jimbo Algebras 6.2.3 Root Vectors and Poincaré-Birkhoff-Witt Theorem .. 6.2.4 Representations with Highest Weights 6.2.5 Verma Modules

157 157 157 161 165 167 168 171 172 173 173 174 175 177 179

Table of Contents XIII

6.3

6.4 7.

6.2.6 Irreducible Representations with Highest Weights 6.2.7 The Left Adjoint Action of Uq (g) The Quantum Killing Form and the Center of Uq (g) 6.3.1 A Dual Pairing of the Hopf Algebras Uq (b + ) and Uq (b_) °P 6.3.2 The Quantum Killing Form on Uq (g) 6.3.3 A Quantum Casimir Element 6.3.4 The Center of Uq (9) and the Harish-Chandra Homomorphism 6.3.5 The Center of Uq (9) for q a Root of Unity Notes

180 181 184

184 187 189 192 194 196

Finite-Dimensional Representations 197 of Drinfeld—Jimbo Algebras 7.1 General Properties of Finite-Dimensional Representations 197 of Uq (g) 197 7.1.1 Weight Structure and Classification 200 7.1.2 Properties of Representations 7.1.3 Representations of h-Adic Drinfeld—Jimbo Algebras. 202 7.1.4 Characters of Representations and Multiplicities 203 of Weights 204 7.1.5 Separation of Elements of Uq (g) 7.1.6 The Quantum Trace 205 of Finite-Dimensional Representations 207 7.2 Tensor Products of Representations 7.2.1 Multiplicities in Tensor Products of Representations 208 211 7.2.2 Clebsch—Gordan Coefficients 212 7.3 Representations of Ùq (g1) for q not a Root of Unity 212 7.3.1 The Hopf Algebra (4 (glr,) 213 7.3.2 Finite-Dimensional Representations of ej/q (g1n ) 7.3.3 Gel'fand—Tsetlin Bases and Explicit Formulas 214 for Representations 217 7.3.4 Representations of Class 1 7.3.5 Tensor Products of Representations 218 7.3.6 Tensor Operators and the Wigner—Eckart Theorem 219 7.3.7 Clebsch—Gordan Coefficients for the Tensor Product Tm 0 Ti 220 7.3.8 Clebsch—Gordan Coefficients for the Tensor Product Tm 0 Tp 221 7.3.9 The Tensor Product Tm 0 T1 for q± 1 0 224 7.4 Crystal Bases 225 7.4.1 Crystal Bases of Finite-Dimensional Modules 226 7.4.2 Existence and Uniqueness of Crystal Bases 227 7.4.3 Crystal Bases of Tensor Product Modules 228

XIV

Table of Contents

7.5

7.6 7.7

229 7.4.4 Globalization of Crystal Bases 230 7.4.5 Crystal Bases of U(n_) 232 Representations of Uq (g) for q a Root of Unity 232 7.5.1 General Results 234 7.5.2 Cyclic Representations 235 7.5.3 Cyclic Representations of the Algebra U,(sli+i ) 237 7.5.4 Representations of Minimal Dimensions 7.5.5 Representations of U E (sli +i ) in Gerfand—Tsetlin Bases 238 240 Applications 242 Notes

8. Quasitriangularity and Universal R-Matrices 8.1 Quasitriangular Hopf Algebras 8.1.1 Definition and Basic Properties 8.1.2 R-Matrices for Representations 8.1.3 Square and Inverse of the Antipode 8.2 The Quantum Double and Universal R-Matrices 8.2.1 The Quantum Double of Skew-Paired Bialgebras 8.2.2 Quasitriangularity of Quantum Doubles of Finite-Dimensional Hopf Algebras 8.2.3 The Rosso Form of the Quantum Double 8.2.4 Drinfeld—Jimbo Algebras as Quotients of Quantum Doubles 8.3 Explicit Form of Universal R-Matrices 8.3.1 The Universal R-Matrix for Uh (S12) 8.3.2 The Universal R-Matrix for U h(g) 8.3.3 R-Matrices for Representations of Uq (g) 8.4 Vector Representations and R-Matrices 8.4.1 Vector Representations of Drinfeld—Jimbo Algebras 8.4.2 R-Matrices for Vector Representations 8.4.3 Spectral Decompositions of R-M atr ices for Vector Representations 8.5 L-Operators and L-Functionals 8.5.1 L-Operators and L-Functionals 8.5.2 L-Functionals for Vector Representations 8.5.3 The Extended Hopf Algebras U(1'(9) 8.5.4 L-Functionals for Vector Representations of Uq (9) 8.5.5 The Hopf Algebras 1-1(R) and UilL (2) 8.6 An Analog of the Brauer—Schur—Weyl Duality 8.6.1 The Algebras fig (soN) 8.6.2 Tensor Products of Vector Representations 8.6.3 The Brauer—Schur—Weyl Duality for Drinfeld—Jimbo Algebras 8.6.4 Hecke and Birman—Wenzl—Murakami Algebras 8.7 Applications

243 243 243 246 247 250 250 254 257 258 259 259 261 264 267 267 269 272 275 275 277 281 283 285 288 288 289 291 293 294

Table of Contents

8.8

XV

8.7.1 Baxterization 8.7.2 Elliptic Solutions

295

of the Quantum Yang-Baxter Equation 8.7.3 R-Matrices and Integrable Systems Notes

297 298 300

Part III. Quantized Algebras of Functions 9.

Coordinate Algebras of Quantum Groups and Quantum Vector Spaces 9.1 The Approach of Faddeev-Reshetikhin-Takhtajan 9.1.1 The FRT Bialgebra A(R) 9.1.2 The Quantum Vector Spaces XL(f ; R) and XR(f ;R) 9.2 The Quantum Groups GL q (N) and SL q (N) 9.2.1 The Quantum Matrix Space Mq (N) and the Quantum Vector Space Clq'T 9.2.2 Quantum Determinants 9.2.3 The Quantum Groups GL q (N) and SL q (N) 9.2.4 Real Forms of GL q (N) and SL q (N) and *-Quantum Spaces 9.3 The Quantum Groups 0q (N) and Spq (N) 9.3.1 The Hopf Algebras 0(0q (N)) and 0(Sp q (N)) 9.3.2 The Quantum Vector Space for the Quantum Group 0q (N) 9.3.3 The Quantum Group SO q (N) 9.3.4 The Quantum Vector Space for the Quantum Group Spq (N) 9.3.5 Real Forms of 0q (N) and Spq (N) and *-Quantum Spaces 9.4 Dual Pairings of Drinfeld-Jimbo Algebras and Coordinate Hopf Algebras 9.5 Notes

10. Coquasitriangularity and Crossed Product Constructions 10.1 Coquasitriangular Hopf Algebras 10.1.1 Definition and Basic Properties 10.1.2 Coquasitriangularity of FRT Bialgebras A(R) and Coordinate Hopf Algebras O(C) 10.1.3 L-Functionals of Coquasitriangular Hopf Algebras 10.2 Crossed Product Constructions of Hopf Algebras 10.2.1 Crossed Product Algebras 10.2.2 Crossed Coproduct Coalgebras 10.2.3 Twisting of Algebra Structures by 2-Cocycles and Quantum Doubles

303 303 303 307 309 310 311 313 316 317 318 320 323 324 325 327 330 331 331 331 337 342 349 349 352 354

XVI Table of Contents

10.2.4 Twisting of Coalgebra Structures by 2-Cocycles and Quantum Codoubles 10.2.5 Double Crossed Product Bialgebras and Quantum Doubles 10.2.6 Double Crossed Coproduct Bialgebras and Quantum Codoubles 10.2.7 Realifications of Quantum Groups 10.3 Braided Hopf Algebras 10.3.1 Covariantized Products for Coquasitriangular Bialgebras 10.3.2 Braided Hopf Algebras Associated with Coquasitriangular Hopf Algebras • • 10.3.3 Braided Hopf Algebras Associated with Quasitriangular Hopf Algebras 10.3.4 Braided Tensor Categories and Braided Hopf Algebras 10.3.5 Braided Vector Algebras 10.3.6 Bosonization of Braided Hopf Algebras 10.3.7 *-Structures on Bosonized Hopf Algebras 10.3.8 Inhomogeneous Quantum Groups 10.3.9 *-Structures for Inhomogeneous Quantum Groups 10.4 Notes

357 359 362 363 365 365 370 376 377 380 382 386 388 390 394

11. Corepresentation Theory and Compact Quantum Groups. 395 11.1 Corepresentations of Hopf Algebras 395 11.1.1 Corepresentations 395

11.1.2 Intertwiners 11.1.3 Constructions of New Corepresentations 11.1.4 Irreducible Corepresentations 11.1.5 Unitary Corepresentations 11.2 Cosemisimple Hopf Algebras 11.2.1 Definition and Characterizations 11.2.2 The Haar Functional of a Cosemisimple Hopf Algebra 11.2.3 Peter—Weyl Decomposition of Coordinate Hopf Algebras 11.3 Compact Quantum Group Algebras 11.3.1 Definitions and Characterizations of CQG Algebras . . 11.3.2 The Haar State of a CQG Algebra 11.3.3 C*-Algebra Completions of CQG Algebras 11.3.4 Modular Properties of the Haar State 11.3.5 Polar Decomposition of the Antipode 11.3.6 Multiplicative Unitaries of CQG Algebras 11.4 Compact Quantum Group C*-Algebras 11.4.1 CQG C*-Algebras and Their CQG Algebras 11.4.2 Existence of the Haar State of a CQG C*-Algebra 11.4.3 Proof of Theorem 39

397 397 398 401 402 402 404 408 415 415 419 420 422 426 427 429 429 431 433

Table of Contents XVII

434 11.4.4 Another Definition of CQG C*-Algebras 435 11.5 Finite-Dimensional Representations of GL q (N) 435 11.5.1 Some Quantum Subgroups of GL q (N) 436 11.5.2 Submodules of Relative Invariant Elements 437 11.5.3 Irreducible Representations of GL g (N) 439 11.5.4 Peter—Weyl Decomposition of 0(GL q (N)) 441 11.5.5 Representations of the Quantum Group Uq (N) 442 11.6 Quantum Homogeneous Spaces 442 11.6.1 Definition of a Quantum Homogeneous Space 11.6.2 Quantum Homogeneous Spaces 443 Associated with Quantum Subgroups 445 11.6.3 Quantum Gerfand Pairs 11.6.4 The Quantum Homogeneous Space Uq (N-1)W q (N) . 447 11.6.5 Quantum Homogeneous Spaces 451 of Infinitesimally Invariant Elements 452 11.6.6 Quantum Projective Spaces 454 11.7 Notes

Part IV. Noncommutative Differential Calculus 12. Covariant Differential Calculus on Quantum Spaces 12.1 Covariant First Order Differential Calculus 12.1.1 First Order Differential Calculi on Algebras 12.1.2 Covariant First Order Calculi on Quantum Spaces 12.2 Covariant Higher Order Differential Calculus 12.2.1 Differential Calculi on Algebras 12.2.2 The Differential Envelope of an Algebra 12.2.3 Covariant Differential Calculi on Quantum Spaces. 12.3 Construction of Covariant Differential Calculi on Quantum Spaces 12.3.1 General Method 12.3.2 Covariant Differential Calculi on Quantum Vector Spaces 12.3.3 Covariant Differential Calculus on CqN and the Quantum Weyl Algebra 12.3.4 Covariant Differential Calculi on the Quantum Hyperboloid 12.4 Notes 13. Hopf Bimodules and Exterior Algebras 13.1 Covariant Bimodules 13.1.1 IA-Covariant Bimodules 13.1.2 Right-Covariant Bimodules 13.1.3 Dicovariant Bimodules (Hopf Birnodules)

457 457 457 459 461 461 462 463 464 464 467 468 471 472 473 473 473 477 477

XVIII Table of Contents 13.1.4 Woronowicz' Braiding of Bicovariant Bimodules 13.1.5 Bicovariant Bimodules and Representations

480

of the Quantum Double 13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules 13.2.1 The Tensor Algebra of a Bicovariant Bimodule 13.2.2 The Exterior Algebra of a Bicovariant Bimodule 13.3 Notes

483 485 485 488 490

491 14. Covariant Differential Calculus on Quantum Groups 491 14.1 Left-Covariant First Order Differential Calculi 14.1.1 Left-Covariant First Order Calculi 491 and Their Right Ideals 494 14.1.2 The Quantum Tangent Space 496 14.1.3 An Example: The 3D-Calculus on SL q (2) 14.1.4 Another Left-Covariant Differential Calculus 498 Oil SL q (2) 498 14.2 Bicovariant First Order Differential Calculi 498 14.2.1 Right-Covariant First Order Differential Calculi 499 14.2.2 Bicovariant First Order Differential Calculi 14.2.3 Quantum Lie Algebras 500 of Bicovariant First Order Calculi 504 14.2.4 The 4D+ - and the 4D_-Calculus on SL q (2) 14.2.5 Examples of Bicovariant First Order Calculi 505 on Simple Lie Groups 506 14.3 Higher Order Left-Covariant Differential Calculi 14.3.1 The Maurer-Cartan Formula 506 507 14.3.2 The Differential Envelope of a Hopf Algebra 14.3.3 The Universal DC of a Left-Covariant FODC 508 14.4 Higher Order Bicovariant Differential Calculi 511 14.4.1 Bicovariant Differential Calculi and Differential Hopf Algebras 511 14.4.2 Quantum Lie Derivatives and Contraction Operators 514 14.5 Bicovariant Differential Calculi on Coquasitriangular Hopf Algebras 517 14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups 521 14.6.1 A Family of Bicovariant First Order 521 Differential Calculi 14.6.2 Braiding and Structure Constants of the FODC F±,z 524 525 14.6.3 A Canonical Basis for the Left-Invariant 1-Forms 14.6.4 Classification of Bicovariant First Order 527 Differential Calculi 528 14.7 Notes

Table of Contents

XIX

Bibliography

529

Index

545

Part I

An Introduction to Quantum Groups

1. Hopf Algebras

The underlying mathematical notion for quantum groups is that of a Hopf algebra. These are associative algebras equipped with additional structures such as a comultiplication, a counit and an antipode. In some appropriate sense, these structures and their axioms reflect the multiplication, the unit element and the inverse elements of a group and their corresponding properties. The purpose of this chapter is twofold. First, the reader unfamiliar with this topic may try to learn the basics of Hopf algebras here and then continue with Chaps. 3 or 4. Subsections marked by * should be omitted by beginners. Secondly, Sects. 1.2 and 1.3 collect general definitions and notation used throughout this book.

1.1 Prolog: Examples of Hopf Algebras of Functions on Groups Probably the best way to take up a new concept and to understand the ideas behind its abstract definition might be to look at well-chosen examples. We shall do so before we enter the path through the wealth of necessary definitions. Our examples will only work with functions on "ordinary" groups, but they provide the motivation for later constructions on quantum groups. To begin with, let G be an arbitrary group and let ..F(G) be the algebra of all complex-valued functions on G with pointwise algebraic operations. The group operations on G allow one to define the following mappings:

- comultiplication A : .F(G) ---) .F(G x G) by (A(f))(gi,g 2 ) := f(gig2), - counit e : ,F(G) ---) C by e( f) := f (e), - antipode S : ..9G) .F(G) by (S(f))(g) := f(g -1 ). Here e denotes the unit element of G. From these definitions it is clear that A and e are algebra hornomorphisms.

(1)

The group axioms lead to the following identities (2)-(4) for the mappings e and S. First, the associativity of the group multiplication implies that (A id) o d = (id ® d) 0 .6,

(2)

4

1. Hopf Algebras

where id is the identity mapping on T(G). Indeed, the definition of d yields (((d 0 id) o d)f)(g i ,g2 ,g3) = f((g i g2)g3) G .F(G x G x G), (((id 0 d) o d)f)(gi,g 2 ,g3) = f (gi(g2g3)) E .9G x G x G).

Since

gi(g2g3) by the associativity law, the two expressions on the right hand sides coincide and (2) follows. Secondly, we have the relation f (eg) = f (ge) = f(g) which can be expressed as (gig2)g3

(E

id) o = (id

E) 0 = id

( 3)

under the usual identifications of CO .F(G) and .F (G) C with .F(G). Define a mapping m : .F(G x G) --* .F(G) by (rnh)(g) := kg, g), h E .F(G x G), and a linear mapping 71: C .F(G) such that T(1) is the unit element of the algebra T(G). Then, thirdly, the relations g-l g — gg -1 =-- e for the inverse g -1 of a group element g yield the identities mo(Soid)od=1)oE=m0(id®,S)04A.

(4)

Indeed, inserting the above definitions of m, d and S we obtain ((mo(idoS)0d)f)(g) = (((idOS)od)f)(g,g) = f(gg -1 ) f(e) = (rloE)(f) which gives the second equality of (4). The first equality follows similarly. Summarizing the preceding, we have seen that the group multiplication, the group unit and the group inverse of G induce mappings 4 E and S of functions on G such that the corresponding group axioms are expressed as identities (2)-(4). On the level of functions the properties of the group G are encoded in the three mappings 4 E and S defined above. One disadvantage is that the algebra .9G x G) of functions on G x G appeared in the definitions of the mappings d and m. In order to work with functions on G alone, we consider the tensor product .F(G) 0 .F(G) as a linear subspace of .F(G x G) by identifying fi O f2 G .F(G) .F(G) with the function (fl O f2 )(gi , g2 ) := fi (gi)f2(g2 ) on G x G. Then we have m(fi O f2)(9) = fi(g)f(g), that is, m(fi f2) is just the product fi f2 of f and f2 in the algebra T(G). After the embedding .F(G) T(G) C .F(G x G) one difficulty still remains: d does not map T(G) to .F(G) .F(G) in general (for example, d(f) .1(G)0.1(G) if f(t) = sin t 2 and G = So it is natural to look for appropriate subalgebras A of .F(G) for which d(A) C A o A and S(A) c A. Such a subalgebra A together with the linear mappings : A -4 A O A, E : A -4 C and S : A A satisfying the above conditions (1)-(4) is called a Hopf algebra (by Definition 4 below). If the Hopf algebra A is sufficiently "large", one can expect that A and its structure maps 4 e and S store enough information about the group G. We consider three important examples of such Hopf algebras A.

1.1 Prolog: Examples of Hopf Algebras of Functions on Groups

5

Example 1 (The Hopf algebra ,F(G) of a finite group). For a finite group G we have .F(G) .F(G) = (G x G), so by the preceding .F(G) is a Hopf algebra. Example 2 (The Hopf algebra Rep (G) of a compact group G). Suppose that G is a compact topological group. Let T be a continuous representation of G on a finite-dimensional complex vector space V. Let C(T) denote the linear subspace of .F(G) spanned by the matrix elements t o (.), i j = 1, 2, • • • , N, of TO relative to some basis of V. Since t o (gig2) = Ek tik(gOtki(g2), gi,g2 G, and t o (e) = 60 , we have ,

d(t o ) =

E k tik t kj

e(tij = .

( 5 )

The vector space C(T) equipped with the linear mappings d : C(T) C(T) 0 C which satisfy the relations (2) and (3) is called a C(T) and e : C(T) coalgebra (by Definition 2 below). Let A = Rep (G) be the linear span of spaces C(T) for all continuous finite-dimensional representations T of G. Since C(Ti ED T2) -= C(Ti) C (T2 ) and C(Ti 0 T2 ) = C(T1 ).C(T2 ) as is easily seen, A is a subalgebra of .F(G). Recall that the contragredient representation Tc of T acts on the dual V' of the space V by (Tc(g)y')(y) = y'(T(g -1 )y), g E G, v E V, y' E V'. If we choose the basis of V' dual to the basis of V, then Tc has the matrix elements qi = S(t). Thus, we get C(Tc) =--- S(C(T)) and hence S(A) C A. Since Zi(C(T)) C C(T) C(T) by (5), we also have d(A) C A® A. Therefore A = Rep (G) is a Hopf algebra. It is called the Hopf algebra of representative functions of G. Since Rep (G) is dense in the algebra C(G) of continuous functions on G by the Peter-Weyl theorem, Rep (G) is sufficiently "large". Let (t 5 )ri_ 1 be a matrix of functions to E .F(G). A simple consideration shows that the functions t o are matrix elements of some continuous finitedimensional representation T of G if and only if they belong to the algebra Rep (G) and satisfy the equations (5). This means that continuous finitedimensional representations of G can be defined completely in terms of the Hopf algebra Rep (G) without referring to the group G itself. Many other concepts of group representation theory can also be formulated in terms which are intrinsic to the Hopf algebra Rep (G) (see Chap. 11). Example 3 (The coordinate Hopf algebras 0(G) of simple matrix Lie groups). Let G denote one of the matrix groups SL(N, C), SO(N, C) or Sp(N, C). Each element g of G is a complex N x N matrix g = (go ). Define the coordinate functions u on G by u(g) gii , g = (go ) E G. For g, h E G we have

L1(u))(g , h) = u ij (g h)

(gh) ii = Ek gikh ki = Ek 4(g)u. 11 (h)

and i4(e) = bo, so that

=

Ek 111

III

and 6(4) =

(6)

6

1. Hopf Algebras

Let A --= 0(G) be the subalgebra of .F(G) generated by the N2 functions j = 1, 2, - - - , N. Since 1 : .F(G) — ■ .9G x G) is an algebra homomorphism, i, we have L\(A) C A 0 A by the first formula of (6). Any element g E G has determinant 1. Hence the function which is a constant equal to 1 on G belongs to A, so A has a unit. Further, it follows that there are polynomials pii in N2 indeterminates such that (g -1 ) ii = pii (g11 , g12 , - • • , gNN ), so that

51 (uii ) (g) = u4(g-1 ) ---- (g -1 ) ii = pii (ul(g) , 74(g) , • - • , u l;,'; (0) = pii (ul, 4 - - • That is, we have S(u i ) e A and hence 51 (A) C A. Therefore, by the discussion preceding Example 1, A = 0(G) is a Hopf algebra. It is called the coordinate Hopf algebra of G. A In all three examples of Hopf algebras the multiplication is commutative (being the ordinary multiplication of functions) and the antipode S is an algebra homomorphism such that S2 = id. These facts no longer remain true for general Hopf algebras and quantum groups. Summarizing and slightly simplifying the above considerations, we have rephrased (certain) groups in terms of Hopf algebras of functions on the groups. This turns out to be the proper perspective that makes it possible to "deform" or to "quantize" the groups by passing to noncommutative algebras. Formally this is done by introducing Hopf algebras depending on say one parameter (usually denoted by q or e h , h being thought of as Planck's constant) which specialize to the Hopf algebra associated with the group at a particular value (usually q = 1, resp. h = 0). The elements of such a deformed Hopf algebra will be viewed as functions on a "quantum group". The language of groups and representations will also be used for the deformed Hopf algebras as an inspiring source of motivation, though there are no groups around.

1.2 Coalgebras, Bialgebras and Hopf Algebras Let us fix some general notation which will be kept throughout the book. The letter K always stands for a commutative ring with unit. If V and W are vector spaces over K, £(V, W) denotes the 1K-linear mappings of V to W and V' is the vector space of all K-linear functionals on V. We set ,C(V) := L(V, V). The symbol 0 means the tensor product OK. We identify K0 V and V 0 K with V by the isomorphisms 1 0 y y, respectively. y and y 0 1 If not specified otherwise, the letter T denotes the flip operator given by T (v 0 w) = w 0 v.

1.2.1 Algebras Recall that an (associative) algebra is a vector space A over K equipped with a mapping (a, b) ab of A x A to A such that

1.2 Coalgebras, Bialgebras and Hopf Algebras

7

a(bc) = (ab)c, (a + b)c = ac +be, a(b + c) = ab + ac, cx(ab) = (aa)b = a(ab)

( 7 )

for all a, b E A and a E K. An element 1 of an algebra A is called a unit of A if 1a = al = a for all a E A. If a unit element exists in A, it is uniquely determined. Throughout this book, by an algebra we always mean an associative algebra with unit element. The following slight reformulation of the algebra definition is the right perspective for its dualization in Subsect. 1.2.2. Definition 1. An algebra (that is, an associative algebra with unit) is a

vector space A over K together with two linear maps m: A A —> A, called A, called the unit, such that the multiplication or the product, and 71 : K id) = 0 (id

m 0 (rn rn (7/

m),

id) = id = m 0 (id

( 8

ri).

)

(9)

Given such an algebra A, ab := rn(a b) is the product of a and b and the mapping n is determined by its value 71(1) E A, which is the unit element of A. Both definitions of an algebra are easily seen to be equivalent. Equation (8) is the associativity law, while (9) says that n(1) is a unit element of A using the identification of IK0 A and A 0 K with A. The conditions in (7) are built into the requirement that rrt:A0A Ais 1K-linear. The associativity (8) of the multiplication m means that the diagram —

m

AoAcgA

id

AoA. (10)

iidom Trt

A 0A

A

is commutative. Likewise, the condition (9) of the unit can be expressed by the commutativity of the following diagram:

IK0A

A 0A

id 0 71

A 0K

jid

KoA

A

.4=>

l id AoK,

where K 0 A <=> A and A 4=> A 0 K denote the canonical identifications. Let A and B be algebras.. A K-linear mapping (i9 : A 13 is called an algebra homornorphism if (p(aa') = (p(a)(p(d) for all a, a' E A and ço(1A) = 16. The two latter conditions can be rewritten as

8

1. Hopf Algebras

There exists a tensor product algebra AO13 whose vector space is the tensor product of vector spaces of A and B and whose multiplication is defined by ms) sc, (id 0 T 0 id). That is, the algebra A 0 13 has the TriA0 B := (mA product (12) (a 0 b)(a' b 1 ) := aal a,al E A, b,b' E B. For each algebra A one can define the opposite algebra A°P. This is an algebra with the same underlying vector space as A, but with the new multiplication m A , := mA 0 T. That is, we have a op b = b • a, where • op and denote the products of A°P and A, respectively. Many algebras are constructed as quotients of free algebras or tensor algebras. Let us recall these notions in the case where K is the complex field C. Let {x I i E II be an indexed set of generators. We denote by I' the union of all sets In, n E No , where 10 consists of a single element o which is not in I. Fort = (i 1 ,i2, • • • ,ik) E /k and j = (ii/i21 . • • I in) E I, E No, we set (i, j) = (i 1 , • • • , ik , j 1 , • - • ,j n ) and xi := x i, • • • x ik • For o E 1 0 and i E Ik, k E No, we put (o, i) = (i,o) := i and s o :--= 1. Then the free algebra C(x i ) with generators x i , i E I, is defined as follows: it is a complex vector space with basis {x I i E /oe } and with the multiplication defined by x i xi E I'. The algebra C(xi ) possesses the following universal property: for any indexed subset {a I i E I} of another algebra A there exists a unique algebra hornomorphism ç :C(x i ) —> A such that ça(xi ) --= ai for all i G I. Next we define the tensor algebra T(V) over a vector space V. Set Vrg° := K and V ®n := V 0 • • V (n times). The direct sum T(V) := @n œ 0 V ®71 of these vector spaces becomes an algebra, called the tensor algebra T(V) over V, with multiplication determined by xn y k := x n yk for x n E V en and yk E V®k. The algebra TM also has a universal property: for every linear map (pi : V -4 A of V to an algebra A, there exists a unique algebra homomorphism ça : T(V) —> A such that ça(v) p 1 (v) for y E V. The universal properties of the algebras C(x i ) and T(V) imply the following fact: for any basis {x j i E II of the vector space V, there is a unique algebra isomorphism ço of the algebras T(V) and C(xi ) such that ço(x) = x i , i E I. Often in this book an algebra A is introduced by stating that it is generated by elements x l , • • • with relations ..,k,x17 f ( 0, k = 1,2, • • ,s. • • • S r) By such a phrase we mean that A is the quotient algebra of the free algebra C(x i ) with generators xi, • • • , Sr by the two-sided ideal I of C(xi ) generated by fk (xi, • • • , xr ), k = 1, 2, • • • , s. For simplicity we shall denote the image of the elements x i E C(xi) in the quotient algebra A = C(xi) also by X. 1.2.2 Coalgebras

Recall that an algebra is a vector space A with two linear mappings m : A® A -4 A and : K —> A such that the diagrams (10) and (11) are commutative. We now dualize this definition by reversing all arrows and replacing all mappings by the corresponding dual ones. In doing so, the multiplication

1.2 Coalgebras, Bialgebras and Hopf Algebras

9

m:A0A >Ais replaced by the comultiplication LI :A >A0A, the unit .71 : IK —> A by the counit E : A —> IK and the diagrams (10) and (11) go into the following diagrams (13) and (14), which have to be commutative: —

—

0 id

A0A0A

(13)

id 0 Zi LI

A0 A K

A c

E

id

A id ®

A 0A

E

A 0K id

id K

A

< >

A

(14)

A 0K

We thus obtain the following "dual" notion to an algebra. Definition 2. A coalgebra is a vector space A over K equipped with two linear mappings 2i : A A 0 A, called the comultiplication or the coproduct, and E : A —> K, called the counit, such that 0 id) (E

= (id

(15)

o

id) 0 = id = (id

E)

(16)

o

Equation (15) is referred to as the coassociativity of the comultiplication 2i, because it dualizes the associativity (8) of the multiplication tn. Proceeding in a similar manner, most concepts on algebras can be dualized to coalgebra notions by giving diagrammatic formulations of the definitions and then reversing all arrows. We only list the outcomes of these dualizations in the following, but the reader unfamiliar with this procedure should carry out a few more examples in order to get used to working with "costructures" . Let A and B be coalgebras. A K-linear mapping cp :A >Bis said to be a coalgebra homomorphism if —

Zi5 0

(cp (,o)

2■ A

and

EA = Es

(p.

The tensor product coalgebra A® B is the coalgebra built on the vector space A0 B with comultiplication ZiA 013 := (id ® T (3) id) o (LIA ® 43) and counit E.A08 EAOEB. The coopposite coalgebra A"P is the coalgebra on the vector space A equipped with the new comultiplication ZIAcop T 0 4A and the counit EA. A coalgebra A is said to be cocommutative if T o LI = 21. A linear subspace B of A is a subcoalgebra if Zi(B) C B 0 B. A K-linear subspace I of A is called a (two—sided) coideal if .A(/) C

/ /

A

and E(/) = {0}.

If I is a coideal of A, the quotient vector space A/I becomes a coalgebra with comultiplication and counit induced from A.

10

1. Hopf Algebras

Next we introduce the so-called Sweedler notation for the comultiplication. If a is an element of a coalgebra A, the element A(a) E A 0 A is a finite sum A(a) =

E aii a2i aii,a2i

(17)

E A.

Moreover, this representation of A(a) is not unique. For notational simplicity, we shall suppress the index i and write the sum (17) symbolically as A(a) =--

E a(i) a(2).

(18)

Here the subscripts (1) and (2) refer to the corresponding tensor factors and the symbol E reminds us that A(a) is actually a sum. Let us define inductively mappings A (n) : A 4 A®n+1 by -

A (n ) := (id 0 A) o A -1 , n> 1,

and A (1) =--

(From the coassociativity (15) it follows that ZA (n) is in fact equal to n 1 compositions of A independently of their order, that is, A (2) = (id0,A) 0,A = (A 0 id) o A, etc.) Then the element A (n)(a) E A®n+1 is denoted by g a( 2) g

(19)

• • • (g) a (n+1) .

One can also replace the sums in (18) and (19) by the symbol E(a) in order to emphasize the element a E A. The reader may check the correct use of this notation by adding the corresponding elements to the sum. The Sweedler notations (18) and (19) turn out to be very useful when working with general Hopf algebras, because they allow one to give elegant shorthand formulations of definitions, formulas and proofs. We strongly recommend the reader to practise its use (for instance, by carrying out proofs in Propositions 1, 5 and others below) in order to become acquainted with this powerful tool. The result will make the effort worthwhile. The coassociativity (15) in terms of the notation in (18) and (19) yields

E(a(1) )(1)

(a( 1 ))( 2)

E a( 1) 0 (a( 2))( 1 ) (a( 2))( 2) = A (2) (a) = E a(1) a(2) a(3) .

a( 2) =

The counit property (16) reads in the Sweedler notation (18) as a=

E ame(a( 2)).

(20)

Proposition 1. Let A be a coalgebra and B an algebra. Then the vector space L(A, 8) of all K-linear mappings of A to 13 equipped with the convolution product defined by

(f * Oa) := (m 8 o (f 0 g)AA )(a) becomes an algebra with unit

77,3

E f (a(i))g(a(2)),

a E A,

(21)

11

1.2 Coalgebras, Bialgebras and Hopf Algebras

Proof. By the associativity of the multiplication in 13 and the coassociativity of the comultiplication in A, we obtain

((f * g) * h)(a) =

E f(a( 1))g(a(2))h(a(3)) = (f * (g * h))(a),

that is, the convolution product (21) is associative. The relation ((r/B 0 6A) * f)(a) =

EA(a(1))f(a(2))

= f (EEA(a(1))a(2)) =

f (a)

by (20) shows that 7p3 o eA is a left unit. It is similarly proved that 716 o EA CI is a right unit. Corollary 2. The dual vector space A' of a coalgebra A is an algebra with

product

(f Oa) := (f * g)(a) =

E f(a( 1))g(a(2)),

Proof. Apply Proposition 1 to the algebra B

a G A,

f,g G A'.

(22) CI

K (over K).

The dual space B' of an algebra B does not become a coalgebra in general by dualizing the multiplication (see the polynomial algebra C[x] for a counterexample), however it does if the algebra 13 is finite-dimensional. The reason is that we always have A' 0 A' C (A A)' and hence mA, : 0 A' A', while B' B' is only a proper subspace of (B BY if 13 is infinite-dimensional, so that z1 (f) f o nit3 may not lie in B' 8'. There are two possibilities to remedy this situation: to replace the dual space B' by a subspace (this will be done in Subsect. 1.2.8) or to enlarge the tensor product (for instance, by taking its completion in some topology). Example 4 (The coalgebra MN (K)' ). Let uii be the linear functional on the algebra MN(K) of N x N matrices over K defined by t4(g) := gii for g = is a coalgebra (gii) G MN(K). Then MN(K)' = Lin fuji I i j = 1, 2, - • - , = E k uik g u.'; and counit 6ii . with comultiplication ,

1.2.3 Bialgebras

A bialgebra is an algebra and a coalgebra, where both structures are compatible in the sense of the following Proposition 3. If A is a vector space which is an algebra and a coalgebra, then the following two conditions are equivalent:

d:A K are algebra homomorphisms. .40 A and e: A (ii) m : A® A A and ïj :DC —) A are coalgebra homomorphisms. Proof. By the above definitions, the assertions of (i) mean that ziom = m.4..4 0(460.6), zion = r1A0A, corn = niKo(e0e),

E077

= r/K 7 (23)

12

1. Hopf Algebras

while the statements of (ii) can be expressed as ,60m = (mom)Q6 A0A , EA0A = Eom, 46on =

( non )046 x , 6K =

eon. (24)

The right hand sides of the first relations of (23) and (24) are both equal to (m m) o (id 0 T 0 id) o (.6 ,6). The second equality of (23) and the third of (24) are equivalent. The two last equations both mean that e(1) = 1. Likewise, the third relation of (23) and the second of (24) are equivalent. (The reader is invited to express the relations of (23) and (24) as diagrams and to check their equivalence in this manner.) LI Definition 3. A bialgebra is a vector space which is an algebra and a coalgebra such that the conditions in Proposition 3 hold. That is, if A is an algebra and a coalgebra, it is a bialgebra if and only if for any a,b E A we have ,6(ab) = ,6(a)46(b), e(ab) = e(a)e(b), 46(1) = 1 01, e(1) = 1.

(25)

Let A and B be bialgebras. By a bialgebra hom,om,orphism of A to B we mean a mapping that is both an algebra and a coalgebra homomorphism. The vector space A 0 B endowed with the tensor product algebra and coalgebra structures becomes a bialgebra. There are three bialgebras A°P, Ac°P and A°P'"P which are obtained from A by taking the opposite of either the algebra or coalgebra structure or of both of them. That is, A°P has the opposite multiplication mAop and the comultiplication of A, A"P has the multiplication of A and the opposite comultiplication .6Acop , and °P'c°P carries the opposite multiplication rnAop and comultiplication .6Acop . (The reader may verify that the above structures are indeed bialgebras.) A linear subspace I of a bialgebra A is called a biideal if it is both a twosided ideal of the algebra A and a coideal of the coalgebra A. The quotient All of a bialgebra A by a biideal I is again a bialgebra with structures inherited from A. We now introduce two concepts of distinguished elements of a bialgebra A. The motivation for this terminology stems from Examples 6 and 7 in Subsect. 1.2.6. A nonzero element g c A is called group-like if A(g) = g 0 g. An element s A is called primitive if .6(x) = 1 + 1 0 x. Proposition 4. Let A be a bialgebra. Then the product of group-like elements is group-like. If x and y are primitive elements of A, then we have e(x) = e(y) = 0 and the element [z, y] := xy — yx is also primitive.

Proof. The first assertion is obvious. Let x and y be primitive elements. Since (id 0 E).6(x) = se(1) + e(x)1 = x by (16) and E (1) = 1, we obtain e(x) = O. Using the fact that .6 is an algebra homomorphism, we get .6(xy) = (x0 1 + 1 0 x)(y 0 1 + 1 0 y)

xy 01+x0y+y0s+ lOsy,,

A(yx)= ys 0 1 -Ey0s-ExOy+10yx, so that Lt ([x, y]) = Li(xy) A(ys) = y] 0 1 + 1 0 [s, y].

1.2 Coalgebras, Bialgebras and Hopf Algebras

13

By Proposition 4, the set of group-like elements of A is a semigroup with unit and the primitive elements of A form a Lie algebra with respect to the

commutator bracket [., • 1. Example 5 (The bialgebras O(IN(K)) and K[x i , x 2 , • • • , xN]). Let A --=-OWN (K)) be the algebra generated by the functions uii from Example 4. That is, A is just the polynomial algebra in N2 indeterminates --= 1,2, • - - , N. Then A becomes a bialgebra with comultiplication and The correspond0 uin; and E(u) counit determined by ,A(u2i ) = ing coordinate algebra of the diagonal matrices in MN (EC) is the polynomial algebra K{x i , x2 , • • • , x N ] in the indeterminates xi = u!. It is a bialgebra with comultiplication = xi 0 xi and counit E(xi ) = 1.

Ek

1.2.4 Hopf Algebras

The following notion is fundamental in quantum group theory. Definition 4. A bialgebra A is called a Hopf algebra if there exists a linear mapping S: A > A, called the antipode or the coinverse of A, such that —

mo(Soid)o

=noe=rrio(idOS)o.

(26)

Clearly, the equations (26) are just the requirement that the diagram A®A lid AOA

A

LI

---- AOA

IS 0 id

177 0 E

S 1m

A

m

A0A

is commutative. In the Sweedler notation (18), the relations (26) say that

E S(a(1))a(2) = e(a)1

(27)

where ,A(a) = a( 1 ) 0 a( 2). The equations (26) can be nicely interpreted by using the convolution product (21) of the algebra £(.4, A). Comparing the formulas (21) and (26) we see that (26) means nothing but that S is the convolution inverse of the identity mapping, that is, S* id = id *S=noe. Thus, a bialgebra A is a Hopf algebra if and only if the identity map of A is invertible in the algebra .C(A,A). Moreover, the preceding characterization implies that an antipode of a Hopf algebra is uniquely determined. The next proposition gives some important properties of the antipode. Proposition 5. The antipode S of a Hopf algebra A is an algebra antihomomorphism and a coalge bra anti-homomorphism of A. This means that S : A > A°P is an algebra homomorphism and S : A Â"P is a coalgebra homomorphism. That is, we have —

14

1. Hopf Algebras

S(ab) = S (b)S (a), a, b E A, 460S=To(S0S)o,6

and S(1) = 1,

(28)

and E0S=E.

(29)

Proof. The proof is a nice exercise in the use of the Sweedler notation. The reader who is unfamiliar with it might first try to carry out the proof itself and then compare it with the detailed presentation below. Using the Hopf algebra axioms and the facts that L and E are algebra homomorphisms, we compute

E S (b(i)e(b(2)))S (a (1)*(2)))

S (b)S (a)

.

s,b(1) s ,a( 1 ),) E _a(2)b(2),

=

E S(b(1)) ,9 (a(1))(a(2)b(2))(1) ,9((a(2)b(2))(2)) E S(b(o)S(a(1))a(2)b(2)S(a(3)b(3)) E S (b (1) )(e(a(1) )1)b (2) S (a( 2)b ( 3)) E a( 1 )

E(b (1) ) S a( 2 )b( 2) = S ab

This proves the first equality of (28). The second one follows at once from the relation mo (Sold) o,6(1) = e(1)1 combined with the facts that 46(1) = 1 01 and E(1) = 1. Similarly, we compute

E S(a) g S(a i ) = = =

E s(a (2) e(a (3) )) ® S(a i ) z_g (S(a(2)) z_g (S(a(2))

E(s(a ( 2) )

0 S(a(1)))(e(a(3))1 0 1) 0 S(a(i)))((a(3)S(a(4))))

s(a( 1 ) ))(a (3) O a(4))(S(a(5)))

E(s(a(2))a(3) E(e(a( 2) )1 0 S(a(i))a(3))((S(a(4))) E(1 0 S(a(1))a(2))G 6 (S(a(3))) E(1 ® E(a(1)) 1 )((S(a(2))) = which is the first relation of (29). The second relation is derived by

E(S (a)) = e (E ame(a( 2)))) = e

S(a( l))a( 2)) = e(e(a)1) = -

LI In the Sweedler notation the first equality of (29) can be rewritten as

E 5(a)( 1) O S(a)(2) = E s(a( 2) ) 0

(30)

1.2 Coalgebras, Bialgebras and Hopf Algebras

15

Let A be a Hopf algebra. Since the bialgebra A°P ,"P has opposite multiplication and comultiplication, the antipode S of A satisfies the relations (26) for A°P'c°P as well. Hence S is also an antipode of A°P'"P and the bialgebra Aop,cop is a Hopf algebra. However, for the bialgebras A°P and Ac°P to be Hopf algebras S needs to be an invertible linear mapping of A. Proposition 6. For any Hopf algebra A the following conditions are equivalent: (i) The antipode S of A is invertible as a linear mapping of A. (ii) The bialgebra A°P is a Hopf algebra. (iii) The bialgebra Ac°P is a Hopf algebra. In this case, the inverse S -1 of S is the antipode of A°P and A'P. Proof. (i)—(ii): The inverse 8 -1 of S is an algebra anti-homomorphism of as S is by Proposition 5. Hence, by formula (28) we have

E S' (a

2 )a( i ) =

=

E s-

A

(a(2))(S -1 o 51)(a(0) = S -1 (E S(a( 1 ))a( 2))

5-1 (e(a)1) = e(a)l

E a( 2)S -1 (a( i)) = e(a)1

for a e A. That is, S' satisfies condition (26) for the bialgebra A°P, so that it is an antipode of A°P. (ii)—qi): Let :5" denote the antipode of A°P . Since S is also an algebra anti-homomorphism of A, we obtain and similarly

,§S(a) =

S (E e(a( 2))a(0) =

e(a(208(a(1))

E(a( 3)(a( 2))),S'51(a( 1)) =

= E(a (2) (e(a(1) )1)

=

E a(3):5"(S(a( 1))a( 2)) =a

and similarly S,§(a)= a for a E A. Therefore, is the inverse of S. The equivalence of (i) and (iii) is proved in a similar manner.

111

Corollary 71 If a Hopf algebra A is commutative or cocommutative, then 52 = id. Proof. In either case we have A = A°P or A == A"P , so A°P or Ac°P is a Hopf algebra and the assertion follows from Proposition 6. 111

The antipode S of a Hopf algebra A is said to be of finite order if Sn = id for some n E N. The smallest such n is called the order of S. Otherwise, one says that S is of infinite order. By a result of D. E. Radford [Radii, the antipode of any finite-dimensional Hopf algebra is of finite order (so, in particular, S is invertible and A°P and Ac°P are Hopf algebras). Further, a result of R. G. Larson and D. E. Radford (see [LaRD says that for any finite-dimensional complex cosemisimple Hopf algebra (see Sect. 11.2 for this notion) we even have 82 = id. The latter solved an old conjecture of I. Kaplansky.

16

1. Hopf Algebras

We shall see later that it is often easy to check the Hopf algebra axioms for a set of generators, so the following simple fact appears to be useful. Proposition 8. Let Ag be a subset of an algebra A which generates A as K be homommphisms and A A and e : A an algebra. Let L : A A an anti-homomorphism of the corresponding algebras. If the S: A coassociativity condition (15) and the counit condition (16) (and the antipode condition (26)) are satisfied for elements in A g , then they are valid on the whole of A and hence A is a bialgebra (resp. a Hopf algebra).

Proof. In the case of zX and e the assertions follow at once from the fact that (46 0 id) 0 4 (id 0 460 0 4 (e 0 id) o 46 and (id 0 e)46i are all algebra homomorphisms. In the case of S it suffices to show that (26) holds for a product ab provided that it is valid for a and for b. In order to do so, we compute E S((ab)( 1 ))(ab)(2) =-- ES(a ( i) b( i ) )a(2) b(2) = ES(b( 1 ) )(S(a ( 1 ) )a (2) )b(2) S(b( 1) )(e(a)1)b( 2)

e(a)e(b)1 = e(ab)1.

E

The other equality of (26) is verified similarly. We close this subsection with a few general definitions and facts. Let A and B be Hopf algebras. A linear mapping (io : A --03 is called a Hopf algebra harnomorphism if it respects the Hopf algebra structures, that is, if (,c) is a bialgebra homomorphism and wo SA = Si30(p. Any bialgebra homomorphism between Hopf algebras is already a Hopf algebra hom.omorphism, that is, the condition (io o SA Sz3 o (io is then automatically fulfilled. A .Hopf ideal of A is a biideal I such that S(I) C I. The quotient A// of a Hopf algebra A by a Hopf ideal I is again a Hopf algebra.

1.2.5* Dual Pairings of Hopf Algebras Let us consider a finite-dimensional Hopf algebra A. Recall that by Corollary 2 the dual vector space A' is an algebra with respect to the multiplication f g(a) (f g g),6(a) E f (a( n )g(a( 2) ). For f C A' we define a functional 461(f) E (A® A)' by ZA(f)(a0b) := (f om)(a0b) f (ab), a, b G A. Since A is finite-dimensional, (A® = A' OA' and so LX(f) E Af ga41 . It is not difficult to show that the algebra A' equipped with the comultiplication IA becomes a Hopf algebra. The antipode, the counit and the unit element of this Hopf algebra A' are given by (S f)(a) = f(S(a)), E (f) == f (1) and 1A' (a) =e(a), respectively. That is, the Hopf algebra A' is obtained by "dualizing" the structure maps of the Hopf algebra A. This situation is generalized in Definition 5. A dual pairing of two bialgeb ras II and A is a bilinear mapping K such that x A (.,.) (Au(f)1a1

a2) = (.f, 040, 2)7 (.6.f2la) =

0 f21 ,6A(a)))

(31)

1.2 Coalgebras, Bialgebras and Hopf Algebras

(f, 1 4) = eu(f),

(1u,a)

17

(32)

e.4(a)

for all f, fi, f2 E LI and a, a l , a2 E A. A dual pairing (•,.) of 14 and A is called nondegenerate if (f, a) = 0 for all f E 14 implies that a = 0 and if (f,a) = 0 for all a E A implies f = O. Proposition 9. If (•,-) is a dual pairing of two Hopf algebras 14 and A, then we have (S(f), a) = (f,S(a)), f E U, a E A. (33)

Proof. Consider the linear functionals F, G and H on the tensor product Hopf algebra Li A defined by F(f 0 a) = (f,a), G(f 0 a) = (S(f),a), H(f 0a) =(f,S(a)), f E 14, a E A. We show that FG = HF =e in the algebra (LI 0 . Indeed, by (31) and (32), we get

FG(f 0 a) =

E(f(1), a(1))(S(f(2))1a(2))

= (E(f)1,a) = e(f

E(f(i)s(f(2)),

a)

a),

that is, FG = e 7 and similarly HF = e. Thus, G = (HF)G = H(FG) = H which yields (33). El

The discussion preceding Definition 5 has shown that for any finitedimensional Hopf algebra A we have a dual pairing of the Hopf algebras := A' and A defined by (f, a) := Pa), a E A, f E A'. Obviously, this pairing is nondegenerate. Let us discuss the above notion with some remarks. First note that in equation (31) the pairing of 14 and A must be extended to one oft' 014 and A® A by setting (fi.o f2, al a2) := (h, ai)(f27a2). Dual pairings of two Hopf algebras are by no means unique. For any couple of Hopf algebras 14 and A there is always the trivial pairing defined by (f, a) := eu (f)eA(a). Further, compositions with Hopf algebra homomorphisms in both variables again give a dual pairing. Let •) be a dual pairing of two Hopf algebras 14 and A Then Iu E (f,a) = 0 for all a E AI and IA := {a E A (f,a) = 0 for all f E /41 are Hopf ideals of 14 resp. A and the pairing (., •) oft' and A induces a nondegenerate dual pairing of the quotient Hopf algebras /4///,/ and Ai/A. From the formulas (31) we see that any dual pairing of two bialgebras is completely determined by the values of the bilinear form (., •) on sets of generators of the algebras 14 and A. This allows one to use shorthand descriptions of pairings by means of matrices. A nondegener ate dual pairing of two Hopf algebras 14 and A stores a lot of information about the links of both structures. (For instance, it follows then from (31) that Li is commutative if and only if A is cocommutative.) In quantum group theory there is a general philosophy which says that then the Hopf algebras LI and A provide dual approaches to the same "quantum object".

18

1. Hopf Algebras

1.2.6 Examples of Hopf Algebras

Three important examples of Hopf algebras related to groups have already been developed in Sect. 1.1. Two others will be described below. The following example is the starting point for our first approach to quantum groups. The quantized universal enveloping algebras studied in Chaps. 3 and 6 are deformations of the Hopf algebras U(g) which are universal enveloping algebras of Lie algebras. Example 6 (The universal enveloping algebra U(g)). Let g be a Lie algebra over K with Lie bracket [., •]. The universal enveloping algebra U(g) is defined as the quotient algebra of the tensor algebra T(g) of the vector space g by the two-sided ideal I generated by the elements x0y—y0x— [x y], where , y E g. That is, U(g) is the free algebra generated by a vector space basis of g subject to the relations [x, y] =-- xy — yx for x, y E g. By its very definition, the algebra U(g) admits the following universal property: given a linear mapping (to : g A of g to another algebra A satisfying

(P(410 (Y)

'10 (Y)40 (x),

xl Y E 91

(34)

there is a unique algebra homomorphism : U(g) A such that (;(3(x) = (p(x) for x E g. This fact is used in proving the following assertion. There is a unique Hopf algebra structure on .6(x) = x 0 1 + 1 0 s,

E(x) = 0,

the algebra U(g) such that S(x) = —x, s E p.

(35)

In order to prove this, we define linear mappings (tc) = 46, E and S of g to the algebras A = U(g) U (g), K, U(g)°P by (35). Condition (34) is obviously fulfilled for the mappings E and S. One verifies (by arguing as in the proof of Proposition 4 above) that d also satisfies (34). Therefore, by the universal property of U(g), these mappings extend to algebra homomorphisms of U(g) to A, denoted still by d, E and S. By Proposition 8, it suffices to check the Hopf algebra axioms on the generators x E g. This is easily done; as an example, we have m o (S 0 id) o d(x) = m(--x 0 1 + 1 0 x) = 0 = e(x) for x E g. By (35), any x E g is a primitive element of the Hopf algebra U(g). In the case K = C it can be shown (see [Bou2]) that the elements of g exhaust all primitive elements of U(g), so the Lie algebra of primitive elements of U(g) is just the Lie algebra g itself. In particular, this gives an intrinsic characterization of the Lie algebra g in terms of the Hopf algebra structure of U(g). From (35) we see that the Hopf algebra U(g) is cocommutative. Let B be either the Hopf algebra Rep (G) of Example 2 for a connected compact Lie group C or one of the Hopf algebras 0(G) of Example 3. Suppose that K = C and g is the Lie algebra of the Lie group G. It is well-known that each element a E U(g) acts as a left-invariant differential operator à on G. If a = x1x2 • - • xn with xl, x2, • • • , x E g, the action of ;a,' on a function f E G'(G) is given by

1.2 Coalgebras, Bialgebras and Hopf Algebras

19

an f)(g) =--

ati

f • • • atn,

exp(tixi) • • • exp(tnxn )).

(36)

ti =•••=tn =o

One can show that there is a nondegenerate dual pairing of the Hopf algebras U(9) and 13 determined by (a, f) := (a n(e), where a C U(g) and f C B. A Example 7 (The group algebra KG). Let KG be the vector space with basis given by the elements of G. It becomes an algebra by extending linearly the product of G. The unit of this algebra is the unit element of G. There is a unique Hopf algebra structure on the algebra KG such that A(g) = g 0 g, e(g) = 1 and 8(g) = g-1 for g E G. The elements of G are obviously group-like. If K = C, it is easily checked that they exhaust all group-like elements of the Hopf algebra KG. The Hopf algebra KG is cocommutative. Let A denote one of the Hopf algebras .F(G), Rep (G) or 0(G) defined in Examples 1, 2 and 3, respectively. Then there is a nondegenerate dual pairing (,•) of the Hopf algebras A and B := KG defined by (f, Ei Ag) f (g i ) , where f E A and Ei Ag iE B. Example 8 (The tensor algebra T ( T)). Let V be a vector space and T(V) the tensor algebra over V defined in Subsect. 1.2.1. Arguing similarly as in Example 6 (replacing thereby the universal property of U(g) by that of VV. )), it follows that there exists a unique Hopf algebra structure on T(V) such that A(v) = v 1 +1 0 v, e(v) = 0 and 8(v) = -v for v E V. This Hopf algebra is cocommutative. For arbitrary elements v1 , v2, - • • , vn E V we have

A(v i • - • Vn ) =

EE

vp ( i ) •••

7)0)

O Vp(k+i)

Vp(n)

7

(37)

k=0 PEPnk 8(V1V2 • • • Vn) = ( - 1) n Vn • " V2V1.

(38)

denotes the set of all permutations of {1, 2, • • • , n} such that 7)(1) < •• • < p(k) and p(k+1) < • • <13(4 The elements of Pn k are called (k, n- k)shuffles. In the cases k = 0 and k = n the corresponding summands in (37) should be read as 1® viv2 • • • vn, and vi v2 • • • Un 0 1, respectively. Formula (37) can be proved by induction on n, and (38) follows immediately from (28). Let V be a Lie algebra g. It is straightforward to check that the ideal I (x y - y x - [x, I x, y E g) is a Hopf ideal of the Hopf algebra T(V). Hence, the Hopf algebra U(g) from Example 6 is just the quotient Hopf algebra T(V)//". In particular, the formulas (37) and (38) remain valid for elements v1, v2, • • - vn E g = V in the Hopf algebra U(9). Let us specialize to the case where V is an abelian Lie algebra with trivial Lie bracket [x, y1 = 0, x,y e g. Then the quotient algebra T(V)II is the symmetric algebra S(V) over V which is shown to be a Hopf algebra. A Here

Pnk

Example 9 (M. E. Sweedler). Let A be the complex algebra with generators x and g and defining relations g2 = 1, x2 = CI and gxg = -x. The elements

20

1. Hopf Algebras

1,g,x,gx form a vector space basis of A. One verifies that the formulas d(g) = g g, ,A(x) = s 01+ g x, e(g) =1, e(x) = 0, 5(g) = g = g-1 and 5(x) = -gx define a Hopf algebra structure on A. It is the noncommutative A noncocommutative complex Hopf algebra of smallest dimension. 1.2.7 *-Structures When we consider *-structures we assume that 1K is the complex field C. By a *-vector space we mean a vector space V over the field C endowed with a mapping a a*, called an involution and denoted by *, such that (av + Ow)* =dv* + T3w* and (v*)* = y for y, w c V and cx, j3 c C. Recall that a *-algebra is an (associative) algebra A (with unit) together with a mapping a a* such that A becomes a *-vector space and (ab)* = b* a* for all a,b E A. Such a mapping is called an algebra involution on A. It follows then that 1* = 1. A coalgebra A is called a *-coalgebra if A is equipped with an involution * such that A is a *-vector space and .A : A A0A is a *-homomorphism. The latter means that d(at) = d(a)* for a C A, where the involution of A 0 A is defined by (a 0 b)* = a* 0 b* . In any *-coalgebra A we have E(at) = a E A. A *-bialgebra is a bialgebra A with an involution for which A is both a *-algebra and a *-coalgebra. A Hopf algebra which is a *-bialgebra is called a Hopf *-algebra. Though the latter definition does not contain a requirement concerning the behavior of the antipode with respect to the involution, the Hopf algebra structure implies the following compatibility relation. Proposition 10. In any Hopf *-algebra A, we have 5(5(a*)*) = a for a E A,

that is, So*oSo*=id.

(39)

In particular, S is invertible with inverse 5_1 = * o S o *. Proof. Put T := * o S o *. Setting a = b* in (27) and applying the involution yields E b (2) T(b (0 ) = 60)1 == E T(b( 2 ))b( i ), b E A. Hence T is an antipode for the Hopf algebra A.°P. By Proposition 6, T = S-1 andsoSoT=So*oSo*=id. Corollary 11. The dual algebra A' of a Hopf *-algebra A is a *-algebra with

involution defined by f* (a) = f (S(a)*), f C A'.

(40)

Proof. Clearly, f* E A' and the involution is anti-linear. From (39) we obtain 0 that (f t ) t = f, while (29) implies that (f g)* = g* f t . Formula (40) provides the motivation for the following definition of dually x A C is called a dual paired Hopf *-algebras. A bilinear form (-, -)

1.2 Coalgebras, Bieigebras and Hopf Algebras

21

pairing of two Hopf *-algebras U and A if it is a dual pairing of the Hopf algebras U and A (see Definition 5) and in addition we have (f *,a) = (f,S(a)*) and

(f, a*) = (S(f)*,a).

(41)

As for the other parts of Definition 5, condition (41) is symmetric in U and A. But it suffices to require one of the relations in (41). Combined with other properties of dual pairings and Hopf *-algebras, the other one will follow. Indeed, by the first equality of (41) and the formulas (39) and (33), we get (fl e)

= ((r)*,a*) = (f*,S(a*)*) = (f*,S-1 (a)) (S -1 (r),a) = (S(f)*,a).

Example 10 (Real forms of complex Lie algebras). Let gEz be a real form of a complex Lie algebra g. Since g is a direct sum of gR and ig R , there is an involution of the complex vector space g defined by (x + iy)* = - X iy, x,y E gR. It extends uniquely to an algebra involution of U(g), and U(g) is then a Hopf *-algebra. Conversely, let * be an involution such that (U(g),*) becomes a Hopf *-algebra. Since g is the set of primitive elements of U(9), this involution leaves g invariant and hence g R = Eg -x= el is a real form of g. That is, we have a one-to-one correspondence between real forms of the Lie algebra g and Hopf *-structures on the Hopf algebra U(g). Example 11 (Real forms of connected complex Lie groups). Let A be the Hopf algebra 0(G) from Example 3 and let GR be a real connected Lie group which is a real form of the complex connected Lie group G. Then for any f E 0(G) there is a unique function f* E 0(G) such that f* (g) = f(g) for all g E GR . Endowed with this mapping *, the Hopf algebra 0(G) becomes a Hopf *algebra. Let us illustrate this in the case G = SL(N,C). For the real form (74) * = GR = SL(N, R) it is obvious that u(g) = u(g) for g E G ,

For the compact real form GR = SU(N) we have (uji )*(g) = u(g) = gii = S(uji )(g) for g E GR and hence (z4)* = S(u). Let g R be the real Lie algebra of GR. It is a real form of the complex Lie algebra g of the complex Lie group G, so U(9) is a Hopf *-algebra by the preceding example. Then the Hopf algebra pairing (., .) from Example 6 (see (36)) is a dual pairing of the Hopf *-algebras U := U(g) and A := 0(G). Let us sketch the proof of the latter assertion. As noted above, it is enough to prove the second relation of (41). Because of (31), it suffices to do so for elements f x E BR . Since exp(tx) E GR for t E R, we then have

d (x,a*) = wi a(exp(tx))

(x,

= (S(x)*, a).

t=0

The previous examples suggest the following terminology. A Hopf *algebra structure on a Hopf algebra A is called a real form of A. Two Hopf *-algebra structures * and *' on A are said to be equivalent if there exists a Hopf algebra automorphism w of A such that w(at) = w(a)*i for all a c A.

22

1. Hopf Algebras

1.2.8* The Dual Hopf Algebra A'

As discussed in Subsect. 1.2.2, the dual space A' of an infinite-dimensional algebra A does not become a coalgebra by dualizing the multiplication of A. However, we shall see that A' contains a largest coalgebra A° and that A° is a Hopf algebra when A is so. Suppose that A is an algebra. As usual, we consider A' 0 A' as a linear subspace of (A 0 AY by identifying f 0 g c A' 0 A' with the linear functional on A 0 A determined by (f g g)(a g b) := f (a)g(b). For f E A' let A(f) be the element of (A A)' defined by A (f)(a b) := f (ab), a, b C A. Let now A° denote the set of all functionals f E A' for which A(f) belongs to A' 0 A', that is, there are functionals • fr, gi, gr E A', r E N, such that f (ab) = Eir=1 f(a)g(b) for all a, b E A. In this case, the element A(f) E A' 0 A' takes the form :\( f) => f O gi. Proposition 12. (1) If A is an algebra, A° is a coalgebra with comultiplication L defined above and counit E given by E(f) = f(1). (ii) If A is a bialgebra, then the coalgebra A° equipped with the convolution multiplication (22) is also a bialgebra. The unit of the algebra A° is then the counit E of the coalgebra A. (iii) If A is a Hopf algebra, then so is A° with antipode given by S(f)(a) = f (S (a)) . (iv) If A is a *-Hopf algebra, then the Hopf algebra A° endowed with the involution ( (0) is a Hopf *-algebra too. Proof. (0: First we have to ensure that A(A°) c A° 0 A°. Let f E A° and A(f) = >J O gi , where the functionals {L} are chosen to be linearly independent. Then we can find elements ai E A such that f(a) = We have gi (ab) = f(a)g(ab) = f (aiab) = fi (aj a)gi (b) which implies that gi C A°. Similarly, A° and so A(f) E A°0A°. Clearly, A° is a linear subspace of A'. The coassociativity of A is derived from the associativity of the multiplication of A by ((A0id)0A(f))(a0b0 c) = f ((ab)e) = f (a(bc)) = ((id 0 A) o A( f )) (a b c). The counit axiom is obvious. (ii): We show that A° is a subalgebra of A'. Let f g E A° with A(f) = gi and A(g) = Ei hi 0 ki . Then we compute f g(ab) = Ei f i hi (a)gi ki (b), so that fg E A°. The compatibility condition of comultiplication and multiplication of A' is easily derived from that of A. (iii): Let A(f) = gi for f c A°. Then we have S(f)(ab) = S(g i )(a)S(f i )(b) and hence S(f) E . We check one relation of the

L

Ei

hc

Ei

Ei

Ei

Ei

antipode axiom (the other follows similarly) by (m o (.5

id)

o

A(f)) (a) = E i (S(Mgi)(a) =

(a) fi (S(a (1) ) )9i (a (2) )

= E f (S(a(l))a(2)) = e(a)f (1) = 1,40 (a)e..40 (f) = ((n o e)(f ))(a) (iv) ia easily verified by using Corollary 11.

1.2 Coalgebras, Bialgebras and Hopf Algebras

23

The coalgebra (resp. bialgebra, Hopf algebra) A' from Proposition 12 is called the dual coalgebra (resp. bialgebra, Hopf algebra) of the algebra (resp. bialgebra, Hopf algebra) A. Equivalent characterizations of A' can be found in [Mon], Lemma 9.1.1, [Sw], Chap. 6, [A], Sects. 2.2 and 2.3, or [Jos], Sect. 1.4. In the Hopf algebra literature, A° is usually introduced as the set of those elements in A' that vanish on a two-sided (or right or left) ideal of finite codimension. Let A be a bialgebra. From the definition of the structure maps of A' it is clear that we have a dual pairing of the bialgebras A' and A given by (f, a) := f (a). If A' separates the points of A, then this pairing (-, •) is fel such that the set (CG )° for the nondegenerate. There are groups G group Hopf algebra CG (see Example 7) contains only multiples of E (see, for instance, Example 2.5 in [A], p. 89). In this case the dual pairing of (CO' and CG is obviously denegenerate. Let us consider a nondegenerate dual pairing (-, .) of two bialgebras (resp. Hopf algebras) 14 and A. By the nondegeneracy of the pairing, we can consider 1,1 as a subspace of A' by identifying f E Li with the functional (L.) E A'. Since (f, aa') = E(f( ,) , a)(f( 2), a') by (31), Li is then contained in A°. Comparing the structure maps of A' with the properties of the dual pairing (see Definition 5) we see that U is even a subbialgebra (resp. Hopf subalgebra) of A°. Similarly, A becomes a subbialgebra (resp. Hopf subalgebra) of U°. When dealing with the duality of Hopf algebras, one is faced with two problems. First, it may not be easy to determine all elements of A' , but one can often find Hopf subalgebras of A°. Secondly, in many cases one obtains a dual pairing quite naturally, but it may be degenerate or it is difficult to prove its nondegeneracy. For these reasons, the concept of dually paired Hopf algebras seems to be the proper setting of duality for quantum groups.

Example 12. Let A = C[x] be the Hopf algebra of polynomials in x with comultiplication determined by A(x) = x01+10x. (This is actually the Hopf algebra U (g) of Example 6 with g = Et.) One verifies that the algebra A' is the algebra C[[h]] of formal power series, where a series f = Er, ah nE CH17,11 corresponds to the functional on A given by f (E k bkx k ) Ek k!bkak. Then it can be shown that A' is just the linear span of functions hneAh, where n E No and A E C. See [PT], [LT] and [Mon], Example 9.1.7, for more details in this matter. 1.2.9* Super Hopf Algebras In the subsequent two subsections we give two variants of the above Hopf algebra definition. For a graded Hopf algebra A the structure maps have to preserve the graded structure and the algebra AOA carries a twisted product rather than the usual tensor product. Definition 6. An No-graded super Hopf algebra is a direct sum A = EDnEN0 An of vector spaces An such that the following conditions are valid:

24

1. Hopf Algebras

(1) A is an algebra such that Am • A n C A m±n for m, n E No and 1 G A0. (ii) A is a coalgebra such that AGA„) c EDi±i=n Ai 0 A i for n E No and e(A n ) = 01 for n ?: 1. : A —■ C are algebra hornomorphisms, where (iii) : A —0 A 0 A and the product • of A ® A is defined by (a a k) • (a'n a') = (-1)knaani a a , ak E Ak

an E An , a, a' E A. (42)

(iv) There is a linear mapping S: A —0 A such that condition (26) holds and S(A n ) C An, ne No. Let A = EDnEN0 An be such an No-graded super Hopf algebra and set Ao = EDn A2 n and Ai = EDn A2n+1) where Ô and I are the two elements of Z2 = Zi2Z. Then we have:

(i)' A is an algebra such that k • Afi C A-k- +t-, for k, f1, E Z2 and 1 E A. (ii)' A is a coalgebra such that A(A fi ) g ED-i+ i- =n Ai ® A3 for n e Z2 and E(Aï) = {O}. WO' : A —0 A A and E : A —■ C are algebra homomorphisms, where A 0 A obeys the product (42) with k, n E {0, 1 } . (iv)' A admits an antipode S such that S(A0) C Ao and S(AI) C A 1 • A direct sum A = A-6 e Ai of vector spaces Ao and AI which satisfies the conditions (i)'—(iv)' is called a Z2-graded Hopf algebra or a super Hopf algebra. Any No-graded super Hopf algebra is in particular a super Hopf algebra. To be precise, Definition 6 is the definition of an No -graded Hopf algebra, but we prefer to refer to it as an N o -graded super Hopf algebra in the following. Remark 1. Let G be an abelian group and A a fixed bicharacter of G, that is, a mapping A:GXG —0 C such that A(n m, k) = A(n, k)Aqm, k) and n+m) = A(k, n)A(k, m) for k,n,m c G. Then the concept of a G-graded Hopf algebra with respect to A can be defined in a similar manner. The Hopf algebra A has to be graded by G and the factor (_,J. ) kn in (42) must be replaced by A(n, k). In the case A(ft, k) = (-1)kn and G = Z2 we recover the Z2-graded Hopf algebra defined above. A Example 13 (Tensor algebra T(V) and exterior algebra A(V)). Let T(V) be the tensor algebra of a vector space V. The quotient algebra of T(V) by the two-sided ideal generated by the elements vOw+w0v, y, w E V , is called the exterior algebra of V and denoted by A(V). Both algebras T(V) and A(V) possess unique No-graded super Hopf algebra structures such that A(v) = v 1 ± 1 y and e(v) = O for V E V. For both algebras the comultiplication and the antipode on products of elements v1, v2, - • • , vn E V are given by Zi(vi v2 • • • vn ) =

E E

(---1)/(P)vp,,, • vp (k)

vp(k+ , ) - • • vp ( n) ,

(43)

v2vi

(44)

k=0 pErnk S(ViV2

vn

)

=

—

W(71+1)121 1/4

where i(p) is the length of the permutation p.

' • •

1.2 Coalgebras, Bialgebras and Hopf Algebras

25

1.2.10* h-Adic Hopf Algebras In this subsection we develop another variant of Hopf algebras Roughly speaking, h-adic Hopf algebras are obtained by taking the ring C[[h]] of formal power series as K and requiring that the algebras and the corresponding tensor products are complete resp. completed in the so-called h-adic topology. First let us describe the structure of the ring Cqh11. The elements of C[[h]] are formal power series f = Enc.° 0 an hn in an indeterminate h with complex coefficients. We define an addition and a multiplication in C[[hj] by setting

f + g := E(an + bn )hn, fg := n=0 n=0

E

arks ) hn ,

(45)

r+s--= n

ncc 0 an hn and g -= where f = ncc bn hn. One easily verifies that these operations turn C[[h]] into a commutative ring with unit. The ring C[h] of all polynomials in h can be considered as a subring of C[[h]]. Note that an element f = E ncc_0 an hn G C[[h]] is invertible in C[[h]] if and only if ao O. In particular, C[[h]] is not a field. Let V be a complex vector space. Then the set V[[h]] of all formal power series f = En°° 0 v n hn with coefficients vn in V is a CEM-vector space with the addition of two elements f, g Ell[h]] defined by the first formula in (45) and the left multiplication of f E C[[h]] and g E V[[h]] given by the second formula in (45). If A is an algebra over C, then AUhl] becomes an algebra over CHhi] with the product of two elements f, g E A [[h ]] defined by the second formula in (45). Next we briefly develop the h-adic topology and its properties. Detailed treatments with proofs can be found in [Boul] and [Kas], Chap. XVI. Let V and W be vector spaces over C[[h]]. The topology on V for which the sets {hn V +v InE No} are a neighborhood base of y E V is called the h adic topology. We shall denote by VW the completion of the tensor product space V Oc[Ehu W in the h-adic topology. Let us make these notions more explicit. Let V and W be complex vector spaces. For any nonzero f = E 0 vn hn E V[[h]] let n(f) be the smallest integer n such that v L O. For f = 0, we set n(0) = +cc. Then d(f, g) defines a metric on V[[h]] and the topology defined by that metric is just the h-adic topology of the vector space V[[1/]]. A sequence f( k) = (k) vn hn , k E N, converges to f = 0 vn hn in the h-adic topology of Vi[h]1 if and only if for any r E N there exists an index kr such that vS1k) = vn for n = 1,2, • • , r and all k . > kr . This characterization implies that in the h-adic topology the partial sums of any f E V [[h]] converge to f and that the polynomial space V[h] is dense in 11[1z11. Further, it follows easily that the vector space V[[h]] over CPU is complete in the h-adic topology and that V[[h]]6W[[h]] = (V ® W ) Rh]] . -

26

1. Hopf Algebras

If the complex vector space V is finite-dimensional, then V[[h]] is canonically isomorphic to V 0 CI[h]]. However, the reader should be aware that for an infinite-dimensional space V the set Vi[h]] is strictly bigger than V0C[Ih]], because the latter contains only sums E77 0 vri hn for which the linear span of all coefficients un, has finite dimension. Now we can define the h-adic variants of algebras, coalgebras, etc. by taking h adic complete vector spaces over C[[h]] and replacing the algebraic tensor products 0 in the corresponding previous definitions by the h adic That is, an h adic .Hopf algebra is a vector space A over completions C[[h]] which is complete in the h-adic topology and endowed with C[[h]l-linear -

-

-

and S : A +

the Hopf algebra axioms with 0 replaced by h-Adic algebras, h-adic coalgebras and h-adic bialgebras are defined similarly. Simple examples of such h-adic structures are obtained as follows: if A is an algebra, coalgebra, etc., then A[[h]] = AÏ:C[[h]] becomes an h-adic algebra, h-adic coalgebra, etc. with structure maps C[[h]]-linearly extended from A. We illustrate the importance of the h-adic completeness by the following Example 14 (Functional calculus in h adic algebras). Let A be an h adic algebra. Let a be an element of A and let f an ti E C[[h]]. The elements sk E nk =0 cn anhn, k E N, form a Cauchy sequence in the h-adic topology, because sk — ski E VA for k, k' > n. Since A is h-adic complete by the definition of an h-adic algebra, the sequence {s n } admits an h-adic limit —

A which satisfy

-

-

Enc°

cn anhn e A. Setting (for instance) f(h) = e h , we obtain an element f(ha) = e ha in A. If b c A commutes with a, we clearly have f(ha) :--=

ehaehb

= e h(a+b) . In particular, (ehar 1 = eh(—a) .

In Sect. 6.1 h-adic algebras will be constructed in terms of generators and relations. In order to clarify what this means, let us consider a set of generators, say x 1 , x2, • • • , xr , and a set of relations, say fk (x i , x27 • • • , xr ) = O. Then the h-adic algebra A with h-adic generators x l , x2, • - - , xr and the set of relations fk(x i , x2 , • • • , xr ) = 0 is defined as the quotient of the h-adic algebra C(x i )[[h]] by the closure in the h-adic topology of the two-sided ideal of C(xi){[h]] generated by the elements fk (xi, x2, • xr ). Being the quotient of the complete metric space C(x i )[{111] by a closed subspace, A is complete in the h-adic topology and hence is an h-adic algebra. A technical argument for the need of h-adic Hopf algebras is that they are necessary for a mathematically rigorous treatment of universal R-matrices for quantized enveloping algebras. A conceptual reason is that they provide the proper setting of deformation theory. In this book the word "deformation" will be used only in the naive sense in order to describe structures depending on (complex) parameters. However, the precise definition is as follows. Let A be a Hopf algebra over the complex field C. An h-adic Hopf algebra Ad over C[[h]] is called a formal deformation of the Hopf algebra A if Ad is mA (mod h) isomorphic to A[N] as a vector space over C[[h]] and ms4, and AAL 44 (mod h). This means that if a vector space isomorphism of

1.3 Modules and Comodules of Hopf Algebras

27

Ad and A[[h]] is fixed, then the multiplication and comultiplication of Ad can

be regarded on Al[h]] and there exist linear mappings m n : A 0 A —> A and zin : A A 0 A for any n NI such that mAd (a

b)

••,

b)h + m2(a b)h 2

mA(a b) + mi(a

i2 (a) h2 +

2i Ad (a) i(a) + A.(a)h

The simplest example of a deformation of a Hopf algebra A is of course the h-adic Hopf algebra A[[h]] itself whose structure maps are obtained as CPUlinear extensions of the structure maps of A. This deformation is called the null deformation of A.

1.3 Modules and Comodules of Hopf Algebras 1.3.1 Modules and Representations

Let A be an arbitrary algebra and V a vector space. By a representation of (pv of A to the algebra A on V we mean an algebra homomorphism L(V) of linear operators on V. That is, we have (p(aa + Ob) = a(p(a)+ 13(p(b), (p(ab) = (p(a)v(b) and v(1) = I for a, b E A and (1, 13 c IK, where I is the identity operator on V. The vector space V is called a left A module if there exists a linear mapping : AO V V, written as v(a 0v) = ai> v, such that (ab) t> y -= a (> (b t> y) and 1 (> y = v for all a, b E A and v c V or, equivalently, -

that o (m 0 id) and

cp 0 (id 0 cp) =

(46)

(p o (770 id) =- id,

where the first equality acts on AO AO V and the second on K0 V V. The map (p is then called a left action of A on V. Sometimes we shall write a.v or simply av for a i> v. Clearly, there is a one-to-one correspondence, given by the formula cp(a)v = (p(a 0 v), between representations and left actions of A on V. Similarly, a right A module is a vector space V with a linear mapping : V 0A V, called a right action of A on V and written as (v ®a) = v aa, such that v (ab) = (y a)
o ((p 0 id) = o (id m)

and (p o (id ® n)

(47)

id

or as the commutativity of the diagrams

VOA0A

id0rn

VOA

trp 0 id 1(P V0A

V

V0K lid 077

v®A

id

V0K j

V.

(48)

28

1. Hopf Algebras

Let V be a left A-module with action ço denoted as ço(a v) = av. We say that a set Ivi j E II of elements vi E V is a left A module basis of V if any v E V is a finite sum v = Ei aivi with uniquely determined elements -

ai E A. A vector space V is called an A bimodule if V is both a left A-module with action av and a right A-module with action va satisfying the compatibility condition (av)b = a(vb) for a, b E A and v C V. We shall freely use standard notions of representation theory of algebras. Let us recall a few of them. An intertwining operator of representations 99v and 99w on vector spaces V and W is a linear mapping A : V W such that kov (a)v = çow(a)Av for a c A and v c V. If V and W are viewed as left A-modules with actions cpv : A ® V V and cpw : A 0 W W, this is equivalent to A o c,ov = 99w o (id 0 A). (49) -

A is then called a morphism of left A-modules. If A is invertible, it is an isomorphism of A-modules. A representation cpv on V is said to be irreducible (and the corresponding A-module V is called simple) if V and {O} are the only invariant subspaces of V. Representations which are direct sums of irreducible ones are said to be completely reducible. If A is a *-algebra and V carries a scalar product, then a representation 99 of A on V is called a * representation if (99(a)v, w) = (I) ,(19(a*)w) for all a E A and v, W E V. Suppose now that A is a bialgebra. Then the comultiplication of A enables us to define a tensor product of representations. Let (p v and 99w be representations of A on vector spaces V and W, respectively. There is a representation ()av ow of A on the vector space V® W, denoted also by (p v 099w and called the tensor product of 99v and 99w, such that -

ço vow (a) := ((ça v 0 (19w ) o 46)(a)

E ,0„(a (1) ) O çow(a( 2)), a

C A.

The preceding formula defines a representation of the algebra A, since we have çovow(1) = (vv 0 ww)(1 0 1) = Iv 0 Iw = Iv o w and wvow(ab) = E vv((ab) ( ,) ) O v w((ab) (2) )

= E wv(a(l))vv(b(i)) cpw(a ( 2) )(pw(b ( 2) ) = vvow(opv.w(b), where we have used the relations (25). If ço is a finite-dimensional representation of a complex algebra A on a complex vector space V, then there is a canonical extension of (Jo to a representation (ph of the h-adic algebra AP]] on the space VE[h]] = VOC[[h]] given by (Ph (E n a n hn ) (v 01(h)) :=

(c,o(a n )v O hn f (h)).

(50)

The representation 9oh is reducible even if the representation (p is irreducible, since the subspace hVHhil is invariant under the operators iph(a), a E

1.3 Modules and Comodules of Hopf Algebras

29

But if the representation 92 of A is irreducible, then the representation (ph of A[[h]] is indecomposable (that is, it cannot be represented as a direct sum of nontrivial irreducible representations). 1.3.2 Comodules and Corepresentations

Dualizing actions of algebras on modules leads to coactions of coalgebras on comodules. Let A be a coalgebra. Definition 7. A corepresentation or, equivalently, a right coaction of A on a vector space V, called then a right A-comodule, is a linear mapping (,(3 :V V 0 A satisfying the relations

((p

id) 0 = (id

A) 0 (p

and

(id

E) 0 = id.

(51)

A left coaction of A on V is a linear mapping (i) : V A V such that (id

(p)

= (A id) () (p

and

(E

(52)

id) 0 = id.

The vector space V equipped with the mapping cp is then called a left Acomodule.

The two conditions of (51) are equivalent to the requirement that the two diagrams

V0A A

A

V

V K

id

V0K (53)

lid OE

V0A

V

V OA

V

are commutative. The diagrams (53) for a right coaction are obtained from the diagrams (48) for a right action by reversing arrows and replacing the multiplication m by the comultiplication A and the unit 7/ by the counit E. Note also that the equations of (51) and (52) which characterize right and left comodules are just the "duals" of the equations of (47) and (46) which define right and left modules, respectively. Example 15 (The comultiplication). Clearly, A is a right coaction and also a left coaction of A on itself. Indeed, in the case (p = A the equations (51) and the equations (52) become the coalgebra axioms (15) and (16). The right coaction LI of A on itself is called the right regular corepresentationof A. A Example 16 (Trivial corepresentation). The linear mapping (p : V --> V 0 A defined by w(v) --= y 0 1 is the trivial corepresentation of A on V. A Next we define some basic concepts on corepresentations. They may be viewed as dualizations of the corresponding notions on representations of algebras or as generalizations of concepts from group representation theory.

30

1. Hopf Algebras

Let (pv and cow be corepresentations of A on vector spaces V and W. A linear mapping A: V W is called an intertwining operator of (pv and (low (and a morphism of the corresponding right A-comodules) if cow o A =

(A

0 id) o (p v .

(54)

The set of such intertwiners A is denoted by Mor (coy, (pw ). We shall write Mor ((pv) for Mor (coy, (ic y ). If there exists an invertible intertwiner A of çov and (pw, then the corepresentations coy and cow are said to be equivalent and A is called an isomorphism of the corresponding A-comodules. A linear subspace U of V is called invariant under (p v if (p v (U) C U0A. The corepresentation (p i/ on V is said to be irreducible if V and {O} are the only invariant subspaces. Other concepts such as subcorepresentations, direct sums, etc. will have their obvious meanings. Let co be a left (resp. right) coaction of A on V. An element y of V is called invariant under ço if co(v) = 1 0 y (resp. co(v) = y 0 1). We shall say that a linear functional f on V is invariant under ço if (id 0 f)(p = fi (resp. (f id)(p = f 1 ). The next proposition characterizes finite-dimensional corepresentations in terms of matrices of elements of A. Proposition 13. Let V be a finite-dimensional vector space with basis e 1 , e2, • • • , ed and let co : V V A be a linear mapping. Let v --= (v ii ) be the d x d matrix of elements vij E A defined by the equations

(p(ei ) =

E

d k--=1

e k vk i , j =- 1, 2, • • , d.

(55)

Then (oo is a corepresentation of A on V if and only if A(Vii)

E

d

k.1

vik O vk i and

6 .1) = 6i j (

Ei,k

i j = ,

1, 2, • • • , d.

(56)

Proof. Since ((v) 0 id) o cio)(ei ) = e i 0 vik O Vki ((id 0 A) o (p)(e i ) = e O d(v ii ) and ((id 0 6) o („o)(ej ) = e i E(vii ), it follows that co satisfies the relations (51) if and only if the equations (56) hold. El

Et

Ei

The elements vij E A given by (55) are called the matrix elements of (p with respect to the basis el , e2 , • • • , ed . A matrix y = (vii ) of elements of A satisfying (56) is called a matrix corepresentation of A. Conversely, let u = be a matrix corepresentation of A and let V be an arbitrary d-dimensional vector space with basis e 1 , e2 , • - ,e d . Then, by Proposition 13, the linear mapping co: V —) V 0 A defined by (55) is a corepresentation of A on V. It is often useful to express notions of corepresentations in terms of their coefficient matrices. As a sample, we shall do this in the case of an intertwiner. Let (pv and (,ow be corepresentations on finite-dimensional vector spaces V and W. Let 7) = (vij) and w = (wii ) be the corresponding matrix corepresentations with respect to some bases. Let A: V W be a linear mapping

1.3 Modules and Comodules of Hopf Algebras

31

and let (a„) be the matrix of A with respect to these bases. Then A is an intertwiner of (pv and (pw if and only if Ei arjvjj = w„asj for all r and j. The set of such matrices (a„) is denoted by Mor (7), w). A matrix corepresentation y = (v ii ) of a Hopf *-algebra with antipode S is called unitary if the matrix elements satisfy the relations S(v ii ) = (yji )*.

Example 17 (Corepresentation of MN (C)' on CN ). Let A be the coalgebra MAC)' from Example 4. The N x N matrix (u i ) of elements uji introduced therein satisfies (56), hence it is a matrix corepresentation of A. By (55), this matrix corepresentation defines a corepresentation of A on CN . A Example 18 (Group representations). Let A be one of the Hopf algebras of functions on a group G from Examples 1, 2 or 3. Then, by the definition of the comultiplication of A, the equations (56) mean that vii(gig2) Ek Vik )Vkj(g2) for all gl , g2 c G and vii (e) = 5. Thus, the matrix corepresentations of A are precisely the matrix representations of G with matrix coefficients in A. Therefore, finite-dimensional corepresentations cp of the Hopf algebra A are in a one-to-one correspondence with finite-dimensional representations T of the group G whose matrix elements belong to A. In order to make this correspondence more explicit, we consider V 0 A as the vector space of V-valued functions on G by identifying an element E vi fi of V 0 A with the V-valued function Ei fi (-)vi on G. After this identification the correspondence (,o T is expressed by the formula

(p(y)(g) = T(g)v,

g E G, V E V.

(57)

Moreover, in the course of this the above notions on corepresentations reduce to the corresponding terms on group representations. Finally, let us discuss the condition that the matrix coefficients of T are in A. Clearly, it is redundant if A .9G) for a finite group G (Example 1). If A = Rep G for a compact group G (Example 2), it ensures the continuity of T. In the case A -= 0(G) of Example 3 this condition means precisely that the matrix elements of T are polynomial functions on G. Such representations of G are usually called rational. A There is also a Sweedler notation for coactions which has similar advantages as for the comultiplication (see Subsect. 1.2.2). If : V 0 A is a right coaction, one writes symbolically cp(v) for the element (p(v) E V 0 E(v (0) ) (0)

v (0)

v(1) ,

VE

(58)

A. Then the first condition in (51) reads as

(v(0) ) ( 1)

1)( 1 ) =

v(0) ® (v(0 ) ( 1 ) ® (v(0)(2)•

This element will be written as E v(0) 0v(1) 0 ,1)(2) . One has a similar notation for iterated applications of A and (p. The second condition in (51) says that

32

1. Hopf Algebras (59)

E V.

The corresponding Sweedler notations for a left coaction : V —> A 0 V are

(„(v)

E „,(_

(id 0 (p) o (p(v) (ZA 0 id) o

(v) =

(60)

V(0) 7

E v( _ 2)

v( _, ) vo,

The reader should keep in mind that terms v( k) with k 0 always belong to A, while v (0) lies in V. Let us assume now that A is a bialgebra. As in the case of representations, we then have a tensor product of corepresentations. If çv : V —> V 0 A and cw : W —+ W 0 A are corepresentations, then it is easily verified that the linear mapping wv o w : V 0 W —> (V 0 W) 0 A defined by

(pvow(v w)

=E v(0)

w(o) v(i) w ( ,) ,

V

E V, w E W,

(61)

is a corepresentation of A on the vector space V 0 W. The corepresentation wvow is also denoted by wv 0 ww. Strictly speaking, we should have taken another symbol for the tensor product corepresentation rather than (p v O (p w in order to avoid confusion with the tensor product mapping (p v O (p w : VOW —+ VOAOW0A which acts as (wv0ww)(v0w) =-- wv(v)0ww(w). But we prefer the notation (pv O ww because it is commonly used in the physics literature. In any case, it will be clear from the context what is meant. The tensor product of corepresentations is associative in the sense that the canonical isomorphism (V W) U f- -? V 0 (W O U) of the tensor products of underlying vector spaces provides an equivalence of the corresponding corepresentations. However, the corepresentations ww and ww wv are not equivalent in general (but they are for our leading examples of quantum groups!). These statements also remain valid for representations. Finally, let us note that the concepts and facts on right comodules developed above carry over to left comodules as well with a few necessary modifications. For instance, (55) must be replaced by (,o(e) = E, Vik O e k . Moreover, if : V V 0 A is a right coaction of a Hopf algebra A on V, then 7/) := (id 0 S) o p : V A 0 V is a left coaction of A on V. 1.3.3 Comodule Algebras and Related Concepts

In many cases a Hopf algebra coacts on an algebra such that the coaction respects the algebra structure. This situation is described by the following notion. We suppose that A is a bialgebra. Definition 8. An algebra X is called a right (resp. left) A-comodule algebra or, equivalently, a right (resp. left) quantum space for A if X is a right (resp. left) A-comodule such that the multiplication m: X O X X and the unit mapq:DC--X of X are A-comodule homomorphisms.

1.3 Modules and Comodules of Hopf Algebras

33

Let us be more explicit about these conditions. Suppose that X is an A. Then algebra and a right A-comodule with coaction (px : X —> X X 0 X is a right A-comodule with coaction çoxox given by (61) and K is a right A-comodule with trivial coaction (A(À) = À 0 1. That in and 77 are id) o (p xox comodule homomorphisms means by (54) that (pxo in = (in çox 0 ij = (71 0 id) o (pK . By (61), the two latter equations can be expressed and as (px(xy) = (px (x)(px(y), X , y E X, and (px(1) = 1 0 1, where the product of X 0 A is defined by (12). That is, we have shown that X is a right Acomodule algebra if and only if the coaction (px : X X 0 A is an algebra homomorphism. A similar statement is valid for left A comodule algebras. -

Example 19 (C[x i ,x 2 ,• • • ,xN] as comodule algebra of 0(G) or 0(MN (C)). Let A be either the Hopf algebra 0(G) from Example 3 or the bialgebra O(MN(C)) from Example 5 and let X be the polynomial algebra C[x i , x 2 , • • • , x N ]. Define algebra homomorphisms (p L : X —> A 0 X and A by (pL(xi) = E i u 0 x i and (pR(xi) = Ei Xj 0 14. Then çoR X X (resp. (pR) is a left (resp. right) coaction of A on X and X is a left (resp. right) A-comodule algebra, or equivalently, X is a left (resp. right) quantum space for A. In typical later applications, A will be the coordinate Hopf algebra of a quantum group and X will be the coordinate algebra of some quantum space. But a quantum space is a virtual object that exists only because of the algebraic structures of the A-comodule algebra X. For this reason, we shall abuse language and use the terms A-comodule algebra and quantum space synonymously. The corresponding notions of *-quantum spaces for a *-bialgebra A are easy to guess. A (left or right) * quantum space for A or, equivalently, a (left or right) A comodule * algebra is a *-algebra X which is a (left or right) A-comodule algebra such that the coaction (px is *-preserving (that is, (px(x*) = (px (x)* for x C X). Although the previous Definition 8 is more abstract and less practical than its reformulation given afterwards, it provides the right setting to develop similar concepts such as A-module algebras, A-comodule coalgebras, A-comodule bialgebras, etc. These notions are defined simply by requiring that all structure maps are A-module or A-comodule homomorphisms, respectively. Using the above formulas (49) resp. (54) and (58) resp. (60), it is not difficult to rewrite these abstract definitions in terms of the Sweedler notation. As a sample, we state these reformulations in the case of A-module algebras. A left A module algebra is an algebra X which is a left A-module such that m : X X —> X and i : • K X are left A-module homomorphisms. If a o x denotes the action of a E A on x E X, the two latter conditions mean that ac' (xy) = E(a(i) x)(a( 2) y) and a r> 1 = e(a)i. -

-

-

-

34

1. Hopf Algebras

Likewise, an algebra X is called a right A module algebra if it is a right A-module with action, denoted by a, such that -

(xy)
(x

a(l))(Y a(2))

and

1
Example 20. Let X be a left A-module algebra for A = U(g). Since d(a) = a 0 1 1 0 a for a E g, we have a r> (xy) = (a x)y x(ar> y) for x,y E X.

That is, each a E g then acts as a derivation on X. Example 21. Let A = CG. An algebra X is a left A-module algebra if and only if the group G acts as a group of automorphisms of X. 1.3.4* Adjoint Actions and Coactions of Hopf Algebras

Let A be a Hopf algebra. For elements a, b E A we define adL(a)b = AdL (a) =

E ambS(a(2)), adR(a)b = E S(a(1))ba(2)

E amS(a( 3))

a ( 2 ) , AdR ((;) =---

7

E a(2) 0 S(a( i ))a( 3).

The mappings adL and adR are called the left and right adjoint actions, respectively, and the mappings Ad', and AdR are called the left and right adjoint coactions, respectively, of the Hopf algebra A on itself. That these are indeed actions and coactions follows from Proposition 14. (i ) adL turns A into a left A-module algebra and adR turns A into a right A-module algebra. (ii) AdL, is a left coaction and AdR is a right coaction of A on itself.

Proof We carry out the proof in the case of adR. Set a 4 b =-- adR (b)a for a,b E A. Clearly, we have a < 1 = a and 1 < b = e(b)1. Further, we compute

aa(bc) = E S((bc)(0)a(bc)( 2) = E S(c (i) )1S(b i) )ab(2) 1c(2) = (a4b)ac (62)

and

(a 4 C(0)(b

c(2) ) =

E S(c( i ))ac(2)S(c( 3))bc( 4) = E S(c( i))ae(c(2))bc( 3)

= E s(cm )abc(2) = (ab) 4 c. These relations show that adR is a right action of A on itself which turns A into a right A-module algebra. LI

In general, the algebra A is not a left (resp. right) A-comodule algebra with respect to the coaction AdL (resp. AdR). This matter will be taken up again in Sect. 10.3. However, A is a left (resp. right) A-comodule coalgebra with respect to the coaction AdL (resp. AdR). Remark 2. If V is a bim.odule of A, then the definitions of adL(a)b and adR(a)b make sense for a E A and b E V and the reasoning of (62) remains valid.

1.3 Modules and Comodules of Hopf Algebras

35

Hence ad', and adR are then left and right actions of A on the bimodule V, A respectively. This general setting will be used in Chap. 13.

Let us consider the above notions in two classical situations. Example 22 (Conjugation representations of groups). Let A be the Hopf algebra 0(G) from Example 3, Then we have

Ad L (f)(g, h) = f (ghg -1 ) and AdR(f)(g,h) = f (h -l gh) and g,h E G. It suffices to verify these formulas on the generators. We shall do this in the case of AdR and compute for f

E 0(G)

AdR(14)(g, h) =

Eic,1 (g)(S (uik )u )(h) = Ek,i gki (h -l ) i kho

=

gh) ii = u13 (h -1 gh).

The corepresentation Ad R of A corresponds by (57) to the conjugation representation of G which is given by (T (h ) f)(g) := f (h - 1 gh).

A

Example 23 (Adjoint representations of Lie algebras). Let A be the Hopf algebra U(9) from Example 6. Since ,i(a) = a 1 + 1 a for a E g, we get adL(a)b = [a, b] and adR (a)b = Ha, b] for a, b E g. That is, on g the mapping A adi, is just the adjoint representation of the Lie algebra g. 1.3.5* Corepresentations and Representations of Dually Paired Coalgebras and Algebras

Let A be a coalgebra and ti an algebra. We suppose in this subsection that U x A

C is a (possibly degenerate) bilinear form such that

(fi f2,(a» = (fif2, a) and (1,a) = e(a), f 1 ,f2 EU, a E A. Proposition 15. If : V on a vector space V, then fo(nv := ((id

/

f) 0 (p)(v)==•-

A is a corepresentation of the coalgebra A , (0) ( f , No),

f E U, v E V,

defines a representation of the algebra Li on V. Proof. By the second equality of (63) and (59), we have

((1)v

No)(1 1

= E v(0) E(v(1) ) = V.

From the first equality of (63) we obtain

43(fif2)v = E v(0) (fi f2, v(1)) =

=E V(0) (fi V(1))(f27 V (2 )

(63)

v(0)(fi f2,(v(i.)»

Eo(fov (0) (f2,v( 1) ) =„b(foo(f2)v.

(64)

36

1. Hopf Algebras

There is an analogous result to Proposition 15 for left A-comodules and right U-modules. It says that any left A-comodule with coaction is also a right U-module with action li)(f)v = ((f 0 id) o

lk)v

E v(0) ( f , v(_ 1)

Note that not every finite-dimensional representation of U is of the form (ià for some corepresentation ça of A (see the end of Subsect. 4.4.2), not even when U and A are Hopf algebras and the pairing (., -) is nondegenerate. We now specialize the preceding to the case when the coactions ça and are given by the comultiplication of A. For f C U and a E A, we define

f.a :=

E a (i) (f,a (2) ) and a.f

(f, a ( 1 ) )a( 2 )

(65)

Proposition 16. A is a bimodule for the algebra U with left action f .a and right action a. f of U on A. If (•,.) is a dual pairing.of two bialgebras U and A, then A becomes a left and right U-module algebra in this manner.

Proof. By Proposition 15 and the corresponding result for left comodules, the vector space A is a left and right U-module. It is a U-bimodule, since (fi .a).f2 =

(f2 , a ( 1) )a( 2) (fi,a( 3 )) = fi .(a.f2 ).

The second assertion follows easily from (31) and (32).

0

Let us turn to another particular case. Suppose that U is the dual algebra A' and the pairing (., •) is the evaluation of functionals (f, a) := f (a), a E A, f E A'. Then the conditions (63) hold by definition (22) of the product of . Therefore, by Proposition 15, each corepresentation ya of the coalgebra A induces a representation çO of the dual algebra A'. Finally, let us suppose in addition that the bilinear form (•, •) is nondegenerate. Then, by considering matrix coefficients with respect to some bases, we easily conclude that the corepresentation cp of A is uniquely determined by the associated representation ça of U. That is, we obtain a one-to-one correspondence between corepresentations of A and certain (!) representations of U. It is not difficult to see that this correspondence preserves standard notions of representation theory such as direct sums, tensor products, etc.

1.4 Notes The notion of a Hopf algebra first occurred in the work of H. Hopf [Hop] on algebraic topology. The basic early monographs are [Sw] and [A]. A modern treatment is [Mon]. Hopf algebras appeared also in the work of G. I. Kac [K] in the 1960s. The concept of a dual pairing was introduced in [Tail. A nice discussion of dual pairings is in [VD1). Only very few interesting examples of noncommutative noncocommutative Hopf algebras were known before quantum groups were invented (see [1(P], [Sw], pp. 69-70, [Pa]).

2. q-Calculus

Quantum groups and their representations are closely related to the socalled g-calculus (g-numbers, g-factorials, g-differentiation, basic hypergeometric functions, and g-orthogonal polynomials). This chapter gives a brief introduction to these topics. The notions and facts developed below will be needed at various places in the book, but they are also of interest in themselves.

2.1 Main Notions of q-Calculus 2.1.1 g-Numbers and g-Factorials For any nonzero complex number q, the q - number [a]q ,

e ah e—ah [

a] q

E.--

[a] :=

=

qq

where q = exp h. Clearly, limq, 1 talq

[[a]],/

[[a]] :=

eh

sinh ah sinh h '

e—h

is defined by

(1)

a. We shall also use the expression

1 — qa

1

a E C,

=q q

(a-1)/2)[,thi

g 1/2 .

(2)

A number of relations of these g-numbers can be derived by using properties of the exponential function. For m, n E N and a, b, c E C we have

[m] =

+

[m + n] = qn [m] +q

3+•••+

(7n 1) ,

[71] = n [m] + qtm[n],

[m — n] = qn [m] — qtm [n] = n [m]

0 = [71] =

[n] ,

[a][b — + [b][e — a] + [c][a b],

[2][n — 1]

[n

2] .

The above equations also remain valid when q is considered as an indeterminate. Then any g-number [ml q , m E Z, belongs to the space Z[g, of Laurent polynomials in q with integral coefficients. Setting q = exp h, we have that [m] q m (mod h). If m E N, then we introduce the g-factorial [mi g ! [m]! by setting

38

2. g-Calculus

[ml! := [1][2] • • • [m],

[OP := 1.

The expression

(a;

:= (1

-

a)(1 aq)(l

-

aq2 ) • • • (1

-

(a; g)0 = 1,

agn -1 ),

(3)

is extensively used in the theory of basic hypergeometric functions and in combinatorics. It is closely related to the g-factorials, since we have n -m(m-1)/2

q2)7n (q2 q2 )7n

[m]!

(4)

`1(1

= 0 if n = m ± 1, m ± 2, •. For q < 1, we From (3) we see that (q- n ; define oc = ( 1_aq3-1 ) . (a; q),„0 := lim (a; ( 5) A- j= 1

Since q < 1, the infinite product in (5) converges for all a E C and defines an analytic function on C. It is clear that

(a; q) n

(a; q),3

(6)

(agn; g) 00 "

The latter relation allows us to define (a; q)

n E N, by

(a; q)„ 1 n := (a; q) (aq- n ; q)„„ = (aq- n ; O n '

( 7)

-

By direct computations one verifies the following useful formulas:

(a; q) N +, = (a; q) N (aq N ; q), (a; g) N a ) (a; ON-r = -1 1-N. ) /Or (a q (a;orN = (a; qr )N (aq; qT) N (aqN;

(a; q)r(aqr (a;

q) N

ON

(8) (9) (9)

(aqr-i ; qr)N7

(a; qMqa -1 ; (aq-N; q)r = (a-lql-r;

(10) _ Nr

q

(11)

where N and r are positive integers. Note that for any a E C we get (a; q -i )n = (a -1 ;ong ----n(n-1)/2

r.

(12)

This equation implies that (.7-2 ; .7 -2) 71 = tnp( - 1)n (1 e)fl q-n(n+3)/2 .

(13)

Next we turn to relations between expressions (a; g) n with different parameters q. We have (a2 ;q2 ) n = (a; g)n(-a; On and, more generally, = (a; g) n (af; q) n (ac2 ; q) n • • - (aer -1 ; g) n , where

e2-ri/r.

2.1 Main Notions of q-Calculus

39

The equalities (8)-(11) for the expressions (a; g) n lead at once to the corresponding relations for g-factorials. Later the following equalities will be used: ( f12N+2. 2\ q Jr -r(r-F2N- 1/2 [N y (14) [N + r]! = ]! (1 q2 )r [N

_ r ]!

[N]!(_ i y ( 1 q 2 ) r nr(r-2N-3)/2 • (

7-

+oo:

By (3), (14) and (15), one derives asymptotics for N

[N

+r!

[N

iir g -r(r-1)/2 ,

]

(15)

2N; q2) r

filn - rqr(r -1 )/ 2 ,

[1■7]!

[NJ!

where 1g < 1 in the first relation and II > 1 in the second one. There are particular properties of the g-numbers when q is a root of unity. Let d be the smallest positive integer such that q d = 1 and let if d is odd

e= d

e = d/2

and

if d is even.

One easily verifies the assertion of the following Proposition 1. If d and e are as above and n E Z, then [ri] q = 0 if and only if n 0 (mod e).

2.1.2 q Binomial Coefficients -

The q-binomial coefficients are defined by the formula

L

n m

=

(TO. (q;

[7nd

[n] q i / 2 !q ( n - m ) "1 / 2

{rni q i / 2![n - m] q i/2!'

-m

(16)

wheren,rnN 0 and n > m. There is a close analogy between the usual binomial coefficients and their g-analogs. Many identities of the usual binomial coefficients have their counterparts for the g-binomial coefficients. For example, [n + 1 1 in

q = [m n- 1 ] q

+e

[n rn

= q n- m+1 [ n L m - 1]

Proposition 2. If v and w are noncommuting variables satisfying the relation vw = qwv, then we have (V ±

w)

n

m] m n-m V

[n

E

[ n1 m=0 LM,

V

Tri

W

7/ 77Z

(18)

1

If q is a primitive p-th root of unity and p is odd, then (v + w)P = vP +w 7 .

Proof. Formula (18) can be proved by induction on n using the first identity in (17). The second assertion follows from (18).

40

2. q-Calculus

Proposition 3. For 0
[n

q(k_8)(m_s) [ k] s

s=o

q

n k [m - s]

(19)

Proof. Consider the free algebra generated by elements v and w subject to the relation vw = qwv. Using formula (18), we expand both sides of the identity = 01+ ok( v+ on-k and compare coefficients of vniwn- m. (7) + Proposition 4. The q- binomial coefficients satisfy the relation

E(_1 )k 71 k

k=1

—flk(k+1)k /2 = 0.

(20)

q

Proof. By the second relation in (17), the left hand side of (20) is equal to

E( _, ) k g _ny k+1)k,2({n - 1

+ q n-k [: 1

k.

q)

where the first summand in the brackets is set to zero if k = n and the second summand vanishes if k = 0. This expression coincides with n-1

E(

q -nyk-wki 2 [ n - 1

E( i ) c q —n(k—ly (k-1)/2

q

k=0

k=1

[n - 1 k-1

which clearly vanishes.

'

El

2.1.3 Basic Hypergeometric Functions

In 1846 the German mathematician E. Heine introduced the series

2çoi (a, b; c; q7 z) =

%---, °° (a; q) (b; q) n, n=0

(c; O n,

zn (q;

(21)

where c 0, -1, -2, • • •. If a or b is a negative integer, then this series is terminating and becomes a polynomial in z. The number q is called the base of the series (21). For < 1, this series converges absolutely if izi < 1. Since limqi (1 qa) (1 = a, the series (21) converges termwise to the classical hypergeometric series

2Fi(a,b; c; z) =

(a)r (b)

c° I-1 (a + 41-'0

zni

r(c + n)

n.

when q 1 and < 1 . If lqj > 1, then using (12) we can transform (21) into the series zol(a', ; ; q 1 , ) with base q - 1

2.1 Main Notions of q-Calculus

41

The function defined by the series (21) is called the basic hypergeometric function (or q-hypergeometric function) 2 (p 1 • It can be analytically continued outside of its domain of convergence. More general basic hypergeometric functions are defined by the series rçOs(ai, a2, • • •

lat.;

bl, b2, • • •

60 .9

b$ ; q3

b2, • • • , bs '

On(a2;q)n -..( a r ; On (( l)nen-1)/2)1+8

(b1;072(b2;

• • -

n (q Z; q)n,

(22)

;

In particular,

rçor-i(ai,a2, - • • ,ar;b1, b2, " ,br-i; q, z) cc n Z (aii q)n(a2; On • (ar; •• -On (br-1; (q; nt2 -0 (bi; q)n(b2;

(23)

Again, if at least one of the numbers al, a2, • • , a,. is a negative integer, then the series (22) terminates. If 0 < II < 1, then the series (22) converges absolutely for all z if r < s and for all lzi < 1 if r = s + 1. It also converges absolutely if q > 1 and z < Ib1b2 • • • b3 1/laia2 • • • ar k If the series (22) does not terminate, then it diverges for z 0 if 0 < II < 1 and r > s + 1 and if iql > 1 and izi > 1b1b2 • bsi/laia2 • ad. 2.1.4 The Function içoo(a; q, z)

By (23), we have

oc (24) n=0

(TOn

From the expressions for i wo(a; q, z) and i(po(a; q, qz) we obtain ivo (a;q,z) i (po(a;

- vioo (a;

q l qz) = (1 —

a)Z

q, z) - a “po (a; q, qz) = (1 -

viagaq; q, z), i ço0(aq; q, z).

These relations yield the equation (1 - z) i wo(a; q, z) = (1 - az) “po(a; q, qz) and so (az; , ) i(poo,; q, qn z). (25) iwo(a; q, z = (z; qn If Iqi < 1, then i çoo (a, q, qn 1 when n cc. Therefore, by (25), 00 00 (a; q) ri 1 azqr (az, 0 00 z = (26) içoo(a, q, 1 zqr (z; q),, (q; n=0 r=0 when Iql < 1. The latter formula has some interesting corollaries. One is

1.400(a; q l z)vpo(b; q, az) = içoo(ab; q, z).

(27)

Formula (26) implies that 1(10 0( 1 ; q7 z) = 1. Therefore, by (27),

içoo(a -i ; q, az) = liwo(a; q, z)} -1 .

(28)

2.1.5 The Basic Hypergeometric Function 2 (1, 1 For many reasons it is useful to have formulas relating hypergeometric functions with different parameters. For the function 2 cp 1 (a, b; c; q, z) the transformation formula

q)(az; 2(P i(a,b; c; q, z) = (c; q)„(z; q)„

„

(29)

holds when WI < 1, zI <1, ibl <1. Indeed, using the formulas (6) and then the equality (26), we get

(a; q) n (b;

2(pia,b;cqz) n=„3 (c; q) n (q; q) ri z

n

(b; q)

(a; q),. (cqn ; q)„

(c; q)

(q- q)n(bq n q) c°

7r0 =

zn

(b; q)„ '12"-, (a; O n zn (c/b; q),bm qmn (c, q) 00 n=0 (q; L s m=0 Ls (q; (b;q)„ x-N c") (c b; q),bm (azqm ; q) (zqm ; q)„ (c; oc rn=0 (q . q) m (b; q)„(az; q)„

(c I b; q),,(z;

(c; q)„(z; 0 00 ,rn =13d (az; q),,(q, q),

bm

This proves (29). In a similar way, one obtains the relation 2(pi (a, b; c; q, z) = (abz (z;

c; q)

24)1 (c/a, clb; c; q, abz

(30)

If a =- q - n, n E No, then we also have (see [GR])

2(Pi(q -n )b;e;q)z) = (— X 2V1(q—

i)n q —n(n-F1)/2

( b ;

(C; 01: 71 7 q —n-F1 c-1 ; q —n-F1b-1 ; ql cqn-F1

Az) .

(31)

There exist several summation formulas for the series 2 w 1 . The most important one is 2 ço i (a,b;

q, el ab) =

(c I a; q)„(c1b; q)„

(c; q)

I ab;

(32)

where ici < labl, II < 1. In order to prove this relation we first use (29) and then (26) and compute 2P1 (a, b; c; q, c I ab)

E (a; q) n (b; n=0

(C; q)n(q;

c \n oLtb)

(b; q) (cl b; q) cc \'-‘ c ") (c ab; q) n bn (c; q) 00 (c I ab; q) c.. (q; q) n (b; q). (c b; q). (c I a; q). (e; 47) 00 (c I ab; q). (b; q).

(c I a; q),,o (c b; q). (e; q) ,,,,o (c ab; q). •

2.1

Main Notions of q-Calculus

43

In the case b = frn the formula (32) can be written as 2491(a, q- n;c;q, qn c I a) =

(c I a; On (c; q)„

(33)

Further, it follows from (31) and (33) that 2401(a, q -n ; c; q) =

(c I a; On (c;

an.

(34)

In what follows, we shall also use the notation 24P1(a7 )3; 'Y; q, z) := 2(,01(e,e;q s ;q, z). 2.1.6 Transformation Formulas

for

(35)

3(p2 and 4403

Transformation relations for the functions 3(092 and 4(p3 are of a great importance for the theory of Clebsch-Gordan and Racah coefficients of the quantum algebra Uq (su2). The fundamental transformation formula for the function 3(192 is 3402(q -n ,a, b; c, q, q) (c I a; On n (c; on

a 3 (122 (q- n,a, d 1 b; aq- n+1 c, d; q,bq I

(36)

A repeated application of this formula leads to the relation 3 (p 2 (q - n,

qn - 1 abcd, bz; bc, dc; q, q)

( b) fl (ac; q)n (ad;

= C)

a)

(bc; q) n (bd; q)n

3(102(47 -n qn-l abCd y az; ac, ad; q, q).

(37)

For the function 4 w3 , we have the transformation formula 4(P3

d, e, f 'q'q

bdC) n (aql -71 e; On (aql - nif;

q -n 7

a, dib, die

4(1°3 (d, aql- n I e, aql- n f ;q' q) (38) which holds when the condition abcq l- n = de f is satisfied. The proofs of (36) and (38) are technically complicated and we refer the reader to the (e; q)n(f ;

monograph [G13].

44

2. g-Calculus

2.1.7 q-Analog of the Binomial Theorem The classical binomial theorem says that (1 1F0( -n; x) for n E N. The expression

2F1 (-n,c,c;

=

(1 - x) 7: := (1 - q -l x)(1 q -2 x) • - • (1 - q - nx) = (q - nx;q) n . can be regarded as a g-analog of (1 - x)'1 . In order to express (1 means of the q-hypergeometric function i vo, we first state that

(z;

En3 =1

j( j -1)/

3 [3] n-1) (

2

x) -

(39)

xrq by (40)

.

Indeed, using (6) and (26), one derives

(z; (zqn;q),,

(z;

Ez =o

(q -71 ; q) i zigin

f

(q;oi

d,0

qn - i)

q

3 ,0

(q;0 3

(q; q)

=

(-1)i (q;

q

x--k cic .Z 3 q"

(q; q):7(q; q)n-j

This gives (40). From (39) and (40) we obtain

(1 -

n n (_xyqr(r-2n-i)/2. xr = Y-7 d-dr=0 r Jq

(41)

Applying first the relation (9) for a = q and then (24), we get

(q -n ; (1 -

x) = 2-dr=0

r

„ X = i c,00 (q - n;q,x). (q; q) r

(42)

Formula (42) is a q-analog of the binomial theorem. Motivated by the equality (42), we define (1 - x) qa := i çoo (g - a; q, x) for arbitrary a E C and II <1. We also define (b - x) qn by setting (b - x) qn := b'1 (l - x I

2.2 q-Differentiation and q-Integration 2.2.1 g-Differentiation For a fixed complex number q z4 1, the q-differentiation operator Dg is defined by

f (x) = Clearly, Dg dlds as q side of (43) we obtain

1(x)

-

f (qx)

(43)

1. Applying the Taylor formula to the right hand

2.2 q-Differentiation and q-Integration oo

1-)n

(q—

D q f (x) n=0

(rt

n

dn+1

dx n-Fi

+

45

(44)

f(x) ,

provided that the expression on the right hand side of (44) exists. One easily verifies that

Dq xa

[[a]]xa -1 , Dq (1 bx) qa= b[[a]](1 bqx) qa -1 ,

a, b E C,

(45)

where [[a]] = (1 - qa)/( 1 q ) = q (a -1) / 2 [a] q 1/2 as defined by (2). For the product of two functions fi(x) and f2 (x) we have

Dqffi(x)f2(x)}

11(5)12(x)

fi(qx) f2(qx)

- q) f2(x)f.h(x) - fi(qx)} + f i(qx) { f 2 (x) - f2 (qx)}

x(1 - 07) Thus we obtain the following q-analog of the Leibniz rule:

{ fl(x) f2(x)} = f2(x)pq fi(x) + fi(qx)Dq f2(x).

(46)

From formula (46) it follows by induction on n that -n(n-1)/2 n x - nq Tri, 1)n (q Dqn f (x) = La ‘"‘

o

[

n (-1)mqm(m-i)/2f(qn-mx).

q

In particular, we have

f (x) = (q — 1) -2 x -2 q - 1 If (q 2 s) — (1 + q) f (qx) + qf (x)} Using this formula one verifies that the function y = 21 (a, b; C; q, x) satisfies the second order q-differential equation

x(qc - qa+ b+ 1 x)Dy

[[c]] (q' [[b + 1]] + q b ja,}1)x) D q y Uanbfly = 0.

This is a q-analog of the second order differential equation for the ordinary hypergeometric function 2 Fi(a, b; c; x). The equality

f (x) = f (a) +

(x

a) pel f (a) + [1 11P

(47)

[[2]]!

q

is a q-analog of the classical Taylor series expansion. Here (x - a) qn is defined in Subsect. 2.1.7, :=-- [ 11] • • • [[m]] and Dqn f (a) = {Dqn f (x)}x--.a. Sometimes, the q-differentiation operator is defined by

PTO

[

5) q b, f (x) = f(q _ 1)x

.

(48)

Definition (48) of the g-differentiation operator corresponds to definition (1) of the g-numbers, because we have Aix' = [n] q xn-I . The g-derivative (48) can be expressed in terms of the g-derivative (43) by /5,/ f(x) = For the operator

A

PO,

where

y := q x.

(49)

the g-analog of the Leibniz rule takes the form

Aff i (x)f2 (x)} = f2 (g -l x)b q h(x)

(50)

h (qx) f 2 (x).

2.2.2 q Integral -

g-Integration will be defined as the inverse operation to g-differentiation. If Dq F (x) = f (x), then we have F(s) - F (qx) = (1 - q)x f (x) and hence j = 0,1,2,- • .

F(qx) F(q3+1x) = (1 - q)q3 x f(qi x),

Summing these relations over j = 0,1, 2, - • - , n - 1, one gets n— 1

F(s) - F (qn x) = (1- q)x

j=0 qi f (q 3 x).

Suppose that 0 < q < 1. Assuming that F (qn x)

F(s) - F(0) = (1 - q)x

F(0) as n ---4 cc we obtain 0 ql f (q 3 x).

Thus, for 0 < q < 1 the q-integral of a function f(s) on the interval [0, c} is defined by OC

f (x)d qx = c(1 - q)

CC

E ql f (q3 = E(x,

-

+1) f (xr)1

(51)

j=-0

where xr

. For the interval [c, oc) the definition of the g-integral is f (x)d q x = c(1 q)

E

f (q -

(52)

j =1 For arbitrary positive numbers a and b we set

fa

a

f (x)d qx = f f (x)d q x - f f (x)d qx.

Putting c = 1 into (51) and (52) and summing these equalities we obtain for the g-integral over an infinite interval the expression 00

r f (x)d q x - (1 - q)

E

q.) f (q ) ).

(53)

i=-00 The formula for g-integration by parts can be derived by g-integrating both sides of (46). In this manner we get

12(x)(Dg fi (x))dqx = 11 (5)12 (5) - f l (qx)(D g f2(x))d qx.

(54)

2.2 q-Differentiation and q-Integration

47

2.2.3 q-Analog of the Exponential Function

There are the following three q-analogs of the exponential function: co

Zn . q; q . n

eq (Z) :=

1z 1 <

(55)

(—z ; q),,,

(56)

= vp0(0, q, z) = (z; 02 ,

ntiO n n(n-1)/2

zn = 0,0( q, —z)

Eg(z) := E n=0

(q;

co

Eg (z) :=

yn

E [ri] !

(57)

n=0

where II < 1. From (55) and (56) we derive that e g (z)Eg (—z) = 1. If the variables x and y commute, then the relations eq (x)e q (y) = eq (x + y) and Eg (x)Eg (y) = Eg (x ± y) are not true in general, but one can show that 00

,72n„

eg (x + y)eV(x)eV(y) =

(1 +

— qn(s +

n=0

.

Multiplying the series of eq (x) and Eg (y) and regrouping the corresponding terms it follows for commuting variables x and y that eq (x)Eq (y) = 1 +

(x+Y) 1 q+ (qi q)i

+..

Proposition 5. If x and y are variables such that xy = qyx, then we have

eq (x)e q (y) =-- eq (x + y) and Eg (x)Eg (y) = Eq (x + y). Proof. The first identity is easily obtained from (18) and (16). The second follows from the first, because the relation eq (z)Eq (—z) = 1 implies that Eg (x y) is the inverse of eq (—x — y) and that Eg (x)Eg (y) is the inverse of eq (—y)e q (—x) = eq (—x y). Using the q-exponential functions we can introduce q-trigonometrical functions by 1

00 (_ i )'l 52n+1

sing x := — (e (ix) —e q ( —ix)) =

2i q

E

n=0

(q;

) 2n + 1

co (_ i )tm 52n 1 cosq x :=-- — (e q (ix) + eq ( —ix)) --= E 2 n=0 (q; q)2n 1 Sin g s := — (Eg (ix) — E g (—ix)) 1 2i Then we have the identities

Cosgx

1 -(Eg (ix) + Eg (—ix)).

sing X Sing x + cosg X Cosgs = 1,

sing x Cosgx = Sing s cosg X.

The functions cosg (ws) and sing (wx) are linearly independent solutions of the second order q-differential equation

w2

Dq2 f

q ) -2 f

= 0.

Similarly, the functions Cosg (ws) and Sing (wx) are solutions of the qdifferential equation Dq2 f (s) + w 2 (1 - q) -2 f (q 2x) = 0. 2.2.4 q-Analog of the Gamma Function The q-gamma function fg (I.) is defined by the q-integral

L

CO

Xv— 1 Eq (— —

OS)Ci q X

=

(58)

q —v(v-1)12 fq

where Re u> 1 and lql < 1. Carrying out the q-integration by parts we get

L

oc

xv -1 Eg (- (1 - q)s)dgs xv = — Eg (- (1 - q)s) -F

[[]] v

r svEq(_(,_ q)s)dqs.

[[v]] o

(59)

From the equality eg (-x)E g (s) = 1 we derive Eg (-x) -) 0 when s +oc. Hence the first summand on the right hand side of (59) vanishes and we get

fg (v + 1) = pErg (v). In particular, if u = n E N, then fq

(n ) =

- 111! = (q; q)n-i

If F(u) is the "ordinary" gamma function, we have rg (v) q / 1. The q-gamma function can also be represented as

Fg (v) = (1 - 0 1-11 (Qq)c°

(q ,q) 00

(60) (I.) when

(61)

where WI < 1. For 1,71> 1 we define fg (v) := For the study of Clebsch-Gordan coefficients of the quantum algebra Ug (512) it is convenient to use the modified q-gamma function given by

î'q(u)

q-(1)--1)(1)-2)/2r42(v).

(62)

This function has the properties î(v) = Aig (v), î'q (n) = [n - 1 ]! , n E N, and Pg (v -I- 1) = [v]fg (v). (63)

2.3 q-Orthogonal Polynomials By means of formula (63) one can show that if y

Pq (v +1)/Pq (v) Consequently, for y

E

Pq (y - n + 1)

E

N, then

-Pq (m + 1)/ 1 q (m).

N, and n

Pq (v + 1)

-m, in

49

E

N we have

( 1) n

Pq (m + n) Fq (m)

(64)

The Stirling formula describes the asymptotic behavior of the classical gamma function. Its analog for the q-gamma function is

pq(v ± 1) q -v(v-i)/2 (1 _ q2\- exp(-Cq )

if

v

+cc,

(65)

where 0 < q < 1 and Cq = EZ`L o ln (1 _ 2(k +1)\) For the quotient of q-gamma functions one has the asymptotic formula

Pq (v

+ a)

q2u-s)/2(1

Pq (y)

2.3 q-Orthogonal Polynomials q-Analogs of classical orthogonal polynomials play an important role for the study of representations of quantum groups and q-oscillator algebras. In this section we consider some important q-orthogonal polynomials in the case when q is a real number. 2.3.1 Jacobi Matrices and Orthogonal Polynomials

Orthogonal polynomials satisfying three-term recurrence relations are connected with linear operators which can be represented by Jacobi matrices. Let {a n c N0 } , { bn I n c No }, {cn I n E No} be complex sequences such that a n, 0 for n c No . To these sequences we associate a linear operator L acting on a Hilbert space with orthonormal basis in), n = 0,1,2, • • by

(66)

Lin) = anin + 1) + bn in) + cn in - 1),

where I - 1) := 0. Suppose that lz) is an eigenvector of L for the eigenvalue z, that is, Liz) = ziz). Then lz) can be written as lz) = E:L 0 pn,(41n) with certain coefficients pn (z). From (66) we get E(pn(z)anin + 1) +p(z)b(n) ±pn(z)enin — 1)) = z Epn( )1 )• n=0

n=0

Equating coefficients of the vector In) we obtain the following three-term recurrence relation for the coefficients pn(z):

cn+1Pn+i(z) + bnpn (z) + an-iPn-i(z) = zpn (z).

(67)

Let us set p_1(z) = 0 and po (z) = 1. Then (67) determines the coefficients Pn(z) uniquely and each p(Z) is a polynomial in z of degree n. Suppose now that the sequences (a n ) and (bn ) are real and en = an _i for n E N. Then L is a symmetric operator. By Favard's theorem, there exists a nonnegative Borel measure t on the real line such that the polynomials pn (z) satisfy the orthogonality relation Pi (A)Pk(A)C 111 (A)

5j k

,

, k = 0 ,1 , 2 , - • -

(68)

One can prove that the symmetric linear operator L always admits a selfadjoint extension, say L, in the Hilbert space .55. If E(.) is the spectral measure of L and WO = (0 I E(A) 10), then (68) holds. If the closure E of the operator L is self-adjoint, then any measure i satisfying (68) is uniquely determined, that is, we have a unique orthogonality relation for the polynomials p72 (z) This is in particular true if the symmetric operator L is bounded. In this case the measure rt has compact support. If the closure of L is not self-adjoint, then L has infinitely many self-adjoint extensions and to every such extension there corresponds a nonnegative Borel measure tt such that (68) is valid. Special cases of the recurrence relation (67) lead to the so-called qorthogonal polynomials. 2.3.2 q-Hermite Polynomials

The recurrence relation 2xp n (x) = Pn+ (x) + (1 — q 7 )Pn_1(x)

(69)

determines the continuous q-Hermite polynomials pn (x) = 117,(xlq). They are given by Hn(Xlq) =

(qi

E (q; q)k(q; q)n-k k=0

e î(n_ -2k)01

where x = cos O. The generating function for these polynomials is Rze io ;

00

001 -2

Ho. -

2x zqn

Hn (x1q) z n

z 2 q 2n ) - 1 = n=0

n=0

7

(q;

where again x = cos 9. If —1 < q < 1, then the polynomials lin (xlq) are orthogonal with orthogonality relation 00

L1 '

1 n (x1q) 11,n(xlq)

(I(1 — 2(2X2 i) qk

q2k)) (1 x2)-1/2dX

k=0

nair(qi

On/(q; Om

•

2.3 q-Orthogonal Polynomials

51

The corresponding self-adjoint operators are related to the position and momentum operators for the q-oscillator algebra (see [ChK] and [K1S]). The polynomials 1-1 7,(xiq) := i- nIIn (ix1q) ,

(70)

i :-= -\/-_,

satisfy the recurrence relation

2xl-t n (siq) = hri+i(xlq) + (qn - 1)h, _ 1 (xlq) .

(71)

For 0 < q < 1, the orthogonality relation for these polynomials is

I

(-1)n(q,q),„a (logq -1 ) 6mn " hn,(sinhulq -1 )h,,(sinhulq -1 ) du --= (q^ 1. q• - 1);. 1 00 (_ qe211. ; 0 00 (_ qe -2u ; 0 00 .

.7

.

The discrete q-Hermite polynomials Hfl (s; q), n c No, are defined by the recurrence relation xpri (x) = Pn+i(x) + q n (1 - qn )pn _ 1 (x).

(72)

In explicit form these polynomials are given by [n/2] ,

,

tq; q),(-1)k q k(k- 1)

I In (x; q) ,-

5

( q2 ; q 2 )k(q;

k=0

n-2 k

7

On-2k

where [n/2] denotes the integral part of n/2. For 0 < q < 1, the orthogonality relation for the polynomials Hn (s; q) is of the form in ( q2j+2 ; 0 00 ( q; q2) 00

00

EIH,n ( qi ; q)11,(q1 ; q) + 11,(-qi ; q) Hn (-qi; OP2

J=1

(q2 ; q2) 00

+1/2 (q; q2 ) 00 11„,(1; q)H„(1; q) = qn(n-1) I 2 (q; On6Trin , that is, these polynomials are orthogonal on the set s = +qi, j E No.

2.3.3 Little q Jacobi Polynomials -

The little q- Jacobi polynomials pfl (x; a, blq) are expressed in terms of the qhypergeometric function

2(P1

by

Pn(x; a, big) =---

2 (P1(q -n ,

abq n+1 ; aq; q, xq).

(73)

One can easily verify that lim q,i pn ((1 - x)12; q&, le lq) = cP4'3) (s), where c is a constant and P4c1'3) (s) is the classical Jacobi polynomial. The polynomials pn (s; a, blq) satisfy the recurrence relation

xpn (s) = AnPn+ 1 (s) where

(An

+ Cn)Pn(S) + CnPn--1(s),

(74)

An, = (—qn )(1 - abqn +1 )(1 - aqn +1 ){(1 - aben +1 )(1 Cn = (-aqn)(1 - q 71 )(1 - bqn ) 1(1 abq 212 )(1 - abq 2n+1 )} -1

The orthogonality relation for the little g-Jacobi polynomials is 00

Epm (qx ; a, blq)p n (qx ; a, bfq)

(bqi q)x

b mn

(aq)s

(q; q)x

o

(75)

hn (a,b; q)'

for 0 < q < 1, where hn (a , b; q) =

(aq\

_n (abq; q) n (1 - abq 2n+ 1 )(aq; q) n,(aq; q) c„,

)

(q; q)(1

abq)(bq; q) n (abq 2 ; q)„.„

That is, these polynomials are orthogonal on a discrete set of points. 2.3.4 Big q-Jacobi Polynomials

Let a,

c, d be fixed complex numbers.

—n 13 7(ler'f3) (X; el dig)

q

32

The functions

a-OA-n-1-1 cx+1 1

_ qa+ldie XI i

c q-07)

n G No,

are polynomials in z , called the big q- Jacobi polynomials. If a, c, d c No, then they satisfy the orthogonality relation

G

(76) R and

fc d

' 3) (x; c, dlq)(qx I c; q),(-qx I d; q)ocl q x = hn6 Tim) 13,,,( a J3) (x; c, dlq)P4" (77)

where 20

hn =

13-1-1. ; q ) a±i ( q; q )n (1

q a-1-13-1-1)( q0-1-1 ; O n (_ q0-i-l ci d;

(

c(1 q)(q; q)

x{(1

( -d/ c; q)

1 (-qc1 d; q) (qa+0+ 1 ;

qa-143-1-2n+1)(qa-1-1; on(_qa-1-1dic; 00-1.

Thus, the big g-Jacobi polynomials are also orthogonal on a discrete infinite

set of points.

2.4 Notes The expressions (a; g) n for q G N have been known since the time of L. Euler. They have been used in combinatorics, Galois theory and finite geometry. The hypergeometric g-series 2 (09 1 was invented in the nineteenth century by E. Heine. The g-integral was introduced and studied by F. H. Jackson [Jki at the beginning of the twentieth century. A good exposition of the theory of basic hypergeometric functions and g-orthogonal polynomials can be found in the book [GR]. A nice survey on q-special functions is given in [Ko5].

3. The Quantum Algebra Uq (512 ) and Its Representations

This chapter is devoted to a detailed study of the Hopf algebra Uq (512 ). It can be considered as a one-parameter deformation of the universal enveloping algebra U(s12) (see Example 1.6) and is the simplest example of the quantized universal enveloping algebras Uq (g) which will be investigated in Chap. 6. Following common terminology in physics, we call Uq (s12 ) a quantum algebra.

3.1 The Quantum Algebras Uq (512 ) and Uh(s12) 3.1.1 The Algebra Uq (512 )

Let q be a fixed complex number such that q O and q2 1. We denote by Uq (s12 ) the algebra (that is, the associative algebra with unit) over C with four generators E, F, K, K -1 satisfying the defining relations 2, —2F7

1

KK -1 = K -1 K =1, KEK -1 =q 2 E, KFK -1

(1) (2)

[E, F] = KK-1

It can be shown by induction that the relations (1) and (2) imply the formulas [E, Fm]

[En , E'2 Fm

EFT'

E = [m]Fm -1 [K ;1 - m ] ,

a En F - FEn = [n]En- 1 [K; n - 1],

( 3) (4)

[n]![mj!

v-•

[r]![n - 7]![m - 71! Fm—rEn—r X [K; n

nt][K; n m 1] - - • [K ; n m

where we set [a] : = [ajg and [K; -

:= (Kqn

-

-

+ 1] n m ,

K -1 q - n)/(q

-

(5)

q- ') for n c Z.

Proposition 1. The set {F1 Km En I m E Z, 1,n E No}, as well as the set {E n K n I m c Z, 1,n E Nob is a vector space basis of Uq (s12). Proof. It follows from (1) and (2) that Uq (512) is spanned by the monomials Fl KmEn , m E Z, 1,n E No. To prove that they constitute a basis of Uq (812) we consider the polynomial ring C[x, y, z, z -1 ) in commuting indeterminates

54

3. The Quantum Algebra Uq (s12) and Its Representations

x, y, z, z -1 satisfying zz -1 =-- z -l z =-- 1. Define linear transformations Liand C —1 of C[x,y,z,z -1 1 by k( x r z S yt) = tj2r xr zs -1- l yt

f'(xr zs y t ) = xr+ 1 z'y t , k( sr z s y t ■) q—28 r s xzy

—1 t-I-1

"

q-1

'J r

z

s yt .

Direct computations show that these transformations satisfy the relations (I) )?, X = E,F,K, K -1 , determines and (2). Therefore, the assignment X a representation of Uq (512) on C[x,y, z, z -1 ]. Since the monomials xrzsyt, s E Z, r, t E No , form a basis of C[x, y, z, z -1 ] and PlfCmkn(1) = xl zmyn , the operators Pi km Pn 771 G Z, 1, n E No, are linearly independent. Thus, the set {Fi Km En m E Z1 1, n E No} is a basis of Uq (512 ). The second assertion is proved similarly. Using Proposition 1 one can show that the algebra Uq (512 ) has no zero divisors, that is, we have ab 0 in Uq (s1 2 ) whenever a 0 and b O. Proposition 2. The quantum Casimir element

Cg := EF +

K' K - ' (q

(21 -

1)2

= FE +

Kq+

(6)

lies in the center of the algebra Ug (s12). If q is not a root of unity, then the center of Ug (s12 ) is generated by Cg . Proof. The second assertion is proved in Sect. 6.3. For the first assertion it is sufficient to verify that Clq commutes with E, F, K and K -1 . Using the relations (1) and (2) this is easily done.

Proposition 3. There exist automorphisms 0 and v = ±1, of the algebra Ug (s1 2 ) such that

a

CVO}, Ti G Z,

0(E) = F, 0(F) = E, 0(K) = K -1 , 11-2n /9,, n ,„(K) = ct,n,v(E) = aKnE, 19,„(F) = va - 1 Proof. First one shows by direct computation that the images of the generators under 0 and 19„, n,, again satisfy the relations (1) and (2). As a sample, we check that t9 t9,, n,„ preserves the relation (2) by computing

19(E)19(F) -19(F)19(E) = v(KnEK - Tiq -2n)F v(K'FKnq -2n)E = v(EF - FE) = (q q -1 ) -1 09(K) -19(K -1 )). Hence there exist algebra homomorphisms 0 and t9 cle ,n,v given by the above formulas on the generators. Since 0 o 0 = id and t9 n, 11 o = id, these homomorphisms are actually automorphisms.

If q is not a. root of unity, it can be shown (using the second assertion of Proposition 2) that the group of automorphisms of the algebra Uq (512) is (see [A1C1). generated by the automorphisms 0 and

3.1 The Quantum Algebras Uq (s12) and Uh(s12)

55

3.1.2 The Hopf Algebra Uq (s12 ) A key property of the algebra Uq (s12 ) is that it carries a Hopf algebra structure. Proposition 4. There exists a unique Hopf algebra structure on Uq (512 ) with comultiplication 4 counit e and antipode S such that

46(E)=E0K+10E, 46(F) F 01 + K-1 0 Fl ,6(K) = K

K7

51(K) =•- K -1 1 8(E) = -EK -1 , S(F)= -KF, e(K) = 1, e(E) = e(F) = O. Proof. First we extend 6 to an algebra homomorphism of the free algebra C(E Fl K 7 K -1 ) to U(s1 2 ) Uq (s12 ). In order to prove that 46 passes to the quotient algebra Uq (s1 2 ) of qE,F,K,K -1 ) it suffices to show that 46 preserves the relations (1) and (2). We shall do this in the case of (2). For the relations (1) the proof is similar and simpler. We have ----- K -1 EF F E

EK -1 ICF + EF ®

-K -1 0FE-K -1 E0FIC-F0E-FE0K K -1 0 (K - K -1 ) ±(K - K -1 ) K

--= {46(K) - 46(K -1 )11(q-

It is proved similarly that 8 extends to an anti-homomorphism of Uq (s12) and

e to an algebra homomorphism of Uq (s1 2 ) to C. By Proposition 1.8, it suffices to verify the Hopf algebra axioms on the generators. For instance, we have (60id)0,6(E)=10 1® E + 10E0K+EOKOK=(id0,6)06(E). The other conditions are checked in a similar manner. El Definition 1. The algebra U(s12) endowed with the Hopf algebra structure from Proposition 4 is called the quantum algebra U q (512)•

Some useful formulas about the Hopf algebra Uq (s1 2 ) are collected in Proposition 5. In the Hopf algebra Uq (s12 ) we have

82 (a) = KaK-1 for all

a E Uq( 51 2),

( 7)

46(E1 KmFn) = I n 1 = E

q r0-8(n-s) []![np(1EI_ rKmn+8F8 0 Er K i-r±m Fn-s

r=0 .9=0

[r]![l - ri![s]![n - s]!

S(Fn K m El ) = —1) 1+n q 1(n-1)-1(1-1) E1 10 -1-m Fn

(8) (9)

Proof. Since 52 is an algebra homomorphism of Uq (s12) by Proposition 1.5, it suffices to check the equality (7) on the generators. We have 82 (E) = --S(EK -1 )= -S(K -1 )8(E)= KEK -1 and similarly 52 (F) = KFK-1.

56

3. The Quantum Algebra

Uq (812)

and Its Representations

By induction one proves the formulas for ,(El) = ,A(E)I and z1(F) = z 1 (F) n using that ,A is an algebra homomorphism. The formula (8) is obtained by multiplying the expressions of 44E1 ) 7 4A(K m ) = K rn ® Km and ,(1'12 ). The relation (9) follows from the expressions for 5(F), S(K) and 5(E) and 0 the fact that S is an algebra anti-homomorphism.

The next proposition deals with Hopf algebra isomorphisms of Uq (s12)• Proposition 6. (i) For any a E CVO}, there is Oa of Uq (512) such that

a Hopf algebra automorphism

V a (E) = aE, IMF) = a-1 F, 19 a (K) = K. Any .Hopf algebra automorphism of Uq (s12 ) is of this form. (ii) Let a be a nonzero complex number. There exist Hopf algebra isomorUq -1(s12) and Oa : Uq (512) phisms (p c, : Uq (s12 ) U_ q (512) such that

çoa (E) = aFK, ço a (F) =

A

, cp c,(K) = K,

zPa(E) = aE, Oa (F) = -a-1 F, 11) 0 (K) = K. Any Hopf algebra isomorphism (io : Uq (512 ) Uq -i(s12 ), resp. Uq (512) U_ q (512 ), is of the form p = cpa, resp. 11) (iii) Two Hopf algebras Uq (512 ) and Up (s12 ) are isomorphic if and only if q is equal to p,-p,p - 1

Proof. That the mappings defined above are indeed Hopf algebra isomorphisms is easily verified on the generators. We now prove the converse assertions and begin with two preliminary results. The first one is that the set of group-like elements of Uq (512 ) is just the set of powers Kn , n E Z. Put K ±1 }. The second one is that .Fq± := E Uq (s12) 0(x) = 1 x x Lin{E,FK,1-K} and = Lin{1-K -1 }. Both results are derived from formula (8) by writing an arbitrary element x E Uq (512 ) as a linear combination of basis elements El K m Fn and comparing coefficients in the expression of ,A(x). It is clear that the set of group-like elements and the sets .Fq± are preserved under Hopf algebra isomorphisms. Now suppose that V) : Uq (s12 ) Up (s12) is a Hopf algebra isomorphism. Since the image CIO of the group-like element K is again group-like, the first result above yields that CK) =--- K n for some n E Z. Similarly, 0 -1-(K) = Km,m E Z. Hence we have CIO = K or 11)(K) = K -1 . If ip(K) = K -1 , then we have C.F:7E) = .F77, . But the latter is impossible, since dim 3 and dim ,Fp- = 1 by the second result. Thus, IP(K) = K. Next we note that the map Ks1C -1 leaves the space .F4,-» invariant and has eigenvalues 1, q 2 , 47 -2 with eigenvectors 1 - K,E,FK, respectively. Therefore, we have either q2 = p2 and CE) --= aE,1,1)(EK) = #FK or q2 =7;1-2 and CE) = aFK,O(FK) = OE for some a, E C. Hence q is equal to p, -p,p -1 or -p-1 . This completes the proof of Now suppose that q = p. Then, as shown by the preceding, tk(K) = K 11;(E) = ûE and

3.1 The Quantum Algebras Uq (512) and Uh(s12)

57

0(FK) = OFK and so 0(F) = OE By (2), 3= eci and hence 1/, = which proves (i). The last assertion of (ii) follows similarly. There is another important variant of the quantum algebra Uq (s12 ) which often occurs in the literature. It is defined as follows. Suppose that q 4 0, 1 and let 6q (512) denote the algebra over C with generators E, F, K K -1 subject to the relations

KK -1 = K -1 K =1, KEK -1 = qE, KFK -1 = q -1F, K2

[E,F]=

(10)

_ K -2

1• g - g-

(11)

Arguing similarly as in the proof of Proposition 4, it follows that this new algebra Oq (512 ) becomes a Hopf algebra with structure maps given by 46(K) = KoK, d(E)=E0K+K -1 0E, d(F)= FOK+K -1 0F, (12)

S(K) = K -1 , 8(E) = -qE, 8(F) = -q -1 F, s(K) =1, E(E) = e(F) = 0. (13) An advantage of the Hopf algebra Ûq (512 ) over Uq (512 ) is that the comultiplications of the generators E and F are given by the same formulas. Using the above formulas and Proposition 1, one can show that there is an injective Hopf algebra homomorphism ço : Uq (s12 ) -> t-71 q (s12 ) determined by cp(E) = EK, (p(F) = KF and AK) = K2 . If we identify Uq (512) with its image under ço, then Uq (512) becomes a Hopf subalgebra of Ùq (512). Note that the algebra t/q (512 ) has four one-dimensional representations (sending K to 1, -1, i, -i and E, F to 0), while Uq (512) has only two such representations. Hence the algebras Uq (512) and 0q (512) are not isomorphic. 3.1.3 The Classical Limit of the Hopf Algebra Uq (512 )

In this subsection we shall discuss how the enveloping algebra U(s12) is related 1 of the Hopf algebra Uq (s12). Recall that U(s12) to the classical limit q is the algebra with generators x+ ,x_,y and defining relations [y, x+ ] = 2x+ , [y, x_1 = -2x_, [x+ , x_} = y and that the Hopf algebra structure of U(s12)

has been described by Example 1.6. Let us formally write q = K = ehH and consider the limit h -> 0. Then the relations (1) and (2) imply (by differentiation at h = 0 resp. by taking the limit h 0) that [H, = 2E, [H, F] = -2F and [E, F] = H. That is, we obtain the relations of U(s12). Similarly, the Hopf algebra structure of U(s12) is recovered from that of Uq (s12) as h -> 0. However, this viewpoint is not quite adequate if we want to compare the theory of Uq (s12) (at least for q not a root of unity) with the classical situation. In order to develop the proper setting we first give a slight reformulation of the definition of Uq (512) that applies also in the case q2 = 1.

58

3. The Quantum Algebra Uq (s12 ) and Its Representations

Clearly, the elements E, F, K, K -1 and the algebra Uq (s12) satisfy the relations

-1(K K-1) o f

G := (q -

[G, = E(qK + q -1 K -1 ) , [G, = -(qK + q -1 K -1 )F, [E,

=G, (q - q -1 )G = K - K -1 .

(14) (15)

Let tIq (s12 ) denote the algebra with generators E, F, K, K -1 , G and relations (1), (14) and (15). The algebras U(s12) and Uq (512), g 2 1, are isomorphic (by mapping E -) E, F -) F, K -) K and G -) (q - q -1 )(K - K -1 )). The Hopf algebra structure of Uq (512) induces a Hopf algebra structure on Oq (s12) which is given by the formulas stated in Proposition 4 and the relations A(G) = G K + K -1 0 G, E(G) = 0 and S(G) ----- -G. It is easy to verify that these formulas endow the algebra Qs12) with a Hopf algebra structure also in the cases q = +1 which were excluded for Uq (s12). We consider the Hopf algebra Oi(s12) as the classical limit of the Hopf algebra tIq (s12)• The two algebras U1 (s12) and U(s12) are closely related to each other. That is, there are canonical algebra isomorphisms

01 (512 ) U(s12 ) CZ2 and U(s12 )

Oi(s12)/(K - 1).

Indeed, putting q = 1 into (1), (14) and (15) we see that K belongs to the center of 01(s12) and that K 2 = 1. Thus the defining relations of the algebra 01 (512) read as [E, = G, [G E] = 2E K and [G, .= -2F K Hence there exists an algebra isomorphism of Oi (s12) and U(s12 ) CZ2 which maps E x + x, F x_ and G -4 yx. Here x is the generator of the two-dimensional algebra CZ2 such that X2 = 1 and we have simply written ab :=- a b for a E U(s12 ) and b E CZ2. That is, the algebra Oi (s12 ) is an extension of the algebra U(s12) by a single central element x satisfying X2 = 1. The appearance of this additional generator x explains why we later have two irreducible representations in any dimension when q is not a - root of unity. Finally, we describe the Hopf algebra structure on U(s12 ) CZ 2 that is induced from 01(s12) by the above isomorphism. Tracing back the corresponding formulas, the structure maps on U(s12 ) CZ2 are given by .

d(x + ) = xs + 01 +10x + , A(x_) = xx_ 01+10x_, d(y) = yO1 +10y,

E(x +) = E(x_) = E(y) = 0, S(x) =

-

xx +, S(x_) = xs_, S(y) = -

-

y.

3.1.4 Real Forms of the Quantum Algebra Uq(512)

Recall from Subsect. 1.2.7 that by a real form of Uq (512 ) we mean this Hopf algebra together with an involution * : U(s12) U(s1) which turns Uq (512)

into a Hopf *-algebra. We do not give proofs in the subsequent discussion. The reader is invited to carry out one or two of them using the following general pattern. One

3.1 The Quantum Algebras Uq (812 ) and (A (s12)

59

first verifies that the corresponding formulas define an involution of the free algebra C(E, F, K, K -1 ). Next one shows that the defining relations (1) and (2) are invariant under this involution, hence it passes to the quotient algebra Uq (512) to define an involution there. Finally, one checks on the generators that Zi and E are *-preserving. Case 1: q E R. For any -y E R, 0, there exists a real form of Uq (s12 ) with involution determined by the equations E* = F* = 1,-1 K -1 E7 K * = K. Using the description of all Hopf algebra automorphisms of Uq (s12 ) from Proposition 6(i) we see that two such real forms with parameters -y and are equivalent if and only if -y-y1 > O. Thus, there are precisely two inequivalent real forms of Uq (s12) in this case. The first one is given by

E* --= FK, F* = K -1 E, K* = K. The corresponding Hopf *-algebra is denoted by Uq (su2) and called the compact real form of Uq (s12). The second one is denoted Uq (su 1 , 1 ) and defined by K* = K. E* = -FK, F* Case 2: Iql= 1. For any fiy C C, = 1, we then have a real form of Uq (512) given by E* =--yE, F* K* = K. But all such real forms are equivalent, so we can assume that -y = 1. This real form is denoted by Uq (s12 (R)). It is clear that the three real forms of Ug (512) defined above correspond to the three real Lie algebras su2 , su i , 1 and s12 (R) of the complex Lie algebra sl(2, C). Note that the classical Lie algebras sui,i and s12 (R) are isomorphic. But their quantum analogs Uq (sui,i) and Uq (s12(R)) are not equivalent, simply because they are defined for different parameter domains. Case 3: q C The following sets of formulas define two inequivalent real forms of Uq (512) which have no classical counterparts:

= Nr-1FK, F* =

Et

K* = K,

= Ft = if* = K. It can be shown that only for the above three parameter domains of q real forms of Uq (s12 ) exist and that the previous list is exhausting. Et

The Hopf algebra 0,7 (512) has, up to equivalence, precisely three real forms. They are denoted by tliq (5112), 0.7 (sui,1 )1 Ûq (s12 (R)) and are determined respectively by the formulas Et

= F, Ft = E, K* = K

for

q E R,

= -F, F* = -E, Kt = K for q E R, Et = -E, F* = -F, K* = K for lqi = 1 Et

3. The Quantum Algebra Uq (s12) and Its Representations

60

3.1.5 The h-Adic Hopf Algebra

U h (s 1 3)

Let h be an indeterminate. We denote by Uh(s12) the (complete) h-adic algebra (see Subsect. 1.2.10) with three generators E, F and H obeying the defining relations

e hH e-hH {H, El = 2E, [H, F] =--- -2F, [E, Fi =

sinh hH sinh h

eh - e -h

(16)

where ehH is the series I hH + OH? 12! + • • • belonging to Uh(s12)• The elements El H" Fn , 1,m, n = 0, 1, 2, • • •, constitute an h-adic vector space basis of Uh(s1 2 ), that is, these elements are linearly independent and the h-adic completion of their linear span coincides with Uh(s12). The element e(H -1) ± e -h(11-1) Ch :=7

EF +

e h(H+1)

= FE +

(eh -e-h)2

e -h(Hd-1)

(17)

(eh - e-h)2

commutes with all elements of Uh(s1 2 ) and generates the center of Uh(s12). There exists a unique algebra automorphism 0 of the algebra Uh(512) such that 0(E) = F, 0(F) = E, 0(H) = -H, 0(h) = -h.

(18)

One can easily prove that there exists a unique h-adic Hopf algebra structure on Uh(s12 ) with comultiplication 4 counit E and antipode S defined on the generating elements by

e(H) e(E) = e(F) = 0,

Ee- hH

s(F)

= _ ehH F.

The h-adic Hopf algebra obtained in this way is called the h-adic quantum algebra Uh(S12). From the preceding formulas we see that the algebra Uh(s12) does not change and that the comultiplication d of Uh(s12) goes into the opposite comultiplication dc°P = o d when h is replaced by -h. We can also define the h-adic analog Ûh(s1 2 ) of the quantum algebra Ûq (s12 ) considered above. It is generated by elements E, F and H satisfying the defining relations

e 2hH [H, = E, [H, = F, [E, = with

eh

e -2hH -

e

h

the h-adic Hopf algebra structure ,A( E) = E e hll

e-hH

,A(F) F (8, e hH

c hH

d(H) = H el+ 1 ® H, e(H) = E(E) = E(F) = 0, S(H) = -H, S(F) = -ehE, S(F) =

F,

3.2 Finite-Dimensional Representations of Uq (s12) for q not a Root of Unity

61

An h-adic Hopf algebra A is called quasitriangular if there exists an invertible element R. in the completion AA (in the meaning of Subsect. 1.2.10) such that 1cc9 (a) = 7ZA(a)R. -1 , a E A, and

(A 0 id)R, = 'R, 13 7Z23 and (id 0 A)R, =

R13R-12

on sAllA(g)A,

where 7Z12 = Ei xi ® yi 0 1, 7Z13 = Ei xi ® 1 (g) yi , 7Z23 — Ei 1 ® xi ® yi for 7?, =-- Ei xi ® yi. The element R. is called a universal R-matrix of A. It is useful for applications in physics, because it yields a solution of the quantum Yang-Baxter equation R12R13R23 = R231'43/712 for any representation of the algebra A. A detailed exposition of this theory will be given in Chap. 8. Here we state only the following proposition which is proved in Subsect. 8.3.1. Proposition 7. The h-adic quantum algebra Uh(512 ) is quasitriangular and admits the universal R-matrix R, = eh ®k

cc , (n+1)/2(1 n -2)n , — v ) En 0 Fri E tin

n=0

q

= eh.

7

[711 q !

(19)

3.2 Finite-Dimensional Representations of Uq (s12 ) for q not a Root of Unity Throughout this section we assume that q is not a root of unity.

3.2.1 The Representations 21,1

If T is a representation of the algebra Uq (512), then the operator T(K) is invertible and we have T(K)T(E)T(K) 1 = q2T(E), T(K)T(F)T(K)

T(E)T(F) - T(F)T(E) =

1 --= q -2 T(F),

T (K) - T (K) -1 1

.

(20) (21)

Conversely, since (1) and (2) are the defining relations of the algebra Uq (512)) any triple of operators T(E), T(F) and T(K) on a vector space V such that T(10 is invertible and (20) and (21) are fulfilled gives rise to a representation T of Uq (512 ) on V. Now let I be an integral or half-integral nonnegative number and let i1J E { +1, -1}. Let 171 be a (21+1)-dimensional complex vector space with basis em , m = -1,-1 + 1, • - • ,l. For notational convenience, we set ei +i = e_i_ i = O. Define operators T1 (E), Tw i(F), Ti (K) acting on V/ by ILA (K)em = wq 2me ni , T,,i(E)e,,, = al - m][1 + m + 1]) 1/2en, + 1,

Twi(F)e,„, = coal + m][/ - in + 1D 112em-11

(22) (23)

62

3.

The Quantum Algebra Uq (s12) and Its Representations

where [n] [n] q (see (2.1)). A simple computation shows that these operators satisfy the relations (20) and (21). Hence they define a representation TA of the algebra Uq (512) on Vi. Apart from the sign factor, the representation TA is built in a similar manner as the (21+ 1)-dimensional representation of the Lie algebra sl(2, C). The same method as in the classical case shows that Ti is irreducible. Therefore, the image Tw i(Cq ) of the Casimir element Cq (see Proposition 2) is a scalar multiple of the identity operator I on V. Since TA(E)ei = 0, we obtain from (6) that TA(Cq ) = co(q1+1 +q -1-1 )(q

(24)

q -1 ) -2 /.

Let Tw ,i and Tw i,i/ be two representations such that (w,1) ( 2,1'). Since q is not a root of unity (!), the spectra of the operators /14 (K) and T,1(K) do not coincide, so the representations T,,i and Tw f,i , are not equivalent. Moreover, by (24), the scalars of T i (Cq ) and T,,/,(C q ) are different. Thus, we have proved Proposition 8. For any I E N0 and co E {1, -1}, Z.,/ is an irreducible representation of the algebra Uq (512 ). If (w, I) (41% the representations and are not equivalent and the values of the operators T„,i(Cq ) and T,,,i,(Cq ) are different.

There is a unique scalar product (a, .) on Vi such that (ern , en ) = 6,n . If q E R, then the operators (22) and (23) satisfy the conditions TA(E)* = TA (F) and T1 (K)* = TA (K ). Therefore, by the definition of the involution of Uq (s112), TA is a *-representation of the *-algebra Uq (su2) on VI. The family TA, w E {-FL -1 } , 1 E N0 , exhausts all finite-dimensional irreducible representations of Uq (512) as shown by Proposition 9. Any irreducible finite-dimensional representation T of (.1q (512 ) is equivalent to one of the representations TA, w E {+1, -1 } , 1 =-- 0, I, • • -.

Proof. Because the underlying vector space V is finite-dimensional, the operator T(K) has at least one eigenvector e, say T (I0e = lie. Since KEr = q 2r Er K by (1) and hence T(K)T(Er)e = perT(Er)e, T(Er)e is either zero or an eigenvector of T(K) with the eigenvalue pq 2r. Since these numbers are pairwise distinct (because q is not a root of unity) and V is finite-dimensional, there exists r E No such that eo := T(Er)e 0 and T(E)eo =-- 0. Let T(K)eo = Aeo. Set F( 8) := F8 /[s]! and e, := T(F (8) )eo, s = 0, 1, 2, • • •. Since KF = q -2 FK , we have T(K)es = A g -28 _e There exists a smallest p E N such that T(F(P))e o 0 and T(F()+ 1) )eo = 0. By (3), for any r E {0,1, 2, • • .} we have

T(E)er =

T(E)T(F (r ) )eo = T(EF (79 )eo /1 7

=

-

qr- -

cir -^ 1

T(F (r—

i)

- (r - 1)

)e0

4711..A1

1

erl•

3.2 Finite-Dimensional Representations of Uq (s12) for q not a Root of Unity

63

Thus, the subspace 171 spanned by the vectors eo , el , • • ,e is invariant under the operators T(E), T(F), T(K). Since the representation T is irreducible, = V and the vectors e0 , e 1 , • • • , ep form a basis of V. Let us find the possible eigenvalues A. Since T(F (P+ 1) )eo = 0, we have {T(E)T(F(P+ 1 )) - T(F (P+ 1) )T(E)}eo = 0, so (q-PA - q pA-1) (47 .7 -1) by (3). Therefore, A = ±qP, and the representation T is given by

(25)

T(E)e, = w[p - r T(F)e, = [r + 1]er-E17 T(K)er wqp-2T

( 26)

1P) 112ei, for w - T([/ - r]!/{l r where w c {±1}. We set l := and er r = -1,-1 +1,- - ,I. Then the formulas (25) and (26) turn into T(E)e's = ([l

s

11[1 _ s ] ) 1i2 e1.9+1,

s +11[1 s]) 112 e's _ i , T(K)e' s = w e e.

T(F)e'3 =

This shows that the representations T and TA are equivalent.

The representations Ti with w = 1 are denoted by T1. They are called representations of type 1. Obviously, the representation TA is the tensor product of T1 and the one-dimensional representation Two . 3.2.2 Weight Representations and Complete Reducibility We begin with a preliminary result.

Lemma 10. Let T be a finite-dimensional representation of Uq (512 ). Then the operators T(E) and T(F) are nilpotent, that is, there is an r E No such that T(E)' = T(FY = 0. Proof. The finite-dimensional representation T takes in some appropriate basis the following matrix form: • ••

O

• •• • ••

(27)

• • •

where TM, T (n) are irreducible representations of Uq (s12) and asterisks mean certain submatrices. By Proposition 9, each representation V ic) in (27) is equivalent to some TA. Therefore, since the operators Tw i(E) and Twi(F) are nilpotent by (22) and (23), there exists in E No such that T(E) t = T (k) (F)m = 0 for k = 1,2, - • • , n. Hence, by (27), T (k) (E) r = T (k) (F)r = 0 for some r. Next we give the analogs for Ug (512) of some notions from classical representation theory of Lie algebras. Let T be a representation of Uq (512) on

3. The Quantum Algebra Uq (512) and Its Representations

64

a finite-dimensional vector space V. For any complex number A, we set VA := { x E V I T(K)x = Ax } . If VA {0 } , then VA is called a weight space and A is a weight of the representation T. The nonzero vectors of V), are the weight vectors. A vector x E V such that T(E)x = 0 and T(K)x = px, is called a highest weight vector and p is called a highest weight of T. If V is the linear span of weight spaces of T, then T is called a weight representation. Then V is the direct sum of weight spaces. In other words, T is a weight representation if and only if the operator T(K) is diagonal with respect to some basis of V. Each representation 71,1 is a weight representation.

Proposition 11. Every finite-dimensional representation of Ug (512) is a weight representation. Proof. Let T be a representation of Uq (s12 ) on a finite-dimensional vector space V. It is known from linear algebra that the operator T(K) is diagonalizable if there exist pairwise distinct numbers a l , a2 , • • • , ak such that

(T(K) a i )(T(K) - a2 ) • • • (T(K) - ak)V = {0}.

By Lemma 10, there exists m E N such that T(F)mV = {0}. Recall that [K; = (K qn K-icn)/(q_

) Put K0 := / and

r-1

Kr =11i=-(r-1) [if; r - m + j],

r = 1, 2, • • • ,m.

We prove by induction on r that T(Fm - r)T(Kr)V = {0 } ,

r = 0, 1,2, • • • ,m.

(28)

For r = 0 this is true because T(F)mV = {0}. Suppose that (28) holds for i = 0,1,2, • • • ,n - 1. Setting B = En Fm 1T1=-11 [K; n - m + j ] , we have T(B)V = {0 } and n-1

B=

E biFm - z (H[K ; i n m + j]) (11[K ; 2i i=o

by (5), where

j=i

-

n

-

m + A) En'

J=1

bi = [n]! [m]!/ [i] ![n i=0

E

il! {m - ij! . Further, we get trri-l-n- 1 [K; - n - m + j]) En-i biFm-i n-i-1

i=0

H 3=0

{K;

- 711 -

ii)

Since T(B)V = {0} as noted above and T(Frn - r)T(Kr )17 = {0 } for r 0,1,2, n 1 by the induction hypothesis, the last equation implies the relation T(Fm - n)T(Kr,)V = {0}. This completes the proof of (28). In the case r = m formula (28) yields ,m-1

T(K m )17 = (II

qiT(IC -1 )T(K 2 q q-1

V = {O}.

3.2 Finite-Dimensional Representations of Uq (512)

for q not a Root of Unity

65

Applying some appropriate power of T(K) we get ( m-1

Hi

{

T(K)

q-j}{T(.10-1-q -3 })V = 101.

The numbers in the set q -1 ,-q -1 , j E Z, are pairwise distinct since q is not CI a root of unity. Proposition 12. Any finite-dimensional representation T of Ul (s12 ) is com-

pletely 7-educible, that is, T is a direct sum of irreducible representations. Proof. Let T be a finite-dimensional representation of Ui7 (512 ) on a space V. We first suppose that T reduces to the matrix form (27) with all T(i) equivalent to a fixed irreducible representation 71,1. Let Vi = {v E V T(E)v = 0 } . From the special form of T it is clear that all v E Vi are highest weight vectors belonging to the same weight. In particular, V1 is a linear subspace of V. Let v4, • • • , vt) be a basis of For each i = 1,2,•• • , k, let Vi be the span of vectors vl := Fav6, s = 0, 1, • • • , 21. Repeating the reasoning of the proof of Proposition 9, it follows that on each space Vi the irreducible representation Twi is realized. We show that the set of vectors via is linearly independent. Indeed, only vectors of the same weight may be linearly dependent. If aivt) = 0 and ai vsi = 0, then the action by the operator T(E) 8 yields so a i = 0 for all i = 1, 2, • • , k. Hence the vectors via are linearly independent Since T is a weight representation by and we have a direct sum 1.7 := Proposition 11, we have V =17. Thus, T is completely reducible. Now let T be an arbitrary representation of the form (27). On the block diagonals the Casimir operator T(q) acts as T(i)(Cq ) = cii. The constants ci are the generalized eigenvalues of the operator T(Cq ). It is known from linear algebra that the space V decomposes into the direct sum V = E V I (T(q) c i )rv = 0 for some r} of of generalized eigenspaces V := TWO corresponding to distinct generalized eigenvalues ej . Since Cq belongs to the center of Uq (512 ) we have

Ei

Ei

ED, V.

ED ; v.;

(T (C q ) - j )T T (a)v = T (a)(T (C - c i ))r v = 0 for a E Uq (512), V E V. That is, each Vi is an invariant subspace for the representation T. By Proposition 8, the restrictions Try; of T are of the form treated in the preceding 1=1 paragraph. Hence Tryi and so T is completely reducible. 3.2.3 Finite Dimensional Representations of 6,7 (512) and Uh(s12) -

The previous considerations carry over with a few modifications to the algebras Oq (512) and Uh(s12) as well. In this subsection we state without proof the corresponding classification results for these algebras. Theorem 13. (i) For each co E { +1, -1, i, -i} and 1 = 1N0 there exists up to equivalence a unique (21+1)-dimensional irreducible representation Tw i of

66

3. The

Quantum Algebra Uq (812) and Its Representations

6,1 (51 2 ). This representation acts on basis vectors ei , j = —1, —1 +1,the representation space by

Tw t(E)em =

— mj[l + in + 1]) 112em+i ,

,l, of

(29)

Twi (F)em = wat + m][1 — m +11) 112 ern _ i , 71,1(10e,, = wren, , (30) where et ±i = e_i_i := O. Any irreducible finite-dimensional representation of ildg (s12 ) is equivalent to one of these representations Tut. (ii) Every finite-dimensional representation of 6,/ (512 ) is completely reducible.

Since Uh (s12 ) is an algebra over C[[h]], we consider representations of Uh(s1 2 ) on linear spaces over the ring C[[h]]. Let I be an integral or halfintegral nonnegative number and let V1 be a complex vector space with a basis ei, i = —1, + 1, • • • ,l. Then the formulas Ti (E )em

Ti(F)em =

=

mi[1+ m+11) 1/2 em+il

+mi[l — m+1])112em _ i , Ti (H)em = 2mem

(31) (32)

define an indecomposable representation T1 of Uh(s1 2 ) on the vector space 11[[h]] (see Subsect. 1.2.10) over the ring C[{111]. (Recall that a representation T is called indecomposable if the underlying space cannot be represented as a direct sum of nontrivial invariant subspaces.) The representation T1 is not irreducible, since any subspace itk Mil is invariant. Proposition 14. Every indecomposable representation of Uh(s12) on a vector space space of the form V[[h]], where V is a finite-dimensional complex vector space, is equivalent to one of the representations

3.3 Representations of Uq (512) for q a Root of Unity

Throughout this section we assume that the parameter q is a primitive p-th root of unity, where p > 3. The representation theory of Uq (s12) is then very different from the case when q is not a root of unity. We set p' = p if p is odd and p' = pI2 if p is even. 3.3.1 The Center of Efq (512 )

Let 3 denote the center of the algebra Uq (512). If q is not a root of unity, we know from Proposition 2 that 3 is generated by the Casimir element Cq defined by (6). However, in the root of unity case the center 3 is much larger. Proposition '15. (0 The elements Ev f , FP' , 10)% IC —P/ belong to 3. (ii) 3 is generated by E", F"', K"', IC —Pf and the Casimir element Cg.

3.3 Representations of Uq (s12) for q a Root of Unity

67

Proof. (j): We give the proof for F"'. By (1), ICFP1 = q-2P1 FP'IC = FP1 K. Since [pi, = 0 by Proposition 2.1, we have EFP1 = FP1 E by (3). Thus, FP' G h. The proof for E"', KP' and 1C -121 is similar. (ii): A proof can be found, for example, in [DCK]. Corollary 16. Any irreducible representation of Uq (512) is finite-dimensional.

Proof. Since an irreducible representation T maps central elements into scalar operators, it follows from Proposition 15(ii) that T(Uq (s12 )) coincides with - 1. Therefore, if y is the linear span of operators T(Er1C8 P), r, s,t < a nonzero vector of the representation space V, then T (Uq (s12 ))v is a finitedimensional invariant subspace, so that T(U q (512 ))v = V. Let 3 0 be the subalgebra of 3 generated by the elements E"', F"', K"', and let J be the two-sided ideal of Uq (s12 ) generated by E"', FP' , 101 - 1. Using the assumption qP = 1 it is easy to calculate that .6(1CP1 ) = 101 1CP', 46(E"') = A(FP) =-

0 1+

0 101 + 1 0 El'', 0 FP'.

Since 8(E"') and S(FP') are multiples of Ell if - P' and 1.01 F/1 , respectively, it is clear from these formulas that 3 0 is a Hopf subalgebra and J is a Hopf ideal of Uq (s12 ). Therefore, the quotient Ugres(512) Uq (s12)/J is a finitedimensional Hopf algebra. Using Proposition 15 it can be shown that the algebra Uq (512) is a 3obimodule with free left (and right) 30-module basis

ErKsF t ,

r, s, t = 0, 1, 2, • • • ,p' -1.

(33)

Hence the elements (33) also constitute a vector space basis of Ur(s12)• 3.3.2 Representations of Uq (51 2 )

In this subsection T denotes an irreducible representation of Uq (512 ) on a vector space V. Then the images T(EP ) and T(FP ) of the central elements EP' and FP' are scalar multiples of the identity operator. Concerning the values of these scalars we distinguish three possible cases. Case 1: T(EP') = T(F/91 ) = O.

In order to find such representations T let us first look at the representations TA from Subsect. 3.2.1. For any (.0 E +1, —1} and 1 E 1N0 the operators TA(E), TA(F) and TA(K) given by (22) and (23) also define a representation Tw i of the algebra Uq (s12) in the root of unity case. Next we construct a three-parameter family of representations Tab), which plays a crucial role in the following. Let a, b, and A be complex numbers such that A 0 and let V denote a p'-dimensional vector space with basis ei , = 0, 1, 2, • • • , p' - 1. One immediately checks that the operators {

68

3. The Quantum Algebra

Uq (512)

and Its Representations

TabA(K)ei = q -2j Aei , Tab),(F)ep f_i = beo, TabA(F)ei = ei+17 2 <

-

1

1

(34)

TabA(E)eo --= aepf_ i , Tab),(E)e i =

ab [i] q

q

47-1

ei_ i , z > 0 1

(35) satisfy the defining relations (1) and (2) of the algebra Uq (512 ). Hence these operators determine a representation Tab), of Uq (512 ) on V. Clearly, we have for any A O. 7-100A (FP ) 7-100A (EP Proposition 17. (0 The representation KA is irreducible if and only if 21 < p' . The representations 71,1, (.4) c { +1, 1}, 1 = 0, 1, - - (p' - 1), are pairwise nonequivalent and satisfy the condition Twi(EP ) = Twi(FP) = O. (ii) The representation Tao, is irreducible if and only if A ±qn for n = 0, 1, • - p' -2. The irreducible representations Ton are pairwise inequivalent. (Hi) The p' -dimensional irreducible representations Too, A = wgP' -1 , and 71 (p,_0 12 for ci) = ±1 are equivalent. Any two other irreducible representations of the form Tw1 and T00), are not equivalent. (iv) Each irreducible representation T of the algebra Uq (s12) such that T(EP') = T(FP) = 0 is equivalent to one of the irreducible representations from (j ) or (ii). -

,

Proof. (0: By the formulas (25) and (26), the action of the representation Tw 1 on the generators E, F, K can be written as

Twi(K)er'

wq21-2reri

ji(E)ert =

T,I(F)er = w{r+11er' ±i ,

where e' 1 = e'21+ := 0 and e ------- 0, 1, • • , 2/, is a basis of the representation space 1/1. First suppose that 21 > p'. Since [pl = 0, Twi(F)ep' _ 1 = [plep' , = O. Hence the subspace W spanned by the vectors e i = 0,1,2, • • • , p' -1, is invariant for the representation Twi . Since W V Twi is not irreducible. Now assume that 21 < p'. It is obvious that Tw i(EP') = T1(F) = O. Since q is a primitive p-th root of unity, all q-numbers [1], [2], • • - [p' - 1] are nonzero. Using this fact the remaining assertions of (i) follow completely similar to the classical case. (ii): The reducibility of the representations Too„ A = ±qn, 0 < n < p' - 2, and the irreducibility of the others are proved as in (i). Suppose that two irreducible representations T00, and Toot, are equivalent. Then there exists an invertible operator B such that BT00),(a) = Toot,(a)B for all a E Uq (s12)Let fen and {e} be the corresponding bases in the representation spaces of T00,\ and Toot,, respectively. Since ker Too),(F) = Ce 'lland ker Tool,(F) Cet, it follows that cet; = Be4' for some c E C. But then we have A./3e1)3 = BT00),(104 = Too p (K)B4 = cT00,..,(104 = citet: = pBe4 so that A = p. (iii): The equivalence of T00A, A = -1 , and Tumpt_ 1 )/2 can be seen at once from the above formulas. Other irreducible representations Too, and Twi cannot be equivalent, because they act on spaces of different dimensions. (iv) is proved in the same way as Proposition 9. ,

,

,

3.3 Representations of Uq (s12) for q a Root of Unity

69

Note that any irreducible representation from Proposition 17 is a representation with highest weight and with lowest weight. Case 2: T(EP 1 )

0 and T(F 1 )

O.

Such irreducible representations are called cyclic. The reason for this name will be seen below. Proposition 18. (i ) The representation Tab), is irreducible and cyclic if and

only if all coefficients ab+[i]q(41-i_A-igi-i)(q_q-i)-1, _ 0,1, ... ,p/_i , (in particular, the parameters a and b) are nonzero. (II) Two irreducible cyclic representations Tab), and Teb , A , are equivalent if A' = q -201/4 and only if a' = a +b-l [i] b' = b and for some i E {0, 1, 2, • • • , p' - 1}. (iii) Any irreducible cyclic representation of Uq (s12) is equivalent to a representation Tab.\ Proof. The assertion of (i) is obvious. (ii): Let Tab), and Ta tbiv be equivalent irreducible cyclic representations. Then the spectra of the operators Tab),(K) and Tat biv(K) coincide, so A' = CI q -2i A for some i E {0, 1, 2, • - • , p' - 1}. For the central elements F' wehav Tab),(FP' ) =

bI, TabA(Cq ) =

+ A -1 q -1 )(q - q -1 ) -2 + ab)I.

Since these values must be the same for the equivalent representation Tatbw we get b= b' and Aq + (q

1)2

+ ab =

(q

q -- 1)2

+ a'b,

that is, a' = a + A -4 qi-1 )(q c l ) -1 . Conversely, a direct calculation shows that for such values of a', b', A' the representations Tab), and Ta t b9,1 are indeed equivalent. (iii): Let T be an irreducible cyclic representation of Uq (512 ) on a vector space V. By Corollary 16, V is finite-dimensional. Hence the operator T(10 has an eigenvector e0 E V, say T(K)e0 = Aeo. We set ei T(Fi )eol i = 0, 1„ •, p' - 1. Since T is cyclic, T(FP') = bI for some nonzero b E C. Then we have T(Firi - i)e i = T(FP)e o = beo 0 and so ei 0, The relations (1) imply that T(K)e 1 = q 2 Ae, i = 0, 1, 2, • • • , p' - 1. Because the numbers q -2i A, i = 0,1,2, • • • ,p' - 1, are pairwise distinct, the vectors ei , i = 0, 1, 2, • - • ,p -1, are linearly independent. Since the Casimir operator (6) acts on V as a scalar operator, there is an a' E C such that T(FE)eo = T(q)e0

11(10q+T(K-1)q-l g g -1

eo = deo.

Therefore, bT(E)e o = T(FP')T(E)e o = T(FP 1-1 )de o = de pt _1, that is, T(E)eo = aépi_i, where a := db -1 . Applying relation (3) we find that

70

3. The Quantum Algebra Uq (s12) and Its Representations

_ A -l qi-i) T(E)e = T(E)T(P)eo = (ab + [i]q

q

_ (7 - 1.

ei-1, > O.

Thus we see that W := Lin{e Ii = 0,1, • • • , p' - 1} is an invariant subspace. Since T is irreducible, W = V. The preceding formulas show that T is Li equivalent to the representation TabÀ• Let us see what is special about the cyclic representations Tao,. The above formulas show that one can move from an arbitrary weight vector to a nonzero complex multiple of any other by repeated application of the operator Tab. (E) (or of Ta bA(F)). This explains why these representations are called cyclic. Recall that in contrast to this property, for the representations T 1.,1, 21 < p', the action of the operator 71„1(E) (or of 71,1(F)) defines a linear order on the weight vectors.

Case 3: T(EP) = 0 and T(FP')

0 or T(EP') 0 and T(FP') = O.

In this case the irreducible representation T will be called semicyclic. We first treat the case where T(EP') = 0 and T(FP') O. Examples of such semicyclic representations are obtained by setting a = 0 in the three parameter family Tao,. More precisely, we have Proposition 19. (i) Too, is an irreducible representation such that Tob),(EP 0 if and only if b = 0 and Tobx(FP') 0 and AP' ±1,0. Two such representations Too, and Tobix, are equivalent if and only if b = b' and A = A'. (ii) Every irreducible representation T of Uq (512) such that T (EP' ) = 0 and T (FP ) 0 is equivalent to one of the representations Tobx b 0, A O. Proof The proof of (0 is straightforward. In order to prove (ii), we repeat the reasoning of the proof of Proposition 18(ii). Since T(EP) = 0 by assumption, the constant a must be zero, that is, T is equivalent to Too,. 0

From the formulas (34) and (35) we see that e o is a highest weight vector for the representation Too, and that any representation Too, with b 0 and AP' ±1,0 has no lowest weight vector. Thus a semicyclic representation T such that T(EP') = 0 and T(FP') 0 always admits a highest weight vector, but no lowest weight vector. Similarly, every semicyclic representation T for which T(EP') 0 and T(FP) = 0 has a lowest weight vector, but no highest weight vector. It is clear from their definition that cyclic representations have neither highest nor lowest weights. Now we turn to the semicyclic representations T such that T(EP1 ) # 0 and T(FP') = 0. Let 0 be the algebra automorphism of Uq (512 ) determined by 0(E) = F, 0(F) = E and 0(K) = K -1 . Then 71' := T 00 is also an irreducible representation. The representation 711:j o, := Too, o 0 on V acts as

1- 1,x (K)ei

= ei+ i, i < p'

1-10,(E)epf_1 = beo ,

71(bx (F)eo =0, T b),(F)ei =[i]q

q q

e,_ 1 , i > O.

-

3.3 Representations of Uq (s12 ) for q a Root of Unity

71

0 if and only if T'(EP') 4 0 Clearly, we have T(EP 1 ) = 0 and T(FP') and r(FP') = 0. Therefore, the assertions of Proposition 19 remain valid if and if the elements Ell and FP' are interchanged. In Tob), is replaced by particular, this shows that any semicyclic irreducible representation T such that T(EP1 ) 0 and T(FP') = 0 is equivalent to one of the representations +1,0. b 0, AP'

To' tIA

not ,

Since the Cases 1-3 exhaust all possible cases for the irreducible representation T, the preceding results can be summarized in the following Theorem 20. Every irreducible representation of Uq (s12 ) has dimension < p' and is a weight representation. The irreducible representations T of dimension < p' are up to equivalence the representations Tw 1, 21 < p' - 1, from Proposition 17. Any irreducible representation of dimension p' is equivalent to one of the representations Ta,b), or T6bA from Propositions 17-19.

Finally, let us return once more to the representation Too), for A := we , w = +1, where n is one of the numbers 0, 1, • - - ,p' - 2. Recall that these representations are not irreducible and the linear span of basis elements e i , n < j < p' - 1, is a nontrivial invariant subspace. Comparing the corresponding formulas we see that the restriction of T00), to this subspace is equivalent to the irreducible representation Tw 1, 21 -= p'-n-1, from Proposition 17. A closer look at the operators Too),(E), To0),(F) and To0),(K) shows that the representation Too), is not completely reducible. That is, the assertion of Proposition 12 is no longer true if q is a root of unity. The existence of finitedimensional representations that are not completely reducible is another new and interesting phenomenon of the root of unity case.

3.3.3 Representations of U(s12 ) In this subsection we determine the irrreducible representations of the finitedimensional quotient algebra U(s1 2 ) = Uq(s12)/ J defined in Subsect. 3.3.1. Clearly, the irreducible representations of Ur (s12 ) are in one-to-one correspondence to those irreducible representations of Uq (512 ) which annihilate the elements EP' , FP KP' -1. Thus, only the representations from Case 1 can give rise to irreducible representations of Up(51 2 ). For notational simplicity let us denote the corresponding representations of Uqr"(s12) and Uq (s12) by the same symbol. From the results developed above we obtain Proposition 21. The algebra Uqres(s12) has the following irreducible repre-

sentations: (i) T1,1 with 0 < 21 + 1 < p if p is odd,(ii) T±1,1, 21 + 1 < p', with 21 even if p and p' are even; (iii) T1,1, 21 + 1 < p' , with 21 even and T_1,1, 21 + 1 < p', with 21 odd if I) is even and p' is odd. Any irreducible representation of U(s12) is equivalent to one of these representations.

72

3. The Quantum Algebra Uq (s12) and Its Representations

3.4 Tensor Products of Representations. Clebsch Gordan Coefficients During the study of representations in the two previous sections only the

algebra structure of Uq (512) was needed. In the subsequent sections we essentially use the comultiplication Uq (s12) by considering tensor products of representations. Throughout Sects. 3.4-3.6 we assume that q is not a root of unity. 3.4.1 Tensor Products of Representations Ti

Let T and T' be representations of the quantum algebra Uq (512) resp. 6,7 ( 1 2 ) on vector spaces V and W, respectively. Recall from Subsect. 1.3.1 that the tensor product TOT' is the representation on the vector space VOW defined by T(a (i) )

(T r)(a) :==

na(2))•

(36)

In what follows, let Th and T1 2 11 1 12 E No , be two irreducible representations of Uq (s12) (resp. Û(s12)). The next result is the Clebsch—Gordan formula on the decomposition of the tensor product into a direct sum of irreducible representations. ,

Proposition 22. If q is not a root of unity, then we have T11

7112D-2 7111-12i e 7111 —121+1 e • - • e T111+121.

(37)

Proof. We carry the proof for Uq (512 ). By Propositions 9 and 12, Th i 1712 is equivalent to a direct sum T = ED m w iTw i of multiples of irreducible representations Ti. The operator (Ti i T/ 2 )(K) = Th (K) T1 2 (K) has precisely the eigenvectors em ®e with eigenvalues q 2 (m+n), where m = —11 , • • • , / 1 and n = —12,• • • ,12 . Comparing with the eigenvalues of T1 (K) and their corresponding multiplicities, it follows that mli = lift -= Ili-121,11i —121+1, • - • '1 1 1+ 121 and m,/ = 0 otherwise.

Note that formula (37) is also valid for the h-adic algebra Uh(s12)In the classical case, the flip operator 7" : VI 1 ® Vi 2 given by r(vi y2 ) = V2 0 v i provides an equivalence of the representations 71 1 T1 2 and 712 0 Th of U(s1 2 ). This is no longer true for Uq (51 2 ), but there is another "natural" operator .h1 1 1 2 that realizes this equivalence at least if q is not a root of unity. We first consider the quasitriangular h-adic Hopf algebra Uh(512) which is equipped with the universal R-matrix R. from Proposition 7. Let 111 1 12

(T1 1 T1 2 )(7?)

and

0 T/2 )(1Z).

to Vi 2 OVi l and, by the relation "P(a)7Z = R.46(a) Then f11 1 12 maps Vt i for a E Uh(s1 2 ), we have

3.4 Tensor Products of Representations. Clebsch-Gordan Coefficients

14 1112( (T11

73

0 T12)(a)) = T(1 0 1112) (R)( 711 1 ® T1 2 )((a))

= T(Tii 0 T/2)( 1R(A(a)) = T(Tii 0 Ti2)(A" P (a)) (Ti 1 0 Ti2) ( 1Z) = (T12 ® 7111)0(0 7- (T11 ® T12 )( R) = ( T12 ® Tii )(41112.

(38)

Since k 1112 is invertible with inverse (Ti i 0 7712)( 7Z -1 ) o 7 - , the operator k1112 realizes the equivalence of the representations 7111 0T12 and 71,0T11 of Uh(S 12) • Using (19) we compute the matrix elements (R1 1 12 )27, of the operator R1 1 12 with respect to the basis {em 0 en } of Vi i 0 Vi 2 . For this reason we abbreviate q = eh. By (31) and (32), we obtain (Ri i i 2 r 1 '2 ni+n,n2-n •.= (e n1+n 0 en 2-n 1 R111 2 I e ni 0 en2) = C(en l -Fn 0 e in 2 -n

I

e WI-10H) /2 En

0 Fn I e 1 0 e in2 ) 7

where c = qn(n+1) I 2 (1 q-2 )([n] q !) -1 . The other matrix elements of R1 1 1 2 vanish. A direct calculation yields -

(Riii2 r n 1,17,n2 ,n2-n =

( 1 _ q-2)n [ t,1 _ niNti + n i + np k -[n]! [11 ± ni]! l i — n i — ni!

1 2 + n2 ]![1 2 - n2 + n]! ([ x

1/2

1/2

q 2 ( n 1 ± n)(n2 -n) qn(n+1)/ 2

[12 - n2 ]![/ 2 + n2 - ni!)

(39)

The expression in (39) is a formal power series in the indeterminate h. But it is also well-defined for fixed q E C, when q is not a root of unity. The corresponding matrix k1112 realizes an equivalence of the representations T1 1 0 T1 2 and 771 2 0 Ti, of the algebra Uq (s12 ) as seen by the following reasoning. Equation (38) holds in particular for the elements e ±hi I 7 E and F of the algebra Uh (s12 ) and for vectors in the complex linear span of the basis elements em 0 en . Now we put q = e h in the matrix entries of it1 1 1 2 65) . Then, by the definitions of the representations and K±1 := e ±hH in ( 3 ,-,‘ Ti for Uh (s12 ) and Uq (512 ) and by the formulas of the comultiplications, (38) is just the intertwining property for the generators of Uq (s12). Hence k1 1 12 intertwines the representations Ti, 0 711 2 and 71/ 2 0 711 1 of Uq (s12 ) and gives their equivalence. A similar result holds for the algebra Oq (s12 ). Let us turn to the special case 1 1 = 1 2 = and denote the operators R1/2,112 and k12,1/2 by R and k, respectively. With respect to the basis e 112 0 e_ 1 / 2 , e_ 1 1 2 0e 1/ 2 , e112 0 e 112) e1 1 2 0 e 1 1 2 these matrices take the form -

-

q

z--

LR

0 0. 1 0 q — q -1 1 0 0

0) 0 0

q

0 1

00 '

0 q - q-1 1 "4 =(q 0 0

1 0

0 001 ' 0 q

(40)

3. The Quantum Algebra Uq (s12 ) and Its Representations

74

3.4.2 Clebsch-Gordan Coefficients Let 1-.7'i denote the subspace of V1 1 0 on which the representation Ti in the decomposition (37) is realized. Let fej l, tek}, {eini } be bases of the vector spaces 14, , Vi2) 1:-// such that the corresponding representations of Uq (s1 2) (resp. Ûq (s12)) are given by the formulas (22) and (23) (resp. (29) and (30)). Since both sets fei

ek I

.= I

{

=- -11,-11 + 1, • • • ,11, k = - 127 - 12 + 11 - /2111/1 - /21+ 1 , - • ,/1 + /2, rn =

1211

-/ +1,—,/}

are bases of VI , 0 1,712 , there exists an invertible matrix C with entries Cq (i i , /2, i; j,k, m) such that (41) The numbers Cq (1 1 ,1 2 ,1; j,k,m) are called the Clebsch-Gordan coefficients (abbreviated, CGC's) of the tensor product T1 1 0 T1 2 . We often write Cq (?,j)

Cq (11,12,1; j,k,m).

In the subsequent treatment we shall see that the main properties of the CGC's known for the classical group SU (2 ) (as described, for instance, in [VKl], Chap. 8) remain valid for the CGC's of td/q (su2 ) as well. In particular, Cq (e,j) -= 0

if j+ k74 m.

an,

m, n = -1, -1 ± 1 , • • • ,1, denote the matrix coefficients of the representation 711 of Uq (512) (resp. Ùq (s12)) with respect to the basis {em } of V/. That is, an are linear functionals on Uq (512) (resp. 6(1 (512)) such that Let

7110e,, =

rn .

(42)

We apply the representation T1 1 0 T1 2 to both sides of (41). On the left hand side, we take into account that acts on eim as Ti does and use the formulas (41) and (42). On the right hind side, we use (42) and the definition of the product in the dual algebra Ug (s12)' (resp. Ùq (512)'). Equating the coefficients of basis elements, we obtain the identity Cq(11 7 12 7 1; j, j7 771 )-tTn

=E

cq ui,12,1;r, s, n)t 7.11) t.(11.92)

for i = -/i, + 1, • • • = -12, -1 2 + 1, - • - , /2. Since the matrix Cq of CGC's is invertible, we derive from this identity that t17.11) t 1( .92) = E

j

cg 17 27 /.r (1) (ii /2 bi Is,n )t mrt• 7 7 7 m)-i(/ 7

(43)

75

3.4 Tensor Products of Representations. Clebsch—Gordan Coefficients

Before we continue the general treatment of CGC`s, let us illustrate the preceding in the case 1 1 = 12 = We abbreviate t11 := t (1/2) -1/2,-1/2' t12 := t (1/2) 4 1/2 t -1/2,1/2' t21 := 1/2,-1/2/ t(22) := 1'1/2,1/2. ,„1 e(i)

= e112

In the above notation, we have

e 1 12 , e 1 = e_ 1 12 e_ 1 12 ,

_ (q -1/2/[2]1/2 )e112

(q 1/2/[211/2\,

1/2 (q 1/2/[2]1/2) e112 0e 1/2

eg

iu 1/2

® e112,

(q -1/2/[2]1/2 \,_ iu 1/2 0

el12.

In order to prove these formulas, it suffices to check that the actions of the representations T1, I = 0, 1, and T112 0 To on both sides are the same. Comparing with (41), we see that Cg @7

= Cq (il

Cq (il i71; Cq

Pf3) = - 1 / 2 / [2] 1 / 2

ou,

1 7 1; 1 1 A 1 0) =

40) = q1/2 /pp./2

1();

1) = Cfq (11110; AI

(11111;

= 1.

Inserting this into (43), we obtain

tiit22 = q(6 0

q -2 t8 0 ) [21,

tg0)/[ 2 ].

ti2t21 =

If we denote the unit element 40 = e of the dual algebra by 1, the two previous equations yield t 11 t22 qt 1 2t2 i = 1. Proceeding in a similar manner we derive from (43) at 1 1 = 1 2 = 1/2 that tlit12 == qt l2 t ll ,

tllt21

t12t21 == t21t12, t11t22

t12t22 == qt22t12,

qt12t21 == 1,

t21t22 == qt22t217 (44) == 1.

t22t11

(45)

These relations define the coordinate algebra of the quantum group SL q (2) which will be extensively studied in the next chapter. In the remainder of this chapter we assume that q is a real number such that q 0, ±1 and we consider the compact real form tig (5u2) of the quantum algebra Ûq (512 ). Then, as noted in Subsect. 3.2.1, any 711 is a *-representation of the *-algebra Ûq (51.12), where the scalar product of the representation space V/ is defined by requiring that the basis {e rn } is orthonormal. Since tig (su2) is a Hopf *-algebra, Th 0 T1 2 is also a *-representation of 6q (5u2 ). Therefore, (37) gives an orthogonal decomposition of the Hilbert space V1 1 01/12 and we can choose the basis {elm } of 171 to be orthonormal. Hence we have CA,1 2 ,1;j,k,m) = (ei

ek , ern ' ).

The unitarity of the matrix Cfq leads to the orthogonality relations for the CGCs: cg (i i, 1 2, 1 ; i7 m

m)Cti(11, 127 II ;

=

bii1

(46)

76

3.

The Quantum Algebra

Uq (512)

and Its Representations (47)

As in the case of SU(2), the CGC's for the quantum algebra (4(5u 2 ) can be calculated by the method of highest weights. Applying the operator T1 (E) to both sides of (41) at m = 1 and taking into account that Tj(E)e = 0 and

E c,(F;i)(ei ® ek) = E

Ti (E)

j±k=m

Cq (ej)Ti(E)(e i ek)

j+k=m

ek + Ciej T1 2 (E)e k ),

Cq (ej){q k (11,(E)ei )

we derive a recurrence relation for Cq (ej), j + k calculations this leads to the solution Cq (.ej)

(-1) 1 1 -i q -(i+1)(11-i)+_1(1+1)/2

1. After some simple

1/2

([11 + j]![12 + - Id!)

-

a'

where j k -= 1 and a is a constant independent of j and k. By the orthogonality relation Ei±k _i ICje;j)1 2 = 1, we find a -2 =

il![12 ±

il2(j+1)(11j)+i(1±1)/2 (11

j±k=1

0112

ki! .

The right hand side can be summed by means of the basic hypergeometric functions 2(101. Indeed, by the relations (2.14) and (2.15), we have [21114/ ± 12 - /11![/1 ± 1 2 ____1g2/10+1) 2451(1 ± 1 2 1 1 ± 1, l -I -12; - 21 1; q27 q2(21+1)), where the notation (2.35) is used. We apply the formula (2.33) to this bypergeometric series. As in the classical case (see [VI(11, Chap. 8), the CGC's Cg (t; j) are uniquely determined only up to a number c(1) of modulus 1, which does not depend on j and k. Hence we can assume that a> O. Thus, a` l = q - ( 1 1+ 1 2 -1 )( 1 + 12 -1 1+ 1 )

(

[21+ 1]![/1 +12 - 11!q211(1+1) + 12 + + 11 - 121![/ - Ii ± 12 ] ! )

1/2

and for j k = 1 we obtain Cq (ii,/2,/;j, k, 1) = (-1)11-i qi{12(12+1)-11(11+0-1(t+1)+2.0+111

[21 + 1P[11 +12 - 1]![11 +j]![12 + k ] ! x

Eli

+ 12 + 1 + 4[1 +11 - 121q1 - 11 + 12P[11 0[12 - kP)

From the relation Ti(E)* =TI(F) and the equality = ([21 ]! [i

m]ql +

i)1/2 dm

112

(48)

77

3.4 Tensor Products of Representations. Clebsch-Gordan Coefficients we derive

where

ek),eb, (49)

ek, TI(F) l- meb = Ni, m (Ti(E) i- m(ei

Cg (t; j) =

j + k m, Ti(E) = (E) Ti,(K) + (.1‘ -1 ) 07112 (E) and Ni on =

Ti i (K), we Cl + mP) 1 / 2 ({2lP[1- 77 ]!) -1 /2 . Setting E1 2 := T1(E) and K1 2 have (E1 1 0 Ki2)((-Kii) -1 E12) = q 2 ((lCi i ) -1 E12 )(E1 1 0 K 12 ). Therefore, the ooperator Erm Ti(E)i - m can be represented in the form -

q -2

(E11

K12 ) k

( ( KO'

(50)

0 Et 2 ) 1-m-k

by Proposition 2.2. Combining (49) and (50) we get r,

1-m

=E

(

raj. [r]![1 - 7• - r]!

r=o

[1 1

[12 + 1 - - rj![11 + + ri! ) 1 / 2 x

(

-

-

[12 - 1 + + r]![11 j - rp

+ ml! k]! [21]![l -

1/2

Cq(11,12,1; + r,k +1- m r, 1 ),

where j k = m and f = rk - j(1 m r). From the latter equation and (48) we obtain the following expression for the CGG's of 0q (su2): (_1)/i - jq0]-([2/ Cq(t

+1][t4) 1 / 2 [/1 + 1 2 — 1]! (t)[1 1 + 12 + 1 + + 0[12 + Id!

' i) x

E

where m --= + k, fi ,âi(t)

(-1)rqr(i+m+ 1)[1 1 + j + rp[1 2 +1 j [r]![1 - m - r]![11 - j - r ]! [12 - 1 + j + r ]!

= 0 2(12 + 1 ) + 12 -

,A(11,1271)

(51)

+1) - 1(1 + 1 ) + 2i(m + 1 )b - 12

± 1]![1

[1 1 + /2 +

1]!

-

/1

+ /0) 1/2

(52)

m]!, (53) + Trtjql j]![il 0[12 ± k]![12 and the summation is over all integral values of r for which the numbers in the q-factorials are nonnegative. Equation (51) is a q-analog of the Racah formula for the CGC's of the group SU (2 ). = [il

By (2.14) and (2.15), the sum in (51) reduces to the hypergeometric series x) := 3 112(a, b, c; d, e; q, x) we 3ç02. Using the notation 3(p2 (g a lqb , e ;q d 7q e ; have

(-1) 11 -j q0146(t)[l +12 - jPgj][21 + Cg(t;61) — [11 — /2 +1]![/ + .12 — 10112 — 1 +

X34P2 12-1+

where fi is as in (51).

+ 1, -/ + rn

11) 1 /2

0[12 + k ]![ 1 — MP

n 2 n2

3 +1 1 —1-12+i ; ‘1 " ) 7

(54)

3.

78

The Quantum Algebra U(s12) and Its Representations

3.4.3 Other Expressions for Clebsch-Gordan Coefficients

A number of other expressions for the CGC's of Oq (5u2) can be derived from (54) by means of the relations (2.36) and (2.37). Setting n

11

j, a

= q2(i+ii+1) , b

q2(j - i+k) , c

q2(i - t -F/2+1),

d = q 2(-1-12)

in (2.36) it follows from (54) that

cq(q ) =-- [1 - 12 + ( X3‘1'2

q /32

()[l + 1 2 - j]! (

[2 1 + 11) 1 / 2

+1 2 - 11! 1 — li ± 12]![/1 11 + 1, — — 12 k —12 1 1 — 12 ± + 1

kJ! — TrZ]

jj![12

]

2 2(k-12)) 7 qq

(55)

where 02 = { 11 ( l l + 1 ) + 1 2( 1 2 ± 1) - 1(1+ 1)+2,114. Inserting here n = 11 - j, a = q

c = q2(i-t i -k+i) , d q 2u i 2 +3 +0

-2(k+1 2 ) , b

-

and using (2.36) we get a q-analog of the van der Waerden formula for the CGC's of SU(2):

go3d(i )( V; j] [2 1 + 1]) h /2 Cg(t;i) = [1- 1 1 - k]! { 11 + /2 — 1 1 11[1 12 4– j }! [11 j _ 11 e

where 03 =

n

12

1-1 1

12 ,

12 -k

3]! [12 + Id!

,2

(56)

+ 12 — 1 )( 1 1 + 12 ± I + 1) + 2(k11 — j12)}. Putting here

k, a = q 2(11 1 4 2) , b =__

d = q 2(1-12+i+1) ,

c.

and applying again (2.36) we obtain

q134 (.0[212]!at, j] [2/ + 11) 1 /2 Cg (t;i) = [1 X 3v-'2

where 04 = n

j, a

12 +

± /2 — 11!1 [1 ± 12 — /11!til

(-k - 1 2 , 1-11 - 12, 1+ 11 - 12 + 1 1-1 + j + -212 2;

jP[12

k]![12 -F kl!

2 2(rn—l)

(57) )

(l 1 — 1+12)(11 + 1-12 + 1) +2(k1 k1 2 j12)}. Setting q - 2(1 - m) , b

q 2(11-pi+i) , c

q 2(12-i+i+i) d

q2(i-1-12)

into formula (54) and using relation (2.36) we derive a q - analog of the JusysBandzaitis formula for

the CGC's of SU(2):

+ :Mt + mi!([21 + 1 ] ) 1 /2 (-1) 11 -iqP5[1 + /2 - j]![11 +12 Cg (t; j) 1]! 1 A(i)(rejj) 1/2 [11 + I2 +

79

3.4 Tensor Products of Representations. Clebsch—Gordan Coefficients X 3 f12 ( M — 1, —/ — li — 1 2 — 1 1 i — 1 1 ; q2 1 q 2(1-1-il —/2-1-1)) 1 -1 - /2 + j, -11 - /2 + m

(58)

where 05 -= 1 1(12 — 11 — 1)(12 + 1 ± 1) + 2i (1 + li + 1) + 2k/11. Letting .41 1 d

n -- = 1 + 11 + 12 ± 1, a --.-. q2(7n -1) , b --= q2(11-12-1) , c =

= q2(-/-/ 2 )

and applying (2.36) we derive

Cq (E;i)

(-1) /1-1 +k q436 [/ + 12 —

=

[4 ± 1 2 — /]![2/P[11

]

AMU-UP/ 2 [1 1 + 12 + 1 + M—

X 3w2

-1 - - /2 — 1, 11 -1 - 12 + j, -21

1 2l 2 2

1 ± 12

2k1

11, a = q -2(ii-F12+/-F1) b = q 2(m—i) , c

(59) )

; q

where 06 = 1(1 + 12 — 11) (1 -I- 11 + 12 + 1) - 212m n

j !( [2l ± 1 ]) 1/2

2k}. If we set here

47 2(-1-12)

d

= 17 -41

and use (2.36), we get

Cq (e; j) =

(-1)/ 2 +k eLl(t)[/ + 12 + j]![21]![1 1 - j]!([21 + 11)112

[11 - 12 ± 11![12 — 1

2

1]!Ge;

-m - 1, -1 -li - 12 — 1, 1 — 12 — 1 2(12+k+1)) <32 ; q )q —1 — 12 — -21 where 07 = 11(li - 1 2 - /)(ii + 1 2 ± 1 + 1)

-

2k(1 + 12 ± 1) ± 2:7121. If we set

ri = l+1 2 -i i , a = q 2(i+m) b = q2(6. i+1) c (1 2(11 - 12 - 7-11+1) d -

-

( 60)

q 2(12+1+i) -

and use again (2.36), we derive a q-analog of the Wigner formula for the CGC's of SU (2 ):

Cq (e; j) =

(-1) 12±ke8 Z1(E)[1 + 12 ± i}![1 Ml![11 j]!([21 + 11)1/2 [li - 12 — MP [/2 — 11 ± 11!([e; 1) 112 /, /1 — + 1, 11 - 12 - 1 —/ — /2 — 1 1 — /2 — 711 -I- 1

X 312

where 08 =--

n=1

• -

22\ ; q

(61)

) 7

{(/ 1 - / 2 - /)(ii + /2 - + 1) + 2j(/ - / 1 ) - 2k(/1 +1)1. Setting

m , a = q2 (ii-i-i2) b = q2(11+12 1+1) c = -

d = q --41

into (60), we obtain a q-analog of the Majumdar formula:

(_1)t .2 +kgo9 tl(e)[2/]!ae; j] [2/ + 1 ]) 11' 2 Cg(t;i)

X

-

2

12

± l]![1 - 1 1 ± 12] 411

-1-

- m]ql + mP[12 + Id!

—7/2 — 1, /1 + /2 — 1 + 1, 11 — 12 — 1 ; 422 , q2) 7

11 1

k + 1, —2 1

(62)

3. The Quantum Algebra U9 (512) and Its Representations

80

where fis = 1-{(11 — 12 — 1)(1 i + 1 2 — 1 + 1) + 2j(1 — li) 2 k( 11 + 1 )}. Using the relations (2.14) and (2.15), formula (56) can be rewritten as

Cq (t; j) = gf33 ,6(t)([e, j][2/ + 11) 1 /2 (_iy q -r(ii+i2+i+i)[/ _11 -k+rri x

[711[11 + 12

-

1

-

rj!{11

j ri![12 + k

rj!{1

-

(63)

12 + j + 71!

In a similar manner, formula (58) can be expressed as

(v; i1121 +1 1)1/2 „ L1 V)

Cq (t; j) =--

XE

[12 - 1c ]! [12 ± k ] ! 12 — 771 —

[r14/ 1 + 12

q[12 +

+ 1 — r +114/1. - j

rly

j -

m

r

]!

•

(64)

-

For formula (60) we obtain the expression

cq t; j ) =

1/2

(_ 1) 1 2 +k gp7zi (t) {/1 j]![1 + TriN1 + 12 + 1 + 11! ( [21 + [11 — + 11!

(

x

i) rqr(12+k) [21

(—

+ /2 -11.

-

—

)

r]![12 + 1 + j

r1!{1 1 +12 +

-

r +11![/ + m - r ]!

(65)

The q-analog of the Wigner formula (61) can be written as rni

cvt; j) = (- 1) 1 2+k es 24.0[1 mpy + 77*

xE

(

14 ( 1[

77.11 1

ii

) 1/2

r

Horgr(hi+12_/,)[/1 j _4_ r ]![ 12 +1+ j

2—

]!

— — m + rifil + m rj!.[rly+1

(66)

Formula (62) can be represented as

1)/2+kiTog

x

2411[21 + 1 ] (1f;i1) 1 / 2 [11 — 1 2 + 1]![1]. +12 — 1]![12 + 0[1

m]!

(-1)r qr(1 1 — +2) [21 — r]![11 + 12 1 + + 12 11 —r![1 —1 — k + 0[1 + m — rj! •

(67)

The expressions for A, in formulas (63)-(67) are the same as in (56), (58), (60)-(62), respectively. Formula (54) for 1 = 0 gives

Cq (41/210; j7 --j, 0) = (_1)h1 For h = l + 1 2 , (56) yields

/25t1 + 12; j,

km)

q ( k l -.5 1 2 )

12h + 1 1 -1/26/112.

3.4 Tensor Products of Representations. Clebsch—Gordan Coefficients

(

X

[210[210[11 + 12 — MP[ii

\ 1/2

[211 + 210[11 - ./14/1 + ..71![/2 - 1412+ kl!)

j k. Setting 1=4

where m

+ 12 + 77*

81

-

12 into (61) we get

Cq (i1,12,/ 1 - 12; j, k,m) = (_1)12+k g (kti+ii2+k) [2/2P[2/1 — 212 + 1-1![/1 [2/1 1P[12 /4/2

\ 1/2

il![/1 +

— 12 +

MP [ii — 12 — 77/P )

7

where m = k. From the last two formulas we easily derive the expressions for the CGC's of the tensor product T10T112 :

q(1,1,1+

= q (')"2([1 ±j + -]1[21 + 1 ]) 1 /2 1

(68)

71 —

+q' ) " 2 ([1 ([I 2j}1[21+1])112 .

(69)

Cq (/

3.4.4 Symmetries of Clebsch-Gordan Coefficients From formula (56) it follows that

Cq (E,j) a- CV/ 1 ,12 ,1;j, k,m) = Cq (12,11,1; - k-j 7 —7n)-

(70)

Equation (58) leads to the formula for permutation symmetry of the CGC's:

kq_k ( [2/ + 1 1 1/2 Cje,j)= (-1)1+11+ Cq (/ 1 12 1 11; m, Tri -k, [211 + 11 )

(71)

The expression (56) is symmetric with respect to the transformation

(1++m 1+—m 1++j k 1+ —j+k

cw,i)

2

2

• '/'

2

2

(72)

where 1+ = 1 1 + 12 and 1 - =11 12. This is the q-analog of Regge's symmetry for the CGe's of the group SU (2 ). Using (2.13), we derive from (51) that -

-

Cg (t,j) = (-1) /1+/2-/Cq -1(l 1 ,12,1;-j,-k,

-

m).

(73)

This formula relates the CGC's of the algebras Oq (5u2) and Oq -1(su2). The classical analogs of the relations (70)-(73) generate a symmetry group for the CGG's of SU(2) which contains 72 elements (see Subsect. 8.2.2 in [ITK1]). The relations (70)-(73) for Ûq (5u2 ) generate a symmetry group for the quantum CGC's also containing 72 elements, but now relation (73) changes the parameter q to q -1- . Along with the above mentioned symmetries, there exist symmetries of the expressions for the CGC's related to the transition I -1 - 1. This substitution leads to expressions of the type [-k]!/[-m , where k,m G N. By (2.64), such expressions have to be replaced by (-1)"[M 1]!I[k+ 1] !. We collect here only a few of these symmetries: ]!

82

3. The Quantum Algebra Uq (s12) and Its Representations Cq (1 1 ,12 ,1; j,k,m) =(-1) 12 ±k Cq (4,12 ,r; j,k,m) - (-1) 11-3 Cq (11,12,1; j,k,m) (-1) 12-1-i Cq (4,12,1; j,k,m) ( -1) 11+12-1 Q11,12,1; k, m)•

3.5 Racah Coefficients and 6j Symbols of

Uq (5u2)

3.5.1 Definition of the Racah Coefficients Consider three irreducible representations Th , T1 2 Ti 3 of Oq(su2). Let lei l, Ifkl, {h m } denote the orthonormal bases of their respective representation spaces such that the formulas (22) and (23) hold. The tensor product of these three representations is associative, that is, under the canonical identification of the vector spaces (1/1 1 0 V1 2 ) 0V1 3 and Vh (V/ 2 0 Vi a ) we have (T1 1 0 71 2 ) 0 Ti 3 =

(74)

(T1 2 T1 3 ).

Proposition 22 yields the decompositions 12+13

T1 1 ®T1 2 =

ØT13

7112

71/23

(75)

/23 = 1 1 2 —1 31

Inserting the latter into (74), we get 1 2+13

1 1+12

ED

(T1 1 ®T1 23 ).

( 711 2 0 7113 = )

123=112 - 131

/12=1/1 -1 21

The carrier spaces of the representations mal bases consisting of the elements /1

E

T1 12

and T1 23 in (75) have orthonor/2

12

E c l lli

i ei

fi ) ,

/3

E E ciit/7,23(fi hk),

bin"

i= -41 3 =-1 2

j=-12 n=-13

respectively, where Caly :=--- Cq (a, b, C; a, 3 ry) denote the CGC's of Ûq (5112). By Proposition 22, we have ,

(1) 1 12+/3

711 12

1113 =

1=1112-131

T1.

(76)

Let {4112(112)21311} be an orthonormal basis of the carrier space of the subrepresentation Ti in (76) such that the formulas (22) and (23) hold. Note that the upper indices indicate the order of the tensor products: first the tensor product 711 1 Th is formed and then the components 7112 of Th O T1 2 are tensored by 111 9 . We have

3.5 Racah Coefficients and 6j Symbols of Uq (su2)

83

112131 (a1m h k ) = Ec!iji21,71,2c171,2(eiofi)ohk,

el ' 12(1 " ) '13 'i -

( 77)

m kp

i,j,k

where i + j = m, m + k = p. 113(123)11 2 Analogously, if {e 11 p i s the corresponding orthonormal basis of 11 +123 then the carrier space of Ti in the decomposition Th 0 T123 = w1=111-1231T1 we have ( f. cd2/3123011/23ç,. ei1,i2/3(23),/ = . (78) Z_d i,j, k

kn

n

j

'46w

where j + k --= n and i + n = p. Since the vectors e

k)

7

1 112(112),13,1 with

Iii - /21 5 112 5Ii +12, 1/12 - /31 < 1 < 112 + 13, p = -1,-1 + 1,- • • ,l,

as well as the vectors ept11213(123)1 with lui -1 231 < < /1+123) 112 -131 <1 23 < 12 + 13, p = -1 7 -1+11

. •

1

1,

are orthonormal bases of the space 0 V1 3 , by Schur's lemma there is a unitary matrix R with entries RA/2/3,112123,1) such that eiii2(/12),/3,/ =

(79) 23 ‘-111123

The complex numbers Rq (11/2/3,112123, 1) are called the Racah coefficients of the quantum algebra (su2 ) or the q-Racah coeffi cients. Note that the matrix R is block-diagonal and that the numbers RA l 2/3, 42123,1) do not depend on the indices i, j, k, m, n, p of basis elements in (77) and (78). The unitarity of the matrix R leads to the orthogonality relations

el

Eit 1 12

= 8/12/1 2 7

(80)

Rq(/1/2/3,/12/23,0Rq(4/2/35/12/3,/). 8 1231/23 '

(81)

23 R4/(1112137 1121237

1)Rq(1112137

421237

1)

By the orthogonality relations for the CGC's, (78) implies that ei (fi ®hk) = EEca.2/3/23 3kn

1 1 231,9,,11,1213(/23) ) 1 .

n ppp ,

(82)

1 23

Since (e i fi) hk = ei 0 (fi 0 hk), (82) can be inserted into (77). Comparing with (79) we obtain

Rq ( 111 2 13 7 11 212 3 7 1) = E E E i.-11 i.--1 2 k=-1 3

06321120112131 012131230111231 "ij k n p7 k p'j k n

(83)

where j + j = m, m + k = p, j + k = n, n = p. Note that (83) holds for each value of the index p. We see from (83) that the Racah coefficients are real if the CGC's are real. .• Recall that the CGC C1111±i vanishes if the triangle condition Iii - 121 < 1 < 1 1 + 12 does not hold. 'Therefore, by (83), RA/2 1 2,11 2123,1) = 0 if at least one of the triples (11, 12 412)7 01 2? 1 31 01 (1 2713,123) and (11,123,1) does not satisfy this condition.

84

3. The Quantum Algebra Uq (512) and Its Representations

3.5.2 Relations Between Racah and Clebsch—Gordan Coefficients In what follows we assume that the CGe's and the Racah coefficients are real. The converse relation to (77) has the form

E

11+12

E

h,, =

(ei fi)

112+13

..,/1/20.),/3,/

.7 m

mk

pw-p

1 12 = 1 1 1 -1 211=1 1 12 - 131

We now insert (79) into this equation. Comparing with the right hand side of (82) and equating the coefficients of epil '1213(123) ' 1 , we obtain C1 2 1 3 1 23 C111231 3,1C5i±k t,j+k,i+j±k

E

==

.131 R,(/1/2/3,112123, 0d.q2./12.0 1,14+3 i+j,k,i+j±k'

112==111 - 121

Using the orthogonality relation (46) for the CGC's one derives ..2/3/23 c hi./ Eci.i/2/12d

• Rq uit2/ 3 ,1121 231 /)C1.121.31 z+3,k,/±3±k)

(84)

where the summations are over the values of j and j such that j + j = const.

3.5.3 Symmetry Relations It is more convenient to formulate symmetry properties in terms of the Wigner 6j symbols which are defined by {

(_i)ii -142-1-13-1-1

11

12

/12

13

1

1 23 }q

( [2112 ± 1] [21 23 ± 11r1/2 IL rP°1:1( 1 1 1 2 13 1121237 1 )•

(85)

As in the classical case, symmetry relations for the Racah coefficients follow from those for the CGe's by means of (83). For this reason, we give these symmetries without proof. The 6j symbol

{ / 1 12 112 , , I, 3

t

1 23 1 .4

is invariant under all permutations of its

columns as well as under simultaneous permutations of / 1 and 1 3, 1 2 and 1. Further, the following q-analog of the Regge symmetry holds:

Ii i

12 1 12

/3 / /23

12 S - 112 1 3 s-1 S - /23 /

S -

where s = -1(1 + 112 + 1 2 + 1 23). The described symmetries generate the symmetry group of the Wigner 6j symbols. This group contains 144 elements. Besides, as follows from (73), we have a new symmetry relation replacing q by q - :

85

3.5 Racah Coefficients and 6j Symbols of Uq (su2)

f l l 12 /3

112

1

=

123 q

11

/2

/3

1

1 121

1 23

As noted in Subsect. 3.4.4, the expressions for the CGC's possess symme1 := -1 - 1. These relations and try properties related to the transitions 1 formula (83) lead to symmetries of expressions for Racah coefficients. Some of them are given by the formulas /12

(_1)m { / 1 1 2 1 12

1 123} q

13

12

13

1

/2

(-1)n { 113

/23 }

1

/12

123} q

1 2 q 1 12 }

(_1)2(112-1-/23) { /1 13

1

l23

1

7

q

where m = 1 2 - 1 12 + 1 23 - 1 and n =2(l i + 1 3).

3.5.4 Calculation of Racah Coefficients The Racah coefficients can be computed by formula (84). Setting j + j = 1 12 and k = 1 / 12 therein we obtain -

Rq (11/2/3,112/23,1) = X

1 (C11122 ,1/31 •12

E

1112112

2 s 3 s/12

C111231 (86) C1213123 — /121,i+/ — /12 213 +/ — /121/

Utilizing here the expression oli/2/

1-1)1e ,i(/ 1 ,12 ,

k

[11

[li + iP[12 +k]![21 + 1]

1) (

-12 + /1!

-11(11 +1)

from Subsect. 3.4.2, where a = - {/2(/2 + and j + k = 12 we get

E q i(i+/,2+2,

{ /1 /112} 1 /23 /3

X

( [1 + /23 - i ] !{ 1 2 + 112 [/23

where m = =

(-1)

j]![/2

/1 + /]! 2 [ii

[/2

-

1(1 + 1) +2j(1 +1)1

) 1/2 C1213123

(87)

- 1 + i]![12 — 1 12 + ip

1 -11 + 12+13 (

[1 + 13 +1 12 ± 1]!1/1 + 1 2 + /2 + /12 ± 11![/1

c(/) := 1(1 + 1), and

/

112]![/1 + 1 23 -

112 ] ![1 1

+1

-

123 41 - 1 1 + 1231! ]

1]!

/23 + 1]![/1 -12 +1121!

[1 - 1 3 + /12 ] 42 1 23 + 11 -1 [12 -

k]!

+ i]]!! j

{c(/2) + c(/23) - c(/3)} c(4) - 1 12(1 +

X.

1/2

1/2

3. The Quantum Algebra 14(s12 ) and Its Representations

86

Substituting the expression (56) for the CGC's into (87) we derive 12 112 13

1

(_1) /—i1+212+6-1 .46(4,12,112)46(11,123,1)

_ ±

123 17

12 + 112]! [12

1121![11

2 (1 2, 13 7 123)2442, 13) 0[1 12 ± 13 ±

I

+

1 ! ]

X r iiri 1123 - 11 ± /iT2 - 13 ± 123P(13 — 1 2 ± 123P

[12

+112 —

z

123 + 1 ]!

11 + 112 ]! [11

( - 1) i±r q1111

(r}![11

i]! [1 3 +123 -1 12 + —

IT [123 +

r]![1 23 —1

[12 — 112 + i — r ]![13 ± 112 — 1 23

+i

where v = c(i) +r(1 i + /23 ± 1) —C(11). Now we change the summation from r and i to k = r and j. The sum over i reduces to 20 1 (-1 1 + k, 1 1 + k + 1; —123 1 + k; q2 , q2 ) and can be summed by using formula (2.34). Putting s = 12 — 112 + k we get —

—

J. 11

12

1

1. /3

X-

-r46 (1(113121112)21(11712311)2 12; 131123)2 112,

112 I

(

v23 }

1 ]! [11

[112 + /3 +

+ 1 1 21I [12

1121![1 1

±

----

+ 123 + 1 ] !

[11 22 i— 1 3 + 123]![13 —

(

1 1 + 1121![11

13,1)

123 + 1

] !

( —1)1212 — si! [11 — 1 2 + 112 + s ! ]

[s]![11 + 12 — 1 12 — s ] !

+123]!

[1 3 + 1 23 — 1 2 ± SP (/2 -1- 1 3 — 1 23 — SP -1 X [123 + 112 — /2 — 1 + s

]!

[123 + 1 1 2 — 1 2 +

(88)

1+5+

where a = 1 1 1 +12+13 and the summation is over all integral values of s for which the numbers in the q-factorials are nonnegative. The sum over s can be expressed in terms of the hypergeometric series 4‘,03(- - ; q 2 , q2 ) . Using first the symmetry relation -

11 131 1 11

112}

2

where I

123

=

( - 1)2N-124-1- 231

12

13

1

112 123 }

= -1 - 1, and applying formula (2.23) for 4(,03(• • • ; q, q) we obtain

f / 1 1 2 112 123 113

=

fq

(

-1)1+12+112+/23

[11 —

2(111 1 27112)2(111 1 2371)24/2; 1 31123)

[1 —13 + 110[12 — /1 ± 112]![/1 , -+ 12 — 11214 1 12 —

2(/12 1 1 3 1 /){212PE12 + 1 12

4w3

/1

/

123P[12

131!

+ /12 + 1 23 + 1]!

1 + 1 23141 — 1 1 ± 1 2314 1 2 + 13 — 123P[12 — 1 3 ±

1 23 ] !

/1 — /2 — /12 7 /3 — /2 — /23, — /1 — /2 — /12 — 1, — /2 — /3 — /23 — 1 2 2

— 212 ) — 12 —112 + 1-123, — 12 - 112 - 1 -a' 1 23 -1

7q

(89) where 4450 3 is related to the function 4V3 as 302 is to 3(,02 (see Subsect. 3.4.3). Other expressions for the Racah coefficients of U q (5u2) can be derived from (88) and (89) by symmetry relations. Symmetries of expressions for

3.5 Racah Coefficients and 6j Symbols of Uq (5u2)

87

Racah coefficients are equivalent to relation (2.38) for the basic hyperge°metric function 4 c,o3 . Applying (2.38) to the right hand side of (89), we obtain distinct expressions for the Racah coefficients of Oq (su2 ) in terms of 44 3- In particular, in this way we get q-analogs of all known expressions for the Racah coefficients of the group SU(2). All these g-analogs are obtained from the corresponding classical expressions by replacing the factorials m! by the g-factorials [m]! and, if the Racah coefficients are expressed in terms of hype rgeometric series, the function 4 F3 (a,b,

c, d;

by

g; 1)

e,

4453(a,

b, c, d;

e,

f, g; g2 q2 ).

Next we give two other relations, which are analogous to (87). Putting k =13 , i + j +k =1 in (86) we find

f1

12

113

1

(_1)112-1-/31-/23q8 23 1/12

}

- 112![l+13 +112 ± 1 ]! ) 1 / 2 \--‘( i)mr(/+/3-1-1)/2 - A(41 1 , 123)(/2, 1 3, 1 23)[21 12 + 1 ]! 112

+ 13

y

X

+

/ 1 + Trt]![/2

1 23 + in]!

1 23 -

—

) 1/2 0.y 2 y

1 + 13 +

y

(90)

1 + 7-rd! 11 — 74[1 2 + 1 — 13 — 7r ] !

where s = {112(112 +1) — 11(11 + 1) — (12(12 + 1) — 21(1 +1) + 2 1 23(123 + 1)1. Another such relation is { 11 13

1 2 /12} 1

1 23

(IS

(_1)1+ 1 1+ 1 2+ 1 3 A(111 v.‘t,3 1 7 v1 23 )A(1 /4 21 vI ) 1 23 )

+

q

— M(1+13+1)/21 " X Zd

([1 + 74[12 +

q

± 123 ±

771,

1]!([1 1 -

1121! [l + 1 3 + 1 12 ± 1]![21,2 ± 1 ]) 1/2 — 13

± /4) 1/2 [1

MN/2

-

—

1

23

+

1 +13 — Tnp)1/2

1 1 1

2 Crril,i%10

(91)

where j = 13 — 1 — in and s is as in (90). We close this subsection by giving the Racah coefficients for some special values of indices. If 112 = 0 in (89), then the triangular conditions yield /1 = 1 2 and / 1 — /2 - /12 0 1 so we get J' 1 1 1 2 13

1

0

(-1) 1 +11 + 123

123 f q — ([21 1 + 1 ] [2 1 3

1 ])1/2 81112 8113.

From (89) and the symmetry relations, for 1 1 + 1 2 = 112 we derive H ( 22bNa 4 + b + d + e + 1P[a + b — d + eP b a+b} ld e f q = (-1)8 [2a + 2b + 1]4d + e — a — Ifia + e — fp

fa

[a + b + d — eP[e + f — aP[d + f — bP[a, — e + fr 1 \1/2 X

[a+e+ f+11![b+d— fp[b—d+ fffib+d+ f +1]!)

7

88

3. The Quantum Algebra Uq (s12) and Its Representations

wheres=a+b+d+e. For the Racah coefficients with

Ii i

to

1

1

1 + 1/2 12 +1/2 f q

+ 12

= (_1)11+12+1+4±4.

/

I

11

t 1/2

1][f/1 + 1 2

1

/

11) 1/2

+ 111212 + 1 ± 1][21 +21 )

[2/2 ± 1 ] [2 1

f

we have the expressions

13 r=

(92)

1

1 — 1/2 1 2 + 1/2 f q

=_- (_1)/1-1-12+t

+ 12+1 -I- 1 ] [1/1 + /2 +

V/2 [212 ± 1 ] [21 + 1][212 ± 1 ± 1 ][21]) [Ii

(93)

3.5.5 The Biedenharn—Elliott Identity The Racah coefficients of Ùq (5U2) satisfy the Biedenharn -Elliott identity

E

123

Rq (111213 ,

112123 1 1 123)Rq (4/23 /4 7 1123 1 234

Rq(/12/3/47/123 1 347

1)Rq (12 1 3 14, 123134 7 /234)

(94)

1) Rq (1112134, 1121234, 1) .

Its proof is similar to the case of the group SU(2). The starting point is the observation that there are five possibilities to decompose the tensor product Th 0 T12 0 T13 0 TN into irreducible representations by Proposition 22: {( 7711

T12) T/31 ® T/41

{T11

( T12 ® T13)} ® T14 ,

(I)

Th {(71/2 T13) 0TO,

(III)

Tti

(T13 T14)},

(IV)

(271 3 0 T14).

(V)

(Th

{T/2

Ti2)

We can go from decomposition (I) to decomposition (IV) by the chain (I)-4 (II) -4(III)—qIV) and also by the chain (I);00— ( IV), using Racah coefficients at each step. Since the final formulas connect the same decompositions (I) and (IV), the corresponding matrices are equal in both cases. Writing down the equality of matrix elements, we get the identity (94). In terms of Wigner 6j symbols the Biedenharn—Elliott identity has the form _ e eb' — a' d d, a' b 1 cfe

E(-1)P[2f + 11

q I 1a

el fe c f f ld

b

q

b d'

f

fe a a' flic d' f ,q

47 9

3.5 Racah Coefficients and 6j Symbols of Uq (su2)

89

where p = a' + b' -e' -d' -a-b-d+e- f. By the orthogonality relation for the Racah coefficients, it follows from (94) that

Rq (aa' f e, d' ) =

E

e,e'

R q (abd, ef, c)R q (c'bd, e'f, d')R q (a' ab, c, e')R q (a' ed,

e,

.

(94')

Putting here d = b = f - and using the expressions (92) and (93), we obtain a recurrence relation for the Racah coefficients. Using the notation bc 3 km

(r) = Rq (r - m, a, b; r k , c; r),

+km

this recurrence relation can be written as

fi(r) = + b - ej [a - b + c + 1j[b kJ[2r --b- k + 1][c + m + 1 ] [21)1 2 [2C + 11[2r + 1] 2 [2c + 21[2r + c m + 21 -1 ,b-1/2,c+1/2 Raj,k+1/2,rn+1/2( r —

1/2

1/2)

a + c] [a + b + c 1][b - lc] [2r - b - k + 1J[e [2b1 2 [2c] [2c + 1] [2r + 1] 2 [2r - e m + 1 1 -1 , b-1/ 2,c-112 1/2) x R j,k+1/2,rn-I-1/2k r

) 11 2

b + c][a b + e + 1llb + k][2r + b - k + 1i[c m + 11 12 [2bi 2 [2c + 1] [2c + 21[2r + 1? [2r - e m] -1 ,b —1 / 2 ,c+1 / 2 /

j,k-1/2,rn-1/2k r

/0 \ i t `i)

eb-a+c][a+b+c+1][b+k][2r+b-k+11[e+mj)

112

[2b] 2 [2c1[2e + 11[2r + 1] 2 [2r + e - m + 1] -1

xRaj,k-1/2,m.-1/2k ' b-1 / 2'c-1/2 ir

1/2). 2

(95)

2

Starting with the Racah coefficients R7 given by (92) and (93), equation (95) allows us to compute inductively all Racah coefficients. Our next result is the relation fa

le

d rl fa b ef lb

d r pia b cl e f = ld e f

(96) where we abbreviated a := -{f (f + 1) + c(c + 1) + r(r + 1)} and (3 :=-{a(a +1) + b(b+ 1) +d(d+ 1) + e(e + 1)}. Note that (96) can be interpreted as an addition theorem for Racah coefficients. In order to prove formula (96) we replace the first and the second Racah coefficients on the left hand side by the expressions (90) and (91), respectively, and use the orthogonality relation for the CGCs. Then the left hand side of (96) coincides with the expression (88) multiplied by q13.

3. The Quantum Algebra U(s12) and Its Representations

90

3.5.6 The Hexagon Relation The expression

1a e

be eh1fb d f =Eel a k g n fq1 e n 1 k gl q

h

k1fefg n fi g ln a b q ' d

(97)

where c = (-1)2n [2n + 11q-tn(n+1)+h(h+1)±d(d+1)±c(c+1)}, will be called the 9j symbol of ejlq (su2). These symbols possess the following symmetry relation permuting the second and the third columns:

bc

la

e

d

h

k

la (_1)0q,(0) e

g

h

c

b

f

d

g

k}

(98)

where 9

9

i (ai +1). ai :a+b-Fe+d+e+f+h+k+g, r(0)=Ea

= J=1

We sketch the proof of (98). First we express both sides in terms of Racah coefficients. Next we insert the expression for the Racah coefficient from (96), with q replaced by q-1 by (86), instead of the first Racah coefficient of the left hand side of (98). Summing over n by using the identity (94') we finally arrive at the right hand side of (98). Writing (98) in terms of Racah coefficients, we obtain the hexagon relation

ni {a c bl {kb d kehf fngf f enf

Ec

a

g

{k C m }{ a m d } {a c b kd m f f eh fhg where c is as in

(97) and

T = ITT1,(M

(99)

g(g + 1) + e(e + 1) + b(b +1)}.

± 1)

3.5.7 Clebsch-Gordan Coefficients as Limits of Racah Coefficients The main aim of this subsection is to show that the CGC's of Ùq (5u2) are the limit cases of the Racah coefficients:

■ rvabc nabc lim I-Lik m kr ) =

+k=

lqi > 1.

In order to prove this formula we first note that the relation

[r +

lim r—,co [r ± 0]

(" -41)/2 7

141> 11

3.5 Racah Coefficients and fij Symbols of Uq (su2)

91

together with (68), (69), (92) and (93) imply that a 1/2 a+m j1 k

r—*oc

'

a 1/2 a+m

(7) = Cj, 'k,j+' k

(102)

> 1/

9

where m = k= . This is the assertion (100) for b = .1. In order to oc in the recurrence derive (100) in the general case we take the limit r c = 1iMr_400 Rcjicm (r), s = j +2k, t = c—b, p = b+ c relation (95). Setting 6.1Pcni and using (101), we obtain

6y1;bccni (t--.9+1)/ 4 (

=

=

[a + b

c][a — b + c+ ii[b k][c + [2b] 2 [2c ± 1.][2c + 2 1

+ 1]

TT/

qC

2

— b + c + 11[b + k][c m +

Ci, 'k+1/2:m+1/2

a b-1/2 c-1/2

j,k+1/2,m+1/2

[2b] 2 [2C + 1 ] {2d

= q (t— s -1)1 4 ( [a + b — c] [a

)

11

2 ,--TA a b-1/2,C+1/2

[2b] 2 [2c + 1J[2c + 2] (P_3)/4 =

q

{b

a + ci [a + b + c + 1 ] [b + k][c + m]

rf- a b-1/2 c-1/2 "ij:k-1/2:m-1/2-

[260] 2 [2C + 11[2C]

(103) On the other hand, formula (86) leads to the identity

R q (abd 7 cf / e)Cc! . b c ryc d

—

ryb

df

Substituting d = b =f — and using the expressions (68), (69), (92) and (93) for special values of CGC's and Racah coefficients, we conclude that (103) remains valid if all qbcm c are replaced by the CGC's qapbcm c . That is, both families of coefficients satisfy the same recurrence relation (103). Since = abc 3km %-/) ,k,rn jkm as shown by (102), it follows that 6'9bc '-'jkm for all values of indices. This completes the proof of formula (100). Relation (100) yields the following asymptotic formula for large r:

f a b ld+r e+r

c

f+r 1 .7

_ir-Fb±cl+eq-112 aoc

([2r + 1][2c + 1]) 1/2 Cf —e,d— f,d—el

iql > 1 1

If iqj <1, then the same reasoning shows that p m pea b c tr.\ = (_i)+b—crta b c k /

r L',144 00

141 < 1 .

By (73), the latter can also be written as Da b c f = f Ya b c k mki k m ■ ti /1 r-4 oc -

lql < 1.

For the Wigner 6j symbols, (104) gives the asymptotic formula

(104)

92

3. The Quantum Algebra Uq (512) and Its Representations

{

(_ i) a+b-cqn2

a f + r}

d+r e+r

q

bc

ir 2c + 11)1/2-f-e,d-f,d--e) (r2 \L-r + 1. JL

where iqj < 1 and r cc. The preceding consideration shows that, as in the classical case, the fiveterm recurrence relation (95) is of fundamental importance for the theory of CGC's and Racah coefficients of the quantum algebra iIdq (su2 ). This relation together with the expressions (92) and (93) for the simplest Racah coefficients determine uniquely all Racah coefficients of 0,2,(5u2 ). By (100) and (104), the Racah coefficients in turn determine all CGC's of Ûq (5u2). We also note the equation lim(-1)

2,[2r+c+f+1P fa b + r c+r1 [2r+b+e]! td e+r f+ r i r+b+d+eT(a,d, f _ e,b f,b - c, c - e)

(105)

for b+e
T (a, d,

, a" , d") =

1 [a' -

( [a + a']!a - a"]! [d- d'}![d d"]! 1 / 2 • [a a']! [a + a"]! [d + d']![d -d"]!)

If b+e>c+f, the limit in (105) vanishes.

3.6 Tensor Operators and the Wigner—Eckart Theorem In this section we define tensor operators transforming under irreducible representations of Ûq (su2) and prove the Wigner-Eckart theorem for these operators. First we recall the corresponding result for compact Lie groups. 3.6.1 Tensor Operators for Compact Lie Groups

Let TT be an irreducible representation of a compact Lie group G acting on a vector space V, and let tc be the matrix elements of Ta with respect to some basis of V. We say that a set Rcr {R,cr I m = 1, 2, • • • , dim Tc, of operators acting on a Hilbert space fj is a tensor operator, transforming under the representation Tic , if there exists a continuous representation T of G on Sj such that

T(g)KT(g 1 )=

E4(g)R„ g E G, m = 1,2, - - • , dim Ta .

dim Tc 1

n=

(106)

Let us see what the latter means in terms of the associated representation dT of the Lie algebra g of G. Let g(t) be a one-parameter subgroup in G with the tangent vector X E g. Setting g = g(t) into (106) and differentiating at t = 0, we obtain

3.6 Tensor Operators and the Wigner-Eckart Theorem

[dT (X), R]

93

En=iT, dim

This equality defines tensor operators in infinitesimal form. Let T = ED, TA , be a decomposition of T into irreducible components. For simplicity we assume that this decomposition has no multiple irreducible representations. Let esA., s = 1,2, • • • , dim TA, be an orthonormal basis in the carrier space of the representation TA. Then the Wigner-Eckart theorem A• asserts that the matrix elements (e I Ram er i ) of the operators RGrm with respect to the basis le ); .1 are of the form (ez

RG;), I erA i) = (A 2 , s I

m;

Here (A i , s a, m; Ai , r) are the CGC's of the tensor product of the representations I', and Li of G (recall that we assumed that this tensor product does not contain multiple irreducible representations in its decomposition) and (Ai IIR' Ai) are the so-called reduced matrix elements of the tensor operator R.'. The reduced matrix elements (A i llR' II Aj ) are independent of the basis elements. Thus, the Wigner-Eckart theorem represents the matrix elements of tensor operators as products of two quantities, where the first is determined by the group structure and the second is independent of the group.

3.6.2 Tensor Operators and the Wigner-Eckart Theorem for tIg (su2) Let Tk be the (2k + 1)-dimensional irreducible representation of Oq (5u2). We say that a set of operators Rk = {R I n = -k, + 1, • • • , k} acting on a Hilbert space 5 is a tensor operator, transforming under the representation Tk if there exists a representation T of Ùq (5U2) on 5 such that

T(K)VT(K)

1 =qnl

(107)

1 T(K) -1 ,

(108)

T(F)R 17c,- q 12 RI7c,T(F) = ([k + ni[k - n +1]) 112 R I,L i T(10 -1 .

(109)

T(E)V, - qn.feiT(E) ([k - n][k + n+1]) 112 R

Note that the coefficients on the right hand sides are just the matrix elements of the operators Tk(K), Tk(E) and Tk(F), respectively. Suppose that there is a subset I C N0 such that the representation T decomposes into an orthogonal direct sum

T= El) LEI

T1

(110)

of irreducible representations fl 1 E I. In other words, the multiplicities of all irreducible representations T„, n E N0 , in the decomposition (110) do not exceed 1. Let {elm } be the corresponding orthonormal basis of the carrier space of the representation Ti in (110). ,

94

3. The Quantum Algebra Uq (s12 ) and Its Representations

Proposition 23. If R k is a tensor operator as above, then there exist numbers (11IIRk III) such that the matrix elements of the operators 1=1,kz. with respect to the bases { em l } are

(et I V, I e m l ) -= Cq(1,k,r;m1 nlmi)(iikiii),

(111)

where Cq (•) are the CGC's of (1q (su2). The reduced matrix elements (11 11Rk are independent of n, m, m'. Proof. We consider the vectors enkim := Rnk leim ). By (107), T(K)eknini (K) R nk ' elm ) q n+m Rnk eknim. Similarly, by (108) and (109), we get 1/2enki,m+i

T(E)e nklm

'flak —nj[k+n+1]) 1 /2enki+Lm ,

v _ m+1 1)1/2enkl,m-1± q- 'flak +nlik _ n+11)1/2 enkt Lm.

T(F)eknim = qn ( v

These formulas show that the representation T acts on the vectors enkim , as the tensor product 7)071k does. Therefore, we can apply formula (41) and express them as sums of vectors transforming under irreducible representations: kl e nm =

1,

cq (I

k 7 1 1 ; Tri, 1 71 1 M1 )énii ±m

(we assume that the CGC's are real). Now we take the scalar product of both sides with the basis vector eim' ,. Using the definition of enkim and the fact that (em li , j ére ) m i r (I liRk iii), we obtain the relation (111), where the numbers CI (PliRk HI) are independent of n, m, m'.

3.7 Applications The aim of this section is to give a short sketch of some applications of representations of Uq (512 ) in nuclear physics.

3.7.1 The Uq (512) Rotator Model of Deformed Nuclei In nuclear physics it was suggested that rotational spectra of deformed nuclei can be described by a q-deformed rotator which corresponds to the Casimir element of Uq (512 ). This rotator is defined by the Hamiltonian H = (2I) 1 C' + E0 , where I is the moment of inertia, E0 is the bandhead energy and C is the quadratic Casimir element chosen such that its eigenvalues on the representations Ti are fIlli + 1]. For real q, q = et', the energy levels of the q-rotator are E(j)

1 1 sinh (p,j)sinh (0(j — EijEj +1] + E0 =

21

21

sinh2p,

1))

+ E0 .

(112)

3.7 Applications

95

Following the physics literature the number I E -1- N0 parametrizing the irreducible finite-dimensional type 1 representations of Uq (512 ) is here denoted by j. If q = p E R, is of modulus one, then one has

1 sin (pj)sin(p(j +1)) E(i) = 21 sinp 2

E0 .

(113)

The expressions (112) and (113) can be represented as E(j) = E0 +Aj(j+1)+B(j(j+1))2 +C(j(j+1))3 + D(j(j+1))4 + - • • , (114)

where the coefficients A, B, C, D depend on p and are expressed in terms of the spherical Bessel functions of the first kind in the case (113) and of the modified Bessel functions of the first kind in the case (112). Since rotational spectra in the first order approximation are described by the usual Lie algebra S1 2 , it is natural to expect p to be near zero. (It was found that p is around the value 0.03.) Empirically it is known that for nuclear rotational spectra the coefficients A, B, C, D have alternating signs starting with positive A. This implies that rotational spectra can be described only by the case (113). Numerical expressions for ground state bands of rare earth and actinide nuclei given by the formula (113) have led to very good agreement with experimental data. There is also satisfactory agreement for the 0- and bands of these nuclei. The variable moment of inertia (VMI) model of rotational nuclear spectra is described by the formula if

E(i) = 2e0

+ 1 ) — —a2 (.7(3 + 1 )) 2 + 62 0(3 + 1 )) 3 173 (30 + 1 114 + • • .) (115)

where e0 is a free parameter of the model corresponding to the ground state moment of inertia and a is related to the softness of the nucleus. Both expressions (114) and (115) have similar properties (alternating signs, successive coefficients falling by about 3 orders of magnitude, etc.). However, the expansion (114) gives better agreement with the empirical values of the ratios AC/(4B 2 ) = 1/10 and A2 D/(24./3 7 ) = 1/280.

3.7.2 Electromagnetic Transitions in the Uq (s12) Model In the case of the VMI model no method has been found for making predictions for the B(E2) transition probabilities connecting the levels of a collective band. The Uq (512) model provides such a link. In rotational bands, the collective model of Bohr and Mottelson gives the transition probabilities described by the formula

B(E2; j

1

5 2 Ici+2,2d = —Q 161. 0 IC,0,-K 1I 2

(116)

96

3. The Quantum Algebra Uq (512) and Its Representations

su(2),

Qg is the intrinsic electric where q+02' 21c is the CGC of the group quadrupole moment and K is the projection of the angular momentum j on the symmetry axis of the nucleus in the body-fixed frame. For the K = 0 bands we have B(E2; j +1-4 j) =

167r — 2 (2j + 3)(2j + 5) •

From the preceding it is clear that the B(E2) values should saturate with increasing value of j. In the case of the Uq (s12 ) model we have to take the corresponding CGC of the quantum algebra Uq (512 ) instead of that of SU(2) in formula (116). Using the explicit form of these coefficients, we find for K = 0 bands that

B (E2;

+1

j) =

5 [3][4] [j +1 Li + 21 2 Q 167r -I?) [21[2j + 2 ] [2j +3] [2j + 4 ] [2j ± 5] ] 2

—

Expanding these functions and keeping corrections of the leading order in p. (it is assumed that the range of values of p is the same as in the description of ground state spectra), one gets

B(E2; j +1 j) =

—

167r

012n —

112(6j 2 ± 22j + 12)) 1 +— 2 (2j + 3)(2j + 5) ( 3

Thus we see that the additional factor depending on p2 leads to an increase with j, while the Bohr—Mottelson model reaches saturation at high j. A discussion of this effect is contained in [BDKL].

3.8 Notes The associative algebra Uq (512 ) first appeared in the paper by P. P. Kulish and N. Yu. Reshetikhin [KuR] devoted to the construction of integrable models of type XXZ with higher spin. Its Hopf algebra structure was found later by E. K. Sklyanin [Sk12]. The universal R- matrix of Uh(s12 ) was discovered by V. G. Drinfeld [Dr2]. The irreducible representations of Uq (812) when q is not a root of unity were constructed in [Skill, see also [Wor2]. The irreducible representations of Uq (812) in the root of unity case are given in [RA]. The expressions for the Clebsch—Gordan coefficients of Uq (su2) were derived in [KR1], [KK], [GKK], [Vak] (see also' [BL], [VK1], Chap. 14, and the references therein). The formulas for the Racah coefficients of Uq (su2) have been obtained in [KR1] and [MK] ( see [BL] and [VK11, Chap. 14). Tensor operators for Uq (812) were introduced by Biedenharn (see [BL]). There is a large variety of applications of the quantum algebra Ugi(s12) in physics. The relations of quantum groups to the theory of integrable models and ideas of applications in quantum field theory are discussed in [Fil, [F2 ] , [AGS], [DV], GoS], [MR]. A good review and extensive references on applications of representations of Uq (512 ) in nuclear physics can be found in

[BDKL].

4. The Quantum Group SL q (2) and Its Representations

In this chapter we investigate the coordinate Hopf algebra of the quantum group S L g (2) and develop its corepresentation theory. It is the simplest example from the series of quantum groups associated with simple complex Lie groups which will be studied extensively in the second part of the book. In this chapter q denotes a fixed nonzero complex number.

4.1 The Hopf Algebra 0(SL q (2)) In Sect. 1.1 we have seen that the algebraic properties of the group SL(2, C) are stored in its coordinate Hopf algebra 0(SL(2)). Let us briefly recall the structure of this Hopf algebra from Example 1.3. As an algebra, 0(52(2)) is the quotient of the commutative polynomial algebra C[uf, u , u, u} in four indeterminates u, u, u?, /J, (the coordinate functions on SL(2, C)) by the two-sided ideal generated by the element ulu3 - 1. On the generators the comultiplication 4 the counit e and the antipode S are given by SOLD = d(uji ) = Ek uik 0 till, E(u) = 6zi, SOLD = u, MUD = 51(q) = ul. The Hopf algebras 0(SL q (2)) introduced below are a oneparameter deformation of this Hopf algebra 0(SL(2)). 4.1.1 The Bialgebra 0(Mq (2))

Let 0 (Mq (2)) be the complex (associative) algebra with generators a, b, c, d satisfying the following relations:

ab = qba, ac = qca, bd = qdb, cd = qdc, bc = cb

(1)

ad — da = (q — q -1 )be.

(2)

In order to shorten formulas we also write ul = a, 14- = b, u = c, u3 = d. Proposition 1. There exists a unique bialgebra structure on the algebra 0(Mq (2)) with cornultiplication zi and counit E such that

A(a) =a0a+b0c, A(b) = a

b + b d,

(3)

A(c)=c0a+dgc, A(d)=c0b+d0d,

(4)

98

4. The Quantum Group SL q (2) and Its Representations

e(a) = E(d) = 1, e(b) = E(c) = 0.

Note that the relations (3)-(5) can be written as L\(u) = E k Uik 0 quii ) = 6, i, j = 1, 2.

(5) 7.L./;

and

Proof. The algebra 0(Mq (2)) is the quotient of the free algebra C(u) with generators uii by the two-sided ideal generated by the elements obtained from the relations (1) and (2). By the universal property of the free algebra C(u) (see Subsect. 1.2.1), there exist algebra homomorphisms A : C(u) —* C(u) 0 C(u)) and E : C(u) —* C such that Lt(u) = E k uL 0 u 3lF and e(u) = 80 . In order to show that A and E pass to algebra homomorphisms of the algebra 0(Mq (2)) it is sufficient to prove that A and E preserve the relations (1) and (2), that is, z(a)z(b) = ai(b) ,i(a), etc. As a sample, we check this for the first relation. By (3) and (4),

A (a) A(b) = aa 0 ab + ab 0 ad + ba 0 cb + bb 0 cd, q21(b) 21(a) ---- aa 0 qba + qba 0 da + ab 0 qbc + bb 0 qdc.

Indeed, by (1) and (2), both terms coincide. Since A and E are algebra homomorphisms of 0(Mq (2)) as just shown, by Proposition 1.8 it suffices to verify the coassociativity and the counit axiom on the generators uii . We have ((A 0 id) 0 A)(U ii ) = >k/U' 0 lit 0 14 = ((id 0 A) 0 A)(uii ) and ((e 0 id) o A)(uij ) = Ek E(U)U 1 / = Iiij = ((id 0 E) 0 A)(uii ). El Definition 1. The bialgebra O (Mq P)) is called the coordinate algebra of the quantum matrix space Mq (2). The elements fa i bi ck di I id, k,1 E Nol form a vector space basis of 0(Mq (2)). The formulas (3) and (4) give the comultiplication A only on the generators. The action of A on products of generators can be calculated by taking into account that A is an algebra homomorphism. We find that 1

=__ \---• [ i 2+1 i jl 2 az-i bt+i 0 at-ici+i A(a21 ) = (a 0 a + b 0 c)21 z__./ q-

7

(6)

/

=

i= - 1

/+ij2

ci-i di±i 0 a\ l- iCi+i 7

( 7)

q-

-I

where l E No and the summation is over integral (resp. half-integral) values of i if I is integral (resp. half-integral). We also have .

1

j =-1

A>0

[1 - 21 2 [ P j q- L i

1

+i -pi g _2

xq-1464+i-i) al-i-ih eci-j+ti ce+i-t4 0, a1-ic1+-1 .

(8)

4.1 The Hopf Algebra 0(SL g (2))

99

We conclude this subsection by giving a reformulation of the six quadratic relations (1) and (2). For this purpose we take the matrix R = (Rtki )i,j,k4=1,2 = 1, Rj = q — q -1 and Rid/ = 0 otherwise. with R11 = RH = q, R 19 =-That is, R is just the matrix (3.40) derived in Subsect. 3.4.1 from the universal R-matrix of the h-adic Hopf algebra Uh(s12). Let us consider the following equations k

041. 111rn =

k

winik .R7/1/77, 7

, n, rn = 1, 2.

( 9)

Inserting the values of R these 16 equations reduce precisely to the 6 relations (1) and (2). The equations (9) are the starting point for the FRT approach to quantum groups which will be elaborated in Chap. 9. ,

4.1.2 The Hopf Algebra 0(SL q (2))

By relation (2), we have ad — qbc da —

(10)

This element of 0(Mq (2)) is denoted Dg and called the quantum determinant. Proposition 2. The quantum determinant A i is a group-like element (that iS, 4i(D,i ) = Dg 0; and e(;) =1) belonging to the center of the algebra 0 (Mq( 2 )).

Proof. The proof is given by direct computation using the relations (1) and 1=1 (2).

If q is not a root of unity, then it can be shown that the center of 0(Mq (2)) is generated by Dg . Since Dg is group-like, the two-sided ideal (Dg — 1) generated by the element Dg — 1 is a biideal of 0(Mq (2)). Hence the quotient 0(SL g (2)) := 0(Mg (2))1(Dg —1) is again a bialgebra. Proposition 3. The bialgebra 0(SL q (2)) is a Hopf algebra. The antipode S of 0(SL q (2)) is determined by

S(a) = d, S(b) = —q -l b, S(c)= —qc, S(d)=. a.

(11)

Proof. Put a' := d, := —q - lb, := —qc, d' := a. One verifies that the elements a', Lo', , d' satisfy the defining relations of the algebra 0 (S L q- (2)). But O(SLq-1(2)) is the opposite algebra of 0(SL g (2)). Hence there exists an algebra anti-homomorphism S: 0(S.L q (2)) 0(SL q (2)) such that S(a) = a', S(b) = b', (c) = c' and (d) = d'. To prove that S is an antipode, by Proposition 1.8 it is enough to cheek the antipode axiom (1.26) on the four generators. We omit this straightforward verification. 0 A direct computation shows that for the algebra 0(SL 9 (2)) the matrices

MO

4. The Quantum Group SL 9 (2) and Its Representations

u

(a

c

b) d

and S(u)

( d -q -l b ) a )

are inverse to each other. This fact is actually equivalent to the validity of the antipode condition (1.26) for the generators a, b, c, d. Definition 2. The Hopf algebra 0(SL q (2)) is called the coordinate algebra of the quantum group SL q (2).

As motivated at the end of Sect. 1.1, we shall think of and treat elements of the Hopf algebra C3(SL q (2)) as functions on the "quantum group" SL q (2). However, this Hopf algebra is not a group Hopf algebra (it is neither commutative nor cocommutative if q 1) and the quantum group SL q (2) exists only in terms of the Hopf algebra 0(SL q (2)) and its structure. Recall that the defining relations of the algebra 0(SL q (2)) are the five equations (1) and the two equations (12)

ad - qbc= da q -l bc = 1.

These are precisely the relations (3.44) and (3.45) which were derived in Subsect. 3.4.1 for the matrix elements of the representation T112 of the quantum algebra Ug (512). A geometric approach to these relations will be given in Subsect. 4.1.3. We now begin to study algebraic properties of the Hopf algebra 0(SL q (2)). Using the formulas (2.17) and (12) one proves by induction on in that

a7724M

7T1

7T1

E

k=0

Li k ]

2 q2km-k2

(bC) k Cram

q-

E k=--0

771 [

q -k2 (bC) k .

k

(13)

q-2

The following result is often used in the subsequent sections. Proposition 4. The set fanbmer, bmcrds space basis of 0(SL q (2)).

I m,r,s E No , n E NI is a vector

Proof. By the relations (1) and (2), this set obviously spans A(SL q (2)). Using

one of the representations ruce° from Corollary 20 below, it is not difficult to show that they are linearly independent. In Subsect. 4.1.5 we use the diamond lemma in order to give another proof of the linear independence. Proposition 5. (i) There are algebra automorphisms 0 and 12, a, fi E CVO}, of the algebra 0(SL q (2)) such that

0(a) = a, 0(b) = c, 0(c) = b, 0(d) d, = aa, V a,p(b) = Ob, V,,, a (c) = 0-1 c, V a,i3(d) = (ii) For fi E CVO}, 191,fi is a Hopf algebra autornorphism of 0(SL q (2)). (iii) There is a Hopf algebra isomorphism p 0(SL q (2)) 0(547 -1(2)) = 0(S L q (2))°P such that p(a) = d, p(b) = c, p(c) = b, and p(d) =-- a.

Proof. These results follow by direct verification of the defining relations.

101

4.1 The Hopf Algebra 0(SL,(2)) 4.1.3 A Geometric Approach to SL q (2)

Matrices with complex entries act as linear transformations on complex spaces. We shall see now that the defining relations (I) and (2) of 0(Mq (2)) occur quite naturally if we think of quantum matrices of Mq (2) as transformations of the quantum plane q Let 0(C.72 ) be the algebra with generators x and y satisfying the relation

xy = qyx.

(14)

We call O(C) the coordinate algebra of the quantum plane C. Let a, b, c, d be elements of an algebra A. We "transform" the vectors

) and (x, y) by the matrix ( a b ) from the left resp. from the right, c d Y that is, we set s

(

(

(x,y) 0 (

a b\ xha Ox+b® y) ( x , ) c d) (

a b (x,/,y,/). ) = (x0a+y0c, x0b+y0d)=: c d

(15) (16)

Proposition 6. Suppose q2 +1 O. The couples (s', y') and (s", y") satisfy (14) if and only if the elements a, b, c,d fulfill the relations (1) and (2). Proof. First suppose that s'y' = qy's' and x"y" = qy"x". The first equation says (a0x+b0y)(e0x+d0y)= q(cOx+d0y)(a0x+bOy), that is, (ac - qca)

0 X 2 + (ad - da + q-l bc - qcb) 0 xy + (bd - qdb)) 0 y2 =

O.

Since the elements X2 7 xy and y2 are linearly independent in O(C), we get ac = qca, bd = qdb, ad - da - qcb + q-l bc = 0.

(17)

Similarly, the relation x"y" = qy"x" leads to ab = Oa, cd = qdc, ad - da - qbc+ q-l cb = O.

(18)

The last relations in (17) and (18) imply that bc = cb. Inserting the latter into (17) and (18) we obtain (1) and (2). The same computations read in reversed order prove the opposite implication. CI In Hopf algebra terminology, the second implication of Proposition 6 yields the following (see also Example 1.19). Proposition 7. 0(Cq2 ) is a left and right comodule algebra of the bialgebra 0(Mq (2)) with left coaction (pL and right coaction WR determined by 'PL(x) =a0s+bOy,

(pL(y)=c0x+d0y,

(19)

wii(x)=x0a+y0c,

(pE(y)=x0b+yod.

(20)

102

4. The Quantum Group SL(2) and Its Representations

Proof. It follows at once from the converse implication of Proposition 6 that O(C) and there are algebra homomorphisms Ç0L : O(C) —) 0(Mg (2))

(PR 0(C O32 ) —) 0(C q2 )00(Mq (2)) such that (19) and (20) hold. To show that (Pi, and çoR are coactions, it suffices to verify the conditions (1.51) and (1.52) for the generators x and y which is easily done. El Since 0(SL q (2)) is a quotient bialgebra of 0(Mg (2)), the assertions of Proposition 7 remain valid for C9(SL q (2)). In order to express the quantum determinant also in geometric terms, we introduce the exterier algebra A(C) of the quantum plane. It is the algebra generated by elements and 77 and relations 0, 772 = 0 1

=

Setting := adO+bOri and i' := c®-Edon, we have eV = (ad—qbc)®&7. In the classical case q = 1, the determinant appears in the first factor, so it is natural to take Dg := ad — qbc as the quantum determinant. 4.1.4 Real Forms of 0(SL q (2)) In the following we list three real forms of 0(SL q (2)). That these formulas give well-defined Hopf *-algebra structures on 0(Slig (2)) can be proved by the method sketched in Subsect. 3.1.3. Case 1: q E R. There is an involution of the algebra 0(SLq (2)) determined by . a. , d ib r „ * = c (21) a* = d, b* = —qc, The corresponding Hopf *-algebra is called the coordinate algebra of the real quantum group SUg (2) and is denoted by 0(SUg (2)). It is called the compact real form of SLq (2). Inserting the formulas (21) for the involution into the relations (1) and (12) for 0(SL g (2)) we see that 0(SUg (2)) is the *-algebra generated by two elements a and c subject to the relations

ac = qca, ac* = qc* a, cc* = c*c, a* a

c* c = 1, aa* q 2 c*c = 1.

(22)

The Hopf *-algebra 0(SUq (1, 1)) defined by the involution a* = d, b* = qc, c* = q

1 b, d* --= a

on 0(SL q (2)) is the coordinate algebra of the real quantum group SUg (1, 1). Case 2: WI = 1. There exists a real form 0(Sli g (2,R)) of 0(SL q (2)) with *-structure given by a* = a, b* = b, c* = c and d* = d.

4.1 The Hopf Algebra O(SL 9 (2))

103

4.1.5 The Diamond Lemma The diamond lemma [Ber] provides a general method in order to prove that certain sets are bases of algebras which are defined in terms of generators and relations. For instance, the Poincaré-Birkhoff-Witt theorem for enveloping algebras U(g) can be derived from it. In this subsection we develop a simplified variant of this lemma. Let C(x i ) be the free algebra with generators x i , x2 , • • • , x,. Setting I := {1, 2, • r}, the set X := Isi i E In of all monomials (see Subsect. 1.2.1 for the corresponding notation) is a basis of C(x i ). We define an ordering "<" on X as follows. Two monomials of different length are ordered according to their length and two monomials of equal length are ordered lexicographically with respect to their indices. Let 11 be a set of pairs (xi , fk), k = 1, 2, • • • , s, where Xi k E X and fk E C(Xi). We assume that 11 is compatible with the ordering "<", that is, each fk is a linear combination of basis elements x i < xik . For x, y E X, let rz ky be the linear map of C(xi ) which sends the basis element xx ik y to xfky and leaves the others fixed. The maps rxky are called reductions. A monomial x E X is called reduced if r(x) = s for all reductions r. Let R be the set of all compositions of reductions. We say that two elements x ik and xi, and three elements s, y, z E X form an overlap ambiguity if xik = xy and xi, = yz. Such an ambiguity is called resolvable if there are r, r' E R such that r(x ik z) = r'(xxii ). Similarly, we say that elements x ik and x ii , k 1, and x, y E X form an inclusion ambiguity if si, = xxik y. It is said to be resolvable if there exist r, E R such that r(s 1 ) = (xxi k ). Lemma 8. Let X denote the algebra with generators x l , x 2 , • - • ,x, and defining relations sik - fk = 0, k = 1,2, • • • ,s. If the set H of pairs (xik , fk) is compatible with the ordering "<" and if all ambiguities are resolvable, then the set of reduced monomials is a vector space basis of X. Proof. This lemma is a special case of Theorem 1.2 in [Ber].

Despite the many definitions the basic idea behind this lemma becomes clear if one thinks of the passage from the free algebra C(x 2 ) to its quotient X as follows. Terms of C(x i ) are reduced to smaller ones according to the order "<" by inserting repeatedly the defining relations x ik = fk. The monomials that remain unchanged during this procedure form a basis of X. The following proof of Proposition 4 shows the usefulness of Lemma 8. A number of similar later results (see Chap. 9) can be proved analogously. Second Proof of Proposition 4. Put x i = b, s2 = a, X 3 = d and x4 = C. Let H be the set of seven pairs (ab, qba), (ad, qbc 1), (ca, q - ac), (cb, be), (cd, qdc), (da, q - bc+1), (db, g - l- bd). Clearly, X is just the algebra 0(S L q (2)). A moment's consideration shows that H is compatible with the ordering "<" and that the monomials biaidkc i with j = 0 or k = 0 are precisely the reduced

104

4. The Quantum Group SL 9 (2) and Its Representations

monomials. Thus, by Lemma 8 and by the relations (1), the proof is complete as soon as we have shown that all ambiguities are resolvable. From the form of the set 11 we see that there are no inclusion ambiguities and we have eight overlap ambiguities. Written as pairs, the latter We are (ad, da) , (ad, db), (ca, ab), (ca, ad), (cd, da), (cd, db), (da, ab) , (da, ad) . check the resolvability of (ad, da) and denote reduction maps by arrows. We have (ad)a -+ (qbc +1)a = qb(ca) + a -4 qb(q -l ac) + a = bac + a, a(da)

a(q -l bc. + 1) =

(ab)c + a -4 q -1 (qba)c + a = bac + a.

That is, there are r, E R such that r((qbc + 1)a) = r' (a(q -l bc + 1)), so (ad, da) is resolvable. The other ambiguities are treated similarly.

4.2 Representations of the Quantum Group SL q (2) Throughout this section we assume that the parameter q is not a root of unity and that A denotes the Hopf algebra 0(SL q (2)). 4.2.1 Finite-Dimensional Corepresentations of O(SL q (2)): Main Results

First we recall that the finite-dimensional irreducible representations of the group SL(2, C) with polynomial coefficients are labeled by nonnegative halfintegers. For I e iN0 , the corresponding representation T1 acts on the vector space of homogeneous polynomials f E C[x , y] of degree 2 1 by (Ti(g) f ) (x y) = f (ax + cy , bx + dy),

(a b) g = SL(2,C). c d

Expressed in the language of Hopf algebras this picture carries over directly to the quantum case. Let 0(C 2 ) 7, be the subspace of 0(C,72 ) formed by the homogeneous elements of degree n. Recall that the right coaction ()OR of the Hopf algebra 0(S.L q (2)) on its comodule algebra Q(C) is, 'by Proposition 7, an algebra homomorphism. Therefore, by (20), 0(Cq2 ),-, is invariant under (pR. Hence the restriction of goR to 0(Cq2 ) 72 is a corepresentation, denoted T7v2, of the Hopf algebra 0(SL q (2)) on the (n +1)-dimensional complex vector space 0(C2q ),-,. We call Tn12 the spin representation of the quantum group SL q (2). Theorem 9. The corepresentations T1, I E 00, are irreducible and pairwise of inequivalent. Moreover, any finite-dimensional irreducible corepresentation 0(SL q (2)) is equivalent to one of these corepresentations. Proof. The proof of this theorem will be given in Subsect. 4.2.5 below.

4.2 Representations of the Quantum Group SL(2)

105

In the subsequent subsections we will develop some harmonic analysis on the quantum group SL q (2). The corepresentations Ti will be studied further in this context. The reader is invited to compare these considerations and formulas with the classical situation (see, for instance, [VI(1], Chap. 6). 4.2.2 A Decomposition of 0(SL q (2))

In this subsection we give a direct sum decomposition of 0(SL q (2)) with respect to the coaction of the quantum subgroup K of diagonal matrices. Let K be the subgroup of diagonal matrices of the group SL(2, C). Obviously, the coordinate algebra 0(K) is the Hopf algebra C[z, z -1 1 of Laurent polynomials in z with comultiplication, counit and antipode determined by A(z) = z z, e(z) =1 and S(z) = z -1 . A quantum subgroup of a Hopf algebra A is defined by a Hopf algebra B together with a Hopf algebra homomorphism q of A onto B. (The corresponding classical picture is as follows: if H is a subgroup of a group G and A and B are Hopf algebras of functions on G and H, respectively, then 0(f) is just the restriction of the function f c A to H.) Clearly, there is a Hopf algebra homomorphism OK of 0(SL q (2)) onto 0(K) such that

= OK (c) = 0, OK (d) = z

OK(a) = z,

1

( 23)

,

so the Hopf algebra 0(K) is a quantum subgroup of the Hopf algebra 0(SL q (2)). We express this by saying that K is a quantum subgroup of SL q (2). Now we define mappings LK : A -4 0(K) 0 A and RK : A -+ A® 0(K) by LK = OK (

id) a A

and

RK = (id OK ) oA.

(24)

Then LK and RK are algebra homomorphisms and also left and right coactions of the Hopf algebra 0(K) on A FE 0(SL q (2)), respectively. In other words, A becomes a left (resp. right) 0(K)-comodule algebra. For rn,n G Z we introduce a subspace A[m, n] of A by setting A{m,n] = {x E A l Lx(x) = zni x, RK(x) =

el.

(25)

We have LK(a) = z Ø a, RK (a) = a z, that is, a E A[1,1 ]. Similarly, b E A[1, -1], c E A[ 1,1], d E A[-1, -1]. Since LK and RK are homomorphisms, -

A[m,n1 • A[p, r] c A[m + p,n + r].

(26)

Consequently, for the basis elements of A from Proposition 4 we have

Vci cir E A[p

-

-

r,1

-

p r], arb1cl E A[r + p

-

1, r + 1

-

(27)

The elements of A[0, 0] are called K biinvariant. By (26), A[0,0] is a subalgebra of A and each vector space A[m,n] is an A[0,0 ] -bimodule. In the case when m n is even we introduce an element enin E A[M, n] by setting -

106

4. The Quantum Group SL q (2) and Its Representations

en,,,, = a (m+n)/2 C (n-m)/2

if

m + n > 0, m < n,

(28)

en,n, = a (m+n)/2b(m-n)/2

if

m

n > 07 m > n7

(29)

ernn = b(m-n)/2 d (-n-rn)/2

if

m + n < 0, m > n,

(30)

en,„ = c(' ) / 2 d( ' - ' ) /2

if

m + n < 0, m < n.

(31)

The next proposition collects the main properties of the spaces A[m, r]. All assertions follow immediately from (27) combined with the fact that the set {drib' cr. , Umcfd 5 } is a vector space basis of A. Proposition 10. (i) A10, 01 is the algebra C[C] of all polynomials in (ii) We have the decomposition

A

0(SL q (2)) =

m, -LEZ

A[m,n].

= —qbc.

(32)

(iii) If m,n E Z and m — n is odd, then A[m, n] = {0}. (iv) If m,n E Z and m n is even, then A[m,n1 = C[(Jeinn, = en,,,C[C]. Moreover, if p(C)enin, = 0 in A[m,n ] for a polynomial p E CM, then p = 0.

4.2.3 Finite-Dimensional Subcomodules of 0(SL q (2)) First we identify the left and right A-comodule algebras O(C) from Proposition 7 with certain subalgebras of A. Let A(a, b) denote the subalgebra of A generated by the elements a and b. Since ab = qba by (1), there is an algebra homomorphism 3R : O(C) A(a, b) such that .19R(x) = a and 19R(y) = b. Proposition 4 implies that VR is an isomorphism. Comparing (3) and (20) we obtain (O R 0 id) o çoR = o 'OR , that is, under the isomorphism 19 R the right coaction çoR of A on 0(C 2 ) goes into the restriction of the comultiplication to A(a,b). Thus, VR provides an isomorphism of the right A-comodule algebra O(C) with right coaction çoR (by Proposition 7) to the right A-comodule algebra A(a,b) with the comultiplication as right coaction. Similarly, the left A-comodule algebra 0(C 2 ) with coaction çaL defined by (19) is isomorphic to the left A-cornodule algebra A(a, c) with an isomorphism 9L such that 19/,(x) = a and 19/,(y) = c. Let 1 E No . The image of 0(Cq2 )21 under the isomorphisms V L, and VR are vi L

Ce 2,

f are given by

where the elements e

1/2 ]

e_2,21 = q-2

f (L) ,

Cfp,

=

,

[ 21 1 12

= it + i j -2 q

e

2it

-2i =

21 i/ 1 2 1+ i _ 2 q 21 1 1/2 [i +

q-

1

(33)

(34)

4.2 Representations of the Quantum Group

SL(2)

107

Since O(C) is invariant under çoi, and (p R , we have

zX:

A.

vi R

_0. A vi L

viL

(35)

Hence VII' is a left A-comodule and ViR is a right A-comodule with coactions denoted by TiL and TiR, respectively. Let 4i1) , i, j = 1, 1 + 1, • • • ,l, be the -

matrix coefficients with respect to the basis {fi(1) } of V TIR( fi(1))

2i (fi(1)\ =

J

)

-

is,

,

(36)

ai •

Since the corepresentations TIR and T1 are equivalent by construction, t also the matrix elements of T1.

t! il i and fi(l) =

Lemma 11. (i )

are

j.

(ii) t ) E A[-2i, —2j]. Proof. (i): By (27),

fi(!) E ,A[21, —2i] and hence LK(e)) = z 21

j t( 1) On the

other hand, (23) implies that OK (f 1 ) = 6i,_iz21 , so LK-(f i(1) ) = z21 t (_1)ii by (36). Comparing both expressions, we get fi(l) = t. Further, si n ce

f__(1)/ = a21 by (34), formula (6) yields 24f--(1)/ )

=

r

21

E ±j] LV

1/2 q

-2

fr

(37)

Comparing (36) with (37) and using (33) we obtain e = t(ii) 1 . (ii): First note that (id RK) o = (46 0 id) o RK by th 'e definition (24) of RK Since fi(1) E A[21, —2M, we obtain from (36) and (25) that (

( 6

o

id) o RK)(f1 1) ) = Zi(fi l) ) ((id

z-23

RK) o 2) (f1 1) ) =

= Ei ,E. f(i)

ii

RK(t).

Hence RK (t) =

t,(01) . Thus, t!

z-2i. Similar reasoning shows that LK(t) = z -2i E A[ 2i, -2j]. CI -

By Proposition 1.13, t

the condition .6(t 1) ) = E lk=-It! ik) Since e!1) = t!! _) 1 by Lemma 11(0, we have

(ell) )

(e 1) =

k t !Ilc)

ei",

(38)

that is, tlit) are the matrix coefficients for the left coaction Tit' as well. The explicit form of t! can be derived if one inserts the definition (33) of e!1) into

(38), then uses (8) and takes on the right hand side the coefficient of eY).

108

4.

The Quantum Group SL q (2) and Its Representations

We close this subsection with two remarkable symmetry properties of the matrix elements 4. The first one will be used in Subsect. 4.3.1. Recall from Proposition 5 that 0 is the algebra automorphism and p is the algebra antiautomorphism of A = 0(84(2)) given by 0(a) = a, 0(b) = c, 0(c) = b, 0(d) =-- d and p(a) = d, p(b) = c, p(c) = b, p(d) = a, respectively.

0(0)

Proposition 12. = t and p(t) = Proof. Clearly, 0 is a coalgebra anti-homomorphism of A. From this fact and the formulas (36) and (38) we get

1 (L 0 0)(e,( 1)) = (T 0 ( 0 0 0) o E fçl) t(-3 1) J 2 = 46(fi ) = = E

eV))

fçl) 3

23

which yields the first assertion. Let a be the algebra anti-automorphism of O(C) such that a(x) = y 1/2

and a(y) = x. We put ak :=- { 2/ 1 . Recalling the definition of the + lc j 2 corepresentation T1 on O(C) 21 we have

q-

Ti(aix1-2 Yi+i ) = aiçoR(x) 1 = ai (x

a + y 0 1-i (x

R(Y) 1±i

b +y d) 1+i = Eai xt- iy1+4 tçizl).

(39)

We now apply the algebra anti-automorphism o- Op of 0(C 2 )21 0A to the last two parts of (39) and replace (i, j) by (-i, -j). Since a k a_ k , the third part of (39) and the first tensor factor of the fourth part remain unchanged. Equating the coefficients of aixi- jyt +i we get = p(tY.23) )• 4.2.4 Calculation of the Matrix Coefficients In this subsection we express the matrix coefficients tLi.) explicitly in terms of the little q-Jacobi polynomials. Since the basis element e 1) of V1L is a complex multiple of we compute the matrix elements 4? by using formula (8). We carry out this calculation in the case where i j < 0 and i > j. Applying formula (13) to the expression ai+i -Pdi ±i-P. in (8), it follows that the first tensor factor of the term at-id+i in (8) is

i( =

y: A>13 k>0

IL

i ji

[

L

i P .11 q-2 1 / +±±

x q p(p+i-j)-(1+j- p)(2te-Fi-j)-Fk 2 (Note that iti) is a complex multiple of tL1) .) k to r =

+ k we reduce this expression to

q -2

(bc)P +k Changing the summation from

4.2 Representations of the Quantum Group SL q (2)

--

(1)

t>

a 23 c23 q (l+3X3 -2

)

p_ii I.IL

11,7'

-2

ri± i _ p

/±i

I)

td

x eil(A+i - i) -2 (l+i)r±r 2

109

I.

-2

r

q

-2

OW .

Let Ir denote the sum over p, on the right hand side of the last formula. Regrouping terms in q-binomial coefficients, we get

p_i i

r

/±i

[ / ±i - fd q - 2

Eq

2,,,(pd_i_i)_2(i+i)r±r2

I

L

/±i

r

j q -2

[ i-j+r 1 47 - 2 IL T - P q -2

I)

This sum reduces to a hypergeometric series of type (2.33) and we have

= (_ nr

(q 2 (i+j) ; q -2 ) r ( 17 -2(1-j+1)

[1 ± i]

( n -2 ; q -2) r ( q -2(i-j+1) ; q -2) r r

‘11

k‘i

Therefore,

t--(/) = a

c

Hi]

q

Li

jj

q- 2

X 2 (101(q 2(1+i) , Q -2(1-i+1) ; q 2(i-i-1) ; Q -2 7

Passing to t

t

i_j

+ ) ( .) - ) a - -

_

( 1 )

23

and using the notation (2.35) for 2 (,0 1 we obtain

[1+i

1

[1 — j

i i q -2

—

X 201

(-

where = -qbc. By (2.73), t polynomials by

/

Li

i

]

q-2

- j,/ - j + 1; i - j + 1; q -2 ,q-2 (),

can be expressed in terms of the little q-Jacobi pi+i(c; q -2(i-j)

=

-

7q2(i+i

(1,-2) 1

(40)

where q -(1+i)U-1

p

j 1 112 Li - j ] g -2 Li - ij q -2 . 1/2

-

(41)

Recall that we assumed that i + j <0 and i > j. In the same way the other cases can be treated. Retaining the notation (41), we then obtain

q2(i+i) I C 2 ) 7

= tçl.) = t !ij ) = Nt

pz

-

17- 2(i+i) I q-2)bi-id,

(C; t7 -2( 2 --j)

q -2(i+j) q-2)ci-jdi+j,

+ < 03 j > j

07 j >

j > 07

j

> j.

(42)

(43) (44)

110

4. The Quantum Group SL,q (2) and Its Representations

4.2.5 The Peter-Weyl Decomposition of 0(S.L q (2))

Let C(T/R) denote the linear span of matrix elements t, i, j = -I, -I + I,. • • ,I. It is clear from Proposition 1.13 that C(T/R) is a subcoalgebra of 0(SL q (2)). Theorem 13. The set ft,g) 1 E N07 i,j =----- -1 1 -1 + 1,. ,I1 is a vector space basis of 0(SL q (2)). The Hopf algebra 0(Sid q (2)) is a direct sum of the coalgebras Mr), that is, 0(SLq(2)) =

ED/E C(71/R).

(45)

Proof. To prove the first assertion, we use essentially the results on the matrix elvments t i( i) established in the previous subsection. From the expressions for

t 7,3(1.) derived there we get t!ii) =

= fri

e-2i,-2i7

(46)

where Fi(39 and Pi(39 are elements of A[0, 01 = CM. It follows from (32), Proposition 10(iv) and (46) that for fixed I the matrix elements 4j1) ,

< j < I, are linearly independent. From (40)-(44) we conclude that the function Fi(39 (4") is a polynomial of degree I - max ( ) • Therefore, the functions Fi(39, = max (lik Iii) + n, n = 0, 1, 2, • • form a basis of A[0, = C[C]. It follows from Proposition 10(iv) and (46) that 00

A[ -2i, -2j] =

C t i(j1) .

Using once more formula (32) the first assertion follows. The second assertion is an obvious consequence of the first one.

(47)

El

The direct sum decomposition (45) is called the Peter-Weyl decomposition of the Hopf algebra 0(S.L q (2)). The coalgebra C(T/R ) carries the full information about the corepresentation TIR T1. All basis elements e l) = t!1) 1 and fi(1) = t(1)/,i (by Lemma 11(i)) belong to C(TIR ). Hence the left coaction Tiri and the right coaction TIR are realized on the coalgebra C(TIR) and are given there by the comultiplication. Theorem 13 is a basic result in the representation theory of the quantum group Sid q (2). A first application of it is the Proof of Theorem 9. We show that TiR is irreducible. Let V 5é {0} be an invariant subspace for TIR and let f = Ei aifi(1) be a nonzero element of V. linearly 0. Since the matrix elements t There is an index r with a,. independent, there are linear functionals Fn on A such that F(t) = binbjr-

111

4.2 Representations of the Quantum Group SL,(2)

We have (id 0 Fn)Till(f) ar f41) E V. Thus, fn(1) E V for each n and hence V = V11 . This proves the irreducibility of T/R . It is obvious that two corepresentations TIR and 71/1,?, I l', are not equivalent, because they act on spaces of different dimensions. Now let T: V V 0 A be an arbitrary irreducible corepresentation on a vector space V. Let F E V', F O. Then the linear mapping A := (Foid)T : V A intertwines the right coactions T and 46. Let P be the projection of

A onto the irreducible invariant subspace Ci(TIR ) = Span { t. i) ,.j = 1, 1+ 1, • • • ,1} of A, given by the decomposition (45). Since 6 - o A = F 0, A is 0 for some 1 and i. Since P o A E not the null operator. Thus, Pj o A Mor (T, TIR ), then ker PI o A and im P 0 A are invariant subspaces for T and respectively. But both corepresentations T and TIR are irreducible, so VI R is that ker I 0 A = {0 } and im PI o A = VIR . Hence 1 o A : V 0 bijective and Ti and TIR are equivalent. —

t dim Mr)

Since for fixed 1 the matrix elements

-

linearly independent and

= (21 + 1) 2 and the right (and .6(t 1) ) = E k ti( ki) t k(ii) we have also the left) coaction of .6 on C(TIR ) is the direct sum of 21+ 1 copies of the irreducible corepresentation TIR (resp. ,

4.2.6 The Haar Functional of 0(SL q (2))

In order to motivate the subsequent definition, we begin with an example.

Example 1. Let A be a Hopf algebra of functions on a group G. Let /I be a measure on G. We set h(f) := f f(g)dp,(g), f E A. (For simplicity we suppress questions on the existence of integrals, but everything works if A = Rep G for some compact group G and i is a Borel measure.) The linear functional h on A is called left- and right-invariant if h(f) f f(g)dm(g)z ---- f f(gog)d(g) = f f(ggo)dtt(g), f E A)

(48 )

for all go E G. In terms of the comultiplication on A these equalities read as h(f)1 = ((id h) o 2)(f) = ((h 0 id) o 2)(f), where 1 is the unit of A. Note that (48) holds if i is the Haar measure of a compact group G. Definition 3. A linear functional h on a Hopf algebra A is called invariant

if it satisfies the condition ((id 0 h) o 20(s) = h(x)I = ((h 0 id) o 20(x), X E A.

(49)

As a second application of Theorem 13 we get a simple proof for the existence and uniqueness of an invariant functional h on the Hopf algebra A = 0(841 (2)) such that h(1) = 1. Since the matrix elements form a vector space basis of A, we can define a linear functional h on A by setting

tly

112

4. The Quantum Group SL, q (2) and Its Representations

h(t;)°0) )

= 1 . and h(t 1) ) = 0,

1 > 0.

(50)

Since il(0) = E k t!kl) t Proposition 1.13, one immediately sees that h is invariant. Conversely, if h is an invariant linear functional on A, then condition (49), applied to x = implies that h(ta) = 0 for I > 0. This proves that there is a unique invariant linear functional h on A satisfying h(1) = 1. This functional will be called the Haar functional of A. Next we want to find the explicit form of the Haar functional h. Let x E A[m, r]. From the definition (24) of LK and RK and from the invariance of h we obtain zmh(x) = h(s)1 =-- h(x)z'. Therefore, h(x) = 0 if (m, n) (0,0). It remains to compute h on A[0,0] = C{(]. We will show that

ta,

1 — q -2 h(C n)-

Let P: A position A

(id

1

n = 0, 1, 2, •

q -2(n+1)

.

(51)

* A[0, 0) be the projection determined by the direct sum decom= EIL ,n A[m, nj. By (3), (4), (13) and (2.17), we have

—

13)

A(Cn ) =

E

2

L Z J q_ 2

eijci ((; q 2) i

Ci( 17 -2( ; q - 2) »

where the expressions (a; O m are given by (2.3). Hence, the first equality in (49) yields

NCI

)

• 1. =

E [1 2 j=--n

/(; q2) i h(v( q -2( ; 2 (1

This is an equality of polynomials in ( we obtain the recurrence formula q -2n

—

qbc. Equating the coefficients of

n

h(( )

n > 1.

Since h(/) = 1 by assumption, the latter implies (51). Formula (51) can be used for the calculation of h(f), where nomial in C. For example, we have

f is a poly-

h(cr ((; q 2) 8 ) = q-2(r+1) (q -2 ; q -2 )r(q -2 ; q -2 )8(q -2 i q -2 )1

(q -2 ; q -2) r±8±1

h(cr(q-2(; 47 —2) s ) =

(52)

q -2) r (q -2 ; 42 -2) 8 (q -2 ; 47 -2 ) I. k"/

(q2; cr2)r±s±i

It follows from (11) that S : A[m, n] —0 A[—n, we get h(S(x)) = h(x), s E 0(SL q (2)). We summarize some of the preceding results in

(5 21)

and S( ( ) = Ç Hence

4.3 The Compact Quantum Group SUq (2) and Its Representations

113

Theorem 14. There exists a unique invariant linear functional h on the Hopf algebra 0(SL q (2)) such that h(1) = 1. It is called the Haar functional of 0(84(2)). The Haar functional vanishes on the space Afm, n] if (m, n) 0 or if r = 0 and (0, 0) and on the basis elements arbkcl and bkddr if r k 1. On the space A[0,0] and on the elements bkck it is given by (51) and by h(b k ck ) = (-1) k q k+1q qg --(k+1)

k E N.

Remark 1. It can be shown (see Theorem 11.13 in Subsect. 11.2.1) that if a Hopf algebra A possesses a Haar functional (that is, an invariant linear functional h satisfying h(1) = 1), then any corepresentation of A is completely reducible. Taking this result for granted, it follows from Theorem 14 that corepresentations of 0(84(2)) are always completely reducible. A

The Haar functional h on 0(84(2)) is not central, that is, we do not have h(xy) = h(yx) in general. (For instance, since q is not a root of unity, O.) The next proposition is a substitute for the h(ad da) = (q q -1 )h(bc) missing centrality. Let 19 be the algebra automorphism 190, 1 from Proposition 5, that is, 19 is determined by the equations I9(a) = = b, = c, I9(d) = q -2 d. In particular, by (27), we have 19(x) = q M+71 X for x E A[m, r]. -

-

Proposition 15. For all x,y E 0(5'4(2)), we have h(xy) = 40(y)x). Proof Since 19 is an algebra homomorphism, it is clear that the assertion holds for y 1 y2 (and all x E 0(SL q (2))) if it is true for Yi and for 1i2. Thus, it suffices to treat the cases y = a, b, c, d. Taking the basis elements for x and using the explicit description of h given in Theorem 14, this reduces to a number of straighforward verifications. We omit the details. 111

4.3 The Compact Quantum Group SUq (2) and Its Representations Recall that the quantum group SUq (2) is described by the Hopf *-algebra 0(SUg (2)) which is just the Hopf algebra C9(SL q (2)) for real q equipped with the *-structure defined by (21). Therefore, the theory developed in Sect. 4.2 applies in particular to SUq (2). But it still remains to deal with all questions and properties related to the *-structure. Throughout this section q is a real number such that q 0, ±1 and A denotes the Hopf * algebra 0(SUq (2)). -

4.3.1 Unitary Representations of the Quantum Group SUq (2) Let us say that a matrix corepresentation y = (vii

A is unitary if

of a Hopf *-algebra

114

4. The Quantum Group SL(2) and Its Representations

E k vLuki = E k Vikt/.4 = bij • 1,

== 1, 2, • • •

d.

(53)

Setting v* = ((v*) ii ) := (v i ), (53) is equivalent to the equations vv = vv* = /. In what follows let TO ) denote the matrix corepresentation (t .i1) ) of A. Proposition 16. For any I E N0 TM is a unitary matrix corepresentation of 0(SUq (2)). ,

Proof. We shall use the algebra automorphism 0 of A which keeps a and d fixed and interchanges b and c. By the formulas (11) for the antipode S and (21) for the involution of 0(SUg (2)), for any n, in, r, s E No we have (anlf-ncrds)* = as (-q -1 (-qc)n dn = (S o 0)(anbm crds).

(54)

Since q is real, it is clear from the construction (compare (8), (33) and (38)) that each matrix element 471) is a real linear combination of terms anbmcrds . Therefore, by (54) and the first relation in Proposition 12, we have (4)* (S o 0) (t 1) ) = S(t). This implies the equations (53), since, by (1 .27),

Ek AS1 (t i( i

= Ek t iç ik) Set k(12).

= bij 1.

A corepresentation T : V -> V A of a Hopf *-algebra A on a finitedimensional Hilbert space V is said to be unitary if there is an orthonormal basis of V such that the corresponding matrix corepresentation of T (see Proposition 1.13) is unitary. A number of equivalent conditions will be given later in Proposition 11.11. Let us define a scalar product on the vector space VIR by requiring that the basis {fi l) } of ViR from (34) is orthonormal. Then, by the previous definition and by Proposition 16, each corepresentation TIR of 0(SUg (2)) on the Hilbert space Vi R in unitary.

4.3.2 The Haar State and the Peter-Weyl Theorem for 0(SUq (2))

Let h be the Haar functional of the Hopf algebra 0(SL q (2)). We introduce two Hermitian forms on A= 0(SUq (2)) by setting (x,

= h(xy*), (x, y) = h(s* y),

x,y E

(SUq (2)) .

(55)

Note that in our terminology scalar products are conjugate linear in the first variable and linear in the second one. We shall see below (see Theorem 17(1)) that (•, and (., .)R are both scalar products on the vector space A. In particular, we have h(ex) = (x,x)/, > 0 for s E A and h(1) = 1. Since a linear functional on a *-algebra with the two latter properties is usually called a state, h is said to be the Haar state of the quantum group SUq (2). From the definition (55) it is obvious that (xz,y)R = (s,yz"`)R and (zs, y)L = z* OL. Moreover, by Proposition 15, (s/ Y)L == (19(1)7x)R.

4.3 The Compact Quantum Group SUg,(2) E;..nd Its Representations

115

The next theorem can be considered as an analog of the classical PeterWeyl theorem for the quantum group SUg (2).

Theorem 17. (i) (-, . )L and (.,.)R are scalar products on the vector space

0(SUq (2)). The Peter-Weyl decomposition (45) of 0(SUq (2)) is an orthogonal decomposition with respect to both scalar products. (ii) The matrix elements tZ),,, of the irreducible core presentations T (1) satisfy the orthogonality relations (a, a) n ,) R = [21 +11-q-lq2n6w6mm/bnn,

tm (1',)n,) L = [2/ + 14-1 Q -2m 611"5mmi 15nn'

(56) (57)

In the proof of this theorem we require the following lemma.

Lemma 18. Let k,1 E

No and let M be a (21 + 1) x (2k + 1) matrix with complex entries. Let At := h(T (1) MT ( k) *) and 'AP := h(T (I) * MT () ). Then - = 0, M' 0 if 1 k and rlt;1 = aI , we have 11;1 = &I for some a, a' E C if 1 =-- k.

Proof. We prove the assertion for the matrix 11//. The proof for .1 is analogous. We denote the matrix T (1) g I by Tr and the matrix I 0 T (I) by T. Using the invariance condition (49) of the Haar functional h, we obtain

T (1) /17/T (k) *

=

(id 0 h)(711) TP MT k) *e ) *)

=

(id 0 h)(T 1) TP M(T I(. k) T4 k) )*)

=

((id 0 h) o L1)(T (1) MT (k) *) = h(T (I) MT (k) *) — .

Since T ( k ) is unitary by Proposition 16 and hence T(k) *T(k ) = I, we have T (1)11;1 - = .11-1T (k ) . Therefore, ti-/-f defines a linear mapping, still denoted by , of 11R to VAR which intertwines the irreducible corepresentations T (1 ) and T (k ) . Hence the invariant subspaces ker it-/-1 and im M are either 101 or the whole space. Thus, we get ker /17/ = 0 if 1 k and fl;/ = al- if 1 = k. (The latter reasoning is just Schur's lemma for Hopf algebras.) El Proof of Theorem 17. (ii): Let Eii be the (2 1 + 1) x (2k + 1) matrix having 1 in the (i, j)-position and 0 elsewhere. Let kij be the corresponding matrix

from Lemma 18. Then (iiii )„ = h(t,(12t s(ki ) *) = (41 2 , t s( )) R . By Lemma 18, we have (412,0)R = 0 if 1 k. Let 1 = k. Since t r(i2t.(1) * E — 2r, 2j — 2i 1 and h(a) = 0 for all a E dti[m,n), (m, n) (0,0), then (t,„(1 2 , t (. ) ) R = 0 for (r,i) 3L (s, j). Since a E AH2r, —2s1, we have 19(4 ) ) = q -2( r+s)4) an d so (t

Hence

,

4,)4

= (3(t)), tT) R = q -2(r+ 8) (47sn) t) R. = 0 if I

k

d if I = k, (i ?

(58)

4. The Quantum Group SL 9 (2) and Its Representations

116

Let now

= h(T(1) Eii T(I) *). By Lemma 18, there exists a constant ai E C such that (kii)ii= h(4t)*) = (tçl) (9)R = ai for all and j (-1 < 2,3 t i3 j < 1). In just the same way, considering gi = h(T(')*Eii T(1)), we find that there exists a constant c

(t

, t)/, = c for all i and j. By (58), we have q -2(i+i ) ai. Consequently, there exists a constant a E C such that a = q2 ia'i = q -2jai for all j and j. Since t!1) e!1) , then by the formulas (33), (13), (2.17) and (52) we have oc_i = q -41 (1-q -2 )/(1-q -41-2 ). Therefore, c = q 21 (1 q2)/(1 q-41-2\) and so E C such that

[21 + 1 ] -1 q2.3 ,

a"z = [21+ 1] -1 q-22 .

(59)

(i ): By (56) and (57), the numbers (a),„ an )L and (tai , tZ)n )R are positive for all real q 0, ±1. Since the matrix elements ta form a vector space basis of A, the equations (56) and (57) imply that (x,x)L > 0 and (x,x) R > 0 for any nonzero x E A. Hence (., -)L and (•, .)R are scalar products on A. The orthogonality of the Peter-Weyl decomposition follows at once from (ii). Let us turn once more to the Haar state. Formula (51) enables us to write the Haar state on C[C] as a q-integral (see Subsect. 2.2.2). For PC) E C[C], we have 1 h(f) =

o

00

f ()d,7-2C = (1 -

1

q -2)

q -2jf( q -2i)

if q

> 11

<

< 1.

(60)

j=0 00

h(f) =-f(q 2 C)c/ 1 2C = (1 _ q 2 ) E Jo i=o

eif( q2i+2)

if

(61)

Indeed, since these formulas are obviously true for f(C) = Cn by (51), they are valid for any f G C[C]. Next we give the scalar product (., •)R on a fixed space A [m, n] . By Proposition 10(iv), A[m, njj = C[C]e rn,„ where the elements emn are defined by (28)-(31). From the formulas (21), (2.17) and (28)-(31) it is easy to compute that emnem % = Omn(C), where Omn (C) is the weight function given by = q (m-n)(2-rn-n)/2((n-m)/2(t.

omn (C) )(m+n)/21 k omn ( ) = q(m-n)(m+n)/2 ( (m-r)/2 ((; q2)(m±,n)/27 ( J,

mn(()

= ((m-n)/2( q -2( ; .7-2)_(m+n)/27

771

In

n > 0 7 m < n, n > 0, m > n,

m + n 0, m n,

m-n c(n-m)/2 (q-2 (;11-2) (m+n)/27 m + n 0, m n. Ç mn(() q In terms of these functions Omn the scalar product (-, .)R for elements of A[m, nj can be derived from the formulas o

(s) Y)R = hth(()Omii(()f2(() * ),

x = fi(Oemn, y = f2(()emn.

(62)

The expression on the right hand side of (62) can be computed explicitly by means of the formulas (52) and (52').

4.3 The Compact Quantum Group SE/q (2) and Its Representations

117

4.3.3 The Fourier Transform on SUq (2)

a complete orthogonal system By Theorem 17, the matrix elements t in 0(SUq (2)) with respect to the scalar product (., •)R. To each element f E 0(SUq (2)) we associate a matrix Po = ( by ja h(ftaii ), E N0, m, n = —1, —1+1, • • • ,1. The mapping f + f( 1) is called the Fourier transform of the Hopf *-algebra 0(SU q (2)) or the Fourier transform on the quantum group SUq (2). Using the formula (56), it is easy to show that the inverse transform is

fa) —

f =E

/E i N.

[21 +

1],1

El

and that the following Plancherel formula holds: [2/ + 11 q

Cf, g)R

1

E. .

— 2i

q f10-M ii

where [21 + 11q is given by (2.1). 4.3.4 *-Representations and the C*-Algebra of 0(SUq (2)) 0(SUq (2)) is in particular a *-algebra, so it is The Hopf *-algebra A natural to look for its *-representations. Let fr be a *-representation of A on a vector space V equipped with a scalar product (-, •). By (22), we have

11*(a)vli 2 + lifr(c)v11 2 = (fr(a) * *(a)v5 v)

(fr (c) * fr(c)v v)

= (fr (a* a + c* c)v v) = (fr(1)v,v) = E V 7 so the operators fr(a) and fr(c) are bounded on V with norms not exceeding 1. This implies that the operators ir. (a*) and 'Fr(c*) are also bounded and have norms less than or equal to 1. Since any x E A is a linear combination of elements ai (c*).1 ck (a*) 1 , it follows that 71-(x) is bounded on V and that there is a constant Cx not depending on the *-representation such that Ilfr(x)II
for

y

11x11 := sup 7r 11 10)11 where ir runs through the *-representations of A on Hilbert spaces. Since 11x11 = sup,T Iiir(x)11 5_ C i , If • Ills finite on A. We shall show that 11x11 = 0 implies x = O. Let us consider the vector space A endowed with the scalar product (•, )1, defined by (55). It is easily verified that ïr(x)y := xy, x, y E A, is a *-representation of A on (A, (-, - )L ). Let 7r be the corresponding *representation on the Hilbert space completion of (A, (- , .) Li). If 11x11 = 0, then

4. The Quantum Group SL q (2) and Its Representations

118

is a scalar product is a norm on A which obviously has the on A, we get x = O. Hence C*-property = 11x11 2 , s E A. Therefore, the completion of A in the norm is a C*-algebra. It is called the C* -algebra of the compact quantum group SUq (2). Next we describe the structure of *-representations of A by means of the following operator-theoretic model. We suppose that q E R, lI <1. Let 0 and Sjo be Hilbert spaces and let y and w be unitary operators on respectively. Let = Er_o &, where 5j := 55o On the Hilbert 0 and space 0 e we define operators a' and e' by

r(x) = 0 and so lir(s)1111 = (x, x)L = O. Because (•,

/

a = v,

ct i (nn )

((1

c' = 0 on

0,

q 2 ) 1/2 771, (1 q 4 ) 1/2 772, ...), cf (1ln) = (qnwiln)

for

(rim ) C Sj.

Then the adjoint operators a'* and c'* act on ij by

a'*(77n) = (0 ) (1 — q 2 ) 1/2770 1 (1 —

47 4)1/2 7711 ...),

c/* (q n ) = (q nw * 17,n ).

Using these formulas one immediately verifies that a' and e' satisfy the relations (22), so there exists a unique *-representation 7r of the *-algebra 0(SUq (2)) on the Hilbert space 0 e such that 71- (a) = a' and 7r(c) = e'.

Proposition 19. Any *-representation of the *-algebra 0(SU q (2)) on a Hilbert space is unitarily equivalent to one of the above form. Proof. Consider an arbitrary *-representation of 0(S/4 (2)) on a Hilbert space E. For notational simplicity we denote the images of a, b, e, d by the same symbols. Throughout this proof we freely use the relations (22) and the polar decomposition a = ulai of the operator a (see, for instance, [RS1). First we show that a* is injective. Let a*ri = O. Then q 2 c*cn = and so lq1 2 11c7/II 11c7/11 2 + 11412 = ((a*a + c*c)77, 17 ) = 11 7711 2 and iql < 1, 11 7711- Since 11c7711 2 we conclude that n = O. Thus, a* is injective and hence uu* = I. Since aa* = ulai 2 u* = ua*au*, the two last equations of (22) imply that ?I

I = aa*

q 2 c*c = ua*au* + q 2 e* = u(I — e* c)u* + q 2 c* e.

Therefore, since uu = I as just shown, we obtain

ue* cu*

q 2 e* e.

(63)

Now let ij be the closure of the linear span of subspaces Z n := dn(ker a), is invariant under the operators a, b, e, d, u, it*. Hence n E No . By (22), its orthogonal complement 0 in E has the same property. Let us consider the restriction of the operators to e3. Since ker a C 55, a is injective on 0 and hence u is unitary on 0. Therefore, by (63), the spectrum of the operator c*c is invariant under multiplication by q-2 . Since ec is bounded and iql < 1, this implies that cc = 0 and so c = 0 on e3. Hence aa* = a*a = I on 0

4.4 Duality of the Hopf Algebras . 14(s12) and 0(SL q (2))

119

by (22). That is, on 0 the operators a and c are of the same form as in the above model. Next we determine the operators on the subspace lj. Let ( E Dn . Then a111(i - a*a)n = = dn.ri with an = 0 and c*c( = c * cd j = q 2nd c* c,J= q2nC. Hence the closure 5j of On is the eigenspace of c*c with eigenvalue 0, q2n and we have li = (1),nec 0 5j,. Here we used the assumption q(q2 -1) because then li, and . 7„, n z4 m, belong to different eigenvalues of c* c and so are mutually orthogonal. For ( C lia, 1a1 2 ( = a* a( = (I c* c)( = ( 1 _ q 2n)( )1/ 2 Since d =- a* = lalu* maps Zn onto by (22) and hence lai( = (1 _ onto Z +1 and uu* = I, it follows that u* is an isometric mapping of &±1 . We have c (64) au* n n = ulake n n = (1 _ q272 ) 1/2 u*n-i 17, ,

-

Further, we have lal 2 c = dac = cda = clal 2 and so laic = clai. Thus, cd = lalcu* = qdc = qialu*c. Since lai is injective on cu*..ui and u*c.., we clalu* obtain cu* = qu*c and hence

Me ri n = qflu*flcll,E 17

(65)

The operator c maps .N = ker lai to itself (because of laic --= Oa') and satisfies ce cc* = I on lio. That is, the restriction of c to lio is a unitary operator on lio. Thus, if we identify li and N via the isometry u*n, we see from (64) D and (65) that the operators a and c on e have the required structure. From Proposition 19 we easily derive the following Corollary 20. For any number u E C, luI = 1, there are two irreducible *representations 7rul and 7ru' of the *-algebra 0(SUq (2)). The *-representation 7rui- is one-dimensional and is determined by the formulas 7 11- (a) = u and 714 1 C) = O. The *-representation 71-u°3 acts on a Hilbert space with orthonormal basis fen I n =- 0, 1,2 • • • } by ci3 (a)en 7ru

2n , 1/ 2

= (1 q )

en-17 7r uc° (c)e, = que, n c No ,

where e_1 0. These *-representations are mutually inequivalent. Any irreducible *-representation of 0(SUq (2)) is unitarily equivalent to one of these representations.

4.4 Duality of the Hopf Algebras Uq (s12 ) and 0(SL q (2)) 4.4.1 Dual Pairing of the Hopf Algebras Uq (s12 ) and 0(SLq (2)) Recall from Example 1.6 that there is a nondegenerate dual pairing of the Hopf algebras U(g) and 0(G) given by formula (1.36) with g = e. Let us specialize this to the case g = 81(2,C) and G = SL(2,C). In terms of the standard generators H, E, F of 81(2, C) satisfying the relations [H, E] = 2E,

120

4. The Quantum Group SL q (2) and Its Representations

a, b, c, d of SL(2, C),

1H, Fj = —2F,[E,Fi =H and the coordinate functions

the dual pairing (,•) of U(s12) and 0(5.42, C)) is expressed by the equations = (F, b) = 1 and zero otherwise.

— (H , a) = (H d)

(66)

The next theorem says that there exists a similar dual pairing in the quantum case. Theorem 21. There exists a unique dual pairing (., Oq (s12) and 0(SL q (2)) such that (K ,

)'

of the Hopf algebras

a)'= q -112 1 ( K , d)= q 1 " 2 , (E ,c)'= (F, by = 1,

= (K, c)'= (E , = ,by = (E , = (F, a)= (F,

(67)

(F, d)= O. (68)

Likewise, there exists a unique dual pairing (-,•) of the Hopf algebras Uq (512) and 0(SL q (2)) such that the nonvanishing pairings between the generators are (K , a) -= q-1 (K, d) = q, (E , c) = (F, b) = 1. (69) If q is not a root of unity, both pairings (•,-) and (.,.) are nondegenerate.

The proof of Theorem 21 is given below. The dual pairings are completely determined by their values on the generators, but for computations it is often useful to know them on the whole algebras. Proposition 22. If q is not a root of unity, the dual pairing (.,-Y of the Hopf algebras 1,q (s12) and 0(SL q (2)) is given by the formulas (K772E71

Fl

dS cr b

t = q (n-r) 2 [

s

n — rl 2 if 0
(K m En Ft as Cr bt) where 7m srnt

,srt imnil

O otherwise, and

= 6rn 6tilim—sn,T,t

M(S+rt)/2C8(n+1)/2 (q2; q2) 1 72; 172) 71 ( 1

q n(n,-1)/2e-1)/2

(1

q 2)i-f-re,

•

Proof. A proof can be given by induction using the relations (1.31) and (1.32). We do not carry out the technical details (see [Koej).

Let us discuss the previous results. First we note that both pairings (., .) 1 if we put formally and (-, •' yield the equations (66) in the limit q K = ehH and q = eh . Further, both pairings become degenerate when q is a root of unity. For instance, if q k = 1, one verifies that the nonzero element a 2k 1 is annihilated by the whole algebra Uq (812) resp. Ûq(812).

4.4 Duality of the Hopf Algebras U(s12) and 0(S.L 9 (2))

121

Let (02 be the embedding of the Hopf algebra Uq (s12 ) into 6q (812 ) defined in Subsect. 3.1.2. Then the two pairings (-, -) and (•, • are related by the formulas (70) (x, y) = (cp(x),19 1 , q _1 / 2(y)Y= ((ço 0 V q 1/2)(x),yY, where x e Uq (s12), y E 0(SL Q.(2)) and Vq 1/2 and V 1 , q -1/2 are Hopf algebra automorphisms of Uq (s12) and 0(SL q (2)), respectively, from Propositions 3.6 and 5. Recall that V1 , q _1 / 2 (resp. 9q 1,2) leaves a, d (resp. K) unchanged and maps b to q -1 / 2 b, c to q 1/ 2c (resp. E to q 1/ 2 E and F to q -1 / 2 F). Combined with (70), Proposition 22 gives also the full dual pairing (., •) on the whole algebras Uq (512) and 0(SL q (2)). Next we turn to the relationship between pairings and *-structures. For real q the bilinear form (-, e)' is a dual pairing of the Hopf *-algebras Ûq (su2) l (sui,i) and 0(SUq (1, 1)). and 0(SUq (2)) and also of the Hopf *-algebras e/q Indeed, it suffices to check condition (1.41) on the generators which is easily done. But (. c ) is not a dual pairing of the Hopf *-algebras Ûq (s1(2,R)) = 1. However, one can define another involution *' and 0(SL q (2,R)) for on 0(SL q (2)) by transposing the involution of tig (512 ) according to formula (1.41) such that (0(SL q (2)), *') becomes a Hopf *-algebra and (•, -) is a pairing of the Hopf algebras ti'q (S1(2 1R)) and (0(SL q (2)),*'). The involution *' is determined by the relations a"' --- a, b"' = q -l b, = qc, d*' = d. Similarly, the bilinear form (•, -) on Uq (s12) X 0(SL q (2)) is not a dual pairing of Hopf *-algebras for all three real forms. But, as just explained, it becomes a pairing of the corresponding Hopf *-algebras when passing to other involutions either of Ug (S12) or of 0(SL q (2)). Proof of Theorem 21. Let B be the subalgebra of Uq (812)' generated by the matrix elements t o , i , j = 1, 2, of the representation T1 1 2 of Uq (s12). Since t o (sy) = Ek tik(S)tki(y) for x, y E Uq (512 ) and t o (1) = 6 0 , it is not difficult to show that B becomes a Hopf algebra with comultiplication, counit and antipode induced by duality from Uq (812 ). (This means that B is a Hopf subalgebra of the dual Hopf algebra Uq (s12)°, see Subsect. 1.2.8.) That is, by construction, we have a dual pairing (x, f) := Ps), X G Uq (s12), f E B, of Uq (512) and B. As shown in Subsect. 3.4.2, the elements t o satisfy the relations (3.44) and (3.45). Since the latter are precisely the defining relations for the generators uii of A := 0(SL q (2)), there exists an algebra homomorphism : B such that Ib(u) = t o , j = 1, 2. The formulas for the comultiplication and the eounit of t o and uii are the same. Thus, i is a bialgebra (and hence a Hopf algebra) homomorphism. Therefore, (x, y) (s ,0(y)) = Cy)(x) is a dual pairing of the Hopf algebras Uq (51 2 ) and A = 0(94(2)). By definition, (s, u) is just the matrix coefficient to (x). Inserting these values from (3.22) and (3.23) at x =-- K, E, F, we obtain the above formulas. The same proof works for the pairing (., 4' as well. Now we prove that the pairings are nondegenerate when q is not a root of unity. This can be derived from (70) and Proposition 22. As an exam-

122

4. The Quantum Group SLq (2) and Its Representations

pie, suppose that (x, y)' = 0 for all x E Ùq (512). We write the element y of 0(S.L q (2)) as a finite sum y = Daimnaienbn +a_ imn dlcmbn). Assume that y O. We choose the smallest rn E No such that oti mn 0 for some I E Z, n E 10 and then the smallest n E 110 such that a/ inn 0 for minimal in. By Proposition 22, (K l EmFn, y) is a nonzero multiple of (li mn which is the desired contradiction. There is another proof of the nondegeneracy based on representation theory. We sketch it in the case of Uq (512 ). For n E N, let An be the linear span of basis elements from A = 0(S.L q (2)) of degree less than or equal to n. Obviously, dim An = Enk= 0 (k+1)2- From the decomposition of the tensor products of representations in Subsect. 3.4.2 it is clear that Bn := 7/'(A) is spanned by From the matrix elements of all irreducible representations Ti,l = 0, • • • , the representation theory of associative algebras (see, for example, [CRD it is known that the set of these matrix elements is linearly independent. Thus, we have dim Bn = Enk=0( k ± 1)2. Hence the restriction of i5 to An coincides with Bn. for n E N and V) is bijective. Therefore, if (x, y) = /p(y)(s) = 0 for all s E Uq (s12 ), then 0(y) = 0 as a linear functional on Uq (s12 ) and so y = O. This proves the nondegeneracy in the second variable. If (x, y) =- 0(y)(x) = 0 for all y E 0(S.L q (2)), then we have in particular that tlnin (x) = 0 for all matrix elements 47, 1 n (see Subsect. 3.4.2) of irreducible representations T1 of Uq (s12). Since the irreducible representations T1 , 1 E IN 7 separate the elements of Uq (s12 ) (this assertion will be proved in a more general context in Subsect. 7.1.5), we conclude that x = O. LI

1,

Corollary 23. If q is not a root of unity, then there is a Hopf algebra isomorphism V) of 0(54.(2)) onto the (Hog) subalgebra of the dual Hopf algebra Uq (s12 )° generated by the matrix coefficients t ij of the representation T11 2 such that 1,1)(uii ) = t u , j =1, 2. Proof. As shown in the above proof,

V) is a Hopf algebra isomorphism.

D

Recall that in the above approach in Sect. 4.1 the relations (3.44) and (3.45) for the matrix elements of the representation T1 12 in the dual algebra Uq (s12 )' have been taken as the defining relations of 0(SL q (2)). Likewise, one can start with the Hopf algebra 0(S.L q (2)) and develop the theory of the Hopf algebra Ug (S12 ) in the dual of 0(S.L q (2)). We do not carry out the details and sketch only the idea. Let K and K -1 be 'the characters on the algebra 0(S.Li g (2)) determined by K(b) = K(c) = K -1 (b) = K -1 (c) = 0, K(a) = K(d) _ q -i , K (d) = K -1(a) = q. Next one checks that there are linear functionals E, F on 0(S.L q (2)) such that E(xy) = E(x)K(y) + E(x)E(y) and F(xy) = F(x)E(y) K -1 (x)F(y) for x, y c 0(S L q (2)) and E(a) = E(b) = E (d) = E(1) = F (a) = F(c) = F(d) = F(1) = 0, E(c) = 1, F(b) = 1. Then E 0(SL q (2)) 1 it is not difficult to verify that the functionals E, F, K, K satisfy the defining relations of the algebra Uq (812). The preceding formulas lead also to the coalgebra structure of Uq (512 ), so that the Hopf algebra Uq (512 ) can be recovered in this manner.

4.4 Duality of the Hopf Algebras Uq (s12) and 0(SL q (2))

123

4.4.2 Corepresentations of 0(.51.4 (2)) and Representations of U(s1 2 ) Since we have a dual pairing of the Hopf algebras Uq (s12) (resp. Oq (s12 )) and 0(SL q (2)), by Proposition 1.15 each corepresentation çc of 0(SL q (2)) gives rise to a representation (pi,' of Uq (s12) (resp. 0q (s12)) by formula (1.64). That is,

we have (ro(x)v = E v(0) (x, v( 0 for s E Uq (s12 ), when (p(v) = E v(0) ® v ( ,) . We will consider the passage from cp to ç in three interesting special cases. For the first two let us assume that q is not a root of unity. First let (p be the corepresentation Till from Sect. 4.2. Since TiR (fi(1) ) = t (.1.) by (36), we obtain i1R(x)f i(1) = E 3. .0 3 ) (s t (1) ). Using the exTheorem 21, one derives that

pressions (40)-(44) for t

ÎR(K)e) = q -2z AU)

fiR(F)fi( i) =

q i+1

+ 1] {l

)1/2 fi(+01

([1 - i + 11[1 +

Comparing these formulas with (3.22) and (3.23) we get the assertion of

tiR

of Uq (s12 ) is equivProposition 24. For any l E 1N0, the representation alent to the representation T1,1 defined in Subsect. 3.2.1.

Because q is not a root of unity, the dual pairing (., •) of Uq (s12) and 0(SL q (2)) is nondegenerate. Therefore, any element y E 0(SL q (2)) can be identified with the functional (•,y) on Uq (512 ). As seen from the above for0(SL q (2)) go into mulas, under this identification the matrix elements t the functionals t E Uq (s12)' defined by (3.42). In particular this justifies denoting both quantities by t i(j1) . By Corollary 23, 0(SL q (2)) becomes a Hopf subalgebra of Uq (sir in this manner. This subalgebra is just the linear span of coefficient functionals t- ) for the representations T1,1 of Uq (s12). Proposition 24 and the Hopf algebra embedding 0(SL q (2)) C Uq (51 2 )° have the following important consequence: the results on decompositions of tensor product representations and on Clebsch-Gordan and Racah coefficients of Uq (s12 ) obtained in Sects. 3.4 and 3.5 remain valid for corepresentations of 0(SL q (2)) when q is not a root of unity. Next let us consider the corepresentation ça = LX of 0(SL q (2)). By Theorem 13, 0(SL q (2)) decomposes as a direct sum of subcoalgebras MIR ), E _21 N0 . As we have seen in Subsect. 4.2.5, the restriction of ça = 41 to MIR ) is a direct sum of 2 1 + 1 corepresentations TIR . Therefore, by Proposition 24 and formula (3.24), the image 43(Cq ) of the Casimir element Cq of Uq (512 ) acts on C(TiR ) as the Multiple of the identity by the complex number (q1 + 1 + q 1 ')/(q - q -1 ) -2 . That is, the Peter-Weyl decomposition (45) of 0(SL q (2)) is just the eigenspace decomposition of the operator c,O(C q ). Further, if ça is a coaktion of 0(SL q (2)) on a right comodule algebra X, it is easy to verify that X becomes a left module algebra of Uq (s12 ) with action

124

4. The Quantum Group SLI (2 ) and Its Representations

0. Let us specialize this to the right comodule algebra O(C) with coaction := çoR given by (20). Then we conclude that O(C) is a left module algebra of Uq (512) with action ço determined by the equations c,OCK)s = qx, (i)(K)y = qy, c,O(E)x = 0(F)y =

0,

= x, cro(F)x = y. Finally, let us note that the one-dimensional representation of Uq (512) with K -1, E 0, F 0 obviously is not of the form "y3 for some corepresentation cp of 0(SL q (2)).

4.5 Quantum 2-Spheres In this section we are concerned with a one-parameter family of quantum homogeneous spaces for the quantum group SL 9 (2). For special parameter values they are quantum analogs of two-dimensional spheres. Throughout this section we assume that q is not a root of unity. 4.5.1 A Family of Quantum Spaces for SL(2)

Let a and O be complex numbers such that (a, f3) (0,0). We denote by Xvo the algebra with three generators x_ i , so , s 1 and defining relations 2 Xo —

(1-

qx i x_ i - q—1x_ixi= 01,

(71)

+ qx_ i x i qxix_ i = (1 q2 )axo , x_ i so _ e xox _ i (1 _ q 2 )cex _ i,

( 72)

sos i

exixo (1 _ q2 )axi.

(73) (74)

Further, let us recall from Sect. 4.2 that the three-dimensional irreducible corepresentation T1 of 0(S.L q (2)) can be given by the matrix corepresentation q,a a2c _1,0111= ) =-1,01 (t),= C

where q' := (1 +

2

q'ab b2 1 + (q + q -1 )bc q'bd ,

qidd d2

( 75)

C 2 ) 1 /2 .

Proposition 25. Each algebra Xgcro is a left quantum space and a right quantum space for A = 0(54(2)) with left coaction çoi, and right coaction (PR determined by

▪

4.5 Quantum 2-Spheres

125

Proof. We sketch the reasoning for çoi, • We first define an algebra homomorphism (pi., of the free algebra C(xi ) with generators s_ i , xo, xi to A0C(x2 ) by (76) and show then that (pL passes to the quotient algebra Xqap of C(si ) by checking that (pi, preserves the relations (71)-(74). We omit these straightforward verifications and mention only the reason why it works: if we rewrite the equations (71)-(74) as Ei ao, ii sisi = 01 and Ei bk , ii x i si = (1 — q2 )faxk, k = -1,0, 1, then the 1 x 3 2-matrix (ao , ii ) intertwines the tensor product corepresentation Ti 0 T1 and the one-dimensional corepresentation To and Ti and T1 . 111 the 3 x 32-matrix (bk,ii) intertwines the corepresentations

Each of the sets {x. 1 x, s6r if I i 7 j E No, k E NI and Ixt,x 11, xt,xi I i,j E No, k E NI forms a vector space basis of Xqap. It can be seen from the relations (71)-(74) that these elements span Xqap. Their linear independence can be derived from Lemma 8. One may also use the algebra homomorphism into 0(S.L q (2)) defined below combined with the fact that the elements cr,, 19" cr ds , m, n, r E No, s E N, are linearly independent. Let us say that two quantum spaces X and Y of a Hopf algebra A are equivalent (or isomorphic) if there exists an algebra isomorphism of X and Y which intertwines the coaction of A on X and Y. Clearly, the quantum spaces Xqa pand X_ 2 0 are isomorphic for any c E C, c 0. (Indeed, under the scaling s i ca l- xi of the generators the defining relations of Xqa p go into that of Xq ,,,,,,2 0 . Hence, there is an isomorphism 0 of both quantum spaces such that e(x i ) = Next we remove one superfluous parameter by choosing a "normal form", of the quantum space Xqa p. In the case denoted by O(S) for P E C when [3 - a2 0 we take numbers p and c such that p z = az (0 _ a 2)-1 1 C2 = fi - a 2 and a = cp. Since then (a, f3) = (cp,c2 (1 + p2 )), the quantum spaces Xqap and Xq ,p , (1+p2 ) are isomorphic. We denote the quantum space Xq,p,( 1+p2), p E C 7 by O(S). If a2 ---- 0, then Xqap is isomorphic to Xqii . Let us denote the quantum space Xqii by Q(S). Now we realize the quantum spaces O(S) as subalgebras of 0(S.L q (2)). For p E C, we put := (1 + q2)-112a2 10

p (l

q -2)1/2 ac q(1 + (12 ) 1/ 2c2

(77)

ba + p(1 (q q-1- )bc) - cd,

(1 + q2)-1/2b2

p(l 472)1/2db

g (1

(78)

q 2)-1/2d2 .

(79)

In the case p =-- oc we set ± -

+ 172)1/ 2 ac, ±- 0 := 1 +

47 -1)bel

:= (1 + q2)112

17

ao

(80)

Note that the elements 1 are linear combinations of matrix coefficients 471: ) . One verifies that the elements of 0(SL q (2)) satisfy the defining relations (71)-(74) of the algebra O(S). Hence there exists an algebra homomorphism : 0(4) -) 0(54(2)) such that 11)(si) = ft. By Proposition

126

4. The Quantum Group SL 9 (2) and Its Representations

31 below, 7,b is injective, so that we can consider O(S) as a subalgebra of Under this identification the coaction çoR 0(S.L 9 (2)) by identifying si with of 0(S 4(2)) on 0(Sq2p ) corresponds to the comultiplication of 0(S4(2)). If the parameters q and p are real, then the algebra 0(S) becomes a *-algebra with involution determined by

=

-

q-lx i , x o* =--- x0 , s = qx_ i . -

(81)

The coactions (pL, and yoR are then *-preserving for the real form 0(SUg (2)), so the *-algebra 0(Sq2p ) is a left and right *-quantum space for the Hopf *algebra 0(SUq (2)). Moreover, the embedding 7,/) of 0(Sq2p) into 0(S.L q (2)) is then a *-homomorphism. We call this *-algebra 0(Sq2p) the coordinate algebra of the quantum 2 sphere S. Finally, let us look at the classical counterpart of the *-algebra 0(Sq2p ). In the limit q > 1, the relations (72)-(74) mean nothing but that the generators xi mutually commute and the equations (71) and (81) read as 4+ 2s i sl = (1+ p2 )1. Taking hermitian variables by setting Yi := xo , V42 x i +x1' and := -xn, the latter becomes the equation y? +0+0 =---That is, in the limit q 1 the *-algebra 0(514L) is just the coordinate *algebra of a 2-sphere in R3 . -

-

4.5.2 Decomposition of the Algebra Q(S) First we proceed in a similar manner as in Subsect. 4.2.2 and decompose the right quantum space 0(Sq2p ) into a direct sum with respect to the coaction of the quantum subgroup K. For this reason, we define an algebra homomorphism RK := (id 0.0(pR : 0(Sq2p ) 0(Sq2p ) 0(K), where (pR is the right coaction of 0(SL q (2)) on 0(St,) and OK is the Hopf algebra homomorphism of 0(Sl q (2)) to 0(K) given by (23). For n E 2Z we set

0(Sq2p )[n]:= Ix E 0 (S:p)

RK(X) = X g znl.

By (76), (75) and (23), we get xo E 0( 51q2p)[0]) xi E 0 ( 51)[2 ] and x i E 0(Sq2p )[-2]. Because RK is an algebra homomorphism, the latter yields that X

i

k

G 0(S2qP)[2i — 2k1 for i, j, k E No.

(82)

As noted in Subsect. 4.5.1, the set {xi i x-4, 44} is a basis of 0(SiL). There-

fore (82) implies that 0 (s:p )[2 n]

el i c [x o ]

= c[x o ixTn

if if

n > 0, n < O.

In particular, thé space 0(Sq2p)[0] of K-invariant elements of 0(S427p) coincides with the algebra C[xol of polynomials in the generator xo. Further, by (82) we have the decomposition

4.5 Quantum 2-Spheres

()(sp )

127

= @,„, ez 0( sp )[2 m 1

Our next result determines the decomposition of the coaction (pR into irreducible components. Theorem 26. The corepresentationçoR of 0(SL q (2)) on O(S) is equivalent to the direct sum of irreducible corepresentationsT1, 1 E N o , with multiplicity 1. That is, we have ÇOR2G

teN0 T1

072 C 9(5 q2 p ) (131 1.0 V1.

(83)

Proof. Let Tirn be the linear span of elements x 1 xx i, j, k E N0 + + k < n. We prove by induction on n that the restriction (,oR,,, of çoR to Wn is equivalent to a direct sum of corepresentations Th 1 = 0,1, • • • n. Obviously, this in turn implies Theorem 26. For n = 0 the assertion is trivially true. Suppose that it holds for some n E No . Since corepresentations of 0(SLq (2)) are completely reducible (see Remark 1 in Subsect. 4.2.6) and any irreducible corepresentation of 0(Sli g (2)) is equivalent to some T1 there exists a çoR-invariant subspace Wn+i of Wn+ 1 such that Wn+ 1 = 14771 Wn+1 and ( 30 FtrWn-F1 at TV . On the other hand, ,

,

i

,

since the basis elements fi(i) (see (34)) of the representation space of Ti TIR are in A[21, —2i] by Lemma 11, we have (id 0 O K )Ti(fi ) = fi(1) Z -2i and Since çoR,n, ED711_ 0 Ti by the induction hypothesis, it so OK (tY)) = 0(Sq2p )[2i1. follows that Ric(14771 ) = (id 0 oic)(pR,n(wn) g By (82), Xnti c Wn-F1 belongs to 0(Sq2p )[2n + 2 ] and up to multiples it is the only element of Wn+ 1 in 0(Sq2p )[2n + 2 ] . Hence there occurs at least one corepresentation Tv in the decomposition (pRI-Wn-F1 ED,, Tv such that 1' > n. But the latter is possible only when çoR [W + 1 Tn+i , because we have dim Wn+i = dim W n+1 _ dim yvn < dim Lin {xi 1 44I j +j ---=m+k= n + 1} = 2n + 1. This completes the induction proof. In the preceding reasoning only corepresentation theory of 0(SL q (2)) was used. Another proof can be given by decomposing the representation gOR of the algebra Uq (512 ) which is induced by the dual pairing from Theorem 21. The key observation in this proof is the fact that the highest weight vectors of (ijii are precisely the complex multiples of xn 1 , n E N0 . El

Remark 2. The following fact from the preceding proof will be used later. Up to multiples, xl 1 and f (1)i are the only elements in the corresponding carrier z 21 spaces of the corepresentation Ti such that (id 0 OK)(pR(x l 1 ) = x i 1 and (id 0 OK )/1(f) = f 0 Z21 , respectively. Moreover, both elements are highest weight vectors for the irreducible representation ti of Uq (512). A The quantum spaces O(S) X q ,p, i+p2 and 0(Sq2 ,_p ) Xq ,_ p, 1±p2 are obviously isomorphic for p E C. As a first application of Theorem 26, we show that these are the only isomorphisms of the quantum spheres O(S).

128

4. The Quantum Group SL q (2) and Its Representations

Proposition 27. Let p, p' E CU{00}. The quantum spaces 0(8,72 ,p ) and 0(Sq2 ,p,) are isomorphic if and only if p = p' or p = -p'.

Proof. Suppose that O is an isomorphism of the quantum spaces 0(Sq2,p ) and Since 9 intertwines the coactions of 0(84(2)), Theorem 26 implies Lin {aci}, where {x i } and {x'i } denote the that 19 maps Vi Lin {xi } to VI! generators of 0(8422 ,p ) and 0(Sq2,p,), respectively. Since the corepresentation T1 is irreducible, it follows that 9(x i ) = Ax for some A E C. From the relations (71)-(74) we see that the one-dimensional representations 7r of the algebra 0 (51472, p) are given by numbers 7r(s i ), i -1,0, 1, such that 7r(s i )7(s_ i ) = 0, 7(x 0 ) = 1 if p = (Do and (q + q -1 )71- (x1)7(x_i) = -1, 7(x0) = p if p E C. Being an algebra isomorphism, 9 preserves the onedimensional representations. Therefore, the previous description implies that 0 (8472 ,00 ) is not isomorphic to 0(8217)P,) for p' E C and that A 2 = 1 and hence 0 = p or p' -p for p, p' E C. As a second application of Theorem 26 we obtain the existence of an ) A linear functional h on the algebra 0(S qP 2 invariant functional on 0(8247P , is called right-invariant if

((h 0 id) o (pR)(x) h(x)1, x E 0(SIL) Proposition 28. There exists a unique right-invariant functional h on 0(4) ) such that h(1) = 1. The functional h vanishes on all subspaces 0(S)[2m], m O.

Proof. The proof proceeds similarly to the case of the Haar functional in Subsect. 4.2.6. By Theorem 26, we can define a linear functional h on 0(Sq2p ) by h(1) = 1 and h(x) = 0 if x E Vi 7 I 0. As in Subsect. 4.2.6, h is rightinvariant and any right-invariant linear functional on 0(8q2p ) with h(1) = 1 is of this form. For x E 0(8,72p )[2n], we have ((h OK)(pR)(x) =-- h(x)z 21 = h(x)OK (1) and so h(x) = 0 if n 0. 0 By Proposition 31 below the algebra 0(Sq2,p ) can be identified with a subalgebra of 0(84(2)) such that the right coaction (pR corresponds to the comultiplication. Then, by the uniqueness assertion of Proposition 28, the right-invariant functional h must be the restriction of the Haar functional of 0(84(2)) to Finally, let us suppose that q is real. We define a Hermitian form (., -) on 0(Sq2p ) by setting

(x, y) = h(xy*), x, y

e

0(S)

.

Because h is the restriction of the Haar state of 0(SUq (2)) to its *-subalgebra 0(Sq2, p ), it follows from Theorem 17 that (-, -) is a scalar product on 0(S,) and that the subspaces 171 are mutually orthogonal with respect to (-, -).

4.5 Quantum 2-Spheres

129

4.5.3 Spherical Functions on Sq2p There exists a basis ei(1) , i = -1,-1 + 1, • • ,l, in the carrier space V/ of the corepresentation T1 in the decomposition (83) such that

E

(e i(i) ) =

e(-1) g t

j

=

-1,-1 +1,•

where t içji) are the matrix elements of the corepresentation 21 of 0(SUg (2)) from Subsect. 4.2.4. By Remark 2 after Theorem 26, we can assume without si 1 . If q is real, then the elements e? ) E O(S) loss of generality that e are called spherical functions on the quantum 2-sphere S. Proposition 29. (i) If O < i < 1, then et(i)

Ni( _ q -20.-Fiw a; g -2) 1

y;

o i ci -2) xL ,

and if -1 <j < O, then = Ar_i _ q 2(i-1) ,(3/a; q -2) 1+ixr pt(;ii,-i) (y; a,01,7 -2 ), (

where 1)4a ' b) (x ; a, OW) are big q-Jacobi polynomials (see Subsect. 2.3.4) and =__ (_ a )l-itp-i)(/+3i+3)/2 [ 2/ 1-

-1/2 _2

1 1

q •-2

a = ( (q ± C 1 y 2 ± p 2 / 4) 1/ 2 ± p/2 , 0 = a ____ p.

(ii) If q is real, the spherical functions e l) satisfy the orthogonality relation 1

(e i( m) , e.(in) ) _ 67nni52 3

q2

[2M 1 _ q -2(2m+1) m -

1 flq( ct ± -273) (0 ± q -2r a ) . q -2 r=1

Proof The proof can be found in [NM].

LI

4.5.4 An Infinitesimal Characterization of 0(S 2 In this subsection we give an elegant description of O(S) as the subalgebra of elements of O(Sli g (2)) which are infinitesimally invariant with respect to a certain twisted primitive element of 6,1 (512)An element z of a bialgebra A is called twisted primitive with respect to a group-like element g of A if A(x) = g -1- gz-i-sOg. If s is twisted primitive with respect to g, then we have S(x)= -g - ixg and E(x) = O. From the formula (12) for the comultiplication of the Hopf algebra Ûq ( 51 2) we see that any linear combination of K -- K -1 , E and F is twisted primitive with respect to the group-like element K of Ûq (s12). We put •

130

4. The Quantum Group SL q (2) and Its Representations -1

q X p := pq 1/ 2,

(K

(84)

IC -1 ) + qE + F, p E C,

XŒ, := K K -1 .

(85)

In the notation of Subsea. 1.3.5, we write x.f = x(2) (f, x ( 1 ) ) for x E 0(SL q (2)) and f E 6(1(512), where (., .) is the dual pairing from Theorem 21. As noted in Subsect. 1.3.5, this defines a right action of the algebra Ûq (512 ) on 0(SL q (2)). Using the formulas (40)-(44) for the matrix coefficients and (67)-(68) for the pairing (., .)' we calculate

t z(j1) .E = V[1+ i + 1] [1 -ijtY1

tg) .F = -,/[1 + i][1

(86)

,,

+ l]tçl) t-1,j •

(87)

Further, let

(88)

fx E 0(SL q (2)) x.X p = 0}

denote the set of right Xp-invariant elements of 0(84(2)). Using the fact that Xp is twisted primitive one immediately checks that 6(19q2p) is a subalgebra of 0(SL q (2)) and 46(6(8L)) C 6(4) 0 0(SL q (2)). Lemma 30. dim fo l for E + No.

(SL) nc(TiR) = 21+1 for 1

E No and 6(4) n c(TiR) =

Proof. Let Ci (T R denote the linear span of matrix elements t )

,

i = -1,-1+

1, • ,l. Clearly, C(T/R ) = - E k t ik (1) t kj i3 =-t C-(T 3 /R )• Since .6(tçl) map z -) x.X p leaves each subspace Ci (T/R) invariant. Moreover, from (86) and (87) we see that Xp acts on the subspaces Ci (TiR), j = -1,-1+ 1, • • • ,l, by the same formulas. In the case p --= co the assertion follows immediately from (86) and (87). Suppose now that p < co. We consider the equation x.Xp = 0 for x ,

3

E. ct•t 1.3(. 1.) E

C (71R ) where ai E C. Inserting the formulas (84), (86) and (87) and comparing the coefficients of tg) , it follows that this equation is equivalent to the recurrence relations (q -2 ql)-y +

with q ---1)1( q -1

i][1 + j + 11 ±

-01 - j + 1] [l + i] = 0 (89)

= ai+1 := 0 for the coefficients di , where we set 1, = q). If ct_i is given, then (89) determines uniquely the coefficients

cti ±i • This procedure yields a solution of the equation a-1+17 ct-t+2) • • • , x.Xp = 0 if and only if ai +i = O. Therefore, since the recurrence relation (89) does not depend on j, it follows that either dim (8q2 p ) nC(TIR) = 21 + 1 or

6(4) nc(TiR) = { 0}.

We find by a direct computation that dim C3(4) n C (T1R ) = 3 and ei(Sq2p) nC(4,2 ) = {0}. In fact, the solutions for the corepresentation are precisely the elements i-19i02ii from (77)-(79), that is,

4.5 Quantum 2-Spheres

(1)

=

.(1) + PO t O,i

131

P lt

(1 + ey112 00 = p an d = q(1 + (12) -1/2 . where 0_ 1 Now we prove by induction on 1 that dim e3(4p) nc(TiR) = 21+1 for integral 1. For 1= 0 this is trivially true. Suppose that it holds for a fixed 1 E N. Take a nonzero element x = E, ceit E 6(8q2 p) nC(T/R ) . By the discussion after Proposition 24 in Subsect. 4.4.2, the product tt„ E 0(5'4(2)) can be computed by means of formula (3.43). In this way we obtain

=

E

"ml =

i,m

Ea

. 0 ,/,1,/±1 m i,m,i+m

1,1,1+1 i+m,1-1-1•

i,m

Thus, xi). E Ci-f-i(Ta i ). Since (5(4p) is an algebra and x E (5(Sq2p) by ashsumption and i E e5(Sq2p), we have x1 1. c (5(Sq2p) nC(T1F+11 ). Because the algebra 0(84(2)) has no zero divisors, x i z4 0 and so ( ( S q2 p) 1 C 1 -FL) {0}. Hence we have dim ô(S) nC(Ta i ) = 2(1 + 1) + 1, which completes the induction proof. Next we show that 6(4p) nc(TiR) = {O} for half-integral 1. Assume the contrary and let 1 be the smallest half-integer such that (5(Sq2p ) n c (TiR) Let x = Ei ai 6,1) E 6(SL)nC(T IR ). {0}. As noted above, 1 1, so l > From formula (3.43) it follows that for k + j <1- 1 the coefficient of x`±'i at (1-1) t s,k+i is ',

I 11-1

,

,-,

r11,1,1-1

as-1P1L

c zo.,/-1 k,j,k-f-j.

Since 1 is minimal, the expression in brackets must vanish and we obtain the system of equations as+10-1C81 1+1 11,111,8 + asOods' ol '

+

as-101c1, 4 W., =

for s = - 1+1,-1+2, • • • , 1-1. Substituting the explicit form of the coefficients from Subsect. 3.4.3 and comparing with the recurrence ' 1 '1-1 0, and C1m,n,m+n relations (89) for a i , we conclude that ai = 0 for all i. Thus we have = 0 and Q(S) nc (TiR) {0}. This contradiction completes the proof. Proposition 31. There is an isomorphism V) of the right quantum spaces O(S) (with respect to the coaction (19R) and e5(Sq2p ) (with respect to the comultiplication) for the quantum group 0(51 4(2)) such that 1b(x) =-= -1,0,1. Here are the elements of 0(54(2)) defined by (77)-(80). Proof. As already noted above, the elements ij are in (5(Sq2 p) and satisfy the relations (71)-(74). Hence there is an algebra homomorphism : 0(4) ej(sq2p) such that 1,b(si ) = ii . One also verifies that A ( ii ) = Ei ti(12. Hence lb intertwines the corresponding right coactions (pR and

A of 0(S4(2)) on 0(51920) and O(S), respectively.

132

4. The Quantum Group SL q (2) and Its Representations

= 1, 1+ I,. • • ,1} gives a The direct sum C(TiR) = EDli= _I Lin {t! decomposition of the corepresentation L FC(T/R) into a direct sum of 21 + 1 irreducible corepresentations 7i. Therefore, by Lemma 30, the comultplication .6 considered as a corepresentation on the vector space 6( 1%2 p) nc(TiR) is equivalent to T1 for 1 e No . Hence there is a direct sum decomposition Ô(S ) = (1)/EN . 17-/- such that 4607, Ti. On the other hand, we have 0(Sq2p ) = vi and (poR [VI Ti by Theorem 26. Since V) intertwines both coactions, '0 maps Vi to 17/ for any 1 e No. Because Ti is irreducible, is either zero or bijective. Assume that C3/4) = { 0}. From Remark 2 in Subsect. 4.5.2 we know that xl 1 E Vi. Hence 7,b(x / 1) = 7,b(x-1) 1 = = 0. By (77) and (80), this is impossible, because the elements aci are linearly independent. Thus, the map lib : Q(S) -+ O(S) is bijective. -

-

4.6 Notes The quantum group SL q (2) and its corepresentation theory were introduced and developed independently by S. L. Woronowicz [Wor2] and L. L. Vaksman and Y. S. Soibelman [VS1]. The approach to SLq (2) via symmetries of the quantum plane is from [Man]. The matrix coefficients of irreducible corepresentations of 0(571q (2)) were determined in [Ko1] and [MMINT]. Our treatment of harmonic analysis on SL q (2) followed mainly the paper [MMN], see also [KV]. The explicit pairing in Proposition 22 is taken from [Koe]. The quantum 2-spheres were invented by P. PodleA [Pod 11. The harmonic analysis on these spheres was developed in [NM] and [DK2]. The infinitesimal characterization of the quantum 2-spheres was discovered by T. Koornwinder (see [DK2] and [K04]). It has been a key result for the construction of quantum homogeneous spaces (see Sect. 11.6).

5. The q-Oscillator Algebras and Their Representations

The q-oscillator algebras are deformations of the harmonic oscillator algebra from quantum mechanics and of its central extension. This chapter gives an introduction into these algebras and their representations. Throughout we suppose that q2 1.

5.1 The q-Oscillator Algebras Acq and

Aq

5.1.1 Definitions and Algebraic Properties Recall that the one-dimensional harmonic oscillator algebra A is generated by two elements a and a+ satisfying the commutation relation (1)

[a, a+1 aa+ a+a = 1.

In the Fock representation the generators a+ and a act as the creation and a+a corresponds annihilation operators, respectively, and the element N to the number operator. From relation (1) we immediately obtain that [N,a+ ] = a+ and [N,

(2)

= —a

in the algebra A. Let Ac be the algebra with three generators a, a+, N and N a+a defining relations (1) and (2). It is clear that the element c belongs to the center of the algebra Ac and that the algebra AC is just the extension of the algebra A by an additiônal generator c which commutes with a and a+. The q-oscillator algebras Aq and A& defined below can be viewed as q-deformations of these two algebras A and AC, respectively. —

Definition 1. The centrally extended q-oscillator algebra Acq is the complex (associative unital) algebra with four generators a+, a, qN q -N and relations [a, a+]

aa+ — qa+ a = q - N N qN = q N q -N = 11 • e a+ = gate, qN a = (rime.

(3)

(4)

It might be necessary to emphasize that there is no element N in the algebra Aeg and that qN and q -N are only symbols for two of the four abstract generators of the algebra A.

134

5. The q-Oscillator Algebras and Their Representations

The defining relations (3) and (4) of the algebra valent to the following relations:

[a,

A& are formally equi-

= q -N Na+= a+ (N + 1), aN = (N +1)a.

( 5 )

Indeed, the second equation Na = a+ (N +1) of (5) implies that f (N)a+ = f (N + 1) for any polynomial f, so that formally gNa+ = a -F q N+1 = qa+qN . Conversely, if we write q = eh and q N = e hN then the relation qN a+ = qa+q N means that e hN a + = eha-FehN. Differentiating the latter at h = 0 yields the equality Na + = (N + 1). The equations aN = (N + 1)a and q N a = q - laqN are related in a similar manner. (Note that for N-finite representations (see Sect. 5.2) such formal manipulations have a rigorous meaning.)- In the limit q -> 1 the three relations of (5) go into the defining relations (1) and (2) of the algebra Ac. For k E Z and a E C, we introduce the notation kNa -F

q a(e)k

and [N + a]

[N + aj q :=

q

N-Fa

q

-N-a

- -1

Using the relations (3) and (4) one easily verifies that the element C := q l-N ({Ni q - a+a) = q 1-N[Ni q q -2N - q -N aa+

(6)

belongs to the center of the algebra A. In the limit q -4 1 this element gives just the central element c = N a+a of the algebra Ac. If q is not a root of unity, then the center of AQC is generated by the element cq . (This can be shown by modifying the proof of Theorem 6.45' below.) The defining relations (3) and (4) of .1t, are not symmetric with respect to q and q-1 . The following algebra Ag can be regarded as a symmetrized version of the algebra A. Definition 2. The symmetric g-oscillator algebra Ag is the complex (associative unital) algebra generated by four elements a+, a, qN , q -N subject to the relations [a, al q aa+ qa+ a =

q -N q N = (I N q -N =

N, 11

[a, a+ ] q - I-a- aa+ - g - a+ a = q N , N + = (la + g isl

(7)

(I N a = 47 -l ag /V . ( 8 )

We shall refer to both algebras ./i& and Ag simply as g-oscillator algebras. From the relations (7) we derive at once that a+a = [N] q and

aa+ = [N +

This in turn leads to

Ina+ =

e-IN +1), and a[Ni q = aa+a = IN + 1)qa.

(9)

5.1 The q-Oscillator Algebras ,.,4& and A q

135

All such formulas are q-deformations of the corresponding formulas for the harmonic oscillator algebra A. The two relations (7) in Definition 2 can be replaced by any two of the four relations (7) and (9). In the limit q -> 1 the relations of the algebra A q reduce to (1) and the relation N = a+ a. Hence the q-oscillator algebra A q can be considered as a q-deformation of the oscillator algebra A. the By definition, the algebra A q is the quotient of the algebra A two-sided ideal J generated by the element aa+ - q - la+a - qN . Since the second condition of (7) can be replaced by the first of (9), the ideal J is also generated by the central element cq . Thus the element cq given by (6) is zero in the algebra A q and the algebra A q has only the trivial center C.1 if q is not a root of unity. Proposition 1. The following sets are bases of the complex vector spaces:

{( a+)k an qmN k n E N, in E Z} fan (a+)k e N I k,n E N, (0, o)}. A q : {(a+)kgmN q m N an fk,n E N, m G Z, (m n)

Tri

Proof. All assertions can be derived from the diamond Lemma 4.8. Proposition 2. (i ) There exist automorphisms 190, n E Z, of the algebras Acq and A q such that

190(a) = Oa,

190 (a+) =

0n (a) = qnN a,

On(a+) =

1a ,

E CI 3

(a) = q r a

ip er,r (a+ )

On N)

aqra+ ,

(iii) There exists an automorphism Ca) = a+ , 0(a+) =a,

0, and On ,

( (r N) = (11\1 7 7- N .

1

0, of Acq deter-

(ii) There are algebra automorphisms (p a,r, a,r E C, a mined by (70 t ,r

D

aq2r q - N

of the algebra A 7 such that

0(q -N ) = -qqN .

Proof. One checks that the images of the generators under i9p, 9„, ço a,, and //) satisfy the defining relations of the algebras Aqc and A q , respectively. Hence the above formulas indeed define algebra homomorphisms. Since the mappings are invertible, these homomorphisms are actually automorphisms. D

If q is not a root of unity, it can be shown that the automorphisms from Proposition 2 generate the groups of algebra automorphisms of Ag and Acq , respectively. We close this subsection by looking at *-structures on the q-oscillator algebras. If the parameter q is real, then both algebras Acq and A g become *-algebras with involutions determined by a* := a+ and (qN )* := qN . In the case iqi = 1 the algebra A is a *-algebra with involution such that a* := a+ and (qN )* := q - N

136

5. The q-Oscillator Algebras and Their Representations

5.1.2 Other Forms of the q-Oscillator Algebra

In this subsection we list some other variants of the q-oscillator algebra that occur in the literature. They are derived by simple formal algebraic manipulations from the algebra A. As abstract algebras they are of course different. But for N-finite representations of the algebra Aqc these formal replacements lead to representations of the corresponding algebras. First, inserting the formal substitution b := q --N/2 a7

b+

a + q - N/2

into (3) and using (4) we obtain the relation bb+ bb = q-2N Replacing formally tr 2 by q in this equation and in (4), we obtain the relations •

[b,b+] bb+

bb _ q N

(I A b+ = qb + giv 7

eb = q - be

Let Ab denote the algebra with generators b, b+ q N N = NqN = 1 . relations (10) and Similarly, putting A := q N/ 2a,

1

(10)

N subject to the

A+ := a+q N/ 2

and then replacing q2 by q, the relations (3) and (4) lead formally to the equations [A, A] q AA + - qA+ A = 1, q N A+

qA+q N q N A

Aq N .

(11)

The algebra with generators A, A+ , qN , q –N and defining relations (11) and q N q -N = q -Nq N = 1 is denoted by AA. Some authors use the name q-oscillator algebra for the algebra with generators A, A+ subject to the relation AA + - qA+ A = 1. A two-parameter deformation of the harmonic oscillator algebra is derived from the q-oscillator algebra Aqc in the following way. First we replace the parameter q by r. Setting

a := r ctN a,

:= a±ra N

(12)

for a E C and q = r2a+1 1 p = r -2+1 in the defining equations (3) and (4) of the algebra A rc we formally obtain the relations N ^ –N , p N a -+ =pa-+p — ± qa-±a=p aa , p N. a=p –1ap N .

(13)

The algebra Apc q with generators a, a+, p N , p -N and relations (13) is called the (p, q) oscillator algebra. Of course, the inverse of the formal substitution (12), that is, a = r - aNii. and a+ = "+ - crN with r 2 = pq, r 4ar = qp - i transforms the two-parameter deformation A back to the algebra Arc . -

, q

5.1 The q-Oscillator Algebras botq and Ag

137

5.1.3 The q-Oscillator Algebra and the Quantum Algebra 6,7 (s12 )

The quantum algebra Ûq (s12) admits various homomorphisms into the qoscillator algebra A. More precisely, we have the following Proposition 3. (i ) For any a E C I there exists the algebra homomorphism A g such that Ta Ùq (s12)

Ta (E) = a, Ta (F) = a±[N 2a]q , T„CK (ii) There is an algebra homomorphism T: (42(s12)

T(E) =

(a+)2 q

a2 , T(F) = q+q-1 7 -

q N-a .

A q determined by

7- (K) . q N+1/2 .

Proof. (i ): As usual, it suffices to check the defining relations (3.10) and (3.11) of the algebra (4(512). As a sample, we carry out this for the relation (3.11). By (8) and (9), we have

[T,r (E),T,,(F)] = aa+ [N 2ajq — a+ [N 2a]q a aa+ [N — 2a]q — a+a[N — 2a 1 ] q [N + 1.]q [N — 2a] q [N]q [N — 2a l] q (q _ q -1)-21 (q N-Fi q -N-1) (qN-2a _ q -N+2a) (q N _ q-N)(Q,N-2a-i _ q -N+201+1)} —

—

2(N-a)

q

-2(N-a)

TcK (K)2 za (K)-2

q

g -1

(ii): The proof follows by similar calculations using the relations (a) 2 a2 = [N]q [N

—

1 ] g , a2 (a)2 = [N + l]q [N + 2] q .

111

Our next aim is to realize the algebra [ji g (s12) in terms of two commuting q-oscillator algebras. In order to do this we need an algebra Aeqxt which is obtained by adjoining formally elements qN/2 and C IV/2 to A g . More precisely, At is the associative unital algebra with generators a, a+, erl 2 and q-N/2 subject to the relations [a, a±jg

q -N :=

-N/2 q N12 = q /V/2 47 -N/2 =

( q -N/2‘2 1

)

[a, a± l q-i

=

17 qN/2a+ = q 1/2 a -F qN 1'2 1

:= (qN/2)2 7 qN12 a = q -112 aq N/2 .

It is clear that the algebra Ag from Definition 2 is isomorphic to the subalgebra of AV generated by the elements a, a+, q N and q -N Let Aeqxt '2 be the tensor product of two q-oscillator algebras A. The corresponding generators of the algebra .A.eqxt' 2 are denoted by a17 e2/2q-N212. q -N1/2 , a2, Note that every element of the set a l , at q± N1 12 4 commutes with any element from a2,4, q± N2/2 ,

138

5. The q-Oscillator Algebras and Their Representations

Proposition 4. There exists a unique algebra homomorphism c,o : Ûq (512) —› Ar t ' 2 such that p(E) = ata2 ,

c,o(F) = 4a1 ,

-. c,o(K) = q(N1N2)12

(14)

Proof. It is enough to verify that the elements (,,o(E, ço(F), (p(K) and ço(IC -1 ) satisfy the defining relations (3.10) and (3.11) of Uq (s12). As an example, we check the relation (3.11). By (9), we obtain (p(E)c,o(F) = ata2a -21- a1 = [N g [N2 + 1] q .

Similarly, (p(F)(p(E) = [Ni + 1]q [N2] q • A straightforward computation shows that [Ni q [N2 + 1] q - [N1 + 1],AN2] q = [N1 N21J q . Hence c,o(E)c,o(F) - ço(F)cp(E) = (v (K)2 _ p(K1)2)/(q _ q --1) The algebra homomorphism ço from Proposition 4 is called the JordanSchwinger realization of the algebra Oq (512)• An algebra homomorphism7,b of a Hopf algebra 7-1 into another algebra X might be a useful tool for the study of the Hopf algebra. First the composition of IP with representations of X provides us with representations of For the Jordan-Schwinger homomorphism c,o : 0,1 (812) —› Aeqx t '2 this will be carried out in Subsect. 5.3.4 where the irreducible representations of Ûq (S12) are realized on the q-analog of the Bargmann-Fock space. Another application is given by the following lemma. It is a slight generalization of Proposition 1.14 and shows that such a homomorphism turns the algebra X into an 7-1-module algebra. Lemma 5. Let 11) : e / be an algebra homomorphism of a Hopf algebra 7-1 into an algebra X. Then X is a left 7-1-module algebra with respect to the left action of 7-1 on X defined by

a> s

adip, L (a)x :=

E 11)(a(0)•s.11)(S(a( 2 ))),

a C 7t, s E X.

(15)

Proof. Since //) is an algebra homomorphism, it is clear that there are left and right actions of 71 on X given by as := 0(a).s and sa := s-7,1)(a), respectively, where the dot means the multiplication of X. By the associativity of this multiplication, X becomes then an 7-1-bimodule. Therefore, as noted in Remark 2 in Subsect. 1.3.4, ado,/, is a left action of 7-1 on X. Since a r> 1 = (a)1 and

(a i ) > s)(a(2)r> y) = E 0(a(1) )s1/0(a(2) )) 0(a(3) )y0 (S(a (4) ))

E 0(a( 1 ))stP(S(a( 2))a( 3))00(a( 4))) = a t> (sy), X is indeed an Ii-module algebra.

0

Lemma 5 applies in particular to the algebra homomorphisrns from Propositions 3 _ and 4. Let us write down the corresponding action in the case of the

5.1 The q- Oscillator Algebras ..Aq and Aq

139

Jordan-Schwinger homomorphism cp. Setting V) = ça in (15) and using (14), (3.12) and (3.13), the left action (15) for the generators of Ûq (512) yields ado,L(E)x = ata2xq(N2

qq(N2

IN11)/2

N1) 12 xat a21

ado,L(F)x = aixq(N2- Ni)/ 2 _ q -l q (N2 _ N1) 12 xa E Aeq xt,2 .

ado,L(K)x = e1-N2)12 5e2-N1)/2 ,

Our next example describes a few irreducible submodules of the module Aext' 2• q

Û (S12)

Example 1. A direct computation shows that the elements 1/2 r 1/2

:=

1/2 r -112 := a2 q

± -N2/2

q

of Aeqxt ,2 satisfy the relations , E,\ 1/2

r 1/2

ado,LkE ir -1/2 - 1/2' adtp,L(E)r1// 2 = O ' ado,L (A

1/2

1/2

1/2

)r i/2. = r_ 1127 ado, L (.1. )r 1/2

This means that r il/i22 and r

=

1/2

07 adtp,L(21.)r +1/2

q

+1 1/2

r+1/2 •

2 form a basis of an irreducible submodule 71112

of the Ûq (s12)-module Aeqxt ' 2 . Similarly, 4/2

r 1/2 .= -a

(N1+1)/2

-1/2 :=

aiq —(N2+1)/2

are basis elements of another irreducible Ûq (s12 )-submodule 71112 of Using the Clebsch-Gordan coefficients of the algebra ei/q (s12 ) from Sect. 3.4 one can find basis elements of polynomials of r 2 and 7' 11 /12/2 transforming

under any representation 11, t E -1N0 , of Ùq (512). For instance, the element 1/2,.1/2 ro := q1/2 r 112 r _ 1/ 2

q -1/2 7,1/2 4/2 1/2 r 1/2

- LIV

.1 V2]

is invariant under the representation ad o,L , that is, ado , L (a)rg = E(a)r8 for all a E Ùq (512). Further, the three elements rI

1 /2 7111/2 / 2 — 1 ii(N1-N2V2a±a2 r 1/2• 1 7 1/24/2 := r 112 r 71/2 =

1 TO

—

(,-1/2 ,1/2 4,4/2 1/2 1 -1/2

1

0-21q

„I/2..1/12/241 11/ 2 ) -

(Ni-N2V2 a + 1 a2

=

[2]

frr-N2-i

form a basis of an irreducible Ûq (s12)-submodule T1 of Airt12 .

+.3/4 [N2])7 A

140

5. The q-Oscillator Algebras and Their Representations

5.1.4 The q-Oscillator Algebras and the Quantum Space

Mo (2)

Let Aext ,4 denote the 4-fold tensor product of the q-oscillator algebra At defined in the preceding subsection. The generators of this algebra Ar,4 are denoted by aii ,ati ,q Nii/ 2 7 q -Nii/ 2 7 where i, j = 1,2. Recall that the algebra 0(Mq 2 (2)) is generated by elements ul, ul,u?,14 satisfying the relations (4.1) and (4.2) with q replaced by q2 .

Proposition 6. There exists an algebra automorphism Aeqxt,4 such that

: 0(Mq2(2))

-y(u1) = a4-0 (N12-N11+N21+N22)1 2 , ,01) = ail-2q (N21-3Ni1-N12+N22)/2

'y(u)

/-y(74) = a-21-0.(N12- 3N,1 - N21+N22)/ 2 a2+2q (N11-3N12-3N21-N22)/2 ± / 1 _ (N22-N11-N12-N20/2 k

-y (Dq - q 3/2 q -(Ni2+N2i+1),,+,,,+ '11'22

q2 )4a2allq

)

,3/2,(Nii+N22+1),+,+

'21'12'

Proof. The proof is given by direct computation. We omit the details.

0

There is also an algebra homomorphism : 0(S.L q (2)) —> Ag such that u(u1) --= a+,

v(14) = q -N

v(u 21) = —q- N-1 , v(4) = (q —q- 1)q-N-la.

5.2 Representations of q-Oscillator Algebras 5.2.1 N-Finite Representations

We shall use the bra and ket notation as is common in physics. If T(N) is an operator on a vector space V and w E C, then the symbol 1w) denotes a vector from V such that T(N)lw) = wiw). Suppose that V is a vector space and T(a), T(a) and T(N) are three operators on V such that the eigenvectors of T(N) span V and the relations [T(a),T(a+)Jg = q-T(N)

T(N)T(a) = T(a+)(T(N)+ I) ,

(16)

T(a)T(N) = (7 1 (N. ) + I) T (a)

(17) are fulfilled on V. Here the operators q±T(N) are defined by q±T(N)1w) q±wIw) when T(N)Iw) = wiw). Then it is clear that the operators T(a), T(a), and T(q±N ) := q±T(N) satisfy the relations (3) and (4) of the algebra A. Hence these operators define a unique algebra homomorphism T of Ai' to the algebra L(V) of linear operators on V, that is, T is a representation of sit: on the vector space V. Definition 3. A representation T of the algebra deq on a vector space V of the above form is called N finite. -

5.2 Representations of q-Oscillator Algebras

141

The following observations are crucial in the following. Proposition 7. Let T be an N -finite representation of Aq and let 1w) be an eigenvector of T(N) with eigenvalue w.

(0 Either T(a+)lw) = 0 or T(a+)jw) is an eigenvector of T(N) with eigenvalue w +1. (ii) Either T(a)lw) = 0 or T(a)Iw) is an eigenvector of T(N) with eigenvalue w —1. (Hi) If T is irreducible, then 1w) is an eigenvector of T(a)T(a) and T(a)T(a). Proof. It follows from (16) that

T(N)T(a)w) = T(a+)T(N)Iw) + T(a)1w) = (w +1)T(a)1w), which proves (i). The proof of (ii) is similar. If T is irreducible, then T(cq ) oil for some a c C and hence

T(cq )1w) = q l- w[w]lw) T(a)T(a)1w) (q i-w {to ]

q -2w )1w )

(q - N )T (a)T (a+ )1w) = ajw).

This implies that 1w) is an eigenvector of T(a)T(a) and T(a)T(a).

El

Now let T be an irreducible N-finite representation of Al,' and let 1w) be an eigenvector of T(N) with eigenvalue w. Concerning the actions of the operators T(a) and T(a±)n on 1w), there are three possible cases:

Case 1: There exists a natural number n such that T(a)n Iw) = O. Then the representation T is called a representation with lowest weight. Case 2: There is a natural number n such that T(a+)fliw) = O. Then T is called a representation with highest weight. Case 3: T(a)njw) # 0 and T(a) w) 0 for any natural number n. 5.2.2 Irreducible Representations with Highest (Lowest) Weights

Throughout this subsection we suppose that q is not a root of unity. The next proposition says that the irreducible N-finite representations of A' with highest weights are parametrized by complex numbers. Proposition 8. To every complex number w there corresponds an irreducible N-finite representation Tit of Aqc with lowest weight. It acts on a vector space V with basis elements 1W + m), M E No , and the operators T(N), T u and Tw+ (a) are given by

T.(N)lw + m) = (w +m)iw + rn), n(a+)1w + m) =1

+ m + 1),

(18)

142

5. The g-Oscillator Algebras and Their Representations

where we have set 1w 1) := 0 and [m] denotes the q-number (2.1). Every irreducible N -finite representation of A ge with lowest weight is equivaient to one of the representations T. Two representations T and Ti-DE, are equivalent if and only if qv" = qw . —

n

Proof. One immediately checks that the operators (N), Tit (a+) and T: (a) defined by (18) and (19) satisfy (16) and (17), hence they indeed determine an N-finite representation T: of Age . We have Tut (a) w) = 0, so T: has a lowest weight. Since q is not a root of unity, one easily verifies that the representation T: has no nontrivial invariant subspace. Thus, is irreducible. Since the formulas for the operators 21: (e), T (at) and 21: (a) depend only on qw, Tit and T, are equivalent if qw = e l . Conversely, if qw qw , then the operators Tw+(qN) and T,(qN) have different spectra, because q is not a root of unity. Hence the representations 71 and Tt-DE, are not equivalent. Let T be an arbitrary irreducible N-finite representation of Age with lowest weight on a vector space V . Then there exist a number w G C and a nonzero vector 1w) E V such that T(N)(w) = w I w) and T(a)lw) = 0. We denote the vector T(a±)miw) by 1w + m), m E N. Since aa+ = qa+ a + q - N , we have

n

T(a)lw + m) = T(aa+)* + m — 1) = qT (a+

(20)

+ m 1) + q - (w -E rn -1) lw + m 1) .

In particular, T(a)lw + 1) = q - luiw). We show by induction on m that

T(a)lw + m) = q - u) [m]fw + m

-

1).

(21)

Suppose that (21) holds for m =-- 2, 3, • • • , n 1. Then, by (20), we obtain T(a)(w + n)

q

n-1

,

qW-1. ( 17

-n+1

r , ) T(a)1w + n 2) +

i

( "u1 -En — 1) + n 1)

= q - w[n]lw + n 1),

so (21) holds for any m E N. Thus, the operators T (N), T(a) and T(a) act on the vectors ( w + m) as described by (18) and (19). Each vector ( w + m) is nonzero. Indeed, otherwise Vm := Lin fw+k) k>m)- is an invariant subspace for the irreducible representation T by (21) and Proposition 7(i), so that Vm = V. This is impossible, since then 1w) is not in V. Similarly, we get V = Lin {1w+ k) I k E No l. Since the (nonzero!) vectors 1w +m) are eigenvectors of T(N) for pairwise distinct eigenvalues (by Proposition 7(i)), they are linearly independent. Thus we have shown that T is equivalent to T. {

Proposition 9. To any complex number w there corresponds an irreducible N -finite representation T; of Acq with highest weight. It acts on a vector space V with basis elements iw m), 1TZ E No, by the formulas Tt; (N)I — m) = (tv m)lw m), Tt;

(Ow -

m

5.2 Representations of g-Oscillator Algebras

(a+)Iw - m) =

143

- m 1),

where 1w ± 1) := 0. Every irreducible N -finite representation of A ge with highest weight is equivalent to one of these representations T;. Two such representations Tu, and Tu7, are equivalent if and only if qw = qwl Note that the action of the central element cq of Age is given by

Ti-DE (cq ) =

[wil. and Tv, (eq ) =

[w + 1]./".

(22)

5.2.3 Representations Without Highest and Lowest Weights

In this subsection we assume that q is not a root of unity. Let a and w be complex numbers such that 0 < Re w < 1. Let V be a vector space with basis lw m), m E Z. A straightforward verification shows that the operators T(N), T(a) and T„,,(a) on V defined by

T,w(N)lw + m) = (w + m)iw + m) T, w (a+)lw T, w (a) jw + in) = (aqm + w [m])lw

in) =1w ± in ± 1) , (23) in - 1)

(24)

satisfy the relations (16) and (17). Hence these operators define an N-finite representation Tc,,„ of the algebra Age on V. For the central element cq we get

T„„,(cq ) = Proposition 10. If ar ±q- w[m]

- a)1".

(25)

0 for all in E Z, then the representation

is irreducible and has neither a highest weight nor a lowest weight. Two such representations Tc,w and Tc'' are equivalent if and only if a = a' and qw = qw . Every irreducible N -finite representation of the algebra A qc without highest and lowest weights is equivalent to one of these representations Tctw Proof. The properties of the representations TQw asserted above follow from the formulas (23) and (24). In order to prove the last assertion, let T be an N-finite irreducible representation of Age without lowest weight and without highest weight. If lx) is an eigenvector of T(N) with eigenvalue x, then it follows from the assertions (i) and (ii) of Proposition 7 that the vectors T (a)T Ix) , T(a)Ix), r E No , s E N, are linearly independent. Since T is irreducible, their linear span coincides with V by Proposition 7(iii). Thus, the spectrum of the operator T(N) is simple and formed by the eigenvalues x rrt, where m E Z. There is an eigenvalue w of T(N) such that 0 < Re w < 1. Let 1w) be a corresponding eigenvector. Let lw -1), 1w), » fw±rn), - be eigenvectors for T(N) such that T(N)lw i) (w i)lw i) and T(a+)lw + i) = iw + +1 )1 i = -1,0,1, 2, • -. By Proposition 7(i), there is a complex number a such that T(a)Iw) = afw - 1). Since T has no lowest weight, a 0. Then we get T(Ow +1) = T(a)T(a+)lw) -= 71(qa+ a + q -N )liv) = (qa +

144

5. The q-Oscillator Algebras and Their Representations

Using similar equalities, we prove by induction on m that

11 (a)ftv m)

=

Oar + 11 (el-

= (aqm

r

+

w EmDlw m

1 ),

W

(771-1) ) ) lw

m

1)

for m E No . That is, the operators T (N), T (a+) and T(a) act on the vectors iw m), m = —1,0, 1, • • •, by the formulas (23) and (24). If 1w)' lw) and lw — 1), then T(a+)Iw 1)' := aiw)' and T(a)17D)' := 1w —1Y. Setting lw —m-1)' := T(a)lw —mY m E No , and using the relation a±a = claa+— N we find by a simple induction argument that

T(a+)Iw mY = („q -m+1 = v/q mi - +1

( q -m+2

c m+4

q m-2))1 w m 1)/

q - w[m — 1 ])1w — m + 1Y.

By assumption, the representation T has no lowest weight. Therefore, the preceding implies that iw — m)' 0 for m E Nci and that aqm q- w[m] L 0 for any m E Z. Setting — m)

:= H 2= 0

-w r •

11 - 1

MY,

it is straightforward to check that the operators T(N), T(a) and T(a) act

on the vectors lw — m), m = —1, —2, •••, by the formulas (23) and (24).

111

Summarizing the preceding, if q is not a root of unity, then the above Propositions 8-10 classify all irreducible N-finite representations of the algebra A& up to equivalence. Let us briefly turn to the symmetric q-oscillator algebra Ag . As noted in Subsect. 5.1.1, the algebra Ag is the quotient of the algebra Aqc by the two-sided ideal generated by the central element eq . Thus, the (irreducible) representations of Ag are in one-to-one correspondence to those (irreducible) representations of ..,4& that annihilate eq . A representation of Ag is called N-finite if the corresponding representation of Aqc is so. Proposition 11. (0 Let w + and w_ denote fixed complex numbers such that qw+ = 1 and qw - ±1 = 1. Then the representations I'd- , T.++ , T -1 and Tu7_ are pairwise inequivalent irreducible N-finite representations of the algebra A g with highest or lowest weight. (II) Let w E C be such that 0 < Re w < 1 and [w]qm q- w[m] 0 for all in E Z. Then T[w ],w is an irreducible N-finite representation of A g . Two such —

—

representations TH, w and Ti.lb., are equivalent if and only if qw = qv) . (iii) Any irreducible N-finite representation of A g is equivalent to one of the representations from (i ) or (ii).

Proof. (i): From (22) we see that the representation Tid„- (resp. T;) of Acq maps cq into the zero operator and so passes to a representation of Ag if and only if 1 (res p. q2(w+1) = 1). Since [w] = 0 (resp. [w + 1] = 0) or equivalently

ew =

5.2 Representations of q-Oscillator Algebras

145

two representations Tit and T:, (resp. Ttr, and T.,) are equivalent if and only if qw = qw1 , there are precisely two equivalence classes with representatives V and T:,+ (resp. Tfi. and T;). Two representations Tit and Ty-7, cannot be equivalent, because Tit has a lowest weight and Tuj, does not. (ii): If T„,,(cg ) = 0, then a = [w] by (25). The other assertions of (ii) follow from Proposition 10. (iii): Each irreducible N-finite representation of A g yields an irreducible N-finite representation of A ge that annihilates the element eq . Hence the assertion follows from the fact that the representations of (0 and (ii) exhaust all representations T from Propositions 8-10 such that T(cq ) = 0. CI 5.2.4 Irreducible Representations of ,..47 for q a Root of Unity

In this subsection q is a root of unity. More precisely, we assume that qP = 1, where p E N is odd and minimal. Lemma 12. The elements (a+)P , (a)P , (qN )P and (q -N )P belong to the center

of A. Proof. It follows from the relations (3) and (4) that a(a + r -=-- [n](a+r -l- q -N + q-n(a+ra,

(a + ) 72,q-N _.,_ eq -N( a +)n .

Putting n = p and using the r6lations qP = 1 and [p] = 0, we see that (a+ )P commutes with the generators and so with all elements of A. The proof for aP, (q N )P and (q -N )P is similar. 0 Definition 4. Let T be an irreducible representation of Al' . T is said to be cyclic if T(a) 0 and T((a+)P) 0. If T(a) = 0 and T((a+)P) 0 or if T((a+)P) = 0 and T(aP) 0, then T is called semicyclic.

Let ,u, and be complex numbers and 0. Let V be a p-dimensional vector space with basis 1m), in = 0,1, • • • ,p - 1. We define operators on V by the formulas T(a+)Im) = Im+ 1), 0 < m < p-- 2, T(a+ )lp - 1) = ej0),

T(a)im) = [m + it]Im - 1), 1 < m < p - 1,

T(a)0) = [ith]Cl iP - 1 ),

TAC(q ±i )im) = q ±(m+A) im) and Ti (a)1M) = IM — 1) , .1 < m < p — 1,

T(a)I0) =

II) — 1) ,

T(a)IM) = [M+ ti+1]1M+1) , 0 in Ç p-2, Tt (a)p-1) = [ IAC 1 10 ), Ti/g (q ±)i m ) = q±(m+o)im).

146

5.

The

q-Oscillator Algebras

and Their Representations

Both sets of operators fulfill the relations (3) and (4), so they define representations Ti.h c and Ti'hc of the algebra A. One easily checks that the representations Tlic and Ti.",c are irreducible. Moreover, if ii, 0 and 0, then the representation Ti..,/ c is equivalent to T[mic -1. The formulas (18) and (19) also define a representation of .Aqc on V when q is a root of unity. But this representation is not irreducible. The subspace Vp spanned by the basis vectors ( w + m), m = p, p ± 1, • - • , is invariant. The quotient space V/Vp is irreducible. The corresponding representation operators Tip (q±N ), Tw (a+) and Tip (a) are then given by Till (q±N )(w + m) = q±("+m ) kv + m), T„(allui + m) = iw + m +1) ,

T(a)1w ± m) = q' [m]lw ± m — 1), where we set I w

—

1) 7---- 1w + p) :---= O. -

Proposition 13. (0 Suppose that qP = 1 for an odd nonnegative integer p. Then every irreducible representation of the algebra Acq is p-dimensional and equivalent to one of the representations T4c for p,, E Cy Oy T4 ,c for E C y e 0 1 or T, w E C. The representations Tm , p. 0, e 0, are cyclic. The representations To ,c , # 0, are semicyclic with lowest weight and the representations 7',: e e 0, are semicyclic with highest weight. The representations Tv, have highest and lowest weights. (ii) The representations 2-1,a and 71/,11, 1.1,11! 0, are equivalent if and only if = and q4 --= qia'±ic for some k = 0, 1, 2, • - • ,p — 1. The representations To ,c (and the representations Ti:,c) are pairwise nonequivalent. The , representations Tip and Tie are equivalent if and only if qw = qw . Proof. (i): The stated properties of the representations T, Tck and Ti, are

immediately verified. Let T be an arbitrary irreducible representation of the algebra .,4 7 on a vector space V. Since T maps elements of the center to scalar operators, it follows from Lemma 12 that T(A) is spanned by the operators T(ai(a+)i (e)k), 1. In particular, we conclude that dim T(A) <00 and il i 1 ikl = 0, 1, • • • ,p hence dim V < co. Let q4 be an eigenvalue of T(q N ) and 10) an eigenvector for this eigenvalue. Since (a+)P belongs to the center of Aqc, there is a complex number such that T((a)P) = C - I. First let us suppose that O. Then each vector 1m) := T(a)m10), m = 0, 1, • • • , p — 1, is nonzero. From the first relation of (4) we obtain that 1, are linearly 7 1 (el )1m) = gli+m im) . Hence the vectors f m), m = 0, 1, • • • , p independent. Since their linear span is invariant under T by Proposition 7(iii), it coincides with the carrier space V. From (3), (4), and (6) it follows that the operator T(a) is given by the formulas T(a)(m) ---= rm E plim 1) , 1 < m p 1 and T(a)10) = [p]r 1 lp — 1). Thus we have shown that T is equivalent to the representation Tg. In the case = 0 one proves analogously that T is equivalent to one of the representations 714 ,c or T. —

—

-

-

-

5.2 Representations of q-Oscillator Algebras

147

(ii): Since the values of two equivalent irreducible representations Tig = E. Further, and TAT at the central element ( a+ )P are equal, we get (qN) must coincide, so that the spectra of the operators Tg(q N ) and T = ±k for some k = 0,1, 2, • • • ,p 1. The proofs of the converse and of the remaining assertions of (ii) are straightforward and will be omitted. It is easy to check that all representations Tg and T, annihilate the element eq and therefore give representations of the algebra A g . For the representation Tu, we have Tw (cg ) = 0 if and only if iew = 1. Thus, only Tw with w =0,p/2 gives a representation of A g . That is, we have Corollary 14. Any irreducible representation of A g is p-dimensional and equivalent to one of the representations Tig, T, To and Tp12 .

5.2.5 Irreducible *-Representations of A.qc and Agq

In this subsection we suppose that q > 0 and q 1. Then, as noted in Subsect. 5.1.1, the algebras A and A g are *-algebras with involutions such that a* = a+ and (qN )* = qN . The next proposition decides which of the irreducible N-finite representations of Acq introduced in Subsects. 5.2.2 and 5.2.3 are *-representations. Recall that a *-representation of a *-algebra A is a representation T on a vector space V, equipped with a scalar product (., .), such that (26) (T(s)v,v') = (v,T(e)v 1 ), s E A, v, E V. Proposition 15. (i) T (resp. T) is a *-representation of d4fi if and only if le > 0 (resp. qw < (ii) If 0 < q < 1, then Tay, is a *-representation of Act? if and only if qw > 0 and a(q q-1 ) + q- w 5_ O. If q > 1, then T,„, is a *-representation if and only if qv' < O and ctqw + (q - q-1)-1 O .

Proof. We carry out the proof of (ii). The proof of (i) is similar. Suppose that is a *-representation. Applied to s = qN , (26) means that the operator T(q N ) is symmetric, so that its eigenvalues qw+m, m c Z, are real. If qw > 0, then applying (26) to s = a, v = 1w + m), = lw + m - 1), we find that aqm + q - w [m] > 0 for all m e Z. This is equivalent to the inequality

+

(q

1 ) l trW > W cr 27n (q

q' ) 1 for all

m E Z.

Since q-- w > 0, this is impossible if q > 1. If 0 < q < 1, it leads to the condition ct(q-q -1 )+q- " < O. If qw 1 and ctqw + (q - q7 1 ) -1 < O. Conversely, suppose that 0 < q < 1, qv' > 0 and ct(q - q -1 ) + q- w < O. Then one easily checks that there is a scalar product (., .) on the space V defined by (w + m + 11w + m + 1) = (ctqm + q - w[rn])(w + mlw + m) and (w + nlw + m) = 0 for m n such that (26) holds for the generators of A.

148

5. The q-Oscillator Algebras and Their Representations

Hence (26) is true for all s E Acq and Ta„ is a *-representation. The case q> 1 is treated similarly. Comparing Propositions 11 and 15 we obtain Corollary 16. Tit and Tj_ are the only N -finite irreducible *-representations of the *-algebra Aq .

The operator T(N) is usually considered as a q-analog of the number operator. Hence it is natural to require that it has a real spectrum. Let us call an N-finite *-representation T of Acq or Ag a physical *-representation if there exists an operator T(N) as in Subsect. 5.2.1 with real eigenvalues. and Taw , 01, of Ai' are not, since there is no real w such that qw < O. The *-representation T6E of Aq is physical, while Tuj_ is not. Thus, by Corollary 16, the q-oscillator V algebra A g admits a unique irreducible N -finite physical *- representation T. It is called the Fock representation and will be studied in Sect. 5.3 below.

n

5.2.6 Irreducible *-Representations of Another q-Oscillator Algebra

In this short subsection we list the irreducible *-representations of another q-oscillator algebra without proof. We assume that q > 0 and q 1. Let 13q denote the algebra with generators b and b+ satisfying the relation

bb + — qb+ b =-- 1.

(27)

The algebra Bq is a *-algebra with involution given by b* := 19+ . Clearly, 5 g is a q-deformation of the q-oscillator algebra A defined by relation (1). Next we introduce three families of *-representations of the *-algebra 13q • Let In) denote elements of an orthonormal basis of a Hilbert space. 1 we define linear operators For arbitrary positive q To(b)In) = [[n]] 1/2 1n — 1), To(b)In) = Rn + 111 1/2 1n + 1), n E No, n E N0 1, where I 1)

Lin {In) acting on the vector space V ly1 (see (2.2)). [[n] ] := (qn _ 1)(q For 0 < q < 1 and q < y < 1, we set

—

-

Ty (01n) = -y4. / 2 in

—

1), Ty (b+) In)

= 7n1/F21 in ± 1 )7 n E Z, -

on V := Lin {In) n E Zl, where Fyn := (1 — q) -1 (qnry + For 0 < q < 1 and (3 E C, IfiI = (1 - q) -1 / 2 , we define operators 4(0 = Vo(b+) =fi

on the one-dimensional vector space V

C.

0 and

5.3 The Fock Representation of the q-Oscillator Algebra

149

One immediately verifies that each of these families of linear operators satisfy the relation (27) and condition (26) for s = b,b+ . Hence they define *-representations To , Ty , TA of the *-algebra Bq on V. It is easily seen that these representations are irreducible and pairwise inequivalent. The representation To admits a unit vector 10) such that To (010) = 0. Therefore, To is called the Fock representation of Bg • Clearly, To goes into the Fock representation of the oscillator algebra A as q -> 1, while the representations T-r and TA- have no counterparts for q = 1. In the case q < 1 the operators To (b) and To (b) are bounded and their norms are given by 11 To(b)li =11 To(b)1I = ( 1

0 -112 .

Both operators To (b) and To(b) are unbounded if q> 1. The operators Ty (b) and T(b) are unbounded for any 7 E [q, 1) .

5.3 The Fock Representation of the q-Oscillator Algebra The most important representation of the q-oscillator algebra is the Fock representation. The aim of this section is to study a q-analog of the BargmannFock realization of this representation and coherent states. Throughout this section we suppose that q> 0 and q 1. 5.3.1 The Fock Representation

V of the *First let us recall the structure of the *-representation T algebra Ag . The corresponding representation operators for the generators are determined by the formulas (18) and (19) with w = 0. If we write 1mY instead of 1m) therein and then 1m) := [m}! -- 1/21mY, these formulas go into the equations

T(a)1n) =

- 1), T(a)ln) = \An + 1] in + 1), T(N)In) = 1)• (28)

Let 55 be a Hilbert space with orthonormal basis 1m), m E No, and let 73 be the dense linear subspace of 55 spanned by the vectors 1m). Then the representation T given by (28) is a *-representation of the *-algebra Ag on V. Indeed, it is straightforward to check that equation (26) holds for the four generators of Aq , hence (26) is valid for all elements of the algebra A. This *-representation T is called the Fock representation of the q-oscillator algebra Ag . Each of the other variants of the q-oscillator algebras mentioned in Subsect. 5.1.2 also possesses a Fock representation. For the algebra Ab it is given by

T(b)in) = [[4 112 1n - 1), T(b)jn)

+ 1]1 112 1n + 1), T(N)In) = ni),

150

5. The q-Oscillator Algebras and Their Representations

where [[n]] = (qn 1)/(q - 1) (see (2.2)). The Fock representation for the algebra AA is given by the same formulas if b is replaced by A and b+ by A. The Fock representation of the two-parameter algebra A; takes the form -

71 (et)In)

= [n]g 2 In

1 )7 71 (a)ln) = [n + 1] 1‘,2 1n + 1) T (N)In) = ni),

where [n]gp = (qn - p - n) (q - p-1 ). Let us return to the Fock representation of the algebra A g . It describes a q-deformed harmonic oscillator with Hamiltonian H = (hw/2)(aa+ + a+a), where h and w are as in case of the "usual" quantum harmonic oscillator. The vectors In) are eigenvectors of this Hamiltonian with eigenvalues

E (n) =

hw

hw sinh (v(n + 1/2)) sinh (v/2)

([ 721+ [n + 1)) -= 2

where q = . In the limit q > 1 these numbers give the eigenvalues of the Hamiltonian of the usual quantum harmonic oscillator. -

5.3.2 The Bargmann-Fock Realization

In this subsection we study a realization of the Fock representation of Ag on a Hilbert space of entire holomorphic functions. Recall that bq is the q-derivative given by formula (2.48). For polynomials f g E C[z] we define

(29)

( f, g) = f(A)g ( z) I z =0,

where 7 = En oen,zn for f = En an zn. Since bq zn = [n]z', it follows that the form (., •) is a scalar product on the vector space C[z] such that the polynomials

un (z) =

Zn

n

VV!'

(30)

1, 2„ • • •

a

constitute an orthonormal basis. Let denote the Hilbert space completion of (C[Z], (-, •)). Since {u n } is an orthonormal basis, the elements of are precisely those entire holomorphic functions f(Z) = E nc% an zn on the complex plane for which Enc.° 0 lan1 2 [n]! < 00. In particular, the q-exponential function (z) = Ec° n=0 zn [n]! (see (2.57)) belongs to and satisfies the equation

a

a

E nc.° 0 un (e)un (Z) = :844 (rz).

(31)

The function .4(z) fulfills the inequality itq (z)l

Ç exp izi, z E C.

(32)

For positive values of x, the function Eq (x) is positive, while for negative x it is an alternating series which oscillates within the bounds (32).

5.3 The Fock Representation of the g-Oscillator Algebra

151

It is clear that there is an isomorphism of the Hilbert spaces S5 and -a such that In) is mapped to un . Under this isomorphism the operators T(a), T(a), T(N) from (28) are transformed into the operators d T(a) z, T(N) = z— dz

T(a) =

(33)

on the space -a. This form of the Fock representation of A g is called the Bargmann Fock realization. In the case of the q-oscillator algebra Ab we define a scalar product (•, •) on C[z] by the formula -

(.fl 9) = f(Dq )9(z)iz=o3

where Dg is given by (2.43). Then the polynomials

vn (z) =

Zn

n = 0,1,2,-. • ,

[[1]] • [[2]] • • • Unfl. Let -ab denote form an orthonormal basis, where [[n ]] ! the Hilbert space which is obtained by completing the space (C [z], (., -)). The Hilbert space -ab consists of all functions f (z) = E 77_0 an zn on the complex plane C satisfying the condition E nc° 0 icen1 2 [[n]i! 1 the exponential function n

E [[n]]! 00

eq ((1 — q) z)

7

n=0

from (2.55) is contained in

ab and we have

E: 0 vn (e)vn M = eq ((1 - 17)). Note that for q > 1 the series defining eq ((1 q)z) converges uniformly and absolutely for all finite z. But eq ((1 q)z) has an essential singularity at infinity. For positive values of z, the function eq ((1 q)z) is positive. It is clear that there exists an isomorphism of the Hilbert spaces 5) and ab such that C1n)) = vn , n = 0,1, 2, • • •. Under this isomorphism the operators T (b ), TO+) and T(N) of the Fock representation are transformed into the operators -

d

T(b) = Dg , T(b) = z, T(N) = zcTz on the space

ab.

152

5.

The q-Oscillator Algebras and Their Representations

5.3.3 Coherent States Let z be a complex number. A unit vector Iz) of the carrier space of the Fock representation of A g is called a q coherent state if -

T(a)z)

zlz).

Any complex number is an eigenvalue of T(a) with unit eigenvector CC

zn

iz) = tig (zT)-1/2 E 17-0 = kg (z)/ 2 -'-'q (zT(a))10), [n]!1/2 n.o

(34)

where tig (zT(a+)) := Er7_0 (zT(a+))n/[n]!. The factor kg (z) 1 / 2 in (34) is only taken in order to ensure that lz) is a unit vector of .F. Note that the operator T(a) is not symmetric and that its eigenvectors lz), z E C, are not mutually orthogonal. More precisely, we have (z t iz) = kg(ztz1)-1/244(z)-1/2.

(zz1).

In the case of the Fock representation of the algebra Ab a q-coherent state is defined by T(b)z) = ziz). For arbitrary z, z' E C we have oc

iz) = eqo - q)z )-1/2 E n=0

z n 2 In)

[[n]]! 1/

and

(z i lz) = eg ((1

-

q)z' z") -1 / 2 eg ((1

-

q).z.) -1 / 2 eg ((1

-

q)z).

It can be shown (see, for example, [DKu]) that in the case 0 < q < 1 the q-coherent states of the algebra Ab have the property that

f where

d(z) =

1

(35)

lz)(zidti(z) =1,

eq ((1 - q)1z1 2 )ev q (-(1

q)qjz1 2 )dq lz1 2 de.

(36)

In equation (36), 0 = arg z and dg lzI 2 means the q-integration from Subsect. 2.2.2. The integration in (35) is over 1z1 2 < 'go and 0 < 0 < 27r, where 770 is the smallest pole of eq ((1 q)z) on the positive real half-line. There exists a similar decomposition for the q-coherent states of the algebra A g . But in this case the measure du(z) is more complicated.

5.3 The Fock Representation of the q-Oscillator Algebra

153

5.3.4 Bargmann-Fock Space Realization of Irreducible Representations of Ûq (S1)

In this subsection we compose the Jordan-Schwinger realization (see Proposition 4) with the Fock representation of the algebra Ar7 2 and develop an approach to the irreducible type 1 representations of Ûq (512) in this manner. The Fock representation T of the algebra Ag"t,2 acts on the Hilbert space ..e 2 := 55 0 ..fj with orthonormal basis 1m, n), m, n = 0, 1, 2, .. •, and is determined by the formulas (28). Let /302 denote the span of basis elements 1m, n). The composition 7r := T o (p defines an infinite-dimensional representation of the algebra 6,7 (5 12 ) by linear operators on the space D° 2 : Ûq(S12)

2_,. Aeqxt,2 2i).

(37)

The decomposition of this representation 7r into irreducible components will be given by formula (38) below. In order to achieve this we first note that the basis elements im, n) of 5Y°' can be represented in the form

T(anm T (ann

V[77/1!

10 7 0).

V[n]!

By (9), we have ± T() am T (an n 7r(E)im,n) = T(a i )T(a2) 10,0) V[m ] ! \/[ni!

T(n am + 1 T( 4)n -1 . 1,n - 1). \/[n -410,0) ,_ am ±i ][01/2im + = ([m +1][n])112 -

/[m+1]!

Similarly, we obtain r(F)Irn,n)

= Orni[n+1])1/2

71(anm 171(4)71±1 -

-

10

m

= ([77 ][n + 1]) 1/2 im - 1 7n + 1 ). Thus, for each I E 111 the linear subspace 171 spanned by the basis elements 1m, n) with m+n = 21 is invariant under the representation 7r. Let us redenote m+n, m and n by 21, i+k and 1 k, respectively. Then the invariant subspace V/ of 55 ° 2 is spanned by the vectors -

elk := 11 + k,1- k) =

T(at)/±k 71(a 2E ) 1-k 10,0), k = -1,-1+1,• • • ,1. V[1 + id! -1[1 -id!

The operators 7r(E), 7r(F) and 7r(K) act on these vectors by the formulas

7r(E)elk = ([1 + k +1 ] [1- kj)1/24+1,

154

5.

The q-Oscillator Algebras and Their Representations r(F)e lk = ([1 + k][1 — k + 1])

112eL 1 ,

q k e 1k . That is, the restriction of T to the invariant subspace 14 is equivalent to the irreducible representation T1,1 of (s12 ) from Subsect. 3.2.3. Thus we have ir(K)e lk

oq

(38)

ici No 711 1.

If we consider the Bargmann-Fock realization on the space 5j° 2 , then 171 becomes the space of homogeneous polynomials in two variables z 1 and z2 degree 2 1 . To the basis element eik E V/ there corresponds the monomial of

a,

z i+k z2/-km 1

k p[i _ k ! )1/2

al.

in These monomials are orthonormal in .F° 2 , since the elements eik are orthonormal with respect to the scalar product (29). By (14) and (33), the operators 7r(E), Tr(F) and 7r(K) are realized on as r(E) = zi bs7( 2) , 7r(F) = z2 bv.) , 7r ( K = g (z1a1-z2a2)/2 , )

a,

where the symbol b (i) means the q-derivative bq (see (2.48)) with respect to the variable zi . Note that the operator q(z 181- ' 282 )/ 2 on is well-defined, because N1 - N2 = Z1 ,91 - z2 02 is a diagonal operator on Finally, the results of this subsection will be used to prove the following

a, al.

Proposition 17. The Jordan-Schwinger homomorphism cp : Ù q (S12) -> A ext , 2 from Proposition 4 is an algebra isomorphism of 6,7 (512 ) into Aex q t '2-"tq •

Proof. Since q is not a root of unity, the dual pairing of the Hopf algebras ejq (s12) and C9(SL q (2)) is nondegenerate. From this it follows easily that for any nonzero element X E Ùq (512) there exists an irreducible finite-dimensional representation T1 ,1 of Ûq (512) such that Ti,/(x) O. (In fact, this assertion is a special case of Proposition 7.21 below.) Now let us suppose that v(x) = 0 for i E Ùq (s12). For the representation 7r = To çc of the algebra Ùq (512) defined by (37) we then have ir(x) = O. From (38) we obtain 711 ,/(x) = 0 for all 1 E so z =-- 0 by the above result. El

IN,

5.4 Notes The q-oscillator algebra and its Fock representation- were introduced by L. C. Biedenharn [Bid] and A. Macfarlane [Macf]. The irreducible representations of q-oscillator algebras for q not a root of unity have been determined by P. P. Kulish [Kul] and G. Rideau [Rid]. The description of q-coherent states and the Bargmann-Fock realization of the Fock representation of Age can be found in the paper [DKu]. The Jordan-Schwinger realization of the quantum algebra td; (812 ) and its irreducible representations were given in [Bid]. Other references, as well as further results, on q-oscillator algebras are contained in [BL], [DKul, [ChK] and [Ge ] . Physical applications of q-oscillators are described in [DKu] and [BDKL].

Part II

Quantized Universal Enveloping Algebras

6. Drinfeld—Jimbo Algebras

The aim of this chapter is to define the quantized universal enveloping algebras, called Drinfeld—Jimbo algebras, and to develop basic algebraic structures and results on these algebras such as the Poincaré—Birkhoff—Witt theorem, braid group actions, Verma modules, quantum Killing forms, quantum Casimir elements, centers and Harish-Chandra homomorphisms.

6.1 Definitions of Drinfeld-Jimbo Algebras 6.1.1 Semisimple Lie Algebras In this subsection we collect general notions, facts and notation on semisimple Lie algebras that will be freely used in the rest of the book. Let g be a finite-dimensional complex semisimple Lie algebra. The operators ad X, X E g, acting on g as ad X(Y) = [X, Y], Y E g, define the adjoint representation of g. A maximal commutative Lie subalgebra b in g consisting of semisimple elements is called a Cartan subalgebra of g. Recall that an element X E g is semisimple if the operator ad X can be diagonalized. Let I) be a Cartan subalgebra of g. Since the operators ad H H E b, are semisimple and commute with each other, the Lie algebra g is a direct sum of joint eigenspaces of these operators: g = b IED

a#0 9a7

( 1 )

where a are nonvanishing linear forms on I), called roots, and gc, are the corresponding eigenspaces, called root subspaces. The Cartan subalgebra b coincides with the eigenspace of the eigenvalue O. The set of all roots a is denoted by 46. The dimension of ij is called the rank of g. Let us collect the main properties of roots and root subspaces:

(i) If a is a root of g, then so is —a. There exist no other roots of the form ca, c E C. (ii) If g a and go, a —0, are root subspaces of g and Ea E g c„, .E13 E go, then [Ea , EA E ga±p if a + '3 is a root of g and [Ea , Ep] = 0 otherwise. For a= we have [Ea , Ep] E

158

6. Drinfeld—Jimbo Algebras

(Hi) All root subspaces g„, a E 4 are one-dimensional. (iv) If a + 3 0, then the corresponding root subspaces g, and go are orthogonal with respect to the Killing form B(X, Y) := Tr (ad X o ad Y). (v) The restriction of B(X, Y) to 4 x 4 is nondegenerate. For every root a there exists a unique element Ha E 4 such that B(Ha , H) = a(H) for all H E 4. Let 4R = E aE RHa . The complex dimension of 4 coincides with the real dimension of 4 R . The Killing form is real and positive definite on 4 R • (vi) The Killing form is nondegenerate on g x g. One can choose elements E±, E g±, such that B(E,, E _ a) = 1. Then, [E„, E._ a] = H. We choose a basis H1, 112 • • • H1 of the Cartan subalgebra 4 such that a(Hk) E R for all a E Lt and take a basis vector E, in each root subspace g, such that {Ea , E_,) = Ha . Then the elements Hi , H2,

•

(2)

Hi I Ea , a E

form a vector space basis of g. We have the commutation relations [Hi , Ea ] = a(Hi)Ea , c E 4

[Hi , Hi ] = 0,

{E a , E_ a] = Ha , a ELX

,

[Ec„Eo] = NasEa+s,

( 3) -0,

(4)

where No = 0 if a + (3 is not a root. A set of elements (2) satisfying the relations (3) and (4) is called a Cartan-Weyl basis of g. We fix an ordered basis H1 , 112 1 - • H1 of the Cartan subalgebra 4. A root a E .6i is called positive (resp. negative) if the first nonzero number in the sequence a(Hi ), a(H2 ), • • • , a(Hi) is positive (resp. negative). The set of positive (resp. negative) roots will be denoted by d + (resp. d_). Let Eac , Ra be the subspace of the dual space 4' for 4. If y E then the formula 7(H) = B(H, H-y ), H E 4, defines uniquely an element Hy E 4 R . There is a one-to-one correspondence -y 4-> 11-y between elements from ffR and 4R. We define a symmetric bilinear form (7, 7') = cB(H-y , 7,7/ E FYR , on IYR x WR , where c is a fixed constant. If a E 4 then the formula 2(7, a)

(a, a)

a,

E

4 /R7

(5)

defines a reflection of the space ifR . Clearly, wa = w_ a . The group W generated by all reflections wa , a E 4 is called the Weyl group of the Lie algebra g with respect to 4. W is a finite group which acts transitively on L. A root a E is said to be simple if it is not the sum of two other positive roots. We state some properties of simple roots: (i) Any positive (negative) root is a linear combination of simple roots with nonnegative (nonpositive) integral coefficients. (ii) The number of simple roots is equal to the rank of the Lie algebra Simple roots are g, that is, it is the dimension I of the Cartan subalgebra linearly independent.

6.1 Definitions of Drinteld Jimbo Algebras

159

-

(iii) If a l , a2, • - at are simple roots of g, then the reflections wai , j = 1, 2, • - • , 1, generate the Weyl group W of g. is not uniquely determined. If Note that the set of simple roots of a l , a2, • • • , ai are simple roots, then wai, wa2, - - • , wai for any fixed w E W are also simple roots. In particular, the set -a i -a 2 , • • • , -at (which corresponds to the basis -I/1 , -H2, • - • , -I11 of 4) is a set of simple roots. e n_, where Formula (1) gives a direct sum decomposition g = n+ ga and n_

Proposition 1. The subalgebras n+ and n_ are maximal nilpotent subalgebras of g. Every maximal nilpotent subalgebra of g can be mapped by an inner n+ and + n_ are maximal autamorphisrn onto n+ . The subalgebras solvable subalgebras in g.

Let us fix an ordered sequence a l , • - • , at of simple roots. Let Eai , • • - , be the corresponding root elements of g. By (4), successive application of the commutator [-, -] gives all root elements Ea , a E A + • That is, the elements Ea, , • • , Ea, generate the Lie subalgebra n + . Analogously, the root elements E_ a„ • • • , E_ a, generate the Lie subalgebra n_. The elements i 1, 2, • • • , I, form a basis of the Cartan subalgebra Thus, the complex semisimple Lie algebra g is generated by the root elements Eai , • • • , Ea,, E,,, • , E„, corresponding to the simple roots al , • - • , at. The 1 x 1 matrix A = (a ii ) with entries

= is called the Cartan matrix of g. It determines the semisimple Lie algebra g uniquely up to isomorphisms. It can be proved that (i) det A 0; (ii) aii are integers, aii
We assume throughout that the length (a, a) of the shortest root a is 2. Then di = (a i , ai )/2 and the matrix DA has the entries (DA) ii = (a i , ai ). We also use the symmetric bilinear form (., •) on x f) defined by (1-1,„1-1,,i ) • It is a multiple of the restriction of the Killing form to x 3 1-1 Let us collect the Cartan matrices for the classical simple complex Lie algebras sl(i + 1,C), so(2i + 1,C), sp(2/,C), and so(2/,C): sl(/ + 1, C): a ii = 2, i = = 0 otherwise;

, 1, a,+1 = a i+Li = - 1, ti = 1, 2, • • • , 1

so (2 / + 1,C): as for sl(/ + 1,C) with ct 1 ,1_1 == -1 replaced by

-

= -2;

sp(21,C): as for sl(/ + 1,C) with ai_1,1 = -1 replaced by ai-1,1 = -2;

1,

160

6. Drinfeld-Jimbo Algebras

so(2/,C): as for sl(/ + 1,C) with = ai, t _ i -1 and ai_2,1 = ai,/_2 = 0 replaced by al_Li = = 0 and ai_2,1 = al,1-2 = 1, respectively. -

The corresponding matrices D = diag (d1 , d2 , • • • , di) are

D = diag (1,1, • • • , 1) D = diag (2, • - • , 2, 1) D = diag • • - , 1, 2)

for

sl(/ + 1, C)

and

for

so(21 + 1, C),

for

sp(21, C).

so(2/, C),

The universal enveloping algebra U(g) of the Lie algebra g is the quotient of the tensor algebra T(g) over g by the two-sided ideal generated by the elements sOy-y0s- y E g. The next result characterizes the algebra U(g) in terms of generators and relations (see [Ser ] for a proof). Theorem 2 (Serve's theorem). Let g be a complex semisimple Lie algebra with Cartan matrix A = (aii ) and simple roots a l , a2 , • • -,ai, I = rank g. Then Ei Ecti , F E,„ H := = 1,2, • • H I, can be chosen in such a way that the universal enveloping algebra U(g) is generated by the elements Ei , Fi , Hi , j = 1,2, • • • , l, subject to the relations

= 0,

= Hi ,

Ei ] =

= 0, i

j,

(6)

[Hi Fi =

(7)

1-a ii

(ad Ei ) l

Ei

E (_1)k(1

k=0

(ad Fi ) l- aiiFi

E ( _, )

k

El-ai,-kE.e i2c

k

= 0,

i

j,

(8)

)

- aii) i-a • • -k " Fi Fi

z

( 9)

k=0

The relations (8) and (9) are usually called the Serre relations. Let U(n) and U(r1) be the universal enveloping algebras of the Lie subalgebras n± and f) of g, respectively. Then the map U(n) U(f)) ® U(n_) D n+ 0 H On_ n+ Hn_ E U(g) is bijective. In particular, we have

U(g) = U(n + )U(1))U(n_) = U(n_)U(13)U(n + ).

(10)

The elements E„, a 24, form a basis of n+ and E_,, a i+, constitute a basis of n_. We choose an order /31 > /32 > - • - > in the set of positive roots, where n is the number of elements in 24. Then the elements Eorl • - - Eor: ri E No , form a basis of U(n). The same procedure yields a basis of U(n). The elements Hi" • • • Hp, si E IN0 , are a basis of U(f)). Theorem 3 (Poincaré-Birkhoff-Witt theorem). The elements E' E2

gyinHv. H12

EPnon

EI'2,82 EPA, r i ,si ,pi E No ,

form a basis of the universal enveloping algebra U(9).

(11)

161

6,1 Definitions of Drinfeld—Jimbo Algebras

W i= Since the Weyl group W of g is generated by the reflections Wa 1, 2, • • ,l, any element w E W is a certain product of reflections ?D i , • • • , wi. Such a representation of w is not unique. A decomposition w = w 1 w 2 • • • is called reduced if the number k is minimal. This number is called the length of w and is denoted by /(w). Clearly, /(ww') < 1(w) + /(w'). ,

Proposition 4. (i) There exists a unique element wo of maximal length in the Weyl group W and l(wo ) is the number of positive roots of g. (ii) Let wo = wi l - • win. be a reduced decomposition of 7E 0 . Define roots n =-- l(wo), by setting 01) 021 • • ' wii • • •

wir_i (air ),

r=1,2,•-•,n,

where air are the corresponding simple roots. Then the roots 01, ,[321• • • On are positive and pairwise distinct. They exhaust all positive roots of g. 6.1.2 The Drinfeld-Jimbo Algebras Uq (g)

Let g be a finite-dimensional complex semisimple Lie algebra of rank 1 with Cartan matrix (aij ) and di (c i ,a)/2. Let q be a fixed nonzero complex number and let qi = gdi . Suppose that q? 1 for j = 1, 2, • • • ,I. Let Uq (p) be the (associative unital complex) algebra with 4 1 generators Ei ,Fi , Ki , Ki-1 , 1 < i < 1, and defining relations

Ki Ki Ki ,

Kï l Ki = 1,

= qia" Ei ,KFK= (7

-

E

r..

(-1)r [[ 1

y [[ 1 —

(13)

Fi ,

Ki-1

Fi E, = 6,3 --T aii ]]

(12)

Eil-aii

(14)

-1

0,

i74j,

(15)

0,

i

(16)

qi ]]

r=0

-r

where n r = [

67

[n]q ! [Tiqqn

= q(r-n)r

rlq!

[fl] q = q2

q-1

Note that the relations (15) and (16) are obtained from (8) and (9) by replacing the usual binomial coefficients by the q-binomial coefficients. Setting q --= eh and K i = the relations (12) and (13) are formally equivalent to (7) and the first relation in (6). Proposition 5. There is a unique Hopf algebra structure on the algebra Uq (g) with comultiplication A, counit e and antipode S such that

6. Drinfeld-iimbo Algebras

162

=

Ki , ,A(Ki-1 ) = KT' + 1 0 Ei , Z1(Fi )

06(Ei) = Ei

E(Ki ) = 1, S(K2 ) = Kr1 ,

Proof.

Let

(17)

Ki-1 ,

0 1 + IÇ 1 Fi ,

(18)

E(E) = E(Fi ) =-- 0,

(19) (20)

-Ei Ki-1 S(Fi) = -Ki Fi.

S(Ei )

A = C(Ei , Fi , Ki , Ki7 1 ) be the free algebra generated by E

Fi , Ki , i = 1, 2, • • • , 1. Then Uq (g) is the quotient algebra kJ, where J is the two-sided ideal generated by the elements corresponding to the relations (12)(16). We proceed as in the proof of Proposition 3.4 and define first algebra homomorphisms L : A 4 Uq (9) Uq (g) and E : A 4 C and an algebra anti-homomorphism S : A 4 Uq (g) such that (17)-(20) hold. Then we verify that 4 E and S preserve the defining relations (12)-(16). We carry out this proof for ,A and the first Serre relation (15). That is, we shall show that -

,

-

-

El

—

(-1y

a •

r=0

,A(E0 1—ai i

—T

(Ej)(Ei) r =0.

(21)

qi

Using Proposition 2.2 and the relation (2.17), we easily verify that

_

q27 k(s-k)[[sid]

vs Z—ak=0

Kik E: - k

E2n.

(22)

qi

Therefore, the left hand side in (21) can be represented as r 1—a ii —r

E E E (_, ),gibr. 1— qi

r=0 s=0 t=0 X

(El

1- a ii r

KTE1-5 )(Ej K i ± 1 0 Ei)(Ef

qi

qi

Kzt -aii —r— t

(23)

where b„ := s(r s) - r t). We split this sum into two parts. The first part contains summands with Ei in the first factor and the second one contains summands with Ei in the second factor. Both parts vanish. Let us check this for the second part. It cal; be written as -

1—aii

-

-

r

aii frr 11

E EE rO s=0 t=0

qi LL s qi

x [[1 -a d - r ] 1 (Ef-Fs 0 Kit-FsEir—sEj E23 —r—t \ t

) •

qi

Changing summation to n = s + t and p = r - s and using the identity

[[p +1) s ll fr p+s sil I qi

- -1

where b = 1

-

1-1- s

rr b n- p_ s s 11 = [[ b p- n 1111 nb 1111 :II , -I-1 qi qi LL -I-1 qi I-1- -J] q, LL -1-1 qi -

we can write the expression (24) in the form

(24)

6,1 Definitions of Drinfeld—Jimbo Algebras

b b—n (n (

)8 [LI]]

b-n1

1)pqn(ra,--p-1)

-1

s

n=0 p=0 8=0

P

qi

Er ® Kr, E)i9

qi

b n

163

qi

.

By Proposition 2.4, the interior sum over s vanishes if n O. The summand for n = 0 reduces to (15), hence it also vanishes. This shows that the second part of the sum (23) is zero. The proof for the first part of the sum is similar. This completes the proof of (21). Since 4 E l and S preserve the relations (12)-(16), they pass to algebra homomorphisms resp. an anti-homomorphism of Ug (0) = Al J. By Proposition 1.8, it is enough to check the Hopf algebra axioms on the generators which is easily done. (It is in fact the same verification as in the case of Ug (s1 2 ) Definition 1. The Hopf algebra Uq (g) of Proposition 5 is called the DrinfeldJimbo quantized universal enveloping algebra (or, briefly, the Drinfeld-Jimbo algebra) corresponding to the Lie algebra g and the complex number q.

Remark 1. One may also define the Drinfeld-Jimbo algebra Uq (g) by omitting first the Serre relations (15) and (16) that are unpleasant to deal with. That is, one first verifies that the algebra Ûq (g) with generators Ei , Fi , Ki , Ki-1 and defining relations (12)-(14) is a Hopf algebra with structure maps given by the formulas (17)-(20). Then one shows that the two-sided ideal of Ûq (g) generated by the elements on the left hand sides of (15) and (16) is a Hopf ideal of Og (g). The corresponding quotient Hopf algebra of (I (g) is then the A Drinfeld-Jimbo algebra Uq (0). Let g be a simple Lie algebra of type A1, Di or E. Then the Cartan matrix (au) is symmetric and we have aii = 2 and a ij = 0 or -1 if i j. Hence, (ai , a i )/2 = 1 for all simple roots ai and all qi coincide with q. In these cases the relations (15) and (16) for aii = -1 take the form

Ei2 Ei (q q -1 )

Ei E,

Ei Ei2 = 0,

1 i2 Fi (q q -1 )FiFj Fi FiFi2 = O.

For many reasons (see, for instance, Subsect. 11.5.1) it is convenient to have also the Hopf algebra Uq (g) for g = gl(N, C). Most of the theory developed later for Uq (g) with a semisimple Lie algebra g remains valid with a few minor modifications for Uq (g1 N ). It is defined as follows. The algebra Uq (g1 N ) is generated by elements K.T 1 , i = 1, 2, - • • , N-1, j = 1, 2, • - • , N, subject to the relations

= K K, K Ki-1 = 1Ç 1 K KiEj Ki-1 =

Ki Fi

Ei

EtF3 F3 Ei =

c

=

1,

1=

Fi

K iK i7+11 — Ki-1 Ki+ i

uti

q g- 1

7

6. Drinfeld-Jimbo Algebras

164

Ei Ej --= Ei Ei , Fi Fi = Fi Fi , Ii - jL2, + Ei±i Et? --= 0,

E?Ei±i

(q + F?Fi±1 - (q+

= 0.

The algebra Uq (gl u ) becomes a Hopf algebra with structure maps given on the generators by

Aw n

Ki+1 K, A(E i ) E, K i Ki-+11 + 1 0 Ei (K) = 1,

A(Fi ) = F 0 1 + K 1 Kid-i 0 F, S(K i ) = S(E)

7

(Ei ) = e(Fi ) = 0,

S(Fi )

The element K 1 K2 - • Ky is obviously group-like and central in Uq (g1 N Hence the two-sided ideal (K 1 K2 -KN - 1) is a Hopf ideal of Uq (gI N ). The quotient Hopf algebra Uq (g1 N )/(KiK2 • - - KN -1) is the extended DrinfeldJimbo algebra Urt(s1N) used in Sect. 8.5. Moreover, the subalgebra of Uq (g1 N ) generated by the elements Ki Ki-+11 , Ei and Fi is a Hopf subalgebra that is isomorphic to Uq (s1N ) with Hopf algebra isomorphism 71 determined by 71(Ki Ki-+11 ) = Ki, 7r(E) = Ei and 7r(Fi ) = F, j= 1,2, - ,N - 1. Let us return to the Drinfeld-Jimbo algebra Uq (g) for a semisimple Lie algebra g. Let ai, • be the simple roots of g and let ).

- -

Q = {Ei ni ai I ni E Z } and For every form À =

Q + = {Ei niai E Q j n, E No}.

Ei ni ai E Q we define

(25)

an element

1(111 KL12 • -

(26)

of the algebra Uq (g). From (13) one derives that

KAFiK A--1 = q -(A ' ) Fi.

(27)

Proposition 6. Let p be the half-sum of positive roots of g. Then we have

82 (a) = K2p aK2-pl ,

a E Ug (9).

(28)

The antipode S is a bijective mapping of Uq (g).

Proof. Since the maps a -> K2paK2-pi and 52 are both algebra homomorphisms of Uq (9), it is sufficient to prove the equality (28) for the generators of Uq (g). This is done by direct verification. Obviously, (28) implies that S is injective. Since a = S(K 2pS(a)K2-p1 ) by (20) and (28), S is surjective. In the case g =-- sl(2,C) the Drinfeld-Jimbo algebra Ug (g) is just the Hopf algebra Uq (s12) studied in Chap. 3. We now introduce the counterpart to the second quantum algebra Ûq (512). Let Ûq (g) be the algebra with generators E • , F Ki , K 1 j .= 1,2, • • • , 1, and the following defining relations: ,

,

Ki Kj = Ki Ki , KiKÏ 1 = Ir1 Ki =-- 1,

(29)

6.1 Definitions of Drinfeld-Jimbo Algebras

K i Ej lCi-1 = qia " i2 Ei , Ki FiKr 1 = qi7 a12/2 Fi , Ei F3 F3 Ei

= 8K A. If?

Kr2

1

qi — qi

El 1 - a,..1 ] r=0 (-1)1[ r 1Fil-a" -r Fj Fir = 0, E

(30) (31)

)

—aii i = __0 1 (-1) r [[ 1 Trazi ]] Eil a "— 1- Ej E r=--0 q.

165

2_ , ..,4 1

(32) (33)

q.

The algebra Ûq (g) becomes a Hopf algebra with structure maps determined on the generators as follows:

,6(1Ci ) = Ki 0 Ki , ,6(1C1 1 ) = Ki-i 0 Ki-i, = Ei 0 Ki + Ki-1 ® E, ,6(Fi ) = Fi 0 Ki + Ki-1 ®F, E(K i ) = 1,

6-(E1 ) = E. WO = 0,

S(Ki) ---= I-Cïl , S(E) = -qi Ei S(Fa ) = -qi-l Fi .

(34) (35) (36) (37)

Similarly as in the case of s12, there is an injective Hopf algebra homomorphism cp : Uq (g) -+ Ûq (g) such that

cp(Ei ) = Ei Ki , (p(Fi ) = Ki-1 Fi , cp(If i ) = IC? . That is, Uq (g) can be considered as a Hopf subalgebra of Ûq (g) if we identify X E Uq (g) with cp(X) E Oq (g). Note that the algebras Uq (g) and Ûq (g) are not isomorphic, because they have different numbers of one-dimensional representations. The quantum algebra Ûq (g) for g = gl(N,C) will be defined and studied in Sect. 7.3. For some investigations it is convenient to treat q as an indeterminate and to work with the field 12(q) of rational functions in q over the field Q of rational numbers. Replacing in the above definition of Uq (g) the base field C by Q(q), we obtain a Hopf algebra over the field K = Q(q). It will be denoted by N(g) and called the rational form of the Drinfeld-Jimbo algebra. 6.1.3 The h-Adic Drinfeld-Jimbo Algebras Uh(g) The Drinfeld-Jimbo algebras Uh(g) are h-adic Hopf algebras over the ring C[A]. Recall that h-adic algebras and Hopf algebras were defined in Subsect.

1.2.10. Let g, (aij ) and di be as in Subsect. 6.1.2, and let h be an indeterminate. Let Uh(g) be the h-adic algebra over the ring C[[h]] generated by 31 elements Ei, Fi, Hi , i = 1,2, • - • , 1, subject to the relations

[Hi , HA = 0,

[Hi , Ei } = aii Ej ,

[Hi , G. =

(38)

166

6. Drinfeld-Jimbo Algebras

Ei Fi -F,Ei = 6,3 e [[1

r=0

dl..

dl..

aii ]]

Z-dr=0

E

dihili —

(39)

'

0,

(40)

qi

(_1)r

[[ 1 - ai1]1

qiFil —aii — r Fi Fir 0,

(41)

where qi := edih . Proposition 7. There exists a unique h-adic Hopf algebra structure on Uh (g) with cornultiplication d : Uh(g) -> Uh(10Uh(g), counit e and -antipode S such that d(Hi ) = 01 -1-1 Hi , (42) A( Ei ) = (43) F edihHi + 1 Ei7

A(Fi) =

01 +

E(Hi ) = e(E) = E(Fi ) = 0,

(44)

S(H i ) = -Hi , S(E) = _Eie-dihHi S(F i ) = —edihHi Fi.

(45)

Proof. The proof is similar to that for the Hopf algebras Uq (g) in Subsect. 6.1.2. Definition 2. The h-adic Hopf algebra Uh(g) is called the h-adic DrinfeldJimbo algebra. One can also define the h-adic Hopf algebra Uh (g) for g = gl(N,C). It is generated by elements Ei, Fi, Hi , j = 1, 2, - • , N - 1, j = 1, 2, • • • , N, subject to the relations H31

0,

[Hi ,

(6,,3

6,,3+1 )E3 ,

[11i,F31= (-6ij 6i ,j+i)F3 , [Ei ,Fj ] = 613

e -h(11.-11.+1)

eh — e —h

and the same Serre relations for Ei and Fi as in the case Uq (g1 N ) with q := eh. The Hopf algebra structure of Uh (gil ) ' is determined by = H®1 + 1 =

Hi ,

A(Ei ) = Ei e h—Hi + 1) + 1

1 + e -h(li1-Hi+ 1)

E(Hi ) =

Ei ,

= E(Fi ) = 0,

S(Hi ) = -H1 , S(E) = -Ei C h(Hi -Hi+ 1) , S(Fi ) = Remarks: 2. The element edih e—dih E C{[N] is not invertible in C[[h]], because its constant term is zero. But the expression on the right hand side of (39) is a formal power series p7,(1/1)hn with certain polynomials pn (Hi ), so it is a well-defined element of the h-adic algebra generated by E1 ,111 ,111 .

En

3. The algebra Uq (e) is formally derived from the h-adic Drinfeld-Jimbo algebra Elh (g) if eh and edihHi are replaced by q and K1 , respectively.

6.1 Definitions of Drinfeld-Jimbo Algebras

167

4. The constant terms in the defining relations (38)-(41) give just the defining relations (6)-(9) for the universal enveloping algebra U(g) according to Serre's theorem. Hence there is an isomorphism

Uh(9)1hUh(g) U(g) of complex algebras. This fact is sometimes expressed by saying that the 0 of the h-adic algebra Uh(g). algebra U (g) is the classical limit h 5. Let us adopt the following notational convention in the h-adic case. For a E C, we write (la for the formal power series e ah E C[[k] and qicilli for the element eadihHi c uh (9 ) . Proposition 6 extends at once to the Hopf algebra Uh (g). That is, we have

512 (a) = e ll2Phae-H2 P h ,

(46)

a E Uh(g)

di ni Hi if 2p = Ei niai. where H2p = As an abstract h-adic algebra, Uh (g) is isomorphic to U(g){{h}j, see Subsect. 1.2.10 for the latter notion. More precisely, we have Proposition 8. There is an isomorphism ço of the h-adic algebra Uh(g) onto U(g)[[h]j which coincides with the identity map modulo h. Further, if cp and (p' are two such isomorphisms of Uh(9) onto U(g)[[h]j, then there exists an invertible element F E U(g)[[hll such that F 1 (mod h) and ço'(a) Fc,o(a)11-1 for a E Uh(g)•

Proof. See [Dr2] or [ShS1, Chap. 11. We give the explicit form of such an isomorphism, denoted by ço, for the algebra Uh(s12) (see [Jim2]). Recall from Subsect. 3.1.4 that Uh(s12 ) is the h-adic algebra generated by elements H, E, F such that [H, E] = 2E, [H, F] = -2F and [E, 1 ] = (ehH _ e -hH)1( e h e -to) . The isomorphism ço of the algebras Uh(s12) and U(s12)[[h]] over C[[h]] is uniquely determined by its action on the generating elements and is given by

ço (H.) = H', ço(F) = F', ço(E) = 2

(cosh h(H' - 1) - cosh 2h [H' -1]2 - 4C' sinh2 h

')

E'.

Here H', E', F' are the generators of U(s12 ) satisfying the relations [H', = 2E', [H' = -2F', [E', = H', and C' = (1/' - 1) 2 + E' F' is the Casimir element of U(s12). 6.1.4 Some Algebra Automorphisms of Drinfeld-Jimbo Algebras

The algebra automorphisms of Uq (812) from Proposition 3.3 have the following counterparts for general Uq (g). Proposition 9. Let c = (Ell • 'El), Ei = ±1, and a = (ai, - • • , ai E C I ai 0. There exist algebra autornorphisms 0 and V ec, of U(g) such that

168

6. Drinfeld-Jimbo Algebras

0(Ei ) = Fi , 60(Fi ) = Ei , 0(K i ) = KT 1 , 1.9„r (Ei ) = az Ez ,

1.9„r (Fi ) = Ez a27-1 Fi , t9,,,r (Ki ) =

Proof. The proof is given by direct verification. Other algebra automorphisms of Uq (g) will be given in Subsect. 6.2.2.

Proposition 10. Let a be as in Proposition 9. There exist automorphisms 0 and 19 c, of the h-adic algebra Uh (g) such that

0(Ei ) = F ,

0(F2) = Ei , 0(Hi ) = -Hi , ai-1F, 19 a (Hi ) = H.

z9 a (Ei ) = ai Ei , 0a(Fi) There eTists a C-algebra automorphism

0(E) = F ,

0(Fi ) = Ei ,

(47)

of Uh (g) such that

OHO = -Hi ,

O(h) = -h.

(48)

There is also a C-algebra anti-automorphism cl; of Uh (g) such that

w(E) = F ,

Lu(Fi ) = Ei ,

co(Hi ) = Hi ,

(.4.0) = -h.

(49)

Below we shall use the Q-algebra automorphism V) of N(9) defined by

0(Ei ) = Ei , 0(Fi ) = F ,

(.1(i ) Ki-1 ,

O(q) =

(50)

6.1.5 Triangular Decomposition of Uq (g) In this subsection we develop an analog of the decomposition (10) for Uq (g). Throughout we assume that q is not a root of unity. Let Uq (n+ ), Uq (n_) and Uq (15) be the subalgebras of Uq (g) generated by the elements E1 , - - El, the elements F1 , • • • , Fi and the elements H1 , • • • , H1 , respectively. The subalgebras generated by E1, - - • , El, H1 , • • • , H1 resp. F1 , • • • , F1, H1 , • • , H1 will be denoted by Uq (b + ) resp. Uq (b_). From (17)-(20) we see that Uq (E)), Uq (b + ) and Uq (b_) are Hopf subalgebras of Uq (g). The action of the elements KA, A E Q, on Uq (g) by conjugation determines a Q-gradation of the algebra Uq (g) and its subalgebras. A nonzero element a E Uq (g) is said to be of degree a = niai , ni E Z, if If),a1VA-1 = for all A E Q. If the element has degrees a and )3, then q() = q( )0) for all A E Q and hence a = 0, because we assumed that q is not a root of unity and the form (-, -) on Q is nondegenerate. Thus, the degree is well-defined. Further, if a is a monomial in the generators with ni factors Ei and mi factors Fi , then it follows from (27) that a is of degree E i (ni - mi )ai . Since such monomials span Uq (g), we have a direct sum decomposition

Uq (g) = (1310EQ Ue(g),

Pqr3 (g) = fa c Uq (g) J KA aKA-1 = q (A,A) a }. (51)

169

6.1 Definitions of D.rinfeld—Jimbo Algebras

Since U(g) • U(g) C U 1 (9), the preceding shows that Uq (g) is indeed a Q-graded algebra. This gradation induces obviously Q-gradations on the subalgebras Uq (n+ ) and Uq (n_.) as well. In particular, we have

U Ug ( LE ) = ED0EQ+:

(n) + , Uq (n_) = ED0EQ+ U 13 (n_),

(52)

where UP(n±) = {a E Uq(n±) I 1‘),a1Ç 1 = q±(A J3) al. Lemma 11.

- - • EP'

(53)

0 for every 1-tuple (n i , • • • ,ni) E

Proof. Let ço be the algebra homomorphism of Uq (g) to Uqi (s12) defined by = 1, = E, ço(Fi ) = F, cp(Ki ) = K and p(E) = c,o(Fi ) = 0, 0 by Proposition 3.1, Ell 0 for any n E N. i. Since c,o(E) = En The algebra Uq (g)®n is Qn-graded. Let d ( r ) be as in Subsect. 1.2.2. From

(18) we easily get dea -1) (Ei ) = Er117,31 1®iØ Ei K: . From this formula it follows that the component of degree (niai, • • niai) of the element 46 (1-1) (4 1 - • • Er) E Uq (g) ®/ is a nonzero multiple of 4 1 0 42 Ifill Ø 4 3 1C111 1q2 ® • • • 0 EtK 1 • • • K' This element is nonzero, since Ell O. Hence EP • • E

.

O.

1=1

Lemma 12. The elements K yl -y E Q, form a vector space basis of Uq (4). Proof. Suppose that E, a-y Ky = 0, where the sum is finite and a..y E C. Using the left adjoint representation adi, of Uq (g) (see Subsect. 1.3.4) we find

adL (E a.7 1C-r )(EP • • • Erl)

-•

-y

.

E 1 = O.

E,y

Since Einl • - • EiN 0 by Lemma 11, we have wy q(7 'Eniaz) =-- 0 for all n i , • • • , ni E N. Since the roots a l , • • • , al form a basis in the dual space 4/ and q is not a root of unity, it is not difficult to show that the latter is only possible if all a..y vanish. Thus, the set {KO is linearly independent. LI From the definition of Uq (n+ ) it is clear that it has a vector space basis consisting of certain monomials EiT • • • Eini , where i i E {1,2, • • • , /} and ni E No. Let us fix such a basis and denote it by {E, r E I}, where I is an appropriate set of indices. Lemma 13. The elements ET K.

Q, form a basis of Uq (b +).

Proof. These elements generate Uq (b + ). We have to show that they are linearly independent. Let a -a ar-y ErK.7 = O. Without loss of generality we assume that all summands of a are of the same Q-degree 0. The summand of degree (0,0) in A(Ear-r Ei.K.y ) must be O. This implies that E ar-y Er K7 --= O. The latter equality can be written as

Ey (L I a"Er lf-y ) Kolfy = O.

170

6. Drinfeld-Jimbo Algebras

By Lemma 12, the expression in the parentheses vanishes. Hence we get ar/ E, = 0 for every 7 C Q. Since {Er I r c I} is a basis of Uq (-n + ), all LI coefficients ary vanish. Let 0 be the algebra automorphism from Proposition 9. We set Fr := 19(Er ). Since Fr Ky := 9(ErK_ I ), it follows from Lemma 13 that the set {Fr Ky I r c I, fy c Q} is a basis of Uq (b_). Theorem 14. The elements Er Ky Fri, r, r' E I, E Q, form a basis of the vector space Uq (g). The mapping Er g Fr/ —> E r IC-y Fr/ gives an isomorphism of the vector spaces Uq (n+) Uq (b) Uq (n_) and Uq (g).

Proof. Since the algebra Uq (g) is spanned by the elements Er Ky Fri, r, r' E I, C Q, it suffices to prove that the elements Er FriKy are linearly independent. Suppose that a finite sum a E arr / Er Fr/Ify is zero, where a rr., / E C. If Er and Fr/ are of Q-degrees A r and —AT!, respectively, then Er Fri.K.7 is of Q-degree — A ra. Without loss of generality we may assume that all summands of a are of fixed Q-degree — A r,. We introduce an order in Q. If a = ni ai and a' = then we write a < a' if E ni n and ni < n'i for the smallest index i such that n i n'i . Let J be the set of r C I for which the degree 7, of the term E r in the sum a is maximal with respect to this order. Clearly, the sum of all terms in ,(a) which have maximal Q-degree in the first tensor factor and minimal Q-degree in the second one must vanish, that is, -y

E EJ,r'EJ'

IC-r,,Fra)(K-1

= 0,

where 7, and -yr, are fixed. Hence Ery,rEJ,r / E.» arr'/( E r iCy 0 Fr' K y ) 0. Since the sets {FriKy } and {E r Ky } are linearly independent, we get a„ , ,y = 0 for r C J and r' E J'. Replacing a by a — arri-t (ErFrIK-y) and repeating this procedure it follows that all coefficients arri,y are zero. D Applying the automorphism 9 to the basis elements in Theorem 14 we obtain the following assertion: The elements Fr ifEr,, r, r' E I, C Q, are also a basis of the vector space Uq (g). The mapping Fr oKy ®Erf —> Fr Ky Erd defines a vector space isomorphism of U q (n_) Uq (1)) Uq (n+) and Uq (9). Remark 6. Recall that Uq (n+ ), Uq (n_) and Uq (b) have been defined as subalgebras of Uq (g). Using the approach sketched in Remark 1 it can be shown that Uq (n+ ) (resp. Uq (n_)) is isomorphic to the algebra with generators E 1 ,. ,E1 (resp. F1 , ... ,F1) and defining relations (15) (resp. (16)) and that the elements Ky , -y E Q 2 form a vector space basis of Uq (1)). These facts hold for any q E C such that q? 1. For such q C C the multiplication map gives vector space isomorphisms of Uq (n+ ) Uq (i)) Uq (n_) and Uq (n_) Uq (b) Uq (n+ ) to Uq (13). However, if q is a root of unity, then the decompositions (51) and (52) are no longer direct sums.

6.1 Definitions of Drinteld-Jimbo Algebras

6.1.6 Hopf Algebra Automorphisms of

171

Uq (e)

In this subsection we suppose that q is not a root of unity. The classification of Hopf algebra automorphisms of Uq (g) is derived from

Uq (g) be a coalgebra homomorphism. Then Lemma 15. Let .11) : Uq (9) there are elements y, G Q + and finite subsets 'il, 'i2, fi, fir2 of the set I from Theorem 14 such that 71b(Ki ) —

,

'CEO —

rti)(Fi ) = where

ar bS 7 Cr

E

E

arE,

E sE/i2

7

(55)

rEJii

ds E C. For r

E

I

(54)

Uf1 and

s

E /2,2 UJ2

we have

E r E Ug"ti (n + ) and F, G Uq- "li (n_ ).

(56)

Proof. By Theorem 14, the element 1/)(Ei ) can be written as a finite sum E ar -yr iE r K7 F r, with ar,fri E C. Since is a coalgebra homomorphism, we have A((E i )) = o A4(E). Using (17) and (18), we expand A(ti)(E i )) and (000) oA(Ei ) in terms of the basis elements Er .K.,Ye. Comparing coefficients we conclude that the term 0(Ei )07/)(Ki ) appearing in the expression op 0) o A(Ei ) must be of the form given by (54) with Er and F, as in (56). Formula (55) is proved analogously. 0 Let A = (a ii ) be the Cartan matrix of the Lie algebra g. A permutation fi of the set {1, 2,• • • , 1} of vertices of the Dynkin diagram of g such that = aii , j , j = 1,2, • • • , 1, is called a diagram automorphism of g. From the explicit form of Dynkin diagrams it is easily seen that the Lie algebras B1, 1 > 2, and C1, 1 > 2, have no nontrivial diagram automorphisms. The Lie algebras A1 and DI, 1 > 4, have precisely one nontrivial diagram automorphism p and p has order 2. The Lie algebra D4 has nontrivial diagram automorphisms of orders 2 and 3. Theorem 16. Let e l , c2, - • • be nonzero complex numbers and let II be a diagram automorphism of g. Then there exists a unique Hopf algebra automorphism V) of Uq (g) such that Y

i) = K12(i), 7P(Ei ) = ci .E12 (i),

CFO

ci7 1 Fii (i).

Every Hopf algebra automarphism of Uq (g) is of this form. Proof. It is easy to verify that each such rti) is a Hopf automorphism. In order to prove the last assertion we apply Lemma 15. Let be a Hopf algebra automorphism of Uq (0). Since 1/) is invertible, the element -yi satisfying = b(K) must be a simple root a p (i). Moreover, p defines a permutation of the roots in the Dynkin diagram of g. By (54), IP(Ei ) is of the form ai.Eii(i) bi.K.y,Fii( i). Since is also an algebra automorphism, we have 1/)(Ki)IP(Ei)C.Ki r 1 = egEi). This equation implies that bi = O. The assertion concerning 0(Fi ) is derived from (55) in a similar manner.

172

6. Drinfeld—Jimbo Algebras

6.1.7 Real Forms of Drinfeld—Jimbo Algebras As discussed in Example 1.10, real forms of the complex Lie algebra g correspond to Hopf *-structures on the Hopf algebra U(g). By a real form of a Drinfeld—Jimbo algebra Uq (g) we mean an involution on Uq (g) such that the Hopf algebra Ug (g) becomes a Hopf *-algebra. As for Uq (512 ), we distinguish three domains of parameter values. Case 1: q E R. In this case the real forms of Uq (g) are described by Proposition 17. Let ti be a diagram automorphism of g and let ci , i = 1, 2, • • , 1, be nonzero complex numbers such that tt2 = id and cp,( i) = There exists a real form of Uq (g) with involution determined by IC: = KA (i) 7 Ei

Fi*

Ci 1

(i)

p (10 .

Two real forms defined by the data ( 1a, c) and (fil, e) are equivalent if and only if there exists a diagram automorphism v such that vkl = jiv and um > 0 for all i satisfying p(i) = Proof. The proof is given by a direct verification. A closer look at the equivalence conditions shows that up to equivalence there are only the following standard real forms:

K: = K1., (i) , EiK = criK p (i)Fti ( i) ,

Fi* = cri E j(i) K ja(1.0 ,

(57)

where p is a diagram automorphism of g such that p 2 = id, ai = 1 for p(i) and ai = ±1 for tt(i) = i. The real form of Uq (g) defined by (57) with p = id and cr = 1, i = 1,2,• • • , 1, is called the compact real form. The compact real form of Uq (s1,,) is denoted by Uq (sun ). Putting p = id in (57) we obtain a real form of Ug (sin ) denoted by Uq (su(Gri, • • .) ai)). Case 2: q = 1. Proposition 18. Let 1u be a diagram automorphism of g and let e i 1,2, • • • , 1, be nonzero complex numbers such that p 2 = id and ci,( i) = 1. Then there is a real form of Uq (g) with involution determined by ,

K: = Kp (i),

E = ci Ei.,( i) 7 Fi* =

F

If q is not a root of unity, then two real forms defined by the data ( 1u,ci ) and (p', c) are equivalent if and only if there exists a diagram automorphism v such that up = jilt'. The equivalence conditions allow us to restrict ourselves to the following standard real forms: (i) 7

Ft:* = Fp

7

(58)

6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules

173

where p is as in Proposition 18. For Uq (slri ) and p = id this real form is denoted by Uq (s1,2 (R)). It is an analog of the universal enveloping algebra U(s1(n, R)).

Case 3: q E

q

Proposition 19. Then the Hopf algebra Uq (g) has real forms if and only if = sp(2n,C). In this case, the corresponding involutions are given by

1‘7 = Ki ,

=

Fi* =

where ci are nonzero complex numbers such that ci = (_1)d. Two real forms with data ci and c are equivalent if and only if eici-1 > 0 for all i. In this case we may restrict ourselves up to equivalence to the real forms K i* = Ki ,

E = (-1)d1 siKi Fi ,

of Uq (sp2n ), where si = 1 for even di and s i

Fi* = for odd di .

Using the form of Hopf algebra automorphisms of Uq (g) obtained in Subsect. 6.1.6 it can be shown that the above lists are exhausting. Theorem 20. If q is not a root of unity, then the Drinfeld-Jimbo algebra Uq (g) has real forms if and only if q E R orlqi = 1 or q E JTR g = sp(2n,C). All real forms of Uq (g) are then described in Propositions 17-19. ,

6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules In this section we construct explicitly a vector space basis for the DrinfeldJimbo algebra Uq (g) in terms of general root vectors. The main technical tool for doing that is an action of the braid group of the Lie algebra g as an automorphism group on the algebra Uq (g). 6.2.1 Braid Groups

If A =-- (aii ) is the Cartan matrix of g, then the numbers aij aii may be equal to 0, 1, 2 or 3. Let mii be equal to 2, 3, 4, 6 when aij aii is equal to 0,1, 2, 3, respectively. Recall (see [Bou2]) that the Weyl group W of g can be defined as the group generated by the reflections wi, w2, • • • , w (corresponding to the simple roots of g) satisfying the defining relations 14. = 1,

= 1, 2, • • • , /, .wiw i wi wi • • • =-- wiwiwiwi • - • , i

(59)

where in the latter equations there are mii w's on each side.

Example 1. The Weyl group W of sl(/ + 1,C) is the permutation group Pi+iIt is generated by elements wi, w2, - • - ,wi with defining relations

174

6. Drinfeld—Jimbo Algebras

7.4 = 1, wiwi+iwi =

and wi wi wi wi if — ji > 1.

A

Definition 3. The braid group associated with g is the group 9.3g generated by elements s i , • - • , si subject to the relations SiSiSiSi

• • • =

SjSiSi Si

(60)

• - -

s's on each side.

where there are

Example 2. The braid group 93g for g = sl(/ + 1, C) has 1 generators and the defining relations

s i si±i si = si±i si si±i and s i si = si s i if

ji > 1.

This group is usually called Artin's braid group on 1 strands.

A

Clearly, a braid group contains an infinite number of elements, because the elements s E Z, are pairwise distinct. The correspondence sri' n, n E Z, j= 1,2, - 1, defines a homomorphism from 939 to Z. If J denotes the normal subgroup of q3g generated by the elements 4, = 1, 2, • , 1, then the quotient group 93g IJ is isomorphic to the Weyl group W of g. Thus, we have a natural homomorphism cp from 93g onto W such that cp(si) = w i . The braid groups 93g have the following important property. ,

Proposition 21. Let w = wi 1 w i2 • - • wi,, k = 1(w), be a reduced decomposition of an element w E W. Then the element s t, = si1 si2 • • • si, of 93g depends only on w and not on the choice of reduced decomposition for w. Proof. The proof is given in [Bou2], Chap. 5, § 1.5, Proposition 5.

111

6.2.2 Action of Braid Groups on Drinfeld—Jimbo Algebras

The importance of the braid group Sg in the present context stems from the fact that it acts on the algebra Uq (9). Theorem 22. To every i, i = 1, 2, - • there corresponds an algebra automorphism Ti of Uq (g) which acts on the generators H E Fi as

—Fi K i ,

Ti (Ki )

T(Ej) =

i

a,3 E

Ti(Fi ) =

r=0

E

(-1raii q

r=0

11;(Fi ).=

r(Ei )(— aii—r)Ei ( Ei )( r),

(_iraii,r(Fi ) (r) Fi

—r) 7

(61) i

j, j1

(62) (63)

where

=

l[n] qi !,

(Fi ) (n)

The mapping si --+ 7; determines a homomorphism of the braid group 93 0 into the group of algebra automorphisms of Uq(g).

6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules

175

Proof. The proof is given by direct and lengthy calculations. First one shows T(F) given by (61)-(63) satisfy the definthat the elements Ti (Ki ), ing relations (12)-(16) of Uq (g). Thus, there exists an algebra automorphism

Ti as stated above. Then one verifies that the automorphisms 7; satisfy the defining relations (60) of the braid group B9. Hence the map s i -+ 1; defines 1=1 an action of 93 9 on Uq (g). The braid group 939 acts also as a group of algebra automorphisms on the h-adic Drinfeld-Jimbo algebra Uh(g). The corresponding algebra automorphisms Ti are given on the generating elements Hi , Ei and Fi , j = 1, 2, • - 1, by the formulas (62) and (63), with qi replaced by e dz h and by

Ti (Hi )= H -

Z(E) =

_FiedihHi Z(F)

e—dihHiEi.

(64)

Clearly, this braid group action induces the classical action of the Weyl group . W on the subalgebra Uh (1)) of Uh(g) which is generated by H1 , • 6.2.3 Root Vectors and Poincaré-Birkhoff-Witt Theorem

In the universal enveloping algebra U(g), there are root elements for every root of g. In Uq (g) we have so far only root elements Ei and Fi corresponding to the simple roots of g. Using the braid group action on Uq (g), we shall define root elements of Uq (g) for arbitrary positive and negative roots of g. Let wo be the longest element of the Weyl group W of g and let wo = , ai be the wi , wi, • - • w ir, be a fixed reduced decomposition of wo. Let a l , simple roots of g. Recall that by Proposition 4 the sequence =

Wii (Cti2), '

/372 = W il • • • W in-1

(ctiTi)

exhausts all positive roots of g. The corresponding root elements of Uq (g) are obtained by Definition 4. The elements

Efir

=

- • • Tir_ i (Eir )

and

Fpr

Ti1Z2 ' • • Tir- I (Fir)

( 65 )

from Uq (g) are called root vectors of Uq (g) corresponding to the roots Or and -A., respectively. Root vectors for the h-adic algebra Uh (0) are defined similarly. The root vectors of the universal enveloping algebra U(g) can be defined in the same way using the Weyl group W instead of the braid group. Then different reduced decompositions of wo give, up to signs, the same root vectors of U(g). This is no longer true for the root vectors (65) of Uq (g).

Example 3. For the Lie algebra g = sl(3,C) there are precisely two reduced decompositions wo = w1w2w1 and wo = w2wiw2 of the longest element wo in W. The corresponding sequences of positive root vectors of Uq (s13) are

176

6. Drinfeld-Jimbo Algebras

E1,

TI(E2)

E2 7

1-2(E1) = — E2E1

— E1E2 + Q-1 E2E11

T1T2(E1) = E27

E1E2,

T2 71(E2) =

Note that Ti(E2) is not proportional to T2(E1).

A

Proposition 23. (i) If w E W and ai is a simple root such that w(c) E L , then T(E) E Uq (n+) and Till (Fi ) E Uq (n_.). All root elements Ef3, and For ,

r =1,2, • • • n, from (65) belong to Uq (n+) and Uq (n_), respectively. (ii) If *xi ) = ai is a simple root, then T(E) = Ei and Tip(Fi ) = F.

(iii) For the root elements Er and For we have If),Eo r KT 1 = q( "r) Epr ,

If ),For lf = q-(43r) F3r .

Proof. The proof is given by direct calculation using the definition of Proposition 4 and the formulas (62), (63) and (65).

3f3 ;

Having root elements in Uq (g) corresponding to all roots of g we can state an analog of the Poincaré-Birkhoff-Witt theorem for Uq (g): Theorem 24. Let Epr , For , r = 1,2, • • • , n, be the root elements from Defi-

nition 4. Then the following set of elements is a vector space basis of Uq (e): • • - For:Kti l • • • Kit ' Ef3s : • • • Ef38117 ri ,si E No , ti E Z.

(66)

Proof. The proof of this theorem can be found in [Ros2] and [Yam] for Ug (sin) and in [Lus] for the general case. Corollary 25. The elements Eroli • • • E

r1, - • • , rn E No, form, a vector space basis of the subalgebra Uq (n+) of Uq (p). Likewise, the set of elements Fka • • • 1114, s,. 7 s i E No , is a vector space basis of Uq (n_). In particular we see that there is a natural one-to-one correspondence between basis elements of Ug (n+) and U(n± ). It can be shown by using Theorem 24 that the algebra Uq (g) has no zero divisors, that is, we have ab 0 in Uq (g) if a 0 and b 0 (see [DC1(]). An analog of the Poincaré-Birkhoff-Witt theorem holds also for the hadic algebra Uh(g) • Theorem 24'. Let E , For , r = 1, 2, • • - n, be the root elements of the

algebra Uh(g). Then the set of elements Fr1 [3].

on lit' • • • 1111 Est'.

• • • F rn

si Si'

ri ti , si E No ,

is a basis of the C{[h]j-vector space Uh(D)• From Corollary 25 we obtain additional information about the decomposition (52) of the subalgebra Uq (ni ). The dimension of the subspace UP (ni ) from (53) is equal to the number of basis elements in Uq (ni) of degree ± 3 . As in the classical case, the latter number coincides with the Kostant partition function K(3). If 3 0, then K (/3) is defined to be the number of partitions

6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules

177

of the linear form f3 into a sum of positive roots of g, where roots may enter into partitions with multiplicities. If = 0, then K(/3) := 0. Thus, we have dim U(n_) = dim U(n) K(/3).

(67)

6.2.4 Representations with Highest Weights Let T be a representation of the Drinfeld-Jimbo algebra Uq (g) on a vector space V. For any function A on the root lattice Q we set VA=IXEVI T(K,)x = A(a)x for all a E Q1,

• KT 1 for a = where K = vial. That is, each VA is a joint • eigenspace of the commuting operators T(Ki ), j = 1, 2, - • • , 1. Definition 5. If VA {0}, then we say that the function A is a weight, the number n-tA = dim VA is the multiplicity of the weight A and VA is a weight subspace of the representation T. The nonzero vectors in VA are called weight vectors. A representation T of Ug (0) is called a weight representation if its underlying space V decomposes into a direct sum of weight subspaces. Weight representations are the most important representations of Ul (g). If q is not a root of unity, then every finite-dimensional representation of Ug (9) is a weight representation (see Subsect. 7.1.1 below). Definition 6. A weight representation T of Uq (g) on a vector space V is called a representation with highest weight if there exists a weight vector E V such that T(K,)eA , = A'(a)eA , , a E Q,

T(E)en' = 0 for i = 1, 2,

, 1 and T(lq (g))eA , = V.

(68)

We then call the function A' on Q a highest weight and the vector eA , a highest weight vector of the representation T. Since Ug (n t ) contains the unit element 1, the first condition in (68) implies that T(Uq (n + ))e iv = Ceiv. The last formula in (68) means that the vector eA0 is cyclic for the representation T. From (68) we easily derive the relations V = T(Ug (n..4)T(UA))T(Uq (n + ))eA , = T(Uq (nAeiy,

(69)

V = E0EQ+ T(Elg- °(n_))eAl.

(70)

Remark 7. The definition of highest weight representations depends on the positive roots and so on the corresponding ordered sequence a l , a2 , - • - , ai of simple roots of g. As noted in Subsect. 6.1.1, we may also take the ordered sequence of simple roots -a i , -a2 , - , -al. In the latter case the elements Ki-1 , Fi play the role of the elements Ki ,Ei in Definition 6. That is, T is a highest weight representation with respect to a l , a2, • • , ai if and only if T o 0 is a highest weight representation with respect to -a1, -a2, • • • I -cti!

6. Drirdeld-Jimbo Algebras

178

where 0 is the algebra automorphism of Ug (9) from Proposition 9. Moreover, A'(a) is a highest weight of T if and only if A'(-c) is a highest weight of T o O. All results of Subsects. 6.2.4-6 and Chap. 7 remain valid (under appropriate reformulations) for highest weight representations with respect to -a l -a 2 , • • , -ai. We shall use this setting in Subsects. 8.4.1, 11.2.3, 11.5.3 and 11.6.4 below. A In the rest of this section we suppose that q is not a root of unity. Proposition 26. Let T be a representation of Uq (g) with highest weight A' on a vector space V. Then the sum (70) is a direct sum and gives the weight subspace decomposition of the space V. We have

T(t.1,7 (n_))e A l ,

where

,u(a)

A'(c).

(71)

All weights of the representation T are of the form pt(a) = ni E No and ai are the simple roots of g.

wher

Proof. Let s E U(n_). By (53), we have

T(K)T(x)e n ' = T(Ifa)T(x)T(K,V)T(If cr )eAr = q

(a)T(x)eAl ,

that is, T(x)en' is a weight vector with respect to the weight q-( ',43)A'(a). Because q is not a root of unity, the sum (70) is a direct sum. Since all forms E Q+ in (70) are sums of simple roots with nonnegative integral coefficients, all weights of T are of the form stated in the proposition. El

A representation of Uq (g) is called a representation of type I if it has

a highest weight of the form Ai(a) =-- q(), where A is a linear form on the Cartan subalgebra 13 of g. We shall prove in Subsect. 7.1.2 that any irreducible representation is a tensor product of a representation of type 1 and a one-dimensional representation. Therefore, we shall restrict ourselves in this chapter to representations of type 1. If T is a representation of type 1 with highest weight A'(a) = q (') , then (with a slight abuse of language) we call A also a highest weight of T and denote T by TA. The corresponding highest weight vector eA , will be denoted by e'A- . By Proposition 26, for any type 1 representation TA we have a direct sum decomposition V = vA_0 , where

_o71 (Uci-13 (11_))eA = {X E V j 71 (1Ca )x =

c, E Uq (4)}. (72) , Note that the functions q(11-0 ) and q (A— /31 ' c') on Q do not coincide if 13 because q is not a root of unity. VA

6.2 Poincaré - Birkhoff-Witt Theorem and Verma Modules

179

6.2.5 Verma Modules The formulas (67) and (72) imply that for a type 1 representation TA with highest weight A on a space V one has the inequality

(73)

dim VA_13 < K(3), where K is the Kostant partition function (see (67)).

Definition 7. Let TA be a type 1 representation of Uq (9) with highest weight A on a vector space VA. The corresponding Uq (g)-module VA is called a Verma module if (74) dim VA_o = K (fi ),

, where VA_o is the weight subspace (72). A Verma module for all fi E with highest weight A is denoted by MA and the corresponding representation of Uq (g) by TT . It follows from (67), (72) and (74) that the map Uq (n_) x —) (x)eA E MA is bijective. That is, a Verma module is freely generated by the action of the algebra Uq (n_) on the vector eA. Any Verma module MA can be realized directly on the vector space Uq (n_). In this case, the unit 1 of this algebra is a highest weight vector and the representation operators are given by the formulas

E (4()),

(75)

i = 1,2,.. •,

(76)

Ff3 E tV(n_),

(77)

TX (K,)1 = q (A 'a) 1, rrV

fr'\1 = Fi ,

TX (Ei )1 = 0,

TX (K)F0 = q (A) Fp,

TX(Fi )Fp = Fi Fp,

(Ei )Fp = Ei Fp,

F, , E Uq-13 (n_).

(78)

Let us explain why the right hand side of the last equality belongs to Uq (n_) and is well-defined. Since the element Ei Fs is contained in Uq (n_)Uq (11)Uq (n+ ), it is a sum of products x_hx +,x_ E Ug (n_), h E Ug (i)) , X + E Uq (n+ ). To every such summand there corresponds a vector of the underlying space Uq (n_) of the representation TX. Namely, if the degree of x + is nonzero, then this vector is TX(x_hx + )1 = O. If the degree of x + is 0 and x_hx + E Uq (n_)Uq (0), the corresponding vector is TX(x_h)1 = TX(x_)TX (h)1. This vector is uniquely determined by the formulas (75)-(78).

Definition 8. A linear form A on i) is called integral if the numbers (A, an, = = 1,2, • • • , 1, are integers. The set of integral linear forms is denoted by P. A linear form A on 4 is called dominant if (A, ai ) > 0 for i = 1,2, • • • ,l. The set of integral dominant linear forms is denoted by P.

Verma modules of Uq (g) have similar properties as in the classical case. Let us describe some of them. Let A E P be such that (A, an > 0 for fixed j E {1,2, • • , / } and put ni := (A, an. Let MA be a Verma module

180

6. Drinfeld—Jimbo Algebras

with highest weight vector eA . Using the relations (14) and repeating the reasoning of Subsect. 3.2.1 we find that

Ei F

l eA = 0,

EiFeA 0,

EiFin l +l en = 01

The vector eA, F+ l eA has the weight A' = A — (ni + 1)ai . We have

TX(Uq (9))eAt TX(Uq (n_))TX(Uq (1)))/I(Uq (n + ))eA, TX(Uq (n_))e x , that is, T (Llq (n_.))ex is an invariant subspace and the restriction of TT to TX(Uq (n...))eAt is a representation with highest weight A' = A — (n i + 1)a i . One can see that this submodule is isomorphic to the Verma module MA'. Thus, we have shown that if a Verma module MA has a highest weight > 0, then it has a Uq (g)-submodule isomorphic to E P such that (A, the Verma module MA'. Moreover, we have (A', an < 0. If A E P+ , then such a submodule MAI exists for every i = 1,2,- • • ,l. In the latter case we have the following stronger result. Proposition 27. Let MA be a Verma module with A E P. Then for every element w of the Weyl group W of g there exists the Verma submodule in MA with highest weight A y, = w(A + p) — p, (79)

where p is the half-sum of all positive roots of g. Every irreducible Uq (g)module contained in MA determines a type 1 representation with highest weight. These highest weights are of the form (79). Proof. The proof of this proposition is analogous to that of the corresponding III assertion in the classical theory [Dix]. 6.2.6 Irreducible Representations with Highest Weights

A Verma module MA has nontrivial Uq (9)-submodules in general. It can be proved that if MA is reducible, then there is always a unique maximal proper Uq (g)-submodule M of MA. That is, any other proper Uq (g)-submodule of MA is contained in M. We denote by LA the representation of Uq (g) determined by the quotient module MA IM. Proposition 28. (0 LA is an irreducible representation. (II) If TA is an irreducible type 1 representation of Uq (g) with highest weight A, then TA is equivalent to LA. In particular, for any linear form A on the Cartan subalgebra I) of g there exists, up to equivalence, a unique irreducible type 1 representation of Uq (g) with highest weight A. (iii) If A E P+ , then the representation LA is finite-dimensional.

Proof. (i) is clear by the mwdmality of M. (ii): Let VA be the underlying space of the representation TA and let eA be its highest weight vector. By Proposition 26, VA is spanned by the vectors TA (Fii. - • - FijeA, n E No. It is easy to see that the operators TA(X),

6.2 Poincaré-Birldioff-Witt Theorem and Verma Modules

181

X E Ug g), act on these vectors by the formulas (75)-(78). Hence the linear mapping T: Uq (n_) -0 VA given by T(Fil • • • Fin ) = TA(Fi i - - • Fir )eA intertwines the representations TT and TA. Since T(M) is an invariant subspace of VA and TA is the irreducible representation of Uq (g) with highest weight A, we have T(M) = 10, so that TA LA. WO: Let i be one of the numbers 1, 2, • • • , 1. As noted before Proposition 27, MA has the submodule MA', A' = A- (ni +1)ai , since A E P± by assump(

tion. The set of weights (counted with multiplicities) of the quotient module MA/MA , is invariant with respect to the element w E W corresponding to the simple root a i . Indeed, the multiplicity no, of a weight A in MA/MA, coincides with rn'Al rnl,;:, where rnI)1, (resp. mn is the multiplicity of A in the module MA (resp. in MA , ). The multiplicities of weights for Verma modules of Uq (g) coincide with the corresponding multiplicities for Verma modules of the Lie algebra g. Therefore, the multiplicities of weights in the Uq (g)-module MA/MA , coincide with those in the quotient MA/MA , of the corresponding Verma modules for g. In the classical case, the set of weights of the module MA/MA , is invariant with respect to the element wi of the Weyl group W. Therefore, this invariance holds in the quantum case as well. Since MA contains the submodule MA I, A' = A - (ni 1)a 1 , for any i = 1, • • • , 1, the set of weights of the representation LA is invariant under the whole Weyl group W. Since MA (and hence LA) has only a finite number of dominant weights, it follows that the set of all weights of LA is finite, that is, LA is finite-dimensional. Proposition 29. Let LA be the representation from Proposition 28 with = mi ai highest weight A E P± and highest weight vector e. Let

be an element from Q+ such that (A, c') > mi , i = 1,2,- - • ,l, and let VA MAIM be the carrier space of the representation LA. Then the linear mapping Uq-13 (n_) D VA(x)e is injective. Proof. By the definition of a Verma module, it is enough to show that 3 is not a weight of the maximal Uq (9)-subnaodule M of MA. This follows from Proposition 27, because no set of weights {Ai, niai ni E Nob w E W, contains 0. D 6.2.7 The Left Adjoint Action of Uq (g)

Recall that by Proposition 1.14 (i ) the left adjoint action adL(a)b =

E a(1)bS(a(2))

defines a representation of the algebra Ùq (g) on itself. In the classical case, this representation is the adjoint action ad(X) b = Xb - bX, X E g, b E U (g), and it decomposes into a direct sum of irreducible finite-dimensional representations over the center of U(g) (see [Dix], Subsects. 2.3.3 and 8.2.4). The last statement is not true for Drinfeld- Jimbo algebras.

182

6. Drinfeld—Jimbo Algebras

An element b E Ûq (g) is called locally finite if the space adL(Ig (g))b is finite-dimensional. Let F be the set of all locally finite elements of t/i g (g). It is obvious that F is a vector space which is invariant under the representation adL and that F decomposes into a sum of finite-dimensional adL-invariant subspaces. Since adL(b)cd = E adL (b( i ))c - adL(b( 2))d, the set F is a subalgebra of 6(g). Throughout this subsection we assume that q is not a root of unity. Our aim is to describe the algebra F. We give a brief exposition for the case Ûq (s12 ) and state the main results for the general case without proofs. Let Tq (r)) denote the multiplicative group of the algebra 69.(rj), that is, Tq (I)) is the group of elements KA, A E Q. Let us begin with the case g = s12 . From the formulas (3.12) and (3.13) for the comultiplication and the antipode of Oq (512 ) we obtain that

ad L (E)a

Ealf -1 — qlf -l aE, adL(F)a = Falf -1 — q -1 1f -l aF,

,

adL(K)a = KaK for any a E

adL(K -1 )a

If -l alf

(80) (81)

6q (512 ).

Proposition 30. The set .F C Ûq (512) is not trivial and

(7q (512).

Proof. Let b = EK. Since adL(E)b = 0 and adL (K)b = qb by (80) and (81), b is a highest weight vector of a weight subrepresentation of adL . A direct calculation shows that adia (F3 )1) = 0, so that b E F. Similarly, c= FK E F. Thus, F is not trivial. Using (80) we easily find that

fl adL(E n )E = (1 - q -2i+1)En±ilf-n. fl

i=1

Therefore, since q is not a root of unity, dim ada/q (s12 ))E = co and so E g _F. 1=1 Proposition 31. S := Tg ()n.F= {K2"

r C No}.

Proof. From the first formula of (80) we derive that ad L (En)Ifs=ll . -A-Lz=1

— qs -2"-2 )En

Therefore, if s 2N0, then we have 1 q8-2i+2 0 for all j E N and hence dim adL(Ûq (sl2 ))K = a), so that 1-(3 F. Suppose now that s = 2r E 2N0. Then the above formula implies that adL(Er+ 1)K2r = 0. Similarly, adL(F 1 + 1)K2r = 0. Further, by ( 81 ), (K-n)K2r = K2 for any n E Z. From the preceding facts and foradL mula (3.5) (more precisely, the corresponding formula for Ûq (s12 )) it follows that the vector space adaq (s12 ))1f2r coincides with the span of elements adL(EiF i ) 2--j < r 1. Thus, If 2r E 1- for r E NO. .1 2r ,

6.2 Poincaré - Birkhoff-Witt Theorem and Verma Modules

183

Let S-1 := fs -1 i s E SI. Clearly, 88 -1 is a subgroup of the group Tq (4). By Proposition 31, the coset space Tq (4)188-1 consists of two elements and we have Tg (4) = SS -I- + SS -I- K -I- . Since EK and FK belong to .F (see the proof of Proposition 30) and K -2 E 8 -1 , FK -1 and EK -1 lie in .FS-1 . Since .F is an algebra and s -i , { K -2r I r E No }, the set .FS -1 is also an algebra. Therefore, the polynomial algebras C[FK-1 ] and C[EK -1 ] are contained in .F8-1 . Hence C[Flf'] • 88 -1 • C[EK -1 ] C ,FS— 1 . Since C[FK -1 ] - T(13) • C[EK -1 ] = tTigf(s12) , we conclude that Ùq (512) = .F8-

1 ± .Fs -1 K -1 .

This equality shows that although the algebra .F does not coincide with Ûq (812) it is nevertheless rather large. Note that 5 -1 n.F contains only the unit element. An explicit description of the set .7. is given by Proposition 32. (i) For any r c No, the set ,F(r) := adL(Ûq (s12))K 2r is an adL-invariant vector space of dimension (r +1)2 with a basis formed by the elements (FK - 1 )mK 2r(EK -1 )n, 0 < m,n
(ii) .F = ED7 0 .F(r) 7---- adL(Ûq (s12 ))S. Proof. (i): Using (80), (81) and the formula (3.5) for the algebra 0q (512), a direct computation shows that the repeated action of the operators adL(E), adL(F) and adL(K) on K2 ' gives all elements (FK - 1 )m K2 r (EK — 1 )72, 1 0 < m,n < r, and that the span of these elements is adL -invariant. D (ii): See [J14. Now we turn to the corresponding results for an arbitrary Drinfeld-Jimbo algebra Og(g) . By (34), (35) and (37), the formulas (80) and (81) then remain valid for any a E Ùq (g) if one adds the index i to the elements E,F,K,K -1 and to the parameter q therein. Let (,4, be the isomorphism from the additive group Q = EL Zai to the multiplicative group Tq (4) defined by ço(ct i ) = K. Theorem 33. Suppose that q is not a root of unity. If .7. is the set of locally finite elements of idlq (g), then we have:

(i) S := Tg (i)M.F = Iço(a) I a E 4P+ n Cn. (ii) The set 88' is a subgroup of TA) and the coset space TAV88 -1 is finite. If K 0 , i = 1,3 2, - - - , m, is a complete set of coset representatives for SS -1- in TA), then Ùq (g) = Eai .FS-1 K(i) . (iii) .7- = adL(Ûq (9))S.

Proof. The proof of these assertions can be found in [JL] or [Jos].

El

184

6. Drinfeld—Jimbo Algebras

6.3 The Quantum Killing Form and the Center of Uq (g) Except for Proposition 34 and Lemma 35 we assume in Subsects. 6.3.1-4 that q is not a root of unity.

6.3.1 A Dual Pairing of the Hopf Algebras Uq (b + ) and Uq (b_)

The following bilinear form (., •) is the key ingredient for the construction of the quantum Killing form on Uq (g) in the next subsection. Proposition 34. There exists a unique dual pairing (-,•) : Ug C of the Hopf algebras Uq (b +) and Uq (b_)°P such that

(K i , Ifj ) =

(at

bii (Ei ,K i ) =0, (Ei , Fi ) = _1

(Kj ,

,ct 2)

xUq (b_)

(82)

qi — qi for j j = 1,2, • •• ,I. Moreover, we have (51(a), S(b)) = (a, b) for a,b E Uq (g) and ,

(aK)„bK 1,) =

(ab),

(a, b) =0 for

for

A,p E Q, a E Uq (b +), b E Uq (b_),

a E U:(b+), b E Uq—(3 (b_), a,

E C2 +, a O.

(83) (84)

Proof. The uniqueness assertion is clear, since any dual pairing of bialgebras is determined by the values on the generators. We prove the existence of the pairing (•, We define linear functionals j 7Pi E Uq (b..4 1 , i := 1,2, • - , 1, by ,

= (q77 1

q —( a"") e(a), a E Uq (n_), 71)i(alf) = 0

for p E Q,

aE

(n_),

qi )

ai .

Then one verifies that the elements cioi and V), of the algebra Uq (b_)' satisfy the defining relations of the generating elements of Uq (b + ) (see Remark 6). Hence there exists an algebra homomorphism 10: Ug (b + ) —> Uq (b_)' such that 410(Ki ) =- cpi and fro(Ei ) = We now define (a, b) := 0(a)(b) for a E Uq (b +), b E Uq (b_). By construction, (82) is fulfilled and we have

(aa' , b) = (COP (a'))(b) =

sto(a)(b (i ) )0(a') (b(2)) =

(a, b(1))(ai l b(2)).

(85) Next we prove that (a, bb') =

(a(i) , bi) (a(2) , b),

It is straightforward to

a E Uq (b+), b,b' E Uq (b—).

(86)

check that (86) holds for a = a1a2 provided that it holds separately for a l and for a2 (and arbitrary b and b'). Similarly, (86) is valid for b = b1b 2 when it is valid for bi and for b2 (and arbitrary a and b'). Hence it suffices to check (86) on the generators which is easily done. By (85) and (86), the bilinear form (., -) satisfies the condition (1.31) for LI = Uq(b+)

6.3 The Quantum Killing Form and the Center of Uq (g)

185

and A = Uq (b_)°P. Since 4P(1) = E by definition, (1.32) also holds. Hence (•,•) is a dual pairing of Uq (b + ) and Uq (b_)°P. LI The formulas (83) and (84) can be verified by induction. Lemma 35. For a E Uq (b+) and b E Uq (b_) we have

ab = E(S(a ( 1) ),b( 1 ) )(a (3) , b(3))b(2)a(2) ba =

(a ( i) ,

(87)

5

(88)

b(1))(S(a(3)), b(3))a(2)b(2)•

Proof. Again one checks that the formulas are valid for a = a 1 a2 (resp. b = b1b2 ) when they hold for al and for a2 (resp. for b1 and for b2 ). Thus it suffices to prove these formulas for the generators. We omit these verifications CI (a sample will be carried out later in the proof of Proposition 8.13). Each of the relations (87) and (88) means in fact that the Drinfeld-Jimbo algebra Ug (9) is a quotient of the quantum double of Uq (b + ) and Uq (b_) (see Subsect. 8.2.1). We shall state this fact later as Corollary 8.15. In what follows we assume that q is not a root of unity.

Lemma 36. If an element a E Uf3 (b_) satisfies the relations Ea = aEi for i = 1,2,• • -, l, then we have a = O. If Fi b = bFi , i = 1,2,• • • , l, for some

bE U:(b+), then b = O. Proof. Let LA be the irreducible representation from Subsect. 6.2.6 with highest weight A E P+ such that the conditions of Proposition 29 are satisfied for 0. Since L A (Eia)e = LA (aEi )e = 0, i =-- 1,2, • • • , l, for a highest weight vector e of the representation LA I the vector LA(a)e generates a proper subrepresentation of LA. Since LA is irreducible, LA(a)e = O. Hence a = 0 by Proposition 29. In order to prove the second assertion, we apply the automorphism 9 from Proposition 9 to the equation Ea = aEi . Proposition 37. For any 3 E Q +, the restriction of the bilinear form (•,.)

from Proposition 34 to r.»q (b +) x Uq-0 (b_) is nondegenerate. Proof. We have to show that (b, a) = 0, a E Uq-0 (b_), for some b E 1./43 (b + ) implies that b = O. This will be proved by induction with respect to the usual ordering of Q +. If is a simple root, then it is true by (82). Let > O and suppose that it holds for all -y E Q+ such that E Q+. Fix b E U:(b + ) such that (b, a) = 0 for all a E Uq-0 (b_). By (83), it is sufficient to assume that b E N3 (n+ ). By the formulas (17) and (18) for the -

comultiplication we can write

AO) =

EOCry<0

b..(1

K-y ),

E U:(n + ) UrY(n+ ),

where bo = 1 b and bo = b 1. Let E Q ± , 0 < and y E U47-- /(b_). By (1.31) and (83), we have

0 = (b,sy) = (A (b ), y

x) = (1)1,(1 K y ), y

(89)

< s x) =

y

s).

(90)

186

6. Drinfeld-Jimbo Algebras

Since for any if' < # the restriction of (•,•) to tr.:(b + ) x Ur(b_) is nondegenerate, so is its extension to a bilinear form on

[V:(b + ) Ufq3-7 (b + )] x Hence it follows from (90) that b.). = 0. Because of (89) this means that = b K0 + 10 b. Applying the formulas (87) and (88) to Fi b and bFi + 1 0 1 0 b, b Ko Ko + 1 b and using the expressions ,i(2)(b) , (2) (Fi )= Fi 0101+Ki-1 0Fi 01+KÏ1 0Ki-1 0Fi , we derive that Fi b = bFi for i = 1,2, • , 1. Thus, by Lemma 36, b = 0. Similar reasoning shows that (a,b) = 0, a E Ut:(b + ), for b E Uq-- f3 (b__) implies that b = 0. In Subsect. 8.3.2 the h-adic counterpart to the above bilinear form (•, .) will be needed. Replacing formally Kr by ehd,H, and q by eh in the relation (Ki , Ki ) = q - (.1) of Proposition 34, we obtain ( e hdi Hi e hdi Hi ) e —h(cl i' aj ) . In order to get this equality we should define hdi di (Hz , Hi ) = But, since h is not invertand so h(Hi , Hi ) --= -(ai , ai ) ible in the ring C[[h]i, there is no value (Hi, Hi) E C[{hl] such that the latter holds. In order to circumvent this difficulty, we shall work with the h-adic Hopf algebra Oh (b_) := C[[h]]• 1 + hU h(b _) instead of Uh(b_). The elements

f1 :=hH1 and

Pi := hFi ,

= 1, 2, - • • ,

belong to Oh(b_) and satisfy

= fii 0 1 +1 OA, The element e -d i fii = 1 + di h) k Ht is also in Oh (b_). Note that ehi i is not in the h-adic subalgebra of Uh(g) generated by Proposition 38. There exists a dual pairing (•,•) : Uh (b + )x0h(b....)

C[[h]]

of the h-adic Hopf algebras Uh(b +) and Oh(b_)°P such that fts ) = di I asi , (113 ,P.,) f or i,

(E3 ,k1 )

0, (E,„ Pi ) - e

h

e d, h

= 1,2,• •-,/.

For the study of the Drinfeld-Jimbo algebras Uq (g) and Uh (g) as quotients of quantum doubles in Sect. 8.3, we shall also use the bilinear form (-, -)/ on Uq (b + ) x Uq (b_) resp. Uh(b + ) x Oh(b....,) defined by

(a,b)' := (S(a),b). Clearly, the form (-, •)/ is then a dual pairing of the Hopf algebras Uq (b + )°P and Uq (b_) resp. of Uh(b+) OP and Oh(b_ ). From the formulas for (., •) we get

(Ki , Kg )' = q(ai'ai) , (K s , Fi )' = (Es , K1 )' = 0, ( 111 ,

fii)/

PO' =

(Es fii )/ ,

(Es , Fi )' =

= 0, (E1, P')'

qi - qi 6is h

= edih —

6.3 The Quantum Killing Form and the Center of Uq (9)

187

6.3.2 The Quantum Killing Form on Uq (g)

Let

A be a Hopf algebra and let p be a representation of A on a vector space

V. We say that a linear functional f on V is invariant with respect to p if f (p(a)y) = E(a) f (y) for all a E A and y E V. A bilinear form (-, .) on A x A is called adj,-invariant if the form (•, .) considered as a linear functional on A 0 A is invariant under the representation adj, 0 adj, of A on A 0 A, that is, (ad L (c(1) )a, adi,(c( 2))b) = e(c)(a, b) , a, b, c E A.

E

It is easily seen that the latter is equivalent to the condition (adj, (Oa, b) = (al ad L(S (c))b) ,

a, b, c E A.

(91)

Let us turn now to the Drinfeld-Jimbo algebra Ug (9). From Subsect. 6.1.5 we recall that the multiplication map a0k Ob —p akb of Uq (n_) 0Uq (1)) 0 Ug (n + ) to Uq (g) is an isomorphism of vector spaces. Hence there exists a well-defined bilinear form (., • )u on Uq (g) x Uq (e) given by

(ailf ,bi, a2K (3b2)u := q-(a '3)12 (S (b1), a2)(b2, 8(a1 ))

(92)

for a l , a2 E Uq (n_), b1, b2 E Uq (n + ) and a, 0 E Q. Here (-, .) is the dual pairing of Uq (b +) and Uq (n_)°P from the preceding subsection. Theorem 39. The bilinear form (•1 .)u on Uq (g) x Uq (g) is adL-invariant,

that is, (ad L (a)b i , b2 )u = (b 1 ,adj,(S(a))b2)u, a, b1, b2 E Uq (0).

(93)

Proof. By Proposition 1.14, adj, is a representation of Ug (9). Hence it suffices to prove (93) for elements Ka , ap E U(n), bp E Uq-0 (n_). First let a = K. We may assume that b1 = vK),S(w), v E U q-e (n_), w E Ueq:Y(n + ), and b2 ---= S(Y')K i,w', y' E Uq-w (n_), /D I E U:(n+ ). A direct computation shows that (adL(a)b i ,b2 )u -= (b i , adi, (S(a))b 2 )u = 60 5 6,-y q -( '°-w) (b1) b2)u. That is, in this case condition (93) is fulfilled. Now let a c U. )3 (n._) and let b 1 and b2 be as before. Using (88) we get adL(a)b i =

E amvKA8(w)8(a( 2)) = E a wK,,8(a (2) w)

= E amvICA8(E(w(i),a(2))(57 (w(3)), a(4))w(2)a(3)) = E (WM a(2)) (8(w(3)), a(4))a(i)v.K),S(a(3))8(w(2)). 7

Since a is a product of the generators Fi and

(94)

188

6. Drinfeld-Jimbo Algebras

Fs ® 1 01 + Kï l K i-1 Fi ® 1

A (3) (Fi )= F2 01 1 01 +

+1Ç 1 KT' KT' it follows that

E on, aC3n ) (K

z3(a) =

where (j3n ) (00/011027 0 and

03)

1+02+03 Ki3-21+03 Ifs-31 0 1),

E WE, I = {(3, i3 , /32,/33) fio +01 + 02 + 03 =

}

a (i3n ) = E

.,) 0 43,7) 0

407.7) 0 48» E Uq-)30 (n_)

O O U(n).

Similarly, for w c U;r(ra + ) we obtain

A(2)( w ) = where (-ym,)

® K-y0

w

(707 71, 72) E Q 3+7 = {( 707 71172)

v

(WO Oh

" (WO r4g) 1,8 V'Y

V-Y W

if

I

0 K-y0-F-y1),

+

+ 72 = 7}

and

(WO E U.: ° (n+ ) 0 U 71 (n+ ) 0 U:2 (n+ ).

2,s

From (94) we conclude that

E

ad L (a)b i =

(4,aK0-21+03 )(s(w -Tsm)K40,77))D,

(19n),(7m),T, 5

(95) where

D := 43;) 0-i1+02+03v1fx1f0 3 8(4,77 ) ) S(wrI ) K1,0 )

;) v s(aV3;0)K0-21 ifx _ ,y0 _ 01 S( v)

q -(9701-H32-H33)-( 19 2,--A-F 19 1+02) al())3

I'srrt.))

with 43,7 ) vS(43,7 ) )1f0-21 c Uq (n_). The expressions in parentheses on the right hand side of (95) vanish unless -yo = fi and 72 = 03. Therefore, (adL (a)bi , b2 ) u =

E

,NO3.) ( A(2) (w), a(i°,7.1)1f 0-- 21+03 0 S -1 (t)') 0 5-1

(4,77 ) ))

(0n),r

x(S-1 (w ') ,43; ) vS (43,7 ) )K0-21 )

=E

eon) (w, s-i(v'43; ))a(if3;.1)K0-21+00( E

e n) (w, (13

(v/43„.r.i))a(10,,,P) (s-i (w/), a ào; )vs(47))),

where N(/3) = -(0101 + f2 + 133) + (1321 + /31 + 02) - (01 - A70/1)/2. By a similar calculation we show that the right hand side of (93) takes the same value. Thus, the assertion is proved for a E The proof of (93) for a E Ue(n+) is completely analogous.

6.3 The Quantum Killing Form and the Center of

U9 (g)

189

Definition 9. The bilinear form (•,.)u defined by (92) is called the quantum Killing form or the Rosso form of the Drinfeld-Jimbo algebra Uq (g). It can be shown that the quantum Killing form (•, -)u is nondegenerate on Uq (g) X Uq (g) if q is not a root of unity. One may also define a bilinear form (-, -Yu on Uq (g) x (g) by

b2 K o S(a2 )yu := q -(a 'f3) /2 (b 2,a1) (b 17a2), where al, a2 to (92) by

E Ug (rt_), b 1 , b2 E Ug (n+ ) and a, 3 E (a,b)u =

(96)

Q. The form (96) is related

S(b)Yu .

Since the adjoint actions a.dL and adR are connected by the formula

adL(S(a))5(b) = S(adR(a)b), the adL-invariance of the form (92) means that the form (96) satisfies the condition (ad', (c)a,b) u = (a, adR(c)b) u . 6.3.3 A Quantum Casimir Element

The Casimir element f2 constructed in this subsection does not belong to the algebra Uq (g). It is an infinite sum of elements from (V( b + )Uq- f3 (n_), f3 E Q + . In order to deal with this infinite sum we first define certain completions CT:(g) of the algebra Uq (g). Consider the vector spaces

U(g)

riseQ± Uq (b±)UP(n T ).

The elements of U:(g) are sequences x = (x0)0 E Q ± of elements xo from Uq (b±)UP(n T ). Let us write such a sequence formally as an infinite sum x = Eo xo. Then, by this definition, the components x0 are uniquely determined by the element x. Since

Uq (g) = ED0EQ+ Uq (b + )Uq- f3 (n_.) by Theorem 14, Uq (g) can be considered as the subspace of U:- (g) formed by the sums x = E 0 so for which all but finitely many terms xo vanish. From the commutation relations of the generators of Uq (g) it follows that for any 0, y E Q + , xo c Uq-(3 (n_) and y-y c Ug(n+ ) the element xoy-y belongs to the sum of spaces Uq (b + )Uq-5 (n_), 1Ij — 1 151 1017 where IcI := Ei ni for a = Ei nicri E Q +.. From this fact one easily concludes that the multiplication of Uq (g) extends canonically to the vector space CT: (g) such that CT:(g) is also an algebra. In a similar way, (-7, - (g) and

(T(g ) i6 - • - &U (g) := U:(g ED

••

ED g)

(n times)

190

6. Drinfeld-Jimbo Algebras

become algebras which contain (40) and U(g) 0 • • • 0 Uq (g), respectively, as subalgebras. By Proposition 37, the pairing (-, .) of Ui:(rt+ ) and UV3 (n_) is nondegenerate. Hence there are vector space bases fan and 1bn of U 13 (n) and fP(n q ± ) 7 respectively, such that (b)3 , a74.3 ) = 6„. Let a be the algebra automorphism of (.4(g) from Proposition 9. We set

00 := Er S(b.)cti°,, 0'0 := 9(00),

a?

c, = Q :=

flo ,

and

Q'

(97) (98)

'V

L--doEQ +

Pq3 (n+ ) is the canonical element with respect to Because Co E Uq73 (rt_) the pairing •), Co and so f20 , Q and Q' are independent of the choice of bases 141 and On. By construction, we have ,f2p E U-3 (b+)U-q— °(rt__) and E Uq— d3 (b__)Ue, (n+), so that Q E N- (2) and Q' E Üq— (g). (g) is called a quantum Casimir element Definition 10. The element 0 E for the Drinfeld-Jimbo algebra Uq (g). The reason for this terminology lies in Proposition 41 below. It says that up to an algebra automorphism 0 1 which corresponds to the identity in the case q = 1, Q commutes with all elements of Uq (e). Lemma 40. The elements C o satisfy the relations

(99)

[1 0 Fi , Co+Q ] = Co(Fi 0 Ki ) - (Fi ® Ki-1 )Co,

[Ei 0 1, Co+ai ] =

(100)

(Ki7-1 Ei ) (Ki ®

Proof. These two relations are proved in the same manner. For this reason, we prove only (99). Since both sides of (99) lie in M —cci (rt__) Uq (9), by Proposition 37 it is enough to show that they coincide when the first tensor factor is paired against an arbitrary element a E Ur' (n+). For the left hand side of (99) we then obtain

E (a, a.P.±'')(Fibr +Qi

Fi )

(E (a, 4+ e" )4±'') Here we used the relation Since

Er (a, a.V)bP,. a_0( 1

.A (a) -----

K.y ),

Fia

= a which follows from (b,

a,.y,

E U;f(n + )

aFi . = 6„.

Uq6 (n+)

71 5 EQ+ lr+ 6 ==i3 A- cri

= E 0u, ap,,„ = for some with ao ,p+ ,„ = 10a, ap +,„, 0 = a0 1, u,v E Ue(n+ ), the pairing of the right hand side of (99) against a gives

6.3 The Quantum Killing Form and the Center of U9 (g)

E

191

(a,F4).K1 1 bf3.1

=E =E 1

_

a 3,)b),(3.1Ci - (244 ari? F1 )1(;-1 1i73.} - (v, )(Ei lfp, Fi ) IC,7 1 bfi3.}

(Ei Fi Kul Ti

(v , afiN173,

uKi

(u, af73.)b,0 C ,} = -

Therefore, the proof is complete as soon as we have shown that

Fia - aFi =(K'v - ulf1 )1(q1 - C 1 ).

(101)

We have 2i (2) (F1 )=-F1 0101+1(27 1 0F101+1C,-1 01C,-1 0F1 and 2k (2)( a)

a123 ( 1 ® "K'Y1 ®

KY1 +1, 2 )

Y1 ,'V2 Y3EQ + 1 1 +1'2 -1-7 3 =0+i , -

where ao,p +ai3O = 1 0 a 0 1, aiai ,p,0 = Ei u 0 1, tZ j3 = 1 0 V O Ei (n+ ) UP (n+ ) 0 UT (nt ). Now relation (101) follows and al, 1 ,),2 ,y3 E immediately from (87) and (88). , c

and cio be the algebra automorphisms of Uq (g) such that 71)(1‘„) = Kc , CEO = E1Ç2 , CFO = KF1 and ço(K,„) = ço(Ei ) = K12E1, ço(Fi ) = F1 Ki2 . Then Proposition 41. Let

Ca)Q= Qa and p(a)Q' = (2'a for

a E Uq (g).

Proof. Since V) is an algebra automorphism, it is enough to prove the first assertion for the generators a = Ki , Ei , Fi . For a = Ki it is obviously true. 0 1) 0 T to both sides of (99) and (100) and Applying the mapping m o summing over E Ch we obtain EiK1211 = QEi and KTFti f? = f/Fi . This means that CE1 )f-2 --= QEi and 0(F1 )(2 = QFi . Applying the automorphism 0 we get the assertion for Q'. Next we apply the element Q' to the study of Verma modules. Let V be the carrier space of a weight representation. We define a linear operator w on V by setting wx = q(A+P ,A±P) x, X E VA, on a weight subspace VA of V. Here, as earlier, p is the half-sum of all positive roots of g. Proposition 42. If MA is a Verrna module with highest weight A, then the

operator TAI (Q')c.4) is a multiple of the identity operator, that is, TX (Q")w = q(A+"+P)

(102)

Proof. Let eA be a highest weight vector of the Verma module MA. Then TX. (U(g))e A = TX. (Ug (rt_ ))eA =

E 0EQ+ Tinu,-0(n_))eA.

192

6. Drinfeld-Jimbo Algebras

If Ff3 E [3 (n_), then A - ‘3. We assert that

en_o := T1 (F)en is

a weight vector for the weight

(TX(r2')w)TT(Fi)eA___p = Tilit (Fi)(TX(f2')w)eA_p

(103)

for any /3 E Q + and any i = 1,2, • • • 1. Indeed, we have

(TX (f2')w)TX (Fi)eA_,3 = TP,f2')T Av. (Fi )ql-Lwe A _ o

=

(Fi KnT (if2')q 4 we A_,(3

TX (Fi )q2(A- f3'a.) qP. TAI (S2')we

(104)

where ju

= (A - - a, +p,A-0-ai+p)-(A-0+p,A-0+p).

Denoting A - 3 by a, we have (a - cei + p, a - cx + p) - (a + p7 a + p) + 2(a ai ) = (ai , ai ) - 2(c, p) = 0, since 2(p, a i )/(ai , (xi ) = 1. Therefore, the right hand side of (104) coincides with (Fi )TX (f2')we A_ p and (103) is proved. The relation (103) yields

(TX (f2')(4))TT (Fo)eA =

TX (110)(TX (r2 ')(4)e A

q (A+p,A+P) TAV ( FogAV V 2 1 )e.A.By (97) and (98),

TX (r2')eA =

(0(0))e A =

(0(r20))e A = TA( (1)e A ---= eA

and the relation (102) follows. Corollary 43. For the irreducible representation LA, realized on the quotient

module MAIM of the Verma module MA by its maximal submodule M, we have L A(r2')w = q(A+PA+P) I. 6.3.4 The Center of Uq (g) and the Harish-Chandra Homomorphism

The aim of this subsection is to give a quantum analog of Harish-Chandra's description of the center of a universal enveloping algebra U(g). Let 3 9, be the center of the Drinfeld-Jimbo algebra Uq (9). Since the elements of 3g commute in particular with all IC,„, a E Q+, it follows from (51) that 3 g, C Uq° (g). By Theorem 24 and Corollary 25, any Z E 3 q is a finite sum

Z

E

r,s

F' • • Fornn kr,sEos: • • •

kr,s

E Uq (i)),

(105)

where 01 , - • , On are the positive roots of g, r = (r1,. • 7 rn)7 = (si, • • 1 Sn) and the summation is over all r, s E Ng such that riOi + • + rflOn s101 + • • • ± snOn• .

6.3 The Quantum Killing Form and the Center of Uq (g)

193

Let Sq denote the center of the algebra (-/ - (g). Any Z E Sq is also of the form (105) with a possibly infinite sum. To define the Harish-Chandra homomorphism for Uq (g) we introduce an Uq ( 0) such that algebra homomorphism : Ug (0)

= g (0), 0 :

and mappings 0 : 3 g

K„,

E Q,

Ug (r)) defined by

(106)

0(Z) = koo 0(Z ) = k oo

where Z is of the form (105). Proceeding as in the classical case (see [Dix]), one proves Proposition 44. The mappings 1/ 0 0 : 3g are algebra homomorphisms.

-)

Ug (r)) and 7 0 0 : Sq

Definition 11. The mappings 7 00 and 'y 0 d• from Proposition the Harish-Chandra homomorphisms.

44

(0)

are called

Put T = E A E2PnQ CICA c uq (0), where P is the set of all integral weights and Q is given by (25). The Weyl group W of g acts on T by wif), = lfA. We denote by T w the set of all W-invariant elements of T. Theorem 45. The Harish-Chandra homomorphisms 7 0 0 and 7 0 0 are injective and their images coincide with the set Tw .

This theorem says that an element K E Uq ((j) is the "k00 part" of a central element Z E 3g if and only if -y(K) E T w . Such an element K E Ug (0) determines uniquely an element Z of 3 g such that (700)(Z) = K. There exists an inductive procedure which allows one to calculate the other summands in

(105). We give the proof of Theorem 45 only for the case of Uq (s12 ). A proof in the general case can be found in [Tan3] (see also [Lus] and PL]). For Uq (512), Theorem 45 can be reformulated in the following form. Theorem 45'. For Ug (s12) we have Sq the Casimir element Cg from (3.6).

= 3 g and the

center 3g is generated by

Proof. We first show that Sq = 3g • Clearly, 3 g = Ug (512) flSg. If Z E

Z=

ni.=0 FT' k m Em ,

km E

Uq (0).

sq , then (107)

Let 19 be the algebra automorphism of Uq (1)) determined by the formula i9(K) = q -1.1 10 n E Z. Substituting the expression (107) into the relation aZ = Za for a = F,E,K, we find by direct calculation that gm+ 1 q -m-1

k m --= ?9(km)

(q

1)2 (q - m K q m K -1 )km+i)

m E N.

(108)

194

6. Drinfeld—Jimbo Algebras

Assume that Z E S and Z g Uq (512 ). By (108), k m 0 for in E N. For n E Z we set Uq (4) n = fk E Uq (1)) I V(k) = qnk}. Then Uq (4) = EBnEz Uq (4) n . For any nonzero element k = EnEZ k (n), k(n) E 140)) n , we define a number

a(k) = max In E Z

k (n) 0} — min In E Z I k (n) 01.

By the definition of V, one has a(V(k m )— k m ) < a(km ). It follows from (108) that a(z9(km ) — km ) = a(km+1)+2. Thus, a(km )> a(km+i) +2 for all m E N. Hence we obtain that a(ko ) > a(km ) + 2m > 2m, m E N. This is impossible, since all a(k) are finite. Therefore, Sq = 3 q and the sum (107) is finite. We now show that 3 q = C[q], where cf, is given by (3.6). Let Z=

m=o Fmk,Ern E 3 q ,

k m E Uq 0).

(109)

Suppose that all Z E 3 q of this form with r < s are in c[q]. Let Z E 3 q be of the form (109) with r = s. As shown above, the elements km have to satisfy the relation (108). Since kr±i == 0, it follows from (108) that kr = a•1 for some c E C. Hence Z — olC qs is a central element of the form (109) with = s — 1, so Z — ozC qs E Crq] by the induction hypothesis. Thus, Z E C[Cq ] and the induction proof is complete. CI For every integral weight A E P we define a character A on Uq (b), that is, an algebra homomorphism A: Uq (4) C, by setting

A(Ka )

A(Ific: • • •

:= K(A)

qn' (''' A) • • • qnL(al'A) = q(A)

for a = n i ai + • • + Nat E Q. Clearly, we have 'y(Ka)(À + p) = &PO. The composition e), = Ao-yo. of the two algebra homomorphisms ofro : 3q (.4(1)) and A : Us7 (4) C is a character of the center 3q) called a central character of Uq (g). The following theorem is the quantum analog of Harish-Chandra's classical result (see [Dix]) on central characters. Theorem 46. For A, p, E P we have E = some w E W.

if and only if A = w(p) for

Proof. The proof can be found in [Rce3].

LI

6.3.5 The Center of Uq (g) for q a Root of Unity In this subsection we describe the center 3q of Uq (z) when q is a root of unity. The proofs of the results stated below are complicated and can be found in the papers [L3] and [DCK], see also [Lus]. We assume that q is a primitive p-th root of unity, where p is an odd integer such that p> di , i = 1, 2, • • • , 1. As it is expected from the case of Uq (512) (see Proposition 3.15), there are many additional elements in the center 3q . Proposition 47. Let Ea and E„, c E 4 be the root elements from Definition 4. Then the elements Et: , Fg:, a E 4 and Kr, j = 1,2, • - • , 1, belong to hg.

6.3 The Quantum Killing Form and the Center of Uq (g)

195

Outline of proof. For Kr the assertion follows immediately from the relations qdiaii Ei and K i Fi K i-1 = q-diao Fi . Similarly, the assertion for and Ft is obtained at once from the defining relations. The proof for the D elements (Ea ) P and (Fa ), a ai , is more involved and is omitted.

Er

We denote the elements Eg, Pc:, and Iff by e a , fa and k , respectively. Clearly, ki-1- G 3 q . For the simple roots ai we write e a, and fa, as ei and fi , respectively. Let 30 be the subalgebra of 3 g generated by e a , fa , a C and k i , k i-1 , i =1, 2, • • • , 1. We also consider the subalgebras 32, 3 0- of 3o generated by the elements k i ,k,7 1 , 1 < i < 1, the elements ea, a E 2i± , and the elements fa , a E 2i+, respectively. The sets 32+ := 383 1-3F and 32- = 30- 38 are also subalgebras of 30 •

2+, 32-

are Hag subalgebras of Proposition 48. (j ) The algebras 3o, 3, 3 Uq (g). The comultiplication of Uq (g) acts on the elements e i , f , k i as

Lt(e) = e i k i + 1 ®e, 21(fi ) = f ®1 + k i-- i

L(k) = k o ki .

(ii) We have 3 = Ug (n±) n3q The multiplication map defines an algebra isomorphism of 30- 038 0 38- onto 30 • (iii) The action of the braid group 23 g on Uq (g) leaves 30 invariant. .

It is quite remarkable that the formulas for the comultiplications of the elements e = Er and fi = FiP are the same as the corresponding formulas (18) for the generators Ei and Fi , respectively. The following theorem determines the structure of Uq (g) in terms of the subalgebra 3 0 of the center 3q . Theorem 49. The algebra Uq (g) is a finite-dimensional linear space over 30 with basis Wrn 1,41 ri 8rL V81 1 < ri ,ti ,si < p. ,3n 11 ,31 '01 Let Spec (3 q ) and Spec (30 ) denote the sets of all characters on the commutative algebras 3 g and 30, respectively. That is, Spec (3 g ) and Spec (30 ) consist of all algebra homomorphisms of 3g and 30 , respectively, to C. These sets are of importance for the study of irreducible representations of Uq (g). For any irreducible representation of Uq (9), the elements of 3 g act as scalar operators. Hence they determine a unique algebra homomorphism of 3 g to C, called the central character of the representation. The following two important results allow us to describe the structure of Spec (30 and Spec (30). •

I-1

Theorem 50. The elements e a , fa, a c L ki , i =-- 1, 2, . • • ,1, are algebraically independent (that is, they do not satisfy a nontrivial algebraic equation with complex coefficients). The center 3g of Uq (g) is algebraic over 3o (that is, every element of 3g fulfills a nontrivial algebraic equation with coefficients in 30 ). ,

Example 4 (Uq (s12 )). The center of the algebra Uq (512) is generated by the elements EP, FP, KP, K -P and Cq , where Cq is the Casimir element (3.6).

196

6. Drinfeld-Jimbo Algebras

It follows from (3.6) that Cq - (q q -1 ) -2 (Ifq+ relation, one easily computes that

. f4= (cq P-1

( 17

--= FE. Using this

1 )2 (Kqi+1 1‹. 1 17 -±4 )) = EpFp

T0

that is,

+ 710: -1 + • • • + where y G C and X = (-1)P(q - q -1 ) -2P(KP

+ X = EPFP, K -P)

G 30-

(110) A

Since the elements e a , fa and ki are algebraically independent, the set Spec (so) is isomorphic to C2 ' x (C x ) 1 , where Cx C\{0}, 1 = rank g and n is the number of roots in A + . Thus, the set Spec (30) is characterized by 2m + 1 = dim g complex parameters. The restriction of characters on 3g to its subalgebra 30 defines a map v : Spec (3q ) -* Spec (30 ). The sets

v -1 (0 =-- {7/ e Spec (3 g ) I v(n) = for E Spec (30 ) are called the fibers of the map v. The second part of Theorem 50 implies that for every point G Spec (so) the fiber v -1 ( ) contains only a finite number of points. Proposition 51. The fibers of y have at most p i points and the generic fiber has precisely pi points, where 1 is the rank of g.

6.4 Notes The quantized universal enveloping algebras were discovered independently by V. G. Drinfeld [Dil] and M. Jimbo [Jim 11. The triangular decomposition of Uq (g) is proved in [Rosi]. The Hopf algebra automorphisms and the real forms of Drinfeld-Jimbo algebras were described in [Tw]. The Poincaré-Birkhoff-Witt theorem is from [Ros2] and [Yarn] for g = slN and from [Lusi in the general case. Braid group actions and general root elements were invented by G. Lusztig [Li] (see also [L2] and [Lus]) and S. Z. Levendorskii and Y. S. Soibelman [LS1]. Verma modules for Uq (g) and the corresponding descriptions of finite-dimensional representations of Uq (g) appeared in [Li]. The adjoint action of Drinfeld-Jimbo algebras was extensively studied in pLi , see also the book [Jos] and the references therein. The existence of a dual pairing of Uq (b+) and Uq (b_)°P was observed by Drinfeld [Dr21. In our exposition we followed the paper of T. Tanisaki ITan31. The description of the quantum analog of the Casimir element is also taken from [Tan3}. The results on the Harish-Chandra homomorphism in Subsect. 6.3.4 appeared in [Ros3]. The center of Uq (g) in the root of unity case was investigated in [L31, [Lus], and [DCK}.

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

Weight representations and Verma modules of Drinfeld—Jimbo algebras Uq (g) appeared in Subsects. 6.2.5-7. The present chapter is devoted to a detailed study of finite-dimensional representations of these algebras. In Sects. 7.1-4 we assume that q is not a root of unity. As we have seen in Sect. 3.3 in the case of Uq (s12 ), the corresponding representation theory of Uq (g) is similar in many aspects to the classical theory. In Sect. 7.5 we investigate representations of Drinfeld—Jimbo algebras Uq (g) in the root of unity case. Then the representation theory strongly differs from the case when q is not a root of unity.

7.1 General Properties of Finite-Dimensional Representations of Uq (g) 7.1.1 Weight Structure and Classification

The aim of this subsection is to prove the following theorem: Theorem 1. Any irreducible finite-dimensional representation of a DrinfeldJimbo algebra Ug (9) is a weight representation and a representation with highest weight. Such a representation is uniquely determined, up to equivalence, by its highest weight. Proof. The proof will be given by several steps stated as propositions. Proposition 2. Every irreducible finite-dimensional representation of a Drinfeld—Jimbo algebra Uq (g) is a weight representation.

Proof. Let T be a nontrivial irreducible representation of Ug (g ) on a finitedimensional vector space V. Since the operators T(Ki ), i --= 1,2, • • • , 1 = rank g, commute with each other, they possess a nonzero common eigenvector. Let V' be a maximal subspace of V on which all operators T(Ki ) are diagonalizable. Then we have dim V' > 1 and V' = V'A I where VA/ = {v E V/ I T(Ki)v = p = (p i , • • • , p i ) . Assume on the contrary that dim V' < dim V. Since T is irreducible, V' cannot be invariant under all operators T (Ei ), TWO, i = 1, 2, • - • , 1. (Indeed, if T(E)v' E V' and T(Fi )vi E V for all weight vectors Arf E V', then V' is an invariant subspace

198

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

of V.) Let y be an element of V' such that T(E i )v is not in V' for some j E {1, 2, • • • , /}. Without loss of generality we can assume that y is a weight vector. Then there is a p, such that y E V. Then we have T(Ei )v 0 and T(E)v, that is, y" T(Ei)v is a T(K i )T (Ei )v =T (Ei )T (If i )v = common eigenvector for all T(K1 ) and all operators T(Ki ) are diagonalizable on V" = V' ® Cy". Since dim V" > dim V', this is a contradiction. Lemma 3. If T is a finite-dimensional weight representation of a DrinfeldJimbo algebra Uq (9), then there exists N E N such that T(E 1 ) -T(E) = 0 -ip E {1, 2, • • • , /}, p N. for all Proof. Let V be the carrier space of T and let V =-- ED V12' be its weight It is enough to show decomposition, where Vo = fy E V 1 T(K i )v = that for every y c V ,i = (p i , • • • , we have T(Eii ) • • • T(Eip )v = 0 for sufficiently large p. Set y' = •T(Eip )v. Then y' c Vp1, where ,

= pti q nkaik . Here nk is the number of T(Ek) = (t4, • • • , p,;) and appearing in the product T(Eii ) • • • T(E). The representation T has a finite number of weights. We denote them by p, p( 1 ), • • - AM. We shall prove that there exists N G N such that for p > N, pt' does not occur in this list. Let /1■ . : s) = /49) 44, i = 1, 2, • • - ,I. We show that there exists a number / } such that i0 E {1, 2, - - • 7

gio

nka t o k

{1 /\% : 01) - •

Let us write the complex numbers q and

A i(0r)

(1)

}.

as q = exp 2i7rv and nkaik

exp (217rvy s) ). Since qi = V aio'i ) / 2 , the equality q i that there is an integer m such that (ai, ai) 2

_ (8) ,, nkaik —

= /1■ 8) implies

m V

-

•

By the definition of aik, the latter writes as Ek nk(ai,ak) = y's) + m/v. Since Ek nk (ai, ak) E Z, we conclude that y i(s) + E Z. Since y t Q7 there exists at most one integer m such that y s) + m/v c Z. Set zi 8) := yi(8) +miv. Assume that for every i E {1, 2, • • • , /} there exists s E {0, 1, • • , r} such that Ek nk (ai, ak) = 4s) with integral 49) . This is a system of linear equations with unknowns n 1 , n2 , • • , ni. Since the l x l matrix consisting of the entries (ai, k) is invertible, for a given 491) , • • • , 4 si) there exists at most one integral solution of this system. However, the number of possible sets fz i(81) - • • , 4 81) } is finite. Therefore, if the integers ni, n2 7 • • ni do not belong to a certain finite set 931, there exists an index io such that (1) is satisfied. Let N := sup(jni I + • • - + Intl) + 1, where the supremum is taken over 931. Then the assertion of Lemma 3 holds.

7.1 General Properties of Finite-Dimensional Representations of Uq (g)

199

Let T be a finite-dimensional representation of Uq (g) on a vector space V. Recall that y E V is a highest weight vector of T if y is a weight vector such that T(Ei )v = 0 for i = 1,2, • • • , 1 and T(Uq (g))v = V.

Proposition 4. Every irreducible finite-dimensional representation of ri g (9) is a representation with highest weight.

Proof. Let T be an irreducible representation of Uq (g) on a finite-dimensional vector space V and let Vo = ker T(Ei ). Let N be the smallest number in N for which the assertion of Lemma 3 holds. Then there exist a vector y E V , / } such that vo = T(Ei, • • - Ei„,_,)v and indices • • -5 iN-1 E {1, 2, and T(E)vo = 0 for i = 1, 2, • • • , 1. Thus, vo E Vo and so Vo {0}. From the formula (6.13) it is clear that Vo is invariant under all operators T(K i ). Since Vo {0}, there exists a common eigenvector e o E Vo for the operators T(Ki ), i = 1,2, • • • , 1. Because T is irreducible, the invariant subspace T(Uq (9))e o is equal to V. Since eo E Vo , we have T(Ei )eo = 0 for all i. The preceding LI shows that eo is a highest weight vector for T.

R

Let w = (wi , • • , we), wi = ±1. Then there is a one-dimensional representation T, of Uq (g) such that

T(E) = T,(Fi ) =0, T(K) =w,

i = 1,2, .., l.

Clearly, every one-dimensional representation of Uq (g) is of this form. If T is an irreducible representation of Uq (g) with highest weight A Z., is an irreducible represen(A i , • • - , Ai), then the tensor product T P i A i , • • • , wi AO. In particutation of Uq (g) with highest weight w - A lar, the irreducible finite-dimensional representation TA with highest weight A _ w (i n _ 7 wig ni is the tensor product of representations Z., and TA, with A' = q". Proposition 5. A = (A i , • - • , Ai) is a highest weight of an irreducible finitedimensional representation T of Uq (g) if and only if it is of the form A = j. q fl= (wiq fl1 ,..., wjq t2) with wi =- ±1 and ni G No .

Proof. Let U(s12 ) be the subalgebra of Uq (g) generated by the elements Ei , Fi , Ki , Ki-I- . It is isomorphic to Uqi (51 2 ). We restrict the representation T to Uqi (s12). Since finite-dimensional representations of Uqi (s12 ) are completely reducible (see Proposition 3.12), this restriction decomposes into a direct sum of irreducible representations of Uqi(s12 ). The highest weight vector e of T belongs to the carrier space of one of these irreducible representations and is its highest weight vector with weight A. By the results of Subsect. 3.2.1, Ai is of the form stated in the proposition. Conversely, let n = (n i , • - • ,ni), ni E No , and w = (w 1 , • • • , we.), w. = ±1. By Proposition 6.28, there is an irreducible representation L r, with highest weight qn. Then Z., L„ is an irreducible representation with highest weight A = w • qn. El Proposition 6. Every irreducible finite-dimensional representation T of Uq (g) is uniquely determined, up to equivalence, by its highest weight.

7. Finite-Dimensional Representations

200

of Drinfeld-Jimbo

Algebras

Proof. Let A = w • qn and A' = w' • qn i be highest weights of the representation T. From the form of eigenvalues of the operators T(K1 ) we see that w = w'.

Since w = w', the irreducible representation qn and q", so qn = qn by Proposition 6.28.

T has the highest weights

Remark 1. By Remark 6.7, we may take the roots -al -a2, • • • , --cti as a set of simple roots of g. In this case, the elements F , i = 1, 2, • • • , 1, correspond to simple roots. Then highest weight vectors are taken with respect to the operators T(Fi ) instead of T(E), and weights A = (A 1 , • • • , A1) and the corresponding weight vectors y are defined by T(Ki-l )v = Av. All results of this chapter are true (under appropriate reformulations if necessary) for this setting. This approach will be used in Sect. 8.4. 7.1.2 Properties of Representations Let T be an irreducible representation of Uq (g) on a finite-dimensional vector space V. Then T is a representation with highest weight A = (A 1 , - • - Ai) and corresponding highest weight vector e. We repeat the main properties of such representations from Subsects. 6.2.5-7: (i) V is spanned by the vectors e and T(F11 ) • • • T(Fip )e, {1, 2, • • - 1 } . Moreover, dim VA = 1.

(ii) The vector T(Fii ) • • • T(Fip )e in V is of weight t = (p i , • -

• • • , ip E

pi) with

aki3 where aki are the entries of the Cartan matrix of g. Every weight of the representation T is of this form. =

The counterpart to Proposition 5 for the algebras Ùq (g) is

Proposition 7. If (0.) = (w i ,. • • ,co), Wk E {1, -1, and n = (n 1 , • • • ,n1) is a dominant integral weight for the Lie algebra g, then A = ce-qn is the highest weight of some irreducible finite-dimensional representation of Oq (g). Moreover, every highest weight of an irreducible finite-dimensional representation of Ùq (g) is of this form. The classical H. Weyl theorem on 4Comp1ete reducibility has the following quantum analog.

Theorem 8. Each finite-dimensional representation of U(g) or of 6, (g ) is completely reducible. Proof. A proof of this theorem can be found in each of the papers [Ros11,

[APW1 and [JL].

Any dominant integral weight n = (n1 , n2 , - • • , ni) corresponds uniquely to a dominant integral form A on the Cartan subalgebra of g by ni = 2(A, ai)/(ai . We shall denote the irreducible finite-dimensional representation T of Uq (g) with highest weight q" by T. and also by TA with a TA are called of type, 1. slight abuse of notation. Such representations Ti,

7.1 General Properties of Finite-Dimensional Representations of U9(g)

201

Since any irreducible representation is a tensor product of a representation T„ and a one-dimensional representation Z.„ we study in the following mainly type 1 representations. The following properties of representations Ta-- TA are similar to those of irreducible finite-dimensional representations of 0. Proposition 9. (i) The Weyl group W of the Lie algebra 0 acts naturally on the set H A of weights of the type 1 irreducible finite-dimensional representation TA of Uq (g). This action of W leaves H A invariant and preserves the dimensions of weight spaces. (ii) If Q' is the element and ci.) is the operator from Subsect. 6.3.3, then q (A+p,A-i-p) we have TA W I )(.4) (Hi) For any type 1 irreducible representation TA of Uq (g) we have TA(Z) = , Z E 3 q , where 3 q is the center of Uq (g) and A A E P+, is the central character of Uq (g) from Subsect. 6.3.4. The central characters separate the finite-dimensional irreducible representations of Uq (g), that is, for any two representations TA' and TAI A' 74 A", there exists an element (Z) 74 Z c 3 q such that Proof. Since TA is equivalent to the representation LA from Subsect. 6.2.6, the proof of (0 is in fact given by the proof of Proposition 6.28(iii). The assertion of (ii) follows from Corollary 6.43. The first part of (iii) follows from the formula A = A oy o 411 (see Subsect. 6.3.4) and from the expression (6.105) for the elements of 3 q . The second part of (iii) is a consequence of Theorem 6.46. 1=1 Let us describe the highest weights A of type 1 irreducible representations of Drinfeld—Jimbo algebras Ug (0) corresponding to the simple Lie algebras sl(/ ± 1, C), so(21 + 1,C), sp(2/,C), and so(21, C). As in the classical case, it is convenient to characterize the highest weights A by the following numbers n = (n i ,n2, • • • ,ni), where ni = mi . Suppose that A Then the relations between the numbers n i and mi are

ni = mi mi+17 = 1, 2, • • -,l, for Uq(sli+1), ni mi — i = 1,2, • • • ,1 —1, n = 2mi for Uq(s 0 21+1), ni = mi — = 1,2, - - l —1, ni = mi for Uq(sP21), ni = mi — mi for Uq (s021)• = 1, 2, - 1, nI = —

Note that ni_i = ni for tlq (so2t) if and only if mj = 0. For Uq (sli+i ) and Uq (sp21 ) all numbers mi are integers, while for Uq (soN) they are all integers or all half-integers. They satisfy the dominantness con-

ditions

m1 m2

• • • > mi + i

m1 > m2 > • • • > mi > 0

for

for

Uq (slz+1),

Uq (80 21 +1 ), Uq(sp2i)

mi > m2 > • • • > mi_i > mi I for

Uq (sow).

(2) (3) (4)

202

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

For g = so2/ and g = sp2i there is a one-to-one correspondence between irreducible finite-dimensional representations of U(g) and collections of numbers (m1, m2, • - , mi) as described above. This is not true for the algebra describe •••, Uq (sli+i ). Two such sets (m i , m2 , • - - ,mi+i ) and the same irreducible finite-dimensional representation of Uq (sli+i ) if and only if there exists an integer m such that fhi m i + m, i = 1, 2, • • • , 1 +1. 7.1.3 Representations of h-Adic Drinfeld-Jimbo Algebras

By Proposition 6.8, the h-adic algebra Uh (9) is isomorphic to U (2)[[h]], where U(g) is the universal enveloping algebra of g. Therefore, as noted in Subsect. 1.3.1, if T is a finite-dimensional representation of the (complex) algebra U(g) on a (complex) vector space V, then formula (1.50) defines a representation U (9)[[h]] on the C[[h]]-vector space V[[h]]. Th of the h-adic algebra Uh(g) Conversely, let V be a finite-dimensional complex vector space and let T' be a representation of Uh (g) on Vh := VDT. Then it is clear that the equation

T(x + hUh(9))(v + hVh) := (x)v + hVh, x E U(g),v

E V,

V such defines a representation T of Uh(9)1hUh(g) _`2 U (g) on Vh/hVh that Th = . It is easily seen that T is irreducible if and only if Th is indecomposable. Thus, we have proved the following Proposition 10. The map T Th determines a one-to-one correspondence between finite-dimensional representations of the complex Lie algebra g and representations of the h-adic algebra Uh(g) on C[[h]]-vector spaces of the form V[[h]], where V is a finite-dimensional complex vector space. The carrier space V[{11 of an indecomposable representation (TA)h of

Uh(g) with highest weight A decomposes into weight subspaces V [[ h]] = e

[[ h]] , V [[h]] 1, = Vi,,

where VI, is the corresponding weight subspace of the representation TA of g on V. In particular, we get dimq[h]] = dimc V. A similar assertion is true for the representations TA of the Drinfeld-Jimbo algebra Uq (g). Proposition 11. A type 1 irreducible finite-dimensional representation TA of U7 (9) acts on a space of the same dimension as the corresponding irreducible representation TA of the Lie algebra g. Moreover, the dimensions of weight subspaces, corresponding to the same weight in these representations of and g, coincide.

Uq (g)

Sketch of proof. For g = sl(n,C) this will be shown in Subsect. 7.3.3. For a general Drinfeld-Jimbo algebra Uq (9) it can be proved by means of Verma MA are the Verma modules of g and Uq (g), respectively, modules. If M with highest weight A, then by (6.74) the dimensions of their weight subspaces, corresponding to the same weight, coincide. We can construct the

7.1 General Properties of Finite-Dimensional Representations of U9 (9)

203

irreducible finite-dimensional representations TA of Uq (g) and of g by means of their Verma modules MA and M. By Proposition 6.28, the representations TA are realized on MA/M and MA/M', respectively, where M and M' are the maximal proper submodules. By Proposition 6.27, the submodule (resp. M') coincides with the sum of all Verma submodules Mn w (resp. A/Au, ) with A w = w(A + p) p, w E W , w 1. This implies that the weight subspaces of M and M', belonging to the same weight, have the same dimensions. Therefore, the dimensions of the corresponding weight subspaces in MAN and MA/M' coincide. El —

Most of the above considerations and facts remain valid almost verbatim for representations of the quantum algebra N(g) over the field Q(q). In this case the representation spaces are vector spaces over Q(q). 7.1.4 Characters of Representations and Multiplicities of Weights

Proposition 11 allows us to define characters of finite-dimensional representations of Uq (g) which characterize these representations up to equivalence. If T is a type 1 finite-dimensional representation of Ug (0) on a vector space V with weight subspace decomposition V = (BoEp Vih , then the function x(T) =--

Et,Ep (dim 174 )ei-i

on the Cartan subalgebra I) of g is called the character of T. Here et is the function on I) defined by el.t(h) = eth(h) , h E b. Recall that the character of a finite-dimensional representation of the Lie algebra g is defined in the same way. Therefore, by Proposition 11, the characters of type 1 finite-dimensional representations of Uq (g) are in a one-to-one correspondence with characters of finite-dimensional representations of g. This leads to Proposition 12. (i ) Type 1 irreducible finite-dimensional representations of Uq (g) are determined uniquely, up to equivalence, by their characters. (ii) The character x(TA) of the irreducible finite-dimensional representation TA with highest weight A is given by the classical Weyl formula x(71/0

=

E (-1)/(w)ewo+P) wE w (_1)1(w) ew(p) E wEw

where W is the Weyl group of g, p is the half-sum of positive roots of g and 1(w) is the length of the element w E W.

Recall that the dimensions of weight subspaces 170 of the underlying space V of an irreducible finite-dimensional representation TA of Uq (g) are called multiplicities of weights p in TA and are denoted by mp,A. By Proposition 11, they coincide with the corresponding weight multiplicities in the irreducible representation of g with highest weight A. There exist several formulas for the calculation of weight multiplicities (see [Hum] or [The]);

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

204

7.1.5 Separation of Elements of Uq (g) The aim of this subsection is to prove the following theorem.

Theorem 13. Let a E Uq (g). If T (a) = 0 for all irreducible finite-dimensional type 1 representations of Uq (g), then a = O. Proof. Assume on the contrary that a nonzero element a E Uq (9) is annihilated by all such representations T. As shown in Subsect. 6.1.5, a can be represented as a=

E

ciiii fiK tL ei,fj E Uq- )31(i) (n_), ei E UP ) (n+ ), c

E CI

where only finitely many coefficients are nonvanishing. Let fio be the maximal O. element in the set of all OW for which there exists a coefficient c For A E P+ 1 let TA be the irreducible tybe 1 representation of Uq (g) on a space VA with highest weight A and highest weight vector e4 . If 0 is the algebra automorphism of Uq (e) from Proposition 6.9, then Tfi := TA 0 O is also an irreducible type 1 representation of Uq (g) on VA. Since 0(Ki ) K and 0(E1 ) = Fi , we have ri (IC ti )e A = qe4 and T(F2 )en = O. Let us denote the vector eji and the vector space VA considered for the representation 711 by e9A and Vi, respectively. Since 71 0 TAI is a direct sum of irreducible type 1 representations by Theorem 8, we have (7107'41)(a) = 0 for all A, E P. By the definition of the comultiplication of Uq (9), L(e) = ei g Kow + b, where b is a sum of terms s yE.i . Therefore, since TA(E i )eA 0, we have VI T Af)(b)(eeA 0 en/) = 0 and so

(T,1

(K)

Since (71

TA1)(ei)(4

eft')

q (i3(i )41) Ttei)e °A 0 eAl

= K 0 K 1,, we obtain

TA1)(K tiej)(e °A

0

= q (f3r4 ) '111)+( "' A+ /3(i)) 11(ei)e9A 0 en/

Further, we have A(fi) = Ki l(1) O fi + d, where d is a sum of terms sFi 0 y. Thus we get

VI

q

Tni)(fiK,ei)(e °A 0 eiti)

(p(i),4/)+(i.1,4I-4+0(i))-(01 (i),-A+00))/t ej ) e6A

O TAI(MeAl + g,

(5)

(recall that is a sum of terms from (11)-4_Fp 0 VA' with 3 < (11)_ 4 _Fp is the weight subspace of VI for the weight -A + 13). Since Po is the maximal weight in the set {OW}, the term (TIOTAi)(fiKpiej)(40eiv) lias a component in (V1)--4-00 0 only for OW = )30 and in this case this component coincides with the right hand side of (5) when g is omitted. Thus, the canonical projection of (Til 0 Tiv)(a)(e9A 0 en') onto (VD- A+0,, 0 VA' is where g

7.1 General Properties of Finite-Dimensional Representations of Uq (g)

A+0.), ( e5)eon ® TA , (f1)en'.

205 (6)

o(i)---00

Since WI TAf)(a)(e °A en,) = 0, this projection is zero. Let N be a positive integer such that fib < M ai +

/3'(i) < N (al + • • • + al)

+

for all j for which there exists a coefficient cito 0 in (6). Let PN be the set of highest weights A for Uq (g) such that 2(A, ai )/(ai , ai ) > N, j = 1, 2, • • • , l. By Proposition 6.29, the mapping e 71(e)e 6A of [430 (n+ ) to (V)-A+00 is bijective. Thus, the vectors 711 (ei )e°A with i3 (..7) = )30 are linearly independent. Therefore, the vanishing of the sum (6) implies that

E 2,12 ciwq (So, A/)+0,,,A , _A+00) _(0

, (, ),_A+00)T A

, (m eA ,

=

( 7)

for all A E PN and all j with OW = Oo• It is shown similarly that for all A' E PN the vectors TA , (f)e4' that occur in (7) with c i 0 are linearly independent. The factorse" /)- (13f (z)t - A-13°) in (7) are independent of If we cancel these factors, we get

E

Citi•q(j.1700

-A)

f q(12,A =

(8)

for all i, j with OW = 00 and all A, A' E PN. For fixed A, we consider the left hand side of (8) as a linear combination of the distinct characters A' q (12 '11') on the semigroup PN. Now we apply Artin's theorem on the linear independence of characters which is also true for semigroups. Thus, all coefficients cii,i g 00° - A) in (8) vanish. Hence c i = 0 for all i, j and j.t with This contradicts the choice of 00 and completes the proof. = The counterpart to Theorem 13 in the h-adic case is Theorem 13'. Let a E Uh(g). If T(a) = 0 for all indecomposable finitedimensional representations T of Uh(g), then a = 0.

Proof. The result follows from the discussion preceding Proposition 10 and from the corresponding result for representations of the universal enveloping algebra of the Lie algebra g (see [Dix], Theorem 2.5.7). D 7.1.6 The Quantum Trace of Finite-Dimensional Representations

If T is a finite-dimensional representation of a cocommutative Hopf algebra A, then we have Tr T(adL(a)b) = Tr

(2))TOS(a(1))) = Tr

E T(a(0)TOS(a(2)))

E TOS(a(1)))71(a(2)) = e(a) Tr T (b )

206

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

for a, b E A, where Tr denotes the "usual" trace of a linear mapping on a finite-dimensional vector space. In this subsection we give a generalization of this relation to the Drinfeld-Jimbo algebra Ûq (g). Let p denote the half-sum of positive roots of the Lie algebra g. Suppose that T is a finite - dimensional representation of the algebra I:41 (g). We define the left and right quantum traces of T by

Trq ,L T(a) = Tr T(aK2p ) and Trq ,R 71 (a)

Tr T(aK2-p1 ), a E Og (g).

The main properties of these quantum traces are given by Proposition 14. (i) For arbitrary elements a,b E Ùq (g) we have Tr q T (ad I, (a)b) = (a) Trq T (b ),

Trq ,LT(ab) = Tr q ,L Ma),

Trq ,RT (adR(a)b) = E(a) Trq ,R T , Tr q ,AT(ab) = Tr q ,RT(ba).

(II) If T is a direct sum of type 1 representations of (4(0, then

T1 q ,LT(1) = Trq,RT(1). Proof. (i ): We prove the first formula. Since adL is an algebra homomorphism by Proposition 1.14, it suffices to verify this equality for the generators a = Ki , Ei , Fi of Oq (9). First we let a = E. Since (p,ai )/(ct i ,ai ) = 1 (see, for example, [Hum]), we have (p, (xi ) = (a i , ai ). Thus, from (6.30) we obtain

K 1 K2E = q-( ai'ai ) /2 + ("i ) Ei Ki-1 K2p = Using the formulas (6.34)-(6.37) and the preceding relation we compute

Trq ,/, 71(adL(Ei )b) = Tr 71((adL(E i )b)K 2p )

= Tr T(Ei )T(bKi-1 1C2p ) qi Tr T(K i-1 )71(bEi K 2p ) Tr T(bK i-1 K 2p )T(Ei ) - qi Tr 71(bEi K2 p )T(K i-1 )

= Tr T(b)71(Ki-l- K2pEi - qiEiK2pKi-1 ) = 0 = E(Ei ) Trq,L T(b).

The proof for the generator a = Fi is timilar. In the case a = K we get Trq , L T(adL (Ki )b) = Tr T(K i )T(bK i-1 K2 p ) = Tr T(bK i-1- 1C2p )T(Ki) = E(K i)Tr q ,LT(b).

The proof of the second formula is similar. The third and the fourth formulas are obvious. (ii): It suffices to assume that T is an irreducible type 1 representation. Then T is a highest weight representation. Let ; be the corresponding rep resentation of the Lie algebra g with the same highest weight. If G is the simply connected connected complex linear group with Lie algebra g, then there is a representation T0 of G which is the exponential of Tg. From the structure of the representations T and Ta it follows that there is an element

7.2 Tensor Products of Representations

207

h G exp Ij C G such that T(K2 ) = TG(h). The Weyl group W of g can be considered as a subgroup of G. If wo is the longest element of W, then we have KV = K -2p = Kw 0 (2p) and hence h = w0h -l wc7 1 (see, for instance, [Me]). Thus we obtain Tr T(K2p) = Tr TG(h) = Tr Tc(wo)Tc(h -1 )TG(wo) 1

= Tr Tc(h) -1 = Tr T(K2p) -1 = Tr T(1C2p ), which gives the assertion of (ii). Clearly, we have Ti)(a) =

Ei Trq,r, Ti(a),

Tr q ,R (IED Ti) (a) =

Trq ,R

Ti (a).

Since 41(1(2p ) = K2p K2 p , it follows at once from the definition of the tensor product T1 0 T2 of two representations T, and T2 (see Subsect. 1.3.1) that Trq ,/, (T1 0 T2)(a) = Trq ,R (Ti 0 T2)(a) —

Trq, Ti(a(1)) Trq ,/, T2(a(2)) E Trq,R T1 (am) Trq,R T2 (a(2)) -

If T is a direct sum of type 1 representations of Og (g), then the number

dimq T := Tr q , L T(1) = Tr QR T(1) is called the quantum dimension of T. If T' and T" are two such representations, then the preceding formulas yield dimq (Ti 0 T2) = (dim q (dim q T2 ) •

Example 1. Let T1,1 be the type 1 irreducible (21 + 1)-dimensional representation of Ùq (512) from Theorem 3.13. Since K2 p = K for t'lq (512), we get 2/ q

dimqTi,/ =

i=0

(21+1)/2

-(21+10

-

q -1/2 _ q -112

= [2 1 + 141/2.

7.2 Tensor Products of Representations The investigation of tensor products and Clebsch-Gordan coefficients for representations of Drinfeld-Jimbo algebras Uq (g) of higher ranks is much more complicated than in the case of (4012 ). The reason is that multiple irreducible representations appear in the decompositions of tensor products. By Proposition 12, the multiplicities of irreducible components in tensor products are determined by the same formulas as in the classical case. We develop these results on multiplicities in Subsect. 7.2.1. Let us assume that all representations in this section are of type 1.

208

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

7.2.1 Multiplicities in Tensor Products of Representations

By Theorem 8, finite-dimensional representations of Uq (g) are completely reducible. Therefore, the tensor product TA 0 TA' of two irreducible finitedimensional representations of Uq (g) decomposes into a direct sum of irreducible components: TA 0 TA/ = (

A A' BO A MA' TA,

( 9 )

where mAA ' denotes the multiplicity of the irreducible representation TA in TA 0 TA'. This multiplicity mAA ' Ai can be expressed in terms of weight multiplicities of one of the representations TA and TA'. In order to give the corresponding formulas we introduce some notation. If p is a weight of some representation of g, let {p} be the dominant weight lying on the orbit Wp. The element w of W for which wp = {p} is denoted by w{4 }. If m the multiplicity of the weight p in the representation TA, then we have (see [K11]) ' A

AA'

77-1,À

E = M ,Ath (det w{ 4± ,v±p}),

(10)

where det w = (-1)/(w) and the summation is over all weights p of TA such that p + A' + p} = A + p. Since any such weight p can be represented in the form w(A + p) A' — p, w E W, formula (10) can be written as {

mA

= \--"‘ wEW

(-1) 1(14 M, Aw ( A±p )_

p

w Ew (-1) 1(w) 74+ 09 _ w (A' -1- P) .

From (10) one derives the following Proposition 15. The decomposition of the tensor product TA 0 TAI of two irreducible finite-dimensional representations TA and TA' of Uq (g) is given by TA 0

=

E

AZEITA

1-1

r 1 AN±P}-P1 14 ±A ±P r{±

(11)

where HA is the set of all weights of the representation TA and 0,, is the number defined as follows: 0, = O if there exists w E W, w 1, such that wv v and 0, det w{,} if no such ilement exists.

Note that the relation w(p + A' + p) = p + A' + p, w 1, means that the linear form { p + A' + p} p is not dominant, hence this linear form cannot be a highest weight and the symbol T{p± A. ±p }_ p has no meaning. But in this case the corresponding coefficient in (11) is equal to 0, so the summand in (11) does not occur. In (11), there are negative and nonnegative coefficients. Adding the coefficients of the same irreducible representation we obtain a nonnegative number which is the multiplicity of this representation. Formula (11) shows that the set of weights of one of the irreducible representations TA and TA' already determines the decomposition (9). Some special cases of the formula (11) are stated separately in Corollary 16. If the form

wA + A' + p is ,dominant for any w

E W, then

7.2 Tensor Products of Representations TA 0

TAI = El) rnA1.4 T

209

( 12)

+A l

where the summation is over all weights p E HA for which p+ A' is dominant. If the linear form w A + A' is dominant for any w E W, then the summation in (12) is over all weights p E HA. In particular, if all forms w A+ A' w E W, are dominant, then to every weight p E HA there corresponds a representation Tp+Ai on the right hand side of (9) and the multiplicity of Tp+Al in TA 0 Tx is equal to the multiplicity of the weight p in TA. If the weights A are given by the numbers ( ) i , - • , Ai) with A i = 2(A, ai )/(ai , ai ), then a direct calculation shows that for the classical complex Lie algebras A1, B1, C1, Di the forms wA+ A' are dominant for all w E W if and only if • + A l,

for A1: for B/ :

1

i

I,

•• + 2Al_i + , • • + 2 A 1- 1 + A i l • • + 2Ai - i + 2Ai,

for C/ :

1CiCI- 1,

• + At_1 + A1,

for D1 :

• • + 2A1-2 + A1-1+

1 < i < I.

If the decomposition (9) is of the form (12), then the set of highest weights A of irreducible representations which occur in (12) is contained in the set HA + A'. This statement is also true in the general case. Proposition 17. The highest weights of irreducible representations in the decomposition (9) belong to the set HA+ A'. Moreover, the multiplicity of an irreducible representation Tp+A, , p E HA, in TA 0 TA' does not exceed the multiplicity of the weight p in TA.

Using the embeddings of irreducible finite-dimensional representations of the Lie algebra g into its infinite-dimensional representations of the principal nonunitary series one proves the following result (see [ATK2], Sect. 4.2). Proposition 18. The multiplicity m/x1A1 of an irreducible representation TA in the decomposition (9) does not exceed the multiplicity of the weight A — A' in TA, where A' is the highest weight of the contragredient representation TA ,

of TN. Moreover,

74A1 ,

74T,

714 ,T■

Propositions 17 and 18 admit the following useful corollary. Corollary 19. (i) If all weight multiplicities of an irreducible representation TA are at most one, then Tx 'appears in the decomposition (9) of any tensor product TA 0 TA/ with multiplicity less than or equal to one. (ii) If the weight multiplicities of one of the representations TA and TA' do not exceed 1, then all multiplicities of irreducible representations in the decomposition (9) are at most one.

210

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

Next we develop formulas for the decomposition of the tensor product of an irreducible representation and the first fundamental representation (also called the vector representation) of a Drinfeld-Jimbo algebra. We characterize the highest weights by the numbers mi given in (2)-(4). Let Tm be an irreducible representation of Uq (g), g =--- A1_1, B1, Ci, Di, with highest weight m (m1, m2, • • - mi) and let T1 be the first fundamental representation of Uq (g) (that is, with highest weight (1,0, • • • , 0)). Then the tensor product Tm 0 T1 decomposes into irreducible components as Tm

=-

(13)

i=1

for Ug WO, where e = (0, • • , 0, 1, 0, • • , 0) (1 is in the i-th place), and as

Tm ® Ti. z.:__

ED

/

„„/

for Uq (8021 + 1) if mi = 0 and also for Uq (sp 21 ) and Uq (5021)• If mi for Uq(sov+i) we have Tm

= Tm ED Wr

(14)

Tm+ei ED giwir i= 1 Tm—ei

i=1

Trn+e, ED W

i=1

0, then

Trn—e •

(15)

•

If for some m ± ei in (13)-(15) the dominantness condition in (2)-(4) is not fulfilled, then the corresponding representation Tm±ei must be omitted. Let Vg) be the set of highest weights of irreducible representations of Uq (g) which are contained in some i--fold tensor product rr, r c No, of the vector representation. Using the decompositions (13)-(15), we derive the following

Proposition 20. If the highest weights are given by the numbers ni := (mil m2, • • -,mi) satisfying the corresponding conditions (2)-(4), then

13 (s1 1) = P, 443(sp2i) = P+1 Vso 21+1 ) = {m E RE I rn E Z}, s43(so21) = fm E P± rn E Z}. Note that in terms of A = (A1, A2 7 • 43 (50 N) are described as 443(so21+

= {A E

I

1 A1), Ai

= 2(A, ai )/(ai , ai ), the sets

Ai E 2Z}, i13(so21) =-- {A

c P+ J Aii + Ai E 2Z } .

If g is a classical simple Lie algebra, a closer look at the proof of Theorem 13 shows that it remains valid if we replace the set of all type 1 irreducible finite-dimensional representations of Uq (g) by the set of representations TA, A E T (g ). Therefore, we get

Proposition 21. Let g be one of the complex simple Lie algebras A1, B1, C1, Di and let a E Uq (g). If TA(a) = O for all irreducible representations TA with A E T (g ), then a = O.

7.2 Tensor Products of Representations

211

7.2.2 Clebsch-Gordan Coefficients

Let TA and TA' be type 1 irreducible representations of Uq (g) acting on finitedimensional vector spaces V and V', respectively. As already noted above, their tensor product decomposes into a direct sum of irreducible representations: MAs A' rr TA 0 TA' = (16) A. • '

A

The corresponding decomposition of the carrier space V 0 V' is V 0 V' = m m" ' Vv 3,8, Twhere m,V, := V8 ED • • • ED V3 (m, times). In order to distinguish different subspaces V, with the same index s we equip them with an additional index r and write V,,., r 1, 2, • - • , m181,A1 . The Clebsch-Gordan coefficients of the tensor product (16) will be defined as for the algebra Uq (512 ) (see Subsect. 3.4.2), but now multiple irreducible representations may appear in the decomposition (16). As in Subsect. 3.4.2, we consider two bases of the space V g V'. The first one consists of the vectors ei e'j , j = 1,2, • • • , dim V, j = 1, 2, • • • , dim V', where fed and {ei } are bases of V and V', respectively. The second is formed by bases {etr} of the subspaces V5r . We suppose that the bases {er}, r = 1,2, • • • , mA/, with fixed s are such that the representations Tit s are given by the same matrices with respect to these bases. Both bases fei and { eskr} are connected by an invertible matrix U with complex entries (e i , e eicr) such that

ekSr =

E

(ei , ej

eksr)ei ei •

(17)

The numbers (ei , e eZT) are called the Clebsch-Gordan coefficients (briefly, the CGC's) of the tensor product TA O TA'. Let q be real. Then we can assume that V and V' are Hilbert spaces and fed, {ei' } and { eZr} are orthonormal bases. Then the matrix U is unitary and

= U* transforms the basis fen into lei 0e'i l. The entries (ei, er) of the matrix U* will be denoted by (er f e i , e'i ). The CGC's (ei , ei' I er) are then equal to the scalar products of basis vectors, that is, we have (e i , ei' eZT) = (e i ei' , er),

(er e i ,e;) = (esr e• e'.) k 3

Since the matrix U of CGC's is unitary, one has the orthogonality relations

E

ij

e'• (eektr I ei , e'.)(e• 3 2' 3

ES,k,r (ei el3

:

eic t` ) =

er)(ei.r I e', e;)

ss'brribkki

(18)

=

(19)

As in the case of the quantum algebra Uq (812 ) we then have the following relations between matrix coefficients of the representations and CGCs:

044'. = 4131

4.—/A ryr k ki

ei i

A, I e ksr »kw (e ki

ar

•e

(20)

, )

212

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

(er I ei ,e1,)t,fli tfli i,(ei , iei, e;;';).

tz.„ =

(21)

For general complex g, the above formulas (20) and (21) remain valid if the numbers (er e, e1) therein are replaced by the corresponding entries of the matrix U -1 . Repeated application of formula (21) leads to the following Proposition 22. The matrix coefficients t of any irreducible representation TA with highest weight A from the set 113(g), described in Proposition 20, are polynomials of the matrix coefficients of the vector representation.

7.3 Representations of Ùq (g1) for 7.3.1 The Hopf Algebra

a Root of Unity

q not

6, (gin )

The aim of this subsection is to introduce the Hopf algebra Oq (gli,). The algebra ti/g (gln ) is generated by elements Ei , Fi , Ki , 1,2, • n - 1, j = 1,2, • • • , n, subject to the relations K 1 K' = KiEj.K.

i-

i

=1,

= ei.d2 q -siti+i /2 Ei7 KiFiK 1 = re-2 Te72

[Ei l Fri =6 2,r

Te.2

g g-1

7

[E1 E] = [Fi , Fi l = 07 ,

E?Ei±i (q + q -1 )E1E1±1E1 + Ei±iE? = F12 F11

27

07

+ F1 + 1 1? = 0.

(q+

The algebra Ûg (g1n ) is a Hopf algebra with structure maps given by

z(E1 ) = Et ® K1K

A(Fi ) = Fi A(K i ) =

1 + K 1 Ki+i E • , +

Ki+i Fil K i , e(E1 ) = E(Fi ) = 0, e(K 2 ) = 1.

8(E1 ) = -qEi ,

(Fi ) =

8(K 1 ) = K1 1 .

For ij/q (gln ) we also have triangular decompositions Oq (gln ) = Uq (n+ ) 0,1 (I) Uq (n...) = Uq (n_)

t/j" Uq (n+ ),

(22)

where Uq (n+ ) and Uq (it_) are the subalgebras of U4(g1) generated by Ei , i = 1, 2,• • • ,n 1, and by Fi , i = 1,2, • • • , n - 1, respectively, and Ùq (t) is generated by Ki , I = 1, 2, • •• , n. Root vectors of Ûq (g1) corresponding to positive and negative roots of the Lie algebra gl(n, C) can be introduced by means of the braid group action

213

7.3 Representations of i,e (gl,n ) for q not a Root of Unity

(see Subsect. 6.2.1). In the present case we prefer to define them explicitly. We set Ei ,i+ i := Ei and Ei+ i,i := Fi . Then the formulas

= Ej+i ,i =

Ei,j-Filq E 1j Ej,j+1 qEi Eiilq-i

t'

i < j, i j,

r'

determine recursively elements Eti and Eii , 1 < i < j < n, of Ûq (gln ) . A direct computation shows that

KiEjk.K.Ïl =q112Eik, KkEikK k-1 = q -112 Eik, that is, the elements Ejk, j L k, indeed have properties of root vectors. If i < k < 1 or i > k > 1, then we have

[Eik,Ek i ] =

K?K -2 - K7 2 K2 k _ 12 k7 g

Eik Ei2ci

(g g -1 )Ek/EikEki

g

Ei2c /Eik ==- REik

Ekdq- 1 O.

7.3.2 Finite-Dimensional Representations of tIs "q (g1n ) Since Ki K2 - - • Kn belongs to the center of Og (gl,i ), the results of Subsects. 7.1.1-4 can be extended to the algebra Ùq (g1) as follows. Theorem 23. Let T be a representation of N(g1„) on a finite-dimensional vector space V such that its restriction to the subalgebra OA) is completely reducible. Then we have:

(0 V is a direct sum of weight subspaces with respect to tilq (1)). (ii) The representation T is completely reducible. (iii) If T is irreducible, then it is a highest weight representation

and the

highest weight subspace is one-dimensional. (iv) If T is irreducible, then its highest weight is of the form ty • qm = (w i rl , • • ,w n eln), where wi E {1, -1, i, -0 and m = (m i , • • • ,m n ) is a highest weight of an irreducible finite-dimensional representation of the Lie algebra gl(n, C). Moreover, every weight of this form is the highest weight of an irreducible finite-dimensional representation of Oq (gln ). (y) If two irreducible finite-dimensional representations of N(g1n ) have the same highest weights, then they are equivalent.

The highest weights of irreducible finite-dimensional representations of gl(n,C) are given by n integers m = (m i , • - • , mri ) such that

mi > m2 > • • • > mn• As above, the irreducible representations Tm of Ùq (g1) with highest weights qm = - • • , qmn) are called representations of type 1. The characters of finite-dimensional representations of Ûq (gln ) can be defined as in Subsea. 7.1.4. They coincide with the characters of the corresponding representations of the Lie algebra gl(n, C) and the analog of Proposition 12 is also true for el'ir(glii ).

214

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

7.3.3 Gelland-Tsetlin Bases and Explicit Formulas for Representations The Gel'fand-Tsetlin bases of carrier spaces of irreducible representations of Og (glu ) are formed by successive restrictions of the representations to the subalgebras Og (g17,_ 1 ), Ûq (gln_2), - • • , iTg (gl i ) U(gl i ). From the character theory we know that the decomposition of the representation Tm of Uq (gli,) into irreducible representations of Og (g1,.,_ 1 ) is the same as for the corresponding representation Tm of gl(n, C). Hence the restriction of Tm , m= (m i ,. • • ,mn ), to Og (gln._ 1 ) decomposes into the irreducible representations Tm,_ 1 7 (mi,n-1/ • • • 7mn- Ln-i), such that 771 1

>

Mi,-1 >

m2

>

>••'>

Mn-1,n-1 > Mn

(23)

and each of these representations enters into the decomposition exactly once. Since the irreducible representations of U(g1 1 ) are one-dimensional, we obtain a basis of the carrier space Vm of the representation Tm of t/q (g1n ) labeled by the Gel'fand-Tsetlin tableaux • •

M =

Ml,n-1 M2,n

M2,n-1

• ••

•••

• •• • •

Mn- 1,n- 1 Mn,n •••

mil where mi,„

(24) mi . The entries in (24) are integers satisfying the betweenness

conditions

m,+1

mii

mi+Li+17

i = 1,2,• • •,j, j = 1,2,• • •,n.

(25)

The set of all tableaux (24), satisfying these conditions, labels the basis elements of the carrier space of Tm . The corresponding basis element will be denoted by 1M), where M is the tableau (24).

Theorem 24. Let q be a positive number and let Tm be the irreducible representation of Ûq (gln ) with highest weiglit m. Then the generators of 64(g1n ) act on the Gelfand-Tsetlin basis of this representation by

k-1

Tm (KOIM) = qak/2 IM), ak =

E mi,k - E i=i

1 < k < n, (26)

i=1

Tin (Ek)IM) = EA(M)IM), Tm(Folm) = E A3k(mk 1
-

3)imk 3),

(27)

1.

Hem M is the GelYand-Tsetlin tableau, obtained from the tableau rnik is replaced by mi,k ±1, and ili(111) is the expression

(24) if

7.3 Representations of 1.19 (g1„) for q not a Root of Unity =

[4,10_1

111(M)

.Fr k- ri &j, kj iii=i

ij,k

—

1/2

7

li,k -11

nioi tli ,k -

215

where tir = mir - i, the positive value of the square root is taken and [m ] is the q-number defined by (2.1).

Multiplying the basis elements by appropriate factors, we obtain from (27) that

Trn(Ek)imy=Ealc(tf)Imiy, T.(Folmy=Equow3Y,

(28)

j=1

where , vriik__ 11ri6i,k-i - ii,k]

_ j,k , ajk (M) =

Hioi [ii,k - ii,k]

IYIk( M ) =

iiioi [.i,k - j,ki

These formulas are valid for any complex q which is not a root of unity. Since the proof of Theorem 24 is long and technically complicated, we give only a sketch of the proof. A complete proof can be found in the paper IUTS].

Sketch of proof of Theorem 24. Let Tm be a type 1 irreducible representation of Og(gl n ) with highest weight m = (mi, - • • , mn ) acting on the vector space Vm . For notational simplicity, we consider Vn, as a left Ûq (g1)-module and write ay instead of Tm (a)v, a E 6,7 (gin ), y E Vm . We denote by 10) the highest weight vector of Vn, (that is, Ei f0) = 0, j =1,2, • - - , n - 1). The left action of iTg (gl,i ) on Vm induces a right action of iTig (g17,) on the dual space V,!._ by yi(Xy) = (eX*)(v), y E Vm , V I E Vrfn , where * is the involution uniquely defined by (Ei )* = Fi and (K)* = K. Let (01 be a vector of Vr'n such that (01F = 0, j = 1,2, • • • , n - 1. For a, b E Og(gin), let ((01a, MO)) denote the value of the functional (Ola E Vr/n on the vector b10) E Vm . We define inductively lowering and raising operators dki and cik by dkk = 1,

dk 1 k-1 = Fk,

Ckk = 1, Ck 1,k = Ek,

dki = (K41K047")Fkdk-i,i (1(41KIT 2 q

-

k-i-i )dk_i,iFk,

cik = ai,k-lEk(la Fi K k-2 q") - —kCi,k-1(1i1+1K where (1q.K .9-2qm) denotes the expression defined by IC; 2qm) •=

1CpC 3-2 qm K j-2 1(92 q -m •

One can verify that the elements dki , i = 1, 2, • • , k, as well as the cik, = 1, 2, • • • , k, commute with each other and satisfy the relations

216

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

(q +q -1 )Fkdk_ Li Fk +

0,

— (q + q -1 )dk_1,iFkdk_1,i + Fkc1 21,_ 17i -= 0, Fkdk-2,i = dk-2,iFk

Fkdk,i= dk,iFk •

For each multi-index r = (ro , r1 , • • • rk- 1 ), ri c No , we set r

Crk. = Cricc:0 • • • drkk, k 1

-

ck

Tk_I

,ro

'k-1 ,k • • • ` -'0,k*

The main relation, used for the construction of the Gel'fand-Tsetlin basis elements 1M) from the highest weight vector, is crk dsk = f5r,s [r

k-1 ri k—i

]!

H

{

H

ll

re-? k--2 /i t-19\ \ -"2-1-1 -"i+t+1V / ,

i=0 p=1 t=1 ri

k—i-1

X ri H

mod A I

i—±2t1-1q t—P±i±ri+t )}

p=1 t=1

where [r]! = {roP[ri]! • • [r

i ]! and Jk is the left ideal of efjg (g1n ) generated

by the elements E1, • • , Ek-1Now with every Gel'fand-Tsetlin tableau (24) we associate the elements AM d rin2-mi d r2n3-1112 • • • dmn-mn-1 n-1

inn M C = Cn-1

where mk 114_1 (7Tti,k Tni,k-1, " • the following facts:

•••C

111 3 2

Mk-1,k — Mk-1,k— 1).

rn2 rn1 c1

(29) One can prove

(i) The weight of the vector dm 10) coincides with the weight of the basis vector corresponding to the Gel'fand-Tsetlin tableau (24). (ii) For any Gel'fand-Tsetlin tableaux M and M' we have

((Ol cml ,dm 10))

Sm, m/N1211 ,

r „c( nk , mk+ i ) andgk (mk, mk +i ) denotes the expression

-1 where N2m - F k=1

[ rn i<3

+i — mk +

[mi,k -

H Il.

— 1P

[Mi,k+1 [Mi,k

rn-Fi

j -

1] ! •

(iii) The vectors 1M) = dm 10) form a basis of the space Vin if M runs over the set of Gel'fand-Tsetlin tableaux for the representation Tm .

Finally, using (29) and the actions of Ek and Fk on dm I0), one proves that the actions of the operators Tm (Ek) and Tin (Fk) on the vectors 1M) are 0 given by the formulas (28).

217

7.3 Representations of r/q (gln ) for q not a Root of Unity 7.3.4 Representations of Class 1

In most applications only the simplest irreducible representations of Ùq (glu ) are needed. An irreducible representation Tm of 6,(gln ) is said to be of class 1 with respect to Ûq (g1_ 1 ) if the restriction of Tm to Ùq (g1_ 1 ) contains the trivial irreducible representation of this subalgebra, that is, the representation , 0). with highest weight (0, Proposition 25. An irreducible finite-dimensional representation Tm of edIg (glri) with highest weight m = (m i , • • • ,m,„) is of class 1 with respect to eldq (gln_ i ) if and only if M2 772 3 =• • • Mn- 1 O. Proof. The assertion follows from (23).

El

If an irreducible representation Tit , := T(0,...,0,v ) of Ûq (gln ) has a highest weight (1,0, - • • ,0,1'), 1 > 0 > l', then its Gel'fand-Tsetlin tableaux are of the form 0

• • •

O

•• . •. •

• ••

• ••

711

0

0

0

0 0

1') ,

(30)

j'

• ••

where 1 > m > j > • • • > 0 > • • • > j/ > m/ > . The operators T11iv(K707 Tui(En-i) and TIV(Fn-1) act on the corresponding basis vectors IMP') by Tit

Tiv(E n_ i )

="

Kn mmim' = tio+r-m-m'oimwre),

(31)

mypn'

milm - + n 1][m - j + l][rn - + n - + n [m m' + n -

2. ]) m m+ 1 m'

frrl! - + 11 [j - mi + n - 3 1D' +n [m m' + n - 2][m - m' + n -

)

(32) T11 i(Fn-1)1Myr) ei-m+1][m-P+n-2][m-j][m-ji +n-31) i m-1 Tre [m - m' + n - 3][m m' + n -

+ (11

+ n 1J[nil - rip m' + n 2llf - m' + 11 ) 4 im3mi;mi_ i). [m-rnt+n-L2llm-m 1 +n - 1] (33)

For representations with highest weights (1, 0, • • • , 0), these formulas turn into To (K)lmrd 0) q (1--rn)/21 . iri 1 d 0) ,

(34)

218

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras Ti,o(En-Optcdo) =

([1 _ in][m _

71,o(Fn—i) i •M;3;° ) = ([1-

1/2 1mrin0+1,0 ), 11) m 1][m -ii)1/21K10-1,o).

(35) (36)

The formulas (31)-(33) show that IM71 ) is an eigenvector of the operator Tuf (K n ) and that the operators Tv (En-i) and Tip (Fn-i) change only the indices j and j' in the tableaux (30). Similarly, the matrix elements of the operators Tip (Kk+i), Tiv(Ek) and Tw(Fk) depend only on the (k+1)-th, k-th and (k -1)-th rows of the Gel'fand-Tsetlin basis elements (30). We see from the formulas (34)-(36) that the matrix elements of the operators Ti,o(Kn) Ti,a(En-i) and T1,0(Fn_1) do not depend explicitly on n. The next proposition shows that this is the case for all irreducible representations of ejg (gin ) with highest weights (mi, m2, • • •, mi, 0, • • • , 0). Proposition 26. If an irreducible representation Tm of t,q (g1n ) has a highest weight ni = (mi,m2,- • • ,mi3 O,• • • ,0) with more than two zeros, then the matrix elements of the operators T,-,,,(Kn ), Trn(E7,1) and Tm (Fn_i) with respect to the Gelfand-Tsetlin basis are independent of n. The corresponding formulas (27) for these operators contain only i summands.

Proof. The proof follows from the expressions for the coefficients A_ 1 (M) in (27) at mi + i = • • • = m n = O. 7.3.5 Tensor Products of Representations

In this subsection we consider tensor products of irreducible representations of E,q (g1n ) for which the multiplicities of irreducible components do not exceed 1. The most important such examples are the tensor products Tm where Tm is an irreducible representation and Ti is the vector representation of Oq (gl ii ). Recall that the decomposition of this tensor product was given in Subsect. 7.2.1.

Other tensor products with multiplicities not exceeding 1 are I'm Tp , where m is any highest weight and Tp is the symmetric irreducible representation with highest weight (p, 0, • • • , 0). From (11) one can derive that T

Tp =

IEDr Trn±r

(37)

where the summation is over all r = (r i , • • , rn ) such that ri E No and ri + - • + rn = p, mi + ri > mi > n12

+

r2 > rn2 > > mn + rn > mn .

(38) If the multiplicities of irreducible components in, the decomposition of a tensor product Tmn. 0 Tm in are at most one, then the CGC's with respect to the Gerfand-Tsetlin bases can be written in the form

42,

n

(M 1M") -=

(39)

219

7.3 Representations of 64,(gln ) for q not a Root of Unity

where mi are the rows of the corresponding Gerfand-Tsetlin tableaux. As in the classical case (see [VK2], Chap. 4), it can be proved that the CGC (39) is a product of the so called Oq (gln_i )-scalar factors (or reduced CGCs) -

Mn- 1 Mn - 2

In n tt ALAn, -1 M n-1 Mn

11/11 1M")=

t/

In n-1 t, M n-2 M n- 2 M n- 1

If

X•••X

( M2

111 2

M2 II)

rn1 m1 1111

(40)

7

which depend only on two rows of the CGC. 7.3.6 Tensor Operators and the Wigner-Eckart Theorem

In Subsects. 7.3.6-9 we suppose that q is a positive number. Let Tm be an irreducible finite-dimensional representation of Ùq (g1) with highest weight m on a space Vm with Gel'fand-Tsetlin basis {IM)}. Suppose that {Rr;vni } is a set of operators acting on a Hilbert space 55 and indexed by the Gerfand-Tsetlin tableaux 1M) of the representation Tm . Let T be a representation of Og (gln ) acting on b, which is a direct sum of irreducible finite-dimensional representations. We say that the set of operators {R741 } is a tensor operator transforming under the representation Tm of Ùq (g1) if for all generators Ei , F K i we have ,

-2)

T (Ei )lr,vni - q(ai-at+i)12R7417(

EA/1(m)R741,T(Ki-iKi+ ,),

(41)

k=1

T(F)R - q (ai - ai+ 1) /2 RZT(Fi ) = EA.(mi-k)RkT(Ki-lKi+1), (42) k=1 q a, /2 Rill T(K)RIAT41 T(Ki7 1) =

(43)

where A(M) and ai a•(M) are as in (26) and (27). Repeating the arguments of Subsect. 3.6.2 one proves the following Wigner-Eckart theorem for the matrix elements of the operators R. Theorem 27. If 55 = EBm, vni is a decomposition of 55 into irreducible subspaces for the representation T and flm' ,M')} are Gelfand-Tsetlin bases of the subspaces Vml , then the matrix elements of the operators Rik' with respect to these bases are expressed in terms of the CGC 1 's of Ùq (g1) by f

(rw, /4' I RR/ 1 m", M") = E(nif ornii m"), f

m'

M

If

M' M M"

•

(44)

)

If

Here Rfl m"), are the so-called reduced matrix elements of the tensor operator which do not depend on M', M, M". The summation index r in

220

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

the tensor product

(44) distinguishes multiple irreducible representations in Tm 0 Tm".

Note that if the tensor product Tm Tml, contains only irreducible components with multiplicities not exceeding 1, then the right hand side of (44) contains only one summand and the index r can be omitted.

7.3.7 Clebsch-Gordan Coefficients for the Tensor Product Tm 011 These CGC's are factorized into products of reduced CGC's of the form

(1,0)

ez mn-i +e3 )' Mn

(0,0)

where 0 = (0, • • , 0). The corresponding reduced CGC's are given by the formulas

ei )

(1,0) (0, 0)

q-1/2(i+1+E m

n- 1

Ei#i

jT

Me . 71)

3

Mn- 1

Fr J

. ,

-

[ in

,n

n-1 2

3±

)

1.

1/2

]

( 45)

njoi[mj,n mi,n —

+ei mn-1 + e3

(fl koi

[Mk,n mj,n-1

k+

[Mk,n mi,n

k + 4 (1di

= 190 -

TT [Mk,n-1 14i

f

M j,n-1

k+ k+

1

1,2 ]

(46) where [m]--=-- [m] q is the q-number (2.1), rri B ,n are the components of the highest weight mn and we have abbreviated V(j i) := 1 if j - i > 0 and .19(j - i) := - 1 if j i < 0. In the case of the tensor product 711 O Tm , the corresponding expressions for the reduced CGC's Ci

Inn-1

+ ei

are obtained from (45) and (46), respectively, if we replace q by q -1 . Both expressions (45) and (46) are special cases of CGC's considered in the next subsection.

7.3 Representations of C4(g1n ) for q not a Root of Unity

221

7.3.8 Clebsch—Gordan Coefficients for the Tensor Product Tm OTI, The corresponding CCC's factorize into products of reduced CGC's of the form

(p, 0) (r, 0)

(i) 0 ) (0,0)

(47)

The expression for the first reduced CGC is (p, 0) (0,0) [Pi,n - kj,n-

x

m.' mnai

=qa-vEll[iii,n i<j 1 ] 1 Fl i
J.

[ii n

-11' -i j,n i j [li,n-1

where ii,n

1 j,n)

Pi,n =

(48)

-

ij

=

mfi,n

1] ! 1/2

1i/ ,n

-j

and n-1

p2

a=

2n

2=1

In order to describe the second reduced CGC in (47) we introduce the notation

spr (mp , mr)

A!

11
1/2

[Mir Mi - j + j

where mn = (m in , • • • , Trim% ). It is not difficult to verify that if mn is the highest weight of an irreducible representation Tm,,, then Snn (Inn, Mn) 2 =

TT

[Lin

(49)

ljni •

For the second reduced CGC in (47) we obtain the expression / mn

m_1

qb/2 [p

Sn,n- 1

ri! Snn( m n i

(Mn Mn-1)

Snn (Mn i Mn )8n,n- 1 (M!r„ Mn 1 _1)

X Sn-1,n-1 (Min-17 Mn-1)Sn-1,n-1(Mn-1) mn-1)

E (- 1)Ei(Mi-ini,n-1) (

n,n-- 1 (m, n7

X Sn2 n - 1 (mn M)S2

(m m)

/ n-1,n-1 f in/ 1,rt- 1 n-17 M)S2 n-1,n-1( m l Mn-1)

where the summation is over all integers m = (mi, - ,mn_i) such that max (Mi+1,n+1 Mi+1,n) 5- mi 5- min (m , _ 1 ,m)

and

(50)

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

222

b= E m-In — min) (m 3n I< j 2 ' n-1

—E

—min)

1) + (p- r) V min

- Mi,n-1)(Mi n-1 -

.(

Ez<3

M.

'

+ 1)

— min)(min

(min

%fit- 1)(M 3,n-1 - M i,n •

TI-

1)•

We sketch the proof of the formulas (48) and (50). If Tpn is the symmetric irreducible representation of Ûq (gln ), then its Gel'fand-Tsetlin tableaux are of the form

o

O P=

Pn - 2

where pn element

0

0

• ••

• ••

\ 1/2

[! Pnl )...

0

0 0

, (51)

> O. With such a tableau we associate the

Pn—i > •

RPn P m-

0

0•••0

0

Pn- 1

0

•••

n

EP= - Pi-1 H

111-1 [Pi - Pi - l P

H

i,n+

i=1

(52)

11.3 j=1

of the algebra Og (g172+1 ). Using the relations Ell,i+1 Ea = unmEr,1EZ i±i and E4_ 1 E171. 1,k = E

[nP[mPe - s)(772-8) si![in .5 Er_-1:10 Eigk E7i1; ]! 8 (n-m+1)

[s]![n sP[m - .51! where j

Erkr s

2 s K i2_; 1

± 1 < k, one proves that the elements (52) satisfy the relations

- i+i Ei, i± iRPn -1 -q -1 K-K7 1 RPn P.--i,"•,pi KK

([Pi+i - Pi][Pi Ei +1

+ 11) 1 / 2 RPn

(53)

Pn ([73i+1

-

Pi ± 1] [p

-pi_1 ] ) 1 / 2 RPn Pn-i 1" - 43= - 1,•")Pi

(54)

This means that for any finite-dimensional representation T of 61q(g1n+1), the operators ) form a tensor operator transforming under the symmetric representation Tyr, of tifq (gln ). Therefore, by Theorem 27 and formula (44) therein, for any irreducible representation Tinn+ , of ùq (g1) we have (M

RPn Pm- 11"*1P1 I M I) = BPI= :Pm -t (Inn+1)(M

P M').

7.3 Representations of ai (gin ) for q not a Root of Unity

223

Here M and M' are the Gel'fand-Tsetlin tableaux given by M =

and M'

Mn+1

mi

=

9 respectively, P is the Gel'fand-Tsetlin tableau (51),

(M I P M') is the Clebsch-Gordan coefficient and B

(mn+ 1) is the reduced matrix element depending on pn ,pn _ i , mn , mn' , where mn and mn' are the first rows in the tableaux M and M', respectively. As noted in Subsect. 7.2.2, the CGC (111 P j M') is equal to the scalar product (M P M'). Since q > 0, Tmn Tpn is a *-representation with respect to the involution of Oq (gln ) given by (Ei )* := Fi and (K)* := Ki , so that PI (Till n Tpn

(M

(En _ i )1141 = ((Tmn Tpn )(Fn _i)(M

P),

Inserting the formulas (26) and (27) we derive the recurrence relation n-1 -

E 3= •

n _1( 11 )(M P 1g

+

-= A (Pn71)q c (M P;-!11 M') An— (Ill n—i)q

Pn - Pn - 1

(Mn—i

P I AP),

where A3n. _ 1 and MZ-I 1 are as in (27) and c = m in + • • • + mnn mi,n-i •• - mn-i,n-i. Setting pn - 1 = pn_ 2 = • • = pi = 0, we obtain a recurrence relation connecting the reduced CGC's (p, 0) (0,0)

and

(p, 0) (0,0)

Inn Mn-1 )

where s takes one of the values 1, 2, • • • , n-1 and mils 1 is obtained from m n _ i if we replace ms,n—i by TY1 8 ,n _ 1 + 1. These recurrence relations determine the reduced CGC's (48) up to a normalization constant. This constant can be computed by using the orthogonality relations for the reduced CGCs. Using (53) one can easily verify that R P,..,0, ... ,0 coincides with ( [p - rPri [! [Pl!

1 r

E s=o

( _i)s q s(r-1—p/2)

[s]! Fr - sp

Er-s RP 7 1(n r ) n-1,n 0,•••,0Es n-1,n 1(7;-1

where p = pn and r -a- pn _ i • From the matrix form of this relation the second reduced CGC in (47) can be expressed in terms of the first reduced CGC's and the matrix elements of the operators Tri,(Enk_ i,n ). The latter matrix elements are given by the formula Mn-1

Tri ,n (EnP _ 1,n )

Mn-2 =

Tinn (EPn ,n_ 1 )

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

224

1 -1) nan-2) ri,ri- 1 (Inn Mn- 1) Sn-1,n-2( 111n (Sn- 1,n- 1 (Inn / _ir Mn- 1)) 2 Sn,n-1 (Inn

- 1) Sn-1,n-2(Inn-1 Mn-2) (55)

where the first rows mn in the Gel'fand-Tsetlin tableaux are omitted and

dn_ i (Mn- 1) Sn-1,n- 1 (Inn- 1 Mn- 1 ) • This leads to the expression (50) for the reduced CGCs. Formula (55) is proved by induction on p. For this, we use the recurrence relation (MI

n-1

Tmn (En in) M) =

AL 1 (M) ( 41 I Trnn (En) MZ 1) 2

x(Mnt i j Tmn (En-i,n) M), where Ani _ 1 (M) is as in (27). In this formula we substitute the expression (55) for the operator Tinr, (E1,) and use the equality

1- x1x2 ". Xn

=

.9=1

(1 —

H x8)

y, - xiyi yi iOs y s

(56)

7

which is well-known in combinatorics (see [Mac]). After some simplifications we obtain then formula (55) for Tir,„. (Er,'

7.3.9 The Tensor Product Tm T1 for e l —0 0 Let us consider Uq (glu ) as a one-parameter Hopf algebra deformation of the universal enveloping algebra U(g1) and assume that q > 0. The CGC's of Uq (gln ) are then functions of the parameter q and we can look at their limits for q 0 and q oo. If q 0, then a direct calculation shows that the reduced CGC's (45) and (46) vanish except for the following cases: =1

liM q -*0 Mn-1

lim

if

Mk,-1 = Mk+1,n,

mn

< k < n,

q-*0(mn-3.

Mn liM q -03 ma-i

=1

where in,t i := mn + ei. Similarly, when q (45) and (46) vanish except for liM q-, 00 •

fl-* 00

-

Mn Mn _

m±i

mn ( Mn

i < k < n,

1

M n- 1

(1, 0) Inn+i

(0, 0) mn-i = =1 if i

if

i = j,

oo, then the reduced CGC's

if

i=1

j +1 or i= j and mi n-1

Min •

7.4 Crystal Bases

225

Each CGC of the tensor product T., 0 Ti is a product of several reduced CGC's of the type (46) and one reduced CGC of the type (45). A closer analysis of these reduced CGC's shows that for fixed j and M' all CGC's ( m7, (1, 0) mrt i) vanish in the limits q 0 and q -* co except for one. ' Thus, by (17), for every basis vector Im M') of the representation space of T +. from (13), there exists a unique basis vector Ilan , M") 01(1, 0), M) oc of the carrier space of Tm„ such that in both limits q -* 0 and q we have M") 01(1,0),M). M') = Now we apply this procedure inductively starting from the tensor product T1 0 71 of two vector representations T1 of Ug (gin ). Let v 1 , v 2 , • - - , v the elements of the Gerfand-Tsetlin basis of T1. Since the irreducible representation Tmn with highest weight mi, = (min 7n2n • • • Innn), ninn 0 , is contained in the tensor product of m = min + • • • + m nn copies of the vector representation T1 , then at q = 0 and at q = oc every Gelfand-Tsetlin basis vector of the representation space of Tnr,„ takes the following simple form: ,

vi l

vi 2 0 • • •

v

,

= M in + • • • + Mnn

E {1, 2, • • • ,n}.

This observation is a starting point for the construction of crystal bases for spaces of finite-dimensional representations of Drinfeld-Jimbo algebras

N(g).

7.4 Crystal Bases The construction of convenient bases in representation spaces is one of the main problems of applied representation theory. For example, Gel'fandTsetlin bases for irreducible representations of the Lie algebras gl(N, C) and so(N, C) have been proved to be a useful tool for a number of applications of representations in theoretical physics. In the preceding section an analog of such bases has been constructed for representations of the algebras eldq (g1N ). However, it is not known how to get Gel'fand-Tsetlin bases for Uq (soN), because there is no Hopf algebra embedding Uq (solv_i) C Uq (soN). This section deals with the construction of Kashiwaras's crystal bases for finite-dimensional representations of the Drinfeld-Jimbo algebra U (0) over the field Q(q) defined in Subsect. 6.1.2. Roughly speaking, these are bases of vector spaces at "q 0" that behave well simultaneously with respect to the action of all generators E i and Fi of U4(9). (These statements are made precise in Definitions 1 and 2 below.) In applications to exactly solvable models in statistical physics the parameter q plays the role of the absolute temperature. Hence crystal bases desribe the behavior of the model at absolute temperature zero which explains the origin of their name. In this, section we give ra brief exposition, without proofs, of the main results on crystal bases. The complete proofs of the results are quite involved

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

226

and can be found in Kashiwara's original paper [Kas2I, see also [Kas1] and [Kas3]. For notational simplicity, we shall use the language of left modules rather than of representations, thus following the terminology used in [Kas2].

7.4.1 Crystal Bases of Finite-Dimensional Modules Recall that Q(q) is the field of rational functions in one variable q with coefficients in the rational field Q. Let A be the subring of 12(q) consisting of all rational functions without poles at q O. For a E A we identify the coset a + qA with the rational number a(0). Then we have A/qA = Q. Definition 1. A basis of a vector space V over Q(q) at q = 0 is a pair (L, B) such that L is a free left A-module satisfying V Q(q) OA L and B is a basis of the Q-vector space L IqL. Let V1 and V2 be vector spaces over Q(q) 'with bases (L 1 , B1) and (L2, B2) at q = 0, respectively. It is easily verified that the pair (L1, B1) ED (L2, B2) (L1 e L2 1 B1 U B2) is a basis of the vector space V1 ED V2 at q = 0 and that (L i , BO 0 (L2, B2) = (L1 OA L27 B1 0 B2) is a basis of 0 V2 at q = 0, where B1 B2 := -01 b2 I bi E Bi b2 E B21. The following example will play a crucial role in what follows.

-IN,

Example 2. For 1 E let V1 denote the (21 + 1)-dimensional simple left Uq' (s12 )-module defined by the formulas (3.22) and (3.23) with w = 1, and

let

h := e1

E V/ be its highest weight vector. Set F(i) := PIN! and fi(1) :--=

F (i) fo for i E No . The vectors 4(1) , j =-- 0, 1, • • • , 2 1 , form a basis of V1 such that

Kf 2(1) =

Efi(1)

[21 - i+l]fi(1) 17 Ffi(1)

(57)

If Li := iED Afi(I) and

B1 := Ifi(1) mod qL1 Ii= 0, 1, • • • , 211,

then the pair (L1, B1) is a basis of the vector space Vi at q = O. Next we define linear operators E j = 1, 2, - • • , rank g, on each finitedimensional type 1 left NW-module V. If g = $12 and V = V1 is as in Example 2, we set Éfi, = fi-1 and = fi+i for i = 0, 1, • • , 21, where f-i f214-1 := O. If V is an arbitrary finite-dimensional type 1 left U.'7 (s12 )module, then we decompose V as a direct sum of modules Vi and define È and Ê on V as direct sums of the corresponding operators for the modules Now let V be a finite-dimensional type 1 left NW-module. The subalgebra u,17 (9) i generated by the elements E, F, Ki, K1 -1 is obviously isomorphic to (812). Hence V is in particular a left U4, (sli)-module and the operators ti and Fi are defined by the preceding. Summarizing, if V = EDA VA is the weight space (k) F,i fk with decomposition, then every element f E VA is of the form f fk E 150-kail Eifk = 0 and we have defined ,

,

N,

Ek

7.4 Crystal Bases

227

tif Ek Fi(k-O fk, pif Ek Fi(k+i) fk. Finally, we can give the definition of crystal bases. Definition 2. A crystal basis of a finite-dimensional type 1 left Uq' (9)-module V is a basis (L, B) of V at q = 0 such that the following conditions hold: L = EBAE p LA and B = UAE p BA where LA = L n VA and BA =

B n (LxIqLx). (ii) t i (L) g L, F2 (L) Ç L, t i (B ) g B U 101, F(B) Ç B U {0 } for = 1,2, • • • ,n = rankly (iii) For any b 1 , b 2 E B and i G {1, 2, • • • , n} we have t ib2 = b 1 if and only if Fb1 = b2. Note that because of t i (L) C L and Pi (L) C L the mappings Et, and Pi pass to the quotient LiqL and define mappings, again denoted by ki and Pi , on LAL, so that t i (b) and Pi (b) are indeed well-defined for b G B. Conditions (i)--(iii) express the "nice" behavior of the basis (L, B) with respect to the actions of the generators Ki , Ei, Fi on V. For instance, if b E B, then either t i (b) = 0 or fi(b) G B by (ii). Further, if fi (b) G B, then b = FE(b) by (iii). The following assertion follows easily from Definition 2. Lemma 28. If (L i , Bi ) is a crystal basis of the N(9)-modules V3, j = 1, 2, • • r, so is the pair (L, B) := (ED i Li ,Lj i Bi ) for the direct sum EDi Vi .

7.4.2 Existence and Uniqueness of Crystal Bases Let us begin with the simplest case g = 512 . The pair (L 1 , B1) from Example 2 is obviously a crystal basis of the q(s12)-module Vi. Since any finitedimensional type 1 q(s12 )-module V is a direct sum of such modules V, it follows at once from Lemma 28 that V has a crystal basis. The existence of crystal bases in the general case is much more subtle. For A E P+ , let V(A) be the irreducible left N(g)-module with highest weight A and highest weight vector ex. We set CX)

L(A)=

E EA1-2

r=0 ,•••,i r =1

B(A) =

{

Pi,

• • • Pir

1 '••

Pi, (eA),

(e),) mod TWO I i , • • • , i r E 1 1 7 2 1' • ' nil.

Theorem 29. (L(A), B PO) is a crystal basis of V(A). Any finite-dimensional type 1 left Uq' (g)-module has 4 crystal basis.

Proof. By Lemma 28, the first assertion implies the second one. It is obvious that B(A) spans the Q-vector space L(A)/qL(A) and that Pi (B) Ç B U 01. The complete proof that (WO, B(A)) is a crystal basis is rather long. It is given in [Kas2) by an induction procedure called the grand loop.

228

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

The following proposition says that crystal bases are unique up to isomorphism. Proposition 30. Let (L, B) be a crystal basis of a finite-dimensional type 1 left (I:, (g)-module V. Then there is an isomorphism -y of V to a direct sum EDA V (A) of highest weight modules V(A) (possibly with multiplicities) which maps (L, B) to the crystal basis (EDA WO, UA B(A)) of EDA V(A).

Proof. A proof is given in [Kas21, p. 478, using Theorem 29. 7.4.3 Crystal Bases of Tensor Product Modules One advantage of crystal bases is their behavior under tensor products. Before we state the corresponding result, we consider an example. Example 3. From Subsect. 3.4.1 we know thai the tensor product left N(812)module 14 0171 12 (see Example 2) decomposes as a direct sum 17/±1/21EDV/-1/2. The vectors

w (1+1/2) =

(1) i

(1-1/2)

2 [

1

fo( 1 / 2)

1

21

(

[21]

_r(1) ® f1(1/2)

1/2) ( )

f

i= 0, 1, - • , 21+ 1,

q[i -I- 1 /1 (1) (1 [2/1 f' +1 ®f0

2)

7

= 0,1, • • , 2/ - 1,

respectively, on which K, E and F act by (57). are bases of 17/ ±1/2 and ii1/2, It follows from these formulas that =

of({1 11'2) ( mod q (L(1) ®L(1/2)

and

= f(i)

))

< 217 41++ 1 /2) =

f (1 /2) ( mod q (L(/) 1

®fP /2)

L(1 1 2))).

(58)

(59)

The pair (L, B), where L = L (1+ 112 ) e L (I -1 / 2) and B consists of the vectors (58) and (59), is a crystal basis of Vi 0 171/2. A

Let (L, B) be a crystal basis of a U(g)-module V. For b e B we set ei (b) = max{k

G NI

E(b)

G B},

O(b) = maxik G NI Pik (b) G Bl.

Theorem 31. (0 Let (L1, B1) and (L2, 132) be crystal bases of two N(9) modules V1 and 172 ) respectively. Then (Li, B1) (L2, B2) is a crystal basis of V1 ® V2. (ii) If bi G B1 and b2 G B2, then for any j e {1 2, • • , n} we have

, ki(b i b2)

b2 for b 1 2ib2 for

Pi (bi 0 b 2)

b2 b1 ®frb2

Oi(b1)

(bi ) <

fi (b2)1 Ei(b2 )1

for (MN) > Ei(b2),7 for MN) -15 4(b2).

7.4 Crystal Bases

229

Sketch of Proof. (i) follows easily from (ii). To prove (ii), it suffices to assume that g = 812 . In this case the assertion can be derived from the formulas for the Clebsch—Gordan coefficients. See [Kas2] for details. 1=1 To a crystal basis one can associate its crystal graph. A directed graph consists of a set of vertices and a set of ordered pairs of vertices, called arrows. A directed graph is called colored if it is equipped with a map from the set of arrows to another set, the so-called set of colors. Let (L, B) be a crystal basis of a N(g)-module V. We define the crystal graph of (L, B) as the colored graph with set of vertices B and set of colors {1, 2, • , n } , n == rank g, such that elements b, b' E B are connected by. an arrow, directed from b to b', with color i if and only if b' = F(b) (and hence b = Éi (b') by Definition 2(iii)). The crystal graph of a crystal basis (L(A), B(A)), A E P± , is a connected graph, because every vertex b E B(A) is obviously connected with the highest weight vector. Suppose that (L, B) is a crystal basis of a type 1 finite-dimensional U(g ) module V which is isomorphic to a direct sum Ep i V(Ai ) of irreducible modules VP4), Ai E P. It can be shown that then the crystal graph of (L, B) is (isomorphic to) the disjoint union of the crystal graphs of all (L(Ai ), B(A)). Thus, the decomposition of V into irreducible components is equivalent to the decomposition of the crystal graph of (L, B) into connected components. From Theorem 31 we can read off the crystal graph of the tensor product V1 0 V2 of two N(g)-rnodules V1 and V2 . Without going further into technical details we only state that it is obtained as the union of tensor products of each component of the crystal graph of (L 1 , B 1 ) with each component of the crystal graph of (L2 , B2 ). Combined with the result of the preceding paragraph, one obtains in this manner an interesting graphic algorithm for the irreducible decomposition of the tensor product V1 0 V2 (see [KM and [Nap. We illustrate this by a simple -

Example 4. The crystal graph of the crystal basis of the N(s12 )-modu1e is fi: 1)

f (1)

f 1) . Hence the crystal graph of V1 0 Vi is hçi l )

0 41)

f(l)

f

fi:(:1 1 )

p) fio)

fil) A l) Therefore, as discussed above, V1 0 Vi decomposes as V2

e Vi ® Vo .

7.4.4 Globalization of Crystal Bases

Let (WO, B(A)) be the crystal basis of the N(g)-module ITN, À E P±. Then -B(A) is a basis of the Q-vector space WNW). In this subsection we

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

230

shall show how to obtain a basis of the Q(q)-vector space V(A) from B(A). Let us fix an irreducible finite-dimensional N(g)-module V(A), A E P±, with highest weight vector e),. There exists a Q-algebra automorphism rl of defined by

N(9)

ri(Ei ) = Ei , rI(Fi ) = Fi ,

?IWO =

, r(q) =

From Proposition 6.28 it follows easily that there is a Q-linear mapping ri : V(A) -4 V(A) such that ri(ae),) = ri(a)e)„ a E N(g), Clearly, A := ri(A) is the subring of functions in Q(q) without poles at q = Do and L(A) = 71(LN) is a free A-module such that V(A) --= Q(q) OA MA). Further, we set VQ(A) =

E (Kg, q -1 1F4n1) • Fi(n r ) eA.

The crucial step for a globalization of crystal bases is Proposition 32. The natural map ( : L(A) an isomorphism of Q vector spaces.

n L(A) n vQ (A)

L(A)IqL(A) is

-

Let B(A) be the set ( -1 (13(A)). Then we have Theorem 33. MA) is a basis of the Q(q) - vector space V(A).

The basis B(A) of V(A) is called a global crystal basis of V(A). One can also prove that FiniV(A) =

Q(q)b, bEB(A),e i

where

Ei

(

ETV(A) =

IED

Q(q)b,

b)]rri

and Oi are as in Subsect. 7.4.3.

7.4.5 Crystal Bases of Uq1 (n_) In Subsects. 7.4.1-4 crystal bases of type 1 finite-dimensional N(g)-modules have been investigated. A similar theory can be developed for the subalgebra U(n) of which is generated by the elements Fi , i = 1,2, • • • , n = rank g. This will be sketched in this subsection. In order to imitate Definition 2 we first need to define the counterpart to the operators ki and P for Uqi(n_). Putting

N( 9)

E(x) :=

- qi7 1 )1Ci (Eix xEi), x E

Uq'

one easily verifies that E(Fi x) = q -( ""al ) FiE(x)+ bijx,

(60)

Ei'(xy) = gi (x)y + cri(x)E:(y)

(61)

where ai (r) := KixiCr i and x, y E Uqi (n....): Equation (61) means that g is a skew derivation of the algebra r.74(n....) with respect to the automorphism ai. -

7.4 Crystal Bases

231

By (60) and (61), we have E(F) =1 and g(Firn') = g2l _m[rn]1 im -1 . Further, the linear mapping g leaves N(n_) invariant and for any x E U(n....) there exists m E N such that (E)m(x) = 0. Using the preceding facts it can be shown (see [Kas2], Subsect. 3.2.1) that any x E U(n_) is of the form

x

nym

) = 0, k E N, Finy m ), m=o with uniquely determined elements ym E N(n_). Then we define E' (x)

= E

k

Ek

(m _ 1) .F14

(Yin) Pi (X) = Fi(In+1) (Yin) m=0 Now Definition 2 can be adapted to the present situation as follows. A crystal basis of Uq' (n_) is a basis of the Q(q)-vector space Uqi(n_) at q = 0 (see Definition 1) such that the conditions 00 and (iii) of Definition 2 are fulfilled. Note that condition (0 of Definition 2 is not needed here. Since obviously EF = I, it follows from (ii) that Pi (b) E B for any b E B. One can prove that Piki is the projection of U(n_) onto Fi • N(n_) with respect to the direct sum decomposition q(n_) = Fi • (.41 (n_) ker.g. Let goo) be the A-linear span of elements Pi , • • • Pir 1 for 7:1) • • , ir E {1,2, • , n } , and let B(oo) be the set of images of these elements in L(oo)/qL(oo). Further, by Proposition 6.28, every irreducible N(g)-module ITN, A E P+, is isomorphic to a quotient of the Verma module MA, which can be realized as a N(g)-module N(n_) by Subsect. 6.2.5. Therefore, there exists a homomorphism 7r), of the U:2 (g)-module U:i (n_) to V(A).

=

1.1(711,=-1

Proposition 34. (0 (goo), BOO) is a crystal basis of Uq' (n_). (ii) The crystal basis (L(A),B(A)) of the Uq' (g)-module V(A) (see Subsect. 7.4. 2) is obtained by L(A) = 71),(goo)) and B(A) = fx E BOO Tr),(b) 01, L(A)1qL(A) is the induced map of 7rA : L(oo) —> where Tr), : L(oo)/qL(oo)

L(A). Also the globalization of the crystal basis (L(oo), B(oo)) of q(n_) proceeds in a similar manner as in the case of N(g)-modules. If U4(n.4 denotes the Q[q, q -1 1-subalgebr a of Uqi(n_) generated by the elements Fi(m) , then it can be proved that the canonical map

: U4(n_) n goo)

nL(°°) —› goo)/qL(oo)

is an isomorphism of Q-vector spaces. Theorem 35. (j ) The set B(oo) := ( - '(B(oo)) is a basis of the Q(q)-vector space N(L.). (ii) For each A E P±, the homomorphism 7r), maps the set of all b E 13(00) such that 'WA(b) 0 to a basis of the Q(q)-vector space ITN. The set B(oo) is called Kashiwara's global crystal basis of Uq' (n_). It can be shown (see [L5] and [GL]) that it coincides with Lusztig's canonical basis which was constructed in [L4] and [L5] using a different approach. A detailed exposition of the latter can be found in the book [Lusl.

232

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

7.5 Representations of Uq (g) for q a Root of Unity In this section we study representations of Drinfeld—Jimbo algebras Uq (g) when q is a primitive p-th root of unity, where p is odd and p > di , i = 1, 2, • • • , I (l is a rank of g). We denote the parameter q in this case by E.

7.5.1 General Results In what follows, we shall use the notation and some facts from Subsect. 6.3.5. Recall that 3, is the center of the algebra U,(g) and Spec (3,) is the set of C. Let Rep (U (g)) denote the set of all all algebra homomorphisms x : 3, equivalence classes of irreducible representations of U,(0). If T is an irreducible representation of U(g) on V, then any T(Z), Z E 3,, is a scalar operator on V, that is, there is a çomplex number XT (Z) such that T(Z) = XT(Z)I. Obviously, XT : 3, C is an algebra homomorphism, called the central character of T. Clearly, equivalent representations have the same central character. Therefore, the assignment T XT defines a map

W: Rep (U ,(g))

Spec (3 6 ).

Proposition 36. Every irreducible representation finite- dimensional.

(62) of the algebra U6 (g) is

Proof. Let IXT be the two-sided ideal of (4(g) generated by the set ker XT {Z — XT (Z)1 I Z c 3, } . Since ker XT C ker T, the irreducible representation T of U,(9) induces an irreducible representation T of the quotient algebra Ur := Ue (9) /IXT . By Theorem 6.49, the algebra Ur is finite-dimensional.

Hence its irreducible representation T is finite-dimensional (see [CR]).

LI

Proposition 37. The mapping W is surjective, that is, for any x E Spec (3,) there exists an irreducible representation T of U,(g) such that xg-, = x. Idea of proof. As in the preceding proof, U := U,(9)/J7 is a finitedimensional algebra, where ix is the two-sided ideal generated by ker x. It can be shown that r./c {0}. Any subrepresentation of the regular representation of U,x gives an irreducible representation, say T, of

U,(g). Since T annihilates ker x, we get xg-, = x. The mapping W is not injective in general. The main properties of this mapping are contained in the following Theorem 38. There exists a nonempty closed proper subset D -of Spec (3,) such that: (i) If x E Spec (3,) and x D, then W -1 (x) consists of one irreducible representation. This representation is of dimension pn , where n is the number of positive roots' of g. (ii) If x E D, then W -1 (x) consists of a finite number of irreducible representations and their dimensions are strictly less than pn

7.5 Representations of Uq (g) for q a Root of Unity

233

Proof. A proof is given in the paper [DCK]. During this proof it is shown that for any X E Spec (3,), x D, the quotient algebra E/c = U,(g)//x is isomorphic to the matrix algebra Mpn (C). This algebra has, up to equivalence, a unique irreducible representation which is of dimension pn

Recall from Subsect. 6.3.5 that the center 3, is algebraic over 30 (that is, each element of 3, satisfies a nontrivial algebraic equation with coefficients in 30) and that the set Spec (30) is isomorphic to C 2n x (Cx ) 1 , where Cx = C\{0}, n is the number of positive roots of the Lie algebra g and 1 is the rank of g. From these facts we conclude that Theorem 38 gives a parametrization of the irreducible representations T of U, (g) such that XT D by 2n +1 = dim g complex parameters. If XT E D, then additional parameters are needed. Note that the set Spec (3,) is a complex algebraic variety of dimension p2n+ 1 . It can be shown (see [DCK]) that D is a proper subvariety. This means that almost all irreducible representations T of UE (g) are of dimension pn • Let us illustrate Theorem 38 by the simplest example. Example 5 (U,(s12 )). By Proposition 3.15, the center 3, of the algebra U,(s12 ) is generated by the elements EP, FP, KP, K - P and C,. (Note that q has been replaced by E.) All irreducible representations of U,(s12) have been classified in Sect. 3.3. Recall that these are the cyclic representations TabA l the semicyclic representations Too, and 714", and the irreducible representations T1 1, 21 < p (see Propositions 3.17-19). All representations TabA, 2 -1(j bA and T„, (p_ i)/2 are p-dimensional. The only irreducible representations of dimensions less than p are T1, 1 < (p-1)/2. For these representations we have T1(EP) = T1(F) = 0, T1 (K') = (ALT and T„i(C,) = wci := w ( 6 21-1-1 + 6 -21-1)( 6 _ 6 -1)-2 1 . The set Spec (3 E ) is determined by the points (s, y, z, c) E C 4 , where T(EP) = T(FP) = yI, T(KP) = zI, T(C) = cI and T is an arbitrary irreducible representation of U,(s12). The 3-tuple (x, y, z) runs over the set C2 x Cx and for each such 3-tuple (x,y,z) the corresponding values c are related to x, y, z by formula (6.110). Thus the subset D c Spec (3,) consists of the points

(0, 0,co,wci),

=

= 0,

1, • • •

5 - 1.

A

Proposition 39. For any nonzero a E U,(0) there exists an irreducible representation T of U(g) such that T(a) 0. Proof. By Theorem 6.49, the algebra U,(0) is a finite-dimensional vector space over 30 with a certain basis, say xi, • • • , xd. Hence we can write a as a = Ei xizi with zi E 30 . Since a 0, there is an index k E {1, • • • , d} such that zk 0. By the remarks after Theorem 6.50, Spec (30) separates the elements of 30. Hence there exists X I E Spec (3 0 ) such that x'(zk) 0. We choose a character X E Spec (3,) such that x' = x on 30. Since D is a proper subvariety of Spec (3,), we can choose x' such that x fl D. Now we show that a has a nonzero image in U,(0)11-x. Assume on the contrary that a E jX. Since Tx is the two-sided ideal of U(g) generated by ker x, it follows then that a can be expressed as a = Ei biyi , where

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

234

bi E (.4 g), yi E 3, and x(Yi) = O. If we represent the elements bi also as bi = Ei xizii with zii E ho, we obtain a --= Ei xz = Eiii sizii yi. Comparing the coefficients of the basis element xi, we get zk = Ei zk i yi . Since x(zk) = xi(zk) 0 and x(yj ) = 0 for all j by construction, this is a contradiction. Hence the canonical image of a in U,(9)//x is indeed nonzero. As noted in the proof of Theorem 38, U6(g)/IX is isomorphic to the matrix algebra Mpn (C). Hence the composition of the quotient map (Mg) U6 (g)//x with the isomorphism U6 (9)/27 Mp n (C) defines a pa-dimensional irreducible representation T of U(g) such that T (a) L O. 1=1 (

Let T be an irreducible representation of U(g) on V. As in the case when q is not a root of unity, one can prove that T is a weight representation, that

is, V = EDA V,,„ where V), {v E V I T(Ki )v = e (A, az ) v, = 1,2, • • • , /1. However, T is not necessarily a highest or lowest weight representation. The problems of classification and explicit descriptions of all irreducible representations of U(g) are only partially solved. For instance, the dimensions of the representations T with )0-, E D are not even known (see [DCKP]). 7.5.2 Cyclic Representations An important class of irreducible representations of the Drinfeld—Jimbo algebras 14(g) are the so-called cyclic representations. Definition 3. An irreducible representation T of (Mg) is called cyclic if T(E) and T(Fi ) j = 1,2, • • • , 1, are nonzero scalar operators.

For Proposition 40 and Corollary 41 we assume that p and the determinant of the symmetrized Cartan matrix (da) have no common prime factor. Proposition 40. Let T be a cyclic representation of (1,40 on a vector space V. Then all weight subspaces of V are of the same dimension and the dimension of V is divisible by p 1 , where 1 is the rank of g. Moreover, T is neither a highest nor a lowest weight representation. Sketch of proof. If VA and VI, are two weight subspaces of V, then A — = kiai, where ki E Z and • • • , cei are the simple roots of g. Set X:ci if ki > 0 and 'Vic' = Ft' for k i < 0. By the definition of a cyclic representation, T(4 1 ./TC 2 • • • 41 ) is an invertible operator from VI, to VA.

E.

Therefore, dim VA = dim V. Since ker T(E) = ker T(F i ) =- {O} by Definition 3, T is not a highest and not a lowest weight representation. _ Finally, we prove that dim V is divisible by 1p. Because V = VA as noted above and dim VA = dim VI, as just shown, it suffices to prove that VA = VI if and only if A — E pQ, where Q is the set of all Ei niai, ni E Z. By (6.13), VA = ym means that Ei diaij kj 0 (mod p), i = 1, 2, • • • , /. Since p is coprime to det (d iao) by assumption, this is true if and only if all ki are divisible by p. Hence dim V is divisible by ,

7.5 Representations of Uq (g) for q a Root of Unity

235

Corollary 41. If T is a cyclic representation of U(g) on a vector space V, then pi < dim V < pn, where 1 is the rank of g and n is the number of positive roots of g.

In order to construct explicitly cyclic representations of t4(g), one uses compositions of certain algebra homomorphisms of U,(g) to an auxiliary algebra 233 chm defined below with irreducible representations of 233q,,• Let 213, denote the algebra with generators x,x -1 ,z,z -1 and defining relations -1 ZX = EXZ, XX = X 1 X = ZZ -1 = Z-1Z =.• 1. In the literature the algebra 213, is occasionally refered to as the q- Weyl algebra.

Let X and Z be the operators acting on a p-dimensional vector space V with basis e l) , e l , • • • , ep _ i by the matrices 0 0

X

1 0

0 1

.••

• .•.

.. •

•

••• •

• • •

1

000 (

1

0

0

• ••

0

0 )

0

Z

0

0 ••

0

•

0

..

E

2

•••

• • •

0

0

( .0 00

••

0

0 0 • • .

•••

ep— 1

that is, Zei = fi e,

Xe i= ei+i ,

where

ep e0 .

(63)

Since EP = 1 by assumption, we have ZX = EXZ and XP = ZP = I. Thus there is an irreduciblerepresentation 7r of 213, on the vector space V such that r(x) = X and 7r(z) Z. For m E N, let ar,,, be the m-fold tensor product of m (commuting) copies of the algebra al,. The corresponding generators of 2/3,, m are denoted by xi and zi , i = 1, 2, • • • , M. Let g = (g, g, • • • gm) and h = (hi, h2, • • • , hm ) be m-tuples of nonzero complex numbers gi , hi . We define an irreducible representation rgh of the algebra 213,, m on the vector space V®m by irg h(xi)=10•••010giX010.•.01, rg h(zi)=10-••01ghiZolo•••01,

where X and Z both act on the i-th tensor factor. 7.5.3 Cyclic Representations of the Algebra Uf(slt+1) Set m := 1(1 + 1)/2. We shall label the generators of 211,,m and the entries of the m-tuples g and h from Subsect. 7.5.2 as xi,, zji and gii , hii , respectively, with i < j, i,j = 1,2, - • 7 1. The main step for the construction of cyclic representations of Wsli+ i) is the algebra homomorphism pri, from

236

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

Proposition 42. Let r = (ri7r27• • ,r i ) and s = (s i , s2, • • • ,si) be two 1tuples of nonzero complex numbers. Then there exists a unique algebra homomorphism pr . : Uf (sli +i ) 90, 5,„ such that prs (Ki) = ri si-1 4zillo zT+11,1 and —1

prs(Ei)=E 1k =i f Prs(Fi) =

faJikZi,k+1 • • • Xi/ 7

E k=1 , -1

-1

X X 2+1-0-1-1—k X 71 2-1-2—k,i+2—k - - • .L, i 1

1

where {z} denotes the expression {z} = (z - z -1 )(f Proof. A proof is given in PDJMM2].

Now we define a representation a -a- o-rsg h of UE (sli+i) on Vom by composing the homomorphism Prs : Uf(s 1/4-1) ME,,, and the representation G(V°m). The representation o-rsgh depends then on 21 + 2m 71-gb : 23 ,m complex parameters ri , si , gii , hii . However, not all of these parameters are independent. It can be shown that the same set of representations arei gh is obtained if we put si = 1, i = 1,2, • • • ,l. Then a o-rgh depends on dim sli±i = 1(1 + 2) complex parameters. Theorem 43. For a generic choice of the parameters r, g, h (that is, except for a set of Lebesgue measure zero in C1(1± 2) ) the representations a = ar gh of UE (sli+ i) defined above are irreducible and cyclic. Proof. The proof can be found in [DJMM2].

LI

Recall that the dimension of the carrier space V (3'4 of any representation argh is pm, where m = 1(1 + 1)/2 is the number of positive roots of the Lie algebra sl(/ +1, C). That is, all irreducible representations o-rgh obtained from Theorem 43 have the maximal dimension (see Corollary 41). For special values of the parameters gii , hii , ri , the representations a are reducible (but not completely reducible in general) and we get invariant subspaces of the carrier space V®/ (1±' ) /2 . Let us treat such an example. Let i and j be integers such that O' < i < p 1 and 1 < j < I - 1. In (63) we used the basis ek, k = 0,1,2,• - • ,p 1, of the space V. Let ei ,i be the vector from the tensor product of j(j + 1)/2 copies of the space V of the form e, = ei ei 0 • • • 0 ei , where the vectors ei belong to the tensor factors V labeled by all pairs (k, n), 1 < k < n < j. We consider the linear • • V of Vol(i+ 1) 12 , where the tensor factors subspace Vi d Ceij 0 V V correspond to labels (k, n) such that 1 < k < n, j
-

= h-1 n+1—kan h'

1
n < jt

(64)

1 < k < n < j.

(65)

7.5 Representations of U,(9) for q

a Root of Unity

237

It can be shown that the system of equations (64) and (65) has a unique solution hk , 1 < k < n < j. The parameters rk, k = j + 1 )• • • ) 1 1 gkn, 1
Since 170,1_i = V®/, it follows easily that the corresponding representation of U,(sln+i ) realized on the invariant subspace 170 ,1- 1 is irreducible and cyclic. It has the minimal dimension pl . 7.5.4 Representations of Minimal Dimensions

Recall from Corollary 41 that the dimensions of cyclic representations of U, (g) are greater than or equal to p l , where 1 is the rank of g. in this subsection we construct for g of type A1, B1 or Ci cyclic representations of Uc (g) of dimensions pi. The main tool for doing this is the algebra homomorphism p obtained in the next proposition. Proposition 45. Let us write (y) for the expression y + y -1 . There exists an algebra homomorphism

Taf,i+1 such that p(K i ) = Ç'z +1 and

p: U,(s114-1)

p(E) --=

-1 X i Xi-I-1 1 -1/2 \ Z1 / ' ( e 1/2 ± 6 -1/2)2 \ f

,I LI \ — 1.-i" 2)

,,, i...,,,-1 ..,.., i±i

--

( f'!2

i -r- f-'!2)2 \

(ii) p : U,(sov+i) -4 213E,1 such that p(Fi ),p(Ei ), p(Ifi ), i < 1, are as in

(0 and

-1

p(E1) =

x1

le -v2 zo,

19( FL ) = I

( 6 1/2 ± 6 -1/2)2 \

p :UE(sP21)

p(Ki) = z i-1 ;

TJ€2,1 such that p(Fi ), p(Ei ), p(K i ), i <1, are as in (i )

and p(E1) =

X

-2

( f8

f -8)2

6 -8

zi)(e-24zo,

P(Fi) =

p(Ki) =

Proof. The proof follows by direct verifications of the defining relations.

0

As in the preceding subsection, we define a representation crgh := rgh O p of (Mg) for g = sli +1 , 80 21+13 SP21• Proposition 46. (i ) The representation agh of Msli +i) is a direct sum of p irreducible representations of minimal dimension p l . The carrier subspaces of these p subrepresentations are just the eigenspaces of the operator C7hg(Z1Z2 " ' Z1+1)-

238

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

(ii) The representations Ugh of U,(sov + i) and U, (sp2i ) are irreducible and of dimension p'. (iii) Any cyclic representation of U,(9), g = sil+1,s021+1, S P21 7 of dimension pi is equivalent to a representation from (j) or (ii), respectively. (iv) The dimension of any cyclic representation of U,(so2i) is strictly greater than p 1 . Proof. The proof is given in [CP2].

All cyclic irreducible representations obtained in parts (i) and (ii) of Proposition 46 have dimension pi . Therefore, by Corollary 41, it follows that for UE (sli + i), UE(so2t±i) and Uf (sp21 ) the number p1 is indeed the minimal dimension of cyclic representations. It can be shown (see [CPC that the minimal dimension of cyclic representations for the algebra U,(so8) is p5 . 7.5.5 Representations of U,(sli+ i ) in Gel'fand-Tsetlin Bases

In this subsection it will be shown that the method of Gel'fand-Tsetlin bases works for the Drinfeld-Jimbo algebra U,(sli +i ) with E a root of unity as well if the parameters are chosen suitably. We fix complex numbers mi ,/ ±1 , i = 1,2, • • • , 1 + 1, and cik, hj k, 1 < j < k < 1, and suppose that the differences h ik - hjk are not integers. Let V be a complex vector space with a basis labeled by the tableaux (M) of the form .•.

M2,/±1

••• •••

77121 -

M14-1 •

/711+1,1+1

• •

Mi- 1,/- 1 • • •

11211

where the first row is fixed by the above numbers and the mij , 1 < i < j < 1, run independently over the values hii, h ii + 1, • • • , h ii + p - 1. Thus, the dimension of the space V is pl (1+ 1) / 2 . Recall that 1(1 + 1)/2 is the number of positive roots of sl(/ + 1, C). Let T(Kk), T(Ek) and T(Fk) be the operators on V determined by T(Kk)1M) = q ak(m)- " +1(m) i M) , ak(M) = E mik -E k

T(Ek)1 AI) =

1

E -- A(M)I 4-3), Cjk

j=1

(66)

k j=1

(67)

where (MV) is the tableau (M) with mik replaced by mik ± 1, ( Fl11 [mi,k+i - mjk - i + j] [mi,k- 1 - mjk - i + i

-

nti [mik___mi , _ i ± Am. _ mik _ i ± ; _ 1]

7.5 Representations of Uq (g) for q a Root of Unity

239

and [m] := [m], is defined by (2.1).

Proposition 47. The operators T(Ek), T(Fk) and T(14) given by the formulas (66)—(68) determine a representation T of U,(sli + i). For generic values of the parameters mi ,i +i , ci k, hik the representation T is irreducible. Outline of proof. In order to prove that the operators (66)—(68) give a representation it suffices to show that they satisfy the defining relations of the algebra UE (sli +1 ). This verification is done either by a long direct calculation or by the method used in Subsect. 7.3.3. As usual, the irreducibility is proved by showing that every nonzero vector of V is cyclic. The corresponding assertion is then derived from the fact that for generic values of mi,1+1, cik, hjk the coefficients in (66) and (67) do not vanish.

Any representation T contains 1(1+1) parameters ci k, hik and the numbers = 1,2, • • ,1+ 1. However, they are not all independent, since if we add to all numbers mik of a tableaux (M) some fixed number, then this does not change the formulas (66)—(68). Thus, the representations T have in fact 1(1+2) = dim sl(/ +1, C) parameters. It can be shown (see [ACh2]) that these parameters are independent. -

Proposition 48. For generic values of the parameters

rni ,i +i , cik, h i k

the

(66)—(68) is cyclic.

representation T determined by formulas

Proof. In order to prove this assertion we calculate the operators T(Ek), T(Fk)' k = 1, 2, • • • ,l. First of all, the matrix element (M' I T(Fk) r I M) can be nonzero only if mi, = m'is for all s < k. For s = k, the k-th row in M' is of the form (Onik 7 Tri2k " m'a) = (Milk — Pi, M2k — P27 • Tri kk — Pk), where Pi + • • • +Pk = r. Using (56), one proves by induction on r that rr

(M I

H

T(Fk)r

Pj

jk

liki [Pli

*JO

+

11 4i

s=1

=1

{r]!

—

= (H i<j

ti k

H

[Pa

1.7k

•

[PO )

ljk

8

112

— 11) 2

s]2

, (69)

j

where iir = Mir — j and rir = m'ir We put r = p in (69). Then all coefficients [p ] ! I [p i ]! • • • [pa vanish except for the cases when one of the pi is equal to p. Hence we get T(Fk)P IM) =

k

P

IT=1 CPik II

j

s=1

i tii,k+1 _ iik + s _ n riik_ii tii,k_i _ iik + s _ 2]

niti

fl[lik —

ljk + S —

11 2

I

2

1m), )

that is, T(FOP is a scalar operator. Since we have (M' I T(Fk) I M) (M I T(EOP I M'), T(Ek) is also a Kalar operator. The preceding formulas

240

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

imply that for generic values of m i,/±1 , ci k, hi k the operators T(Fk)P and 1=1 T(Ek)P are nonzero scalar operators, so the representation T is cyclic. One can also construct the representation T when some of the numbers hik are equal to each other. Then T becomes reducible and we can find subrepresentations of T with dimensions smaller than pi( 1± 1)/ 2 . For example, let all mi k, j > 2, in (M) be equal. Then we obtain an irreducible representations T' of Uf (sli+i ) of dimension pi given by nifk)IM) =irk - ak+i 1M) and r(Fk)1 M ) = c k ( - [mk+ i m k + 1 ] [mk--1 mkpoimi c 1), --

-

r(Ek)IM) = c k 1 ( [mk F1 -

-

-

rak][mk i - mk -

-

1])1/2imi-c1),

where ak is as in (66) and mi,k+il mi,k) mi,k-i are denoted by 7 k+11 mk, 7 k -i, respectively. Each mi takes the values m i = hi, hi + 1, - • • , hi + p - 1. The representation r depends on 21 + 1 paravneters hi, i = 1,2, • , I + 1, and ci , = 1,2, - • • , /.

7.6 Applications The aim of this section is to sketch in a short review how Drinfeld-Jimbo algebras and their representations are used in physics. The Drinfeld-Jimbo algebras Uq (s1„) have been used in order to replace the unitary groups SU (n) and their irreducible representations in describing flavor symmetries of hadrons, in particular, of vector mesons (see [Gay]). In this case, the algebra Uq (s17,±1 ) is considered as a dynamical symmetry. The latter realizes the necessary breaking of flavor symmetries up to exact isospin symmetry Uq (su2 )/ and produces some g-analogs of mass relations. This application of Uq (s1.) uses the generators corresponding to nonsimple

root elements. The well-known octet mass sum rules of Gell-Mann and Okubo are 3mA +m

= 2(mN +m5)

(70)

rn = 4M K •

(71)

for baryons (1/2)± and 37nci

for vector mesons 1 - . They have been derived on the basis of an SU(3) symmetry breaking. If instead the quantum algebra Uq (s13 ) with g = e."16 is used (note that then [2] q = ±-4), the g-analog of formula (70) becomes mN ±

1+0 2

2 M- = 0MA ±

-

9-

0

6 ME•

(72)

The equality (72) gives a surprisingly good mass formula and holds with an accuracy of 0.22%. For a comparison, the Gell-Mann—Okubo relation (70) is satisfied only with an accuracy of 0.57%.

7.6 Applications

241

The q-analog of the mass relation (71) in the case of three flavors takes the form 421 mu,. + (2131 - - 1) m p = 2- mKt. (73)

[ 2

[31

In the case [3]q = [2],i, this formula simplifies and reads as mu,. +mp = 2mK.. Setting m u,. = m0 , one recognizes here the nonet formula of Okubo. This relation agrees with the corresponding experimental data (up to errors of the experiment and of an averaging over isoplets). Applying the same approach in the cases of more flavors, new mass sum rules (higher analogs of Okubo's nonet mass relation (73)) are obtained that are also consistent with experimental data. These mass formulas are closely related to the Alexander polynomials of certain torus knots. One of the main physical motivations for the representation theory of Drinfeld-Jimbo algebras in the root of unity case originates from the chiral Potts model. This is a solvable lattice model built on solutions of the quantum Yang-Baxter equation whose spectral parameters live on certain algebraic curves of genus greater than 1. These solutions can be derived as intertwiners of tensor product representations of the Drinfeld-Jimbo algebra for the affine Lie algebra of 512 with q a root of unity (see [BSt]). Attempts for extending these results to the case of 513 have been made in [BK] and [DJMM1]. It turned out that some properties of the chiral Potts model are closely related to the underlying structure of nonhighest weight representations of Drinfeld-Jimbo algebras. This understanding has proved to be very fruitful and allowed the construction of infinitely many new exactly solvable models as generalizations of the chiral Potts model (see [BK], PJMM11). The free parameters which characterize these nonhighest weight representations appear in the form of spectral parameters in these generalizations. Among others, Drinfeld-Jimbo algebras have been invented for the construction of solvable models of statistical mechanics. In this context, the parameter q appears as the temperature, and the case q = 0 corresponds to the absolute temperature zero. For this reason, the passage to q = 0 is called "crystallization" (with a naive faith that any material is crystallized at temperature zero). The theory of crystal bases was used for the computation of 1-point functions for the 6 vertex model and its generalizations [JMM0], [KKMMI‘11\1]. The 1-point functions are basic macroscopic quantities that describe the multiphase structure of a given lattice model of statistical mechanics. For the two-dimensional solvable models there is a method for computing the 1point functions, called the corner transfer matrix method. It reduces the twodimensional statistical sums of the 1-point functions to the one-dimensional statistical sums over certain paths. In the vertex models given by the Rmatrices of the Drinfeld-Jimbo algebras, the parameter q behaves like the temperature, and the sum over the paths leads to series expansions at low temperature for the 1-point functions.

242

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

Under certain conditions the set p of paths can be identified with the crystal basis B of a NW-module. The sets of paths corresponding to the multiphases are in a one-to-one correspondence with the crystals of certain irreducible highest weight N(3)-modu1es. As a result, closed expressions of the one-dimensional statistical sums (and of the 1-point functions) in terms of the string functions of the corresponding affine Lie algebras have been derived in [KKMMNN]. Each line of a two-dimensional lattice bears a finite-dimensional module. Multiple lines are equipped with the tensor products of the associated modules. In this manner, the termodynamic limit refers to an infinite tensor product. An exact treatment of this infinite product is quite difficult. It turns out to be tractable when only the situation at q = 0 is considered. The result is then summarized as an isomorphism between the set of paths P and the crystal B.

7.7 Notes The classification of finite-dimensional irreducible representations of DrinfeldJimbo algebras for q not a root of unity is due to G. Lusztig [L1] and M. Rosso [Rosi]. Among others, Theorem 8 on complete reducibility of finitedimensional representations was proved in these papers. Theorem 13 on separation of elements by irreducible representations is from [Lus]. We followed the proof given in [Jan]. Theorem 24 on irreducible finite-dimensional representations of the algebra Uq (gli ) is stated without proof in [Jim3]. It is proved in [UTS]. The general formulation of the Wigner-Eckart theorem for Drinfeld-Jimbo alge bras is given in [BT1 (see also [K121 and [RiS]). Expressions for the ClebschGordan coefficients of tensor products Tm 0 Tp of Uq (gln ) have been derived in [AlS]. Clebsch-Gordan coefficients of Uq (gln ) were also studied in [GLI3], [Gou] and [K13]. The results of Subsect. 7.3.9 are taken from [DJM]. Crystal bases of representations of Drinfeld-Jimbo algebras were invented and developed by M. Kashivara [Kas1], [Itas2]. More material on crystal bases can be found in [Jan], [JMM], [Kas3], [K1\11 and [Van. For an application of crystal bases to representations of classical Lie algebras see [Na]. Basic results on irreducible representations of Drinfeld-Jimbo algebras Uq (n) for q a root of unity have been obtained by C. De Concini, V. G. Kac and C. Procesi [DCK], [DCKP]. The results of Subsect. 7.5.2 on cyclic representations have been taken from the paper [DJMM2]. A recursive method for the construction of such representations was developed in [Schn1], [Schn2]. Proposition 46 was obtained in [CP2]. Gelland-Tsetlin bases for irreducible representations of Uq (gl.„) for q a root of unity were constructed in [ACh2]. Further results on representations in the root of unity case can be found in [ACh1], [Bee], [DCL] and [L3].

8. Quasitriangularity and Universal R-Matrices

A quasitriangular Hopf algebra is a Hopf algebra equipped with a so-called universal R-matrix which induces solutions of the quantum Yang-Baxter equation on their representations. The Drinfeld-Jimbo algebras Uh(g) are h-adic quasitriangular Hopf algebras. This emerges from Drinfeld's quantum double construction. The universal R-matrix of Uh(g) allows one also to introduce the so-called L-functionals. Suppressing mathematical subtleties they provide an alternative approach to the Drinfeld-Jimbo algebras which turns out to be very useful for applications (see Sects. 11.5-6 and 14.5-6). This chapter is devoted to a detailed study of these topics. The universal R-matrices, the R-matrices and the main L-functionals for the vector representations of Drinfeld-Jimbo algebras are explicitly determined.

8.1 Quasitriangular Hopf Algebras 8.1.1 Definition and Basic Properties Recall that a bialgebra A is cocommutative if Ac°P = A, where Ac°P = T o LI and T is the flip. Now we consider bialgebras A which are cocommutative up to conjugation by an element R. E A 0 A. Definition 1. A bialgebra (resp. Hopf algebra) A is called quasitriangular if there exists an invertible element R. of A 0 A such that

A cop( a) = RA(a).R.-1 (46 id)(7Z) = Ri3R23)

(id

a E A,

46)(7Z) = Te13R.121

(1) (2)

where R42 = 1Z23 = Ei 1 ® si Yi xi ® yi 0 1, 7Z 13 = Ei xi ® 1 ® for R. --= Ei xi ® y. An invertible element REAOA satisfying conditions (1) and (2) is called a universal R matrix of A. A quasitriangular bialgebra (reap. Hopf algebra) with universal R-matrix R. is said to be triangular if Ei yi 0 xi. R21 = R.-1 , where R.21 = ) -

The preceding definitions carry over almost verbatim to h-adic bialgebras and Hopf algebras. The only modificatiOn is that the element R. then belongs to the h-adic completion AA of A A.

244

8. Quasitriaingularity and Universal R-Matrices

We rewrite (1) by means of the Sweedler notation. If R, is of the form R, = Ei xi ® yi and A(a) = a(i) a(2), then (1) is equivalent to E i,(a) a(2) xi ® a( i )Yi =E i,(a) x a

® yia(2), a E

A.

The relations (2) mean that

E(xi) (,) (,) (2) Ei E xi ® ( Yi)(,)

yi =

xi 0 xi 0 yi yi ,

(yi )(2) =

xi xi yj yi •

(5)

Example 1. The element R, = 1 0 1 is a universal R-matrix for any cocom-

mutative bialgebra.

A

Example 2 (Sweedler's Hopf algebra). Let A be the Hopf algebra from Example 1.9. For any complex number a, the element

'R(a) := -1 (101+10g+g01-g0g)+ 2 (x0x -x0gx+gx0x+gx0gx) 2 2 is a universal R-matrix of A (see [Rad2] for details). Important examples of quasitriangular Hopf algebras will be obtained later in Subsects. 8.2.2 and 8.3.2. Leg numbering notations such as R13R237 R12) etc., will often be used in the following. The lower indices always refer to the numbers of factors of a tensor product on which the corresponding element, matrix or transformation

acts.

A be a bialgebra. Recall that A°P (resp. Ac°P) denotes the bialgebra with coalgebra (resp. algebra) structure of A but with opposite product (resp. Let

coproduct). Proposition 1. Let R. be a universal R-matrix of a bialgebra A. R21 1-ZA(a) = d(a)'R21 /Z for all a E A. (ii) 7Z2-11 is also a universal R-matrix of A. (iii) 7Z21 and R1-21 are both universal R-matrices for the bialgebras A°P and A"P . In particular, if a bialgebra A is quasitriangular, then so are A°P and Ac°P Proof. (0: The flip transforms (1) into A(a) = 7Z2lAc 0 P(a)R;11 , so that

A(a)R.2 1 R. =-- 1:121 46c°P (a)R, = 7Z21/ZA(a). (ii): From A(a) = R.21.6 c0P(a)Rill we get Ac°P(a) = RVA(a)R 21 . Since A is an algebra homomorphism, we have ((A 0 id)(7Z))-1= (A id)(R-1) . Therefore, by (2), (A id)(7Z -1 ) = R.i31 7:1d. Permuting the order of tensor factors, we derive that (id 46)(1Ç1 l ) = = (re1')13(R:2-11 )12. Likewise, (id A)(R -1 ) = Rer21 7Zr31 leads to the relation (LI® id)(RV) = ( 1Z7 211 )13(Rii1 )231 that is, RV is indeed a, universal R-matrix for A. (iii) is proved similarly.

8.1 Quasitriangular Hopf Algebras

245

Some basic properties of the universal R-matrix are described in Proposition 2. Let A be a quasitriangular bialge bra with universal R-matrix Te. Then we have (6) R-127?-13R023 = R-237?-137?-121 (e

id)(7Z.) = (id 0 E. )(7?,) = 1.

( 7)

If A is a Hopf algebra, then we also have (S

id)(R) = 1?, -1 ,

(S

(id 0 S)(R. -1 ) ----

S )( ?) = R.

(8)

Proof. Using the first equality in (2) twice and relation (1) we obtain R-12R13R23 = R12(/.1 0 id)(R) =(i6c°P id)(R) - R-12 =

id)( ,6Ø id)(74Tel2 =(r 0 id)(1 13R-23)-R-12 = R-23R-137?-12.

From (e 0 id) 0,6 = id and (2) one derives that

R. = (e 0 id 0 id)(46 id)(Te)

0 id 0 id) (R43R-23) = (e id)(Te) - RI

that is, (E. 0 id)(R) = 1. The second part in (7) is proved analogously. Since m(51 0 id)(a) = E(a)1 by the antipode axiom, we have (m id)(S 0 id 0 id)(,6 0 id) (TO

= E. 0 id)(R.) = 1. (

(9)

Since S is an algebra anti-homomorphism, (9) implies that 1 = (m id)(S 0 id 0 id)(TenTe23) = (S0 id) (R.)-R., so that (S 0 id) (R) = R.-1 . Using the fact that R. is a universal R-matrix for A°P (by Proposition 1), similar reasoning yields the second equality of (8). Finally, the third equality follows by

(S S) (7Z) = (id 0 S)(51 id)(R) = (id 0 S)(7Z -1 ) =

(10)

Definition 2. The relation (6) is called the quantum Yang-Baxter equation (abbreviated, the QYBE).

Let it = Ei yi . Define mappings OR and OR from the dual bialgebra A° to A by setting (f) =

>2 f(x)y,

R (f) =Ei xif(07

f

E

A°.

From relations (2) one easily proves the following assertions. Proposition 3. Let R. be a universal R-matrix of a quasitriangular bialge bra A. Then 1,1PR. : A° -4 aht." P and 1,b lz : A°P are bialgebra homomorphisms. That is, 7,IPR. : A° 4 A is an algebra homomorphism and a coalgebra antihomomorphism and OR : A° -.4 A is an algebra anti-homomorphism and a coalgebra homomorphism. -

246

8. Quasitriangularity and Universal R-Matrices

8.1.2 R-Matrices for Representations If Tv and Tw are representations of a bialgebra A, one may ask whether or not the tensor product representations Tv 0 Tw and Tw 0 Tv are equivalent. We shall see that for quasitriangular bialgebras this is indeed true and the equivalence is realized by an operator derived from the universal R-matrix. Suppose that A is a quasitriangular bialgebra with universal R-matrix R. = Ei xi ® yi . Let Tv : A L(V) and Tw : A L(W) be representations of A on vector spaces V and W, respectively, where L(V) denotes the algebra of linear operators on V. We define linear mappings Rvw and _hv w by (Tv 0 Tw )(R.) and k'vw :=

° (TV

(11)

Tw)(R,).

That is, by definition we have w)

_hvw (y w) = T O (Tv 0 Tw)(R,)(v

The inverse of I4vw acts as .hv- w i (w

Ei Tw (yi )w Tv (xi )y.

= (Tv 0 Tw)(7?. -1 )(v w).

Proposition 4. (i ) The mapping 14vw defined by (1 1) provides an equivalence of the representations Tv OTw and Tw 0 Tv of A. (ii) For any triple T T, Tu of representations of A on vector spaces V, W and U, respectively, we have

fivow,u = (ftvu O idw)(idv O ft wu ),

(12)

.ñv,w ®u = (idw O ftvu)(ftvw O Wu))

(13)

(fiwu idv)(idw 0 ftvu)(ftvw O idu) = (id u O itvw)(ftvu O idw)(idv O ftwu)•

(14)

Proof. Since ftvw is invertible as noted above and

ftvw ( Tv

Tw)(a)

=

(T 0 (Tv 0 TW)( 7Z)) ( TV ® TW)(A(a)) T O ((Tv 0 Tw)(7ZA(a))) cop T O ((Tv 0 TW)(A (a))(Tv Tw)(R))

=

(Tw Tv0)(.6(a))(1- o (Ti,' 0 Tw))(R,) (Tw Tv)(a)ftvw, ,

=

ftvw realizes an equivalence of the representations Tv 0 Tw and Tw 0 T. Equation (12) is a consequence of the first formula in (2). Indeed, we have -hVOW,LI

(T O (TV 0147 0 Tu))(R) = (T12,3 0 (Tv®w O Tu)(( 21 0 id)(R))

T12,3 ° (TV 0 TW O 710(R-13R-23))

Tu(Y0i)OT(j) ® Tw(x) (Ei Tu(y i )

Tv(si) 0 id

(itvu O idw)(idv O fiwu),

Tu(N) O idv

Tw

8.1 Quasitriangular Hopf Algebras

247

where 1-12,3 (v w u) = u 0 y 0 w. Similarly, (13) follows from the second formula in (2), and (14) can be derived from the QYBE (6) or by combining El (12) and (13).

Let us consider (14) in the special case when V = W = U and Tv = Tw = T. Setting R := _hvv , (14) reads as

(15)

fl12ii23f112 = ii23-h121123-

Equation (15) is called the braid relation for the linear transformation IL It is not difficult to check that an arbitrary linear transformation R satisfies the quantum Yang—Baxter equation (6) if and only if the transformation TOR fulfills the braid relation (15). In this sense the QYBE and the braid relation are equivalent. Therefore, by Proposition 4(ii), any transformation Rvir = To-hvy = (Tv 0Tv)(R.) satisfies the quantum Yang—Baxter equation (6). That is, the universal R-matrix R. gives rise to a solution Rvir of the QYBE for every representation Tv of the quasitriangular .bialgebra A. Since solutions of the QYBE are usually referred to as R matrices, this fact is the reason why R. is called the universal R-matrix of A. Finding all solutions of the QYI3E turns out to be a difficult task. To a large extent, quantum groups were invented as structures which allow one to produce such R-matrices. -

8.1.3 Square and Inverse of the Antipode Recall that by Corollary 1.7 the square of the antipode of a cocommutative Hopf algebra is the identity map. For quasitriangular Hopf algebras this is not longer true, but 82 is still an inner automorphism as stated in Proposition 5. Suppose that A is a quasitriangular Hopf algebra with universal R-matria R. Then u := rn(S Ø id)(R, 21 ) is an invertible element of A 52 )(R.21). The element uS(u) = S(u)u is central. with inverse n -1 =-- rri(id The antipode S of A is invertible and we have 52 (a) = uau -1 and S -1 (a) = u -1 S(a)u for all a E

A.

Ei

Proof. Let R. = x 0 yi. We first prove that 52 (a)u = ua for a E A. By (1) we have the relation (R, 1 ) ,6(2)( a ) = (21cop id)2i(a) (R. 0 1). It gives

E

i(a)

xi a( i)

yia( 2) a( 3) =

E

a( 2)x i amyi a( 3).

(16)

,

We now apply the mapping id 0 8082 to both sides and multiply the terms in reversed order. By (1.20), (1.27) and (1.28), the two sides of (16) yield

E E

s 2 (a(3))S(yia(2))sia(i)

(a)

S2(a(3))S(a(i)yi)a(2)si i,

E i, S(a(2)5(a(3)))5(Yi)xia(1) E . S(yi)s ia = ua, E S2(a(3))S(yi)S(a(o)a(2)xi E. S (a)S(y i )x i = S2(a)u, () a

i(a) ,

2.

248

8. Quasitriangularity and Universal R-Matrices

respectively, so that 82 (a)u = ua. Let y = m(id S2 )(7Z21 ) = Ei yi s2(s z ). Setting R. -1 = Ei x! -1) y, -1) and using the relations (S S)(R.) = R. and (S id)(R.) = R.-1 , we derive

vu =

E

yi S2 (x 2 )u = i

E

yi uxi =

S(y i )S(y i )xi S(x i ) =

E.. yi S(yi)xix i

i: i ,is(y ç-1))s( yi )six!-1).

Applying id 0 5 to the identity = Ei x i s!' multiplying again in reversed order we obtain vu =

yiy -1) = 1 0 1 and

E 2,3 S(y -1) )S(y i )si x! -1) = 8 (1)1 =1.

Similarly,

UV = E . uyi S2 (xi ) = E . S2(yi )uS2 (xi )

E. S(y i )uS(xi) =N

yi uxi = 1.

By the preceding we have shown that y is the inverse of u and that 52 (a) = uau-1 , a E A. Since S2 (u± 1 ) = uu±lu-i = u±1 we have 52

(u' au) = 5 2 (u -1 )5 2 (a)82(u) = u-

= a, a E A,

so that A = (A). Applying S to the relation ua = 82 (a)u we get S(a)S(u) = S(u)8 3 (a) = S(u)uS(a)u-1 and so S(a)S(u)u = S(u)uS(a) for a E A. Since S(A) = A, S(u)u belongs to the center of A. Setting u = S(a), the latter identity yields uS(u)u = S(u)uu and hence uS(u) = S(u)u. The mapping T defined by T(a) := u-1 S(a)u is the inverse of S, since ST (a) = S ( L - S (a)u) = 8(u)52 (a)S(u -1 ) = S(u)uau-1 8(u -1 ) = a,

T S (a) =- u-1 S 2 (a)u =

= a.

Note that the square of the antipode for the Drinfeld-Jimbo algebras has been described as an inner automorphism in Subsect. 6.1.2. The central element uS(u) is occasionally called the quantum Casimir element of the quasitriangular Hopf algebra A.

Remark 1. A Hopf algebra A is said to be quasi-cocommutative if its antipode is bijective and if there exists an invertible element R. E AOA satisfying (1). By some slight modifications of the first part of the above proof (use m o (5-1 o id)(1.2-11 ) as y) it follows that the formula 52 (a) = uau -1 , a E A, is valid for quasi- cocommutative Hopf algebras. Corollary 6. (id 0 8-1 )( 1 ) =

249

8.1 Quasitriangular Hopf Algebras

Proof. The third identity of (8) implies that (S -1 0 S-1 )(7Z-1 ) = Applying S-1 0 S -1 to the first formula of (8), we get (id 0 5 -1 )(7?) =(5-1 0 s-1)(R-1) = Te-1 .

Proposition 7. If A, R, and u are as in Proposition 5, then the element w := uS(u -1 ) is group-like and

_ __--1 a E A. S4(a) = w aw 7 Moreover, Lt(u) = (7Z2 1 7?) -1 (uou) and ,i(S(u)) = (R217 Z- ) -1 (S(u) S(u)). Proof. We first prove the relation Zi(u)7Z2 1 7Z. = u u. By (1), we have

.6(u)R211Z = E zi(S(Yi))Zi(xi)l-Z217-Z = E(5 0 S)(2P P (M))Zi(xig-Z2iTe E(5 s)(46c0P(o)R,217m(xo.

Defining a right action a of the Hopf algebra A°4 on A0 A by (a0b)

4

Pc 0

:=

(S S)(Y)(a 00X, a,b A 7 X 7 -17 E A0 A,

and using the formulas (2) we can rewrite the previous relation as Zi(u)7Z217e = 7 -21 <1 (R12R13R23R,14R24)•

By the QYBE, the right hand side is equal to R.21 ci (7e237-e13R12Re14R24)• Since (id 0 S -1 )(7-e) = TZ-1 by Corollary 6, we derive 7?-23

R-21

E.i,3 S(yi )yi (S

Thus,

xixi = (S ® id)

E.

.

2,3

S -1 (yi )yi x z xj

id )( R-2-11R21 ) = 1 0 1.

TZ21 (R23R13) = (1 0

R21 < (R,23 7Z13R12R14)

1) < 1 13 =

Ei S(yi )si 0 1 = u 1 and hence

-= (u 0 1)(E . . xixi 0 S(y)y) i,3 _ (u 0 1)(id 0 S)

E.2.,3. xis; ® S-1 (yi )yi

= (u 0 1)(id 0 S)(7Z -1 7Z) = u 0 1,

so that Zi(U)12iTZ = (U 1) 4 R24 = (il 1)(1 U) =

u.

(17)

Because A(a)1Z217Z = 1.217Z.6(a) by Proposition 1(0, this yields L(u) = (R.211Z) -1 (u u). From the latter relation and the third formula of (8) we obtain

250

8. Quasitriangularity and Universal R-Matrices

A(S(u)) =

(S

5')(T o d)(u) = (S

SKr

(R-21 R Y (u u)) -

'

= (S® 5)(u 0 u)(S S)( 1Z7 -21) -1 (G9 SRu u))(RaiTe) 1 . Therefore, by (17), we have

A(w) =A(uS(u -1 )) = (u

u)(R2iR,) -1 1-Z2 i TZ(S(u - 1 ) S(u -1 )) = w w

that is, w is group-like. Since 512 (8(a)) = S(uau -1 ) hence 512 (b) = S(u -l )bS(u), b E A, we get 514 (b) S(L -1 )ubu -1 51 (u) =

S(u 1 )S(a)S(u) and 83 (u -1 )82 (b)513 (u) = 1=1

8.2 The Quantum Double and Universal R-Matrices The quantum double is one of the most celebrated Hopf algebra constructions. We develop this method here for general skew-paired bialgebras. A number of other approaches to the double will be treated in Sect. 10.2. In the present context the quantum double is used in order to derive explicit formulas for the universal R-matrices of the h-adic Drinfeld-Jimbo algebras.

8.2.1 The Quantum Double of Skew-Paired Bialgebras If not stated otherwise, A and 13 are bialgebras in this subsection.

Definition 3. A bilinear mapping a- :Ax,6-1C is called a skew pairing of the bialgebras A and 5 if o- (.,.) is a dual pairing of the bialgebras A and 5°P, that is, for all a, a' E A and b, b' E B we have -

cr(a , 1) = E(a),

• a- (aa' , b) =

(18)

o (1, b) = E(b),

E o- (a, bw)o- (a' , b (2)) , cr(a, bb') = E a- (a (2) , b)cr(a (1 ) , b').

(19)

The mapping a- is said to be convolutigp invertible (or, briefly, invertible) if there exists another bilinear mapping17- : A x C such that o -T1 = Wo = 6A. 0 c5 , that is, for all a E A and b E 13 we have

E u(a( 1 ) 7 b( 1 ))cr(a(2) 7 1)(2)) = E-1,--(a(1) ,b(o )u(a(2) , b(2)) = EA(a)E 8 The inverse of a- is always denoted by 7. It is easily seen that ô- is a skewpairing of A and B if and only if 721 is a skew-pairing of 13 and A, where 21 (b, a) := Tr(a l b). A skew-pairing a- of A and 13 is invertible if A is a Hopf algebra or if B is a Hopf algebra with invertible antipode. In these cases the inverse Fr is given by Tr(a, b) =

(S (a), b), resp. cr(a l b) = a- (a, 51-3- (0) ,

a E A, b E B.

(20)

8.2 The Quantum Double and Universal R-Matrices

251

We verify, for instance, the second formula. By (18) and (19), we have

E or(a0),8 -1 (b0)))(70(2),b(2)) = E 0- (a, b(2) 8 -1 (b0))) = E a (a, 51-1 (b(i)S(b(2)))) = o - (a, e(b)1) = e(a)e(b). Similarly we get

E o- (a( i ), b(o)a(a(2), 5 -1- (b ( 2)))

=

Proposition 8. (i) Let A and B be bialgebras equipped with an invertible skew-pairing o: Ax B --- C. Then the vector space BOA becomes an algebra with product defined by

(b 0 a)(b I 0 a') =

E by(2) 0 a( 2)a/ (7(a( i ), b'm )o- (a (3), N3) )

(21)

for a, a' E A and b, b' E B. With the tensor product coalgebra structure of B 0 A (that is, with the coproduct ZiB e,A(b 0 a) = E b(i) ® a(1) 0 bp) 0 a(2) and counit E50A(b 0 a) = ei (b)eA(a)), this algebra is a bialgebra. (ii) If A and B are Hopf algebras, this bialgebra is a Hopf algebra with antipode S given by SO 0 a) = (1 0 SA(a))(88(b) 01). Proof. First we prove the associativity of the product. Since a is a dual pairing of A and 130P, Ti is a dual pairing of A°P and B. Using appropriate properties of a and 7, we compute that both expressions ((b 0 a)(b' g al) (b" 0 a") and (b 0 a)((b' 0 a') (V 0 a")) are equal to

E bb/(2) q3) 0 a(3)4a n7(a(i), 11(1) )7 (a (2) , q2) )a (a (4) , 114) )a (a (5) , b'(3) ) xa-- (a4 ) , b'( '1) )0. (a'(3) ,q5) ) .

This proves the associativity. It is obvious that (b0a)(101) = (101)(b0a) = b 0 a and that er3 0}1 is an algebra homomorphism. We verify that Zi 80.4 is an algebra homomorphism by computing 430..4(b0a)ZiB0.4(bi gai ) =7---

(b( i ) Oa(i) ) (Ni) (0 1(0 00(2) ea(2) )(Y(2) 0 (1/(2 ))

= E b (1) N2) 0 a(2)a4 ) 0 b(2)N5) e a(5)d(2) 7(a(i),194 ) )a(a(3) 7 1/(3)) X -3 (a(4) ) b i(4)) 0"(a(6), NO

= E b ( 1 ) N2) 0 a(2)a4 ) 0 b(2)1/(3) 0 a(3)d(2)F-(a(i), b4))47(a(4))1/(4)) = z180A((b 0 a)(b' 0 a')).

Finally, we show that SO 0 4) := (1 0 SA (a))(SB(b) 01) is an antipode if A and B are Hopf algebras. Using (21) we get

252

8. Quasitriangularity and Universal R-Matrices

E s(b (1) 0 a(l))0(2) a(2)) E(1 SA(a(1)))(SB(b(1)) 1)(b(2) 0

1)(1 0 a(2))

= E(1 SA(a(1)))(518(b(1))b(2) 0 1)(1 0 am)

= e(b)

E ® SA(a( 1) )a (2) = e(a)e(b)(1 0 1)

and similarly E(b(l) 0 a(OS(1)(2) a(2)) = e(a)e(b)(1 0 1).

Definition 4. The bialge bra from Proposition 8 is called the generalized quantum double of the bialge bras A and B with respect to the skew-pairing a or, briefly, the quantum double of A and B. It is denoted by D(A, B; a) or simply by D(A, B) if no confusion can arise. Let us discuss some basic properties of the quantum double. From the corresponding definitions in Proposition 8 it is clear that both mappings B b b 0 I E D(A, B; a) and A a —+ I 0 a E B; a) are injective bialgebra homomorphisms. Let us identify b 0 1 with b and 1 0 a with a. Then B and A are subbialgebras of D(A, B; a). Note that the expression b a -=- (b 0 1)(1 0 a) = b0a is different from ab (1 0 a)(b 0 1) in general. By (21), we have the following cross commutation relations between elements a 1 0 a e A and b b 0 1 E B in the algebra D(A,B; a):

ab =

E a(a(l) , b( i ))b( 2 )a(2)o- (a( 3 ), b(3)),

E o- (a( i ), bm)a(2)b(2) == E bma(00- (a(2)1b(2)).

(22) (23)

In fact, as an algebra the double D(A,B; a) is the free algebra generated by the algebras B and A with cross relations (22) or, equivalently, (23). As a coalgebra, D(A, B; a) is just the tensor product coalgebra of B and A. We give another isomorphic variant of the quantum double. Since Tr21 is an invertible skew-pairing of B and A, as noted above, the quantum double D(B A; 21) is also well-defined. The bialgebra D(B, A;5-2 1 ) is realized on the vector space A 0 B. Its coalgebra structure is inherited from the tensor product of the coalgebras A and 5 an its algebra structure is given by the product (a

b)(a' b') =

E ad(2) 1)( 2) b' a(do) , b (1 ) )5- (d(3) , b(3) )

for a, a' E A and b, b' E B. The cross commutation relations of b and a..a01.e.A in the algebra D(B, A; -5-21 ) are

ba = > cr(a(l), b(o)a(2)b(2) 7f(a(3) b(3)

(24) 1 0 b e 13

(25)

In the literature both variants D (A , B; a) and D(B, A; F21) occur as generalized quantum doubles of the bialgebras A and B. The bialgebras D(A, B; a) and D(B A; 21) are indeed isomorphic with isomorphism 9 defined by

8.2 The Quantum Double and Universal R-Matrices

0(b (8) a) =

E a(2)

b(2)cr(a( 1 ), b( 1 ))(7(a(3), b(3)),

a E A, b E B.

253

(26)

Remark 2. All quantum doubles occurring in this book are of the form D(A, B; a) with a skew-pairing a- of A and B and built over the vector space B 0 A. The reader should notice that there are also variants of the double in A the literature (see [Jos]) where the tensor factors are interchanged. Let us specialize the preceding to the following situation. We suppose that A is a Hopf algebra with invertible antipode and let p(f, a) denote the evaluation (f, a) := f (a) of the functional f E A° at a E A. Since the antipode of A is invertible, A°P and A"P are Hopf algebras by Proposition 1.6 and (Ar)COP and (AwP)° so are (A°P )r) (A°)°P as well by Proposition 1.12. As noted in Subsect. 1.2.8, p is a dual pairing of A° and A. Hence p is a skew-pairing of (A°)'°P and A, and by (20) its convolution inverse Ti is given by Ti(f, a) = (f, 5 1 (0. Therefore, by Proposition 8, the quantum doubles D((A°)c°P , A; p) and D(A, (A°)C°P ; Tt21 ) are well-defined Hopf algebras. Further, since p is a dual pairing of A° and A, v := P21 is a skew-pairing of A and (A°)°P with convolution inverse 7 such that P(a, f) = (f, 8(0 (again by (20)). Thus, the corresponding quantum doubles D (A , (A°)°P ; v) and D((A°)°P, A; 1721) are also well-defined. Each of the four Hopf algebras D((A0 )'op I A; if) 7

(A' ) c°1 iT121) I DO, (A7P ;

D((A7 P 1A;P21)

is called a quantum double of the Hopf algebra A. Although their products look quite different at first glance, all four Hopf algebras are in fact isomorphic. It was already noted above that the first and second ones and the third and fourth ones are isomorphic with isomorphism 0 given by (26). One immediately verifies that SAD 0 id is an isomorphism of the Hopf algebras (A°)°P ; v) and D(A, (AO)COP ; 71 21 ). The reader should notice that 7121 is in general, because the convolution inverses of p and different from v = i1 21 are formed with respect to different pairs of Hopf algebras! In Subsect. 8.2.2 we shall use the double D(A, (Alc°P; 7121 ), so we restate here its structure explicitly. By definition, D(A, (A°)" ; 7121) is the quantum double D(A, B; ci) with product (24) and B = (A0 )Cop a = 712i . By (20), the convolution inverse of T./.21 is /22 1 (a, f) = (f, a). Let us adopt the convention that the Sweedler notation AA. (f) = E fo) f(2) refers to the comultiplication of A°, so that AA0),(f) E f(2) fm. Then the Hopf algebra (A43 ) 09 ; 7i 2i ) is realized on the vector space A' 0 A with product

Cf 0 a) (g 0 b) =

E f g(2) O ab (g(3), am) (gm, SA 1 (a(3))),

(27)

unit E 0 1, counit 1 0 e, comultiplication d(f 0 a) =

f(2) a( i )

f(1)

a(2)

and antipode

S(f

= (1 0 S.V(f))(SA(a) 1).

(28)

8. Quasitriangularity and Universal R-Matrices

254

Finally, let us note that the previous definitions and results carry over almost verbatim to h-adic Hopf algebras. The only modifications are that cr : A13 C[[h]] is C[[h]i-linear and that the quantum double D (A, 13) is built over the completion 56A of 13 0 A in the h-adic topology. 8.2.2 Quasitriangularity of Quantum Doubles of Finite-Dimensional Hopf Algebras

Suppose that A is a finite-dimensional Hopf algebra. Let {e j — 1, 2, - • • ,71} and {fi f i = 1, 2, • • • , 771 be bases of the vector spaces A and A' A°, respectively, such that (fi , ei ) We retain the notation from Subsect. 8.2.1.

Theorem 9. The quantum double D(A) :--= angular Hopf algebra with the universal R.matrix

(AT' ;T-121) is a quasitri-

= E i (e 0 ei) 0 (h 0 1) E D(A) 0 D(A).

(29)

Proof. Since the antipode of the finite-dimensional Hopf algebra A° is always bijective ([S w], Corollary 5.1.6), (.A°)C°P is a Hopf algebra. Therefore, by Proposition 8, D(A) Ea D (A , (A°)C°P; 11 21 is a Hopf algebra. The main part of this proof consists of showing that R. is a universal R-matrix of D(A). First we prove that (2i id)(R) = R13R23. Since le i ] and {fij are dual bases, we have a = E(fi , a)ei and g = E(g, ei )fi for a c A and g E A°. Using these identities we compute the expression (gh, a) in two possible ways and obtain )

Ei

(g, ei,( 1 ))(h, ei,(2))(fi, a) =

Ei73. (g, e) (h, ei)(f,fi, a)

for a E A and g, h E A°. Since the pairing (,•) of A° and A is nondegenerate (because A is finite-dimensional), the latter equation implies that

Ei E

E

0 e i,( i )

E

ei,( 2) fi 0 1 --= E i e

ei 0 E ei fifi 0 1.

This means that id)(1Z) Te137Z23e'The relation (id is derived in a similar manner.

,i)(R.) =: R,13R12 -

Next we show that R. is invertible. More precisely, we prove that

:=

E

S (e i )) (fi 0 1)

(30)

is the inverse of R. in the algebra D(A) 0 D(A). The relation R. -1'R, = 1 0 1 is obviously equivalent to the equation 2,7

S(ei )ej fi fi = 1

e.

(31)

Pairing the second tensor factor of the left hand side against a E A and using the identity b = E i (fi ,b)e i we get

8.2 The Quantum Double and Universal R-Matrices

255

E S(ei)ei(fi7a(0)(fi,a(2)) = E S(E i (fat a(o)ei) E(fi,a)ei = E S(a(o)a( 2) = E(a)1. This implies (31) and so the relation RT 1 R. = 1 0 1. It is proved similarly that RR -1 = 101. Our next aim is to prove that 'R.z1(g 0 a) = 2icciP (g a)R, for g c A° and a c A. Inserting the definitions of the R-matrix (29), the product (27) and the coproduct (28) of the quantum double D(A) we get T?.,6(g 0 a) =

=

Ei » ei)(g(2 ) 0 a(0) o (f o

1 )0(1) O a(2))

E g(3) 0 ei,(2)a(i) 0 fag(i) O a(2)(gmlea,(0)(9(2)18-1(ei,(3)))

>2(g(1) a(2))(E o ei) o

(32)

a(1))(fi 0 1)

(g(2)

= ® a)re = AcciP(g Ea E g(1) O a(4)e 0 g(2)fi,(2) O a(2)(fa,(3), a(1))(fi,(1), S' (a( 3))). (33)

It suffices to prove that the expressions (32) and (33) are equal when paired in the third tensor factor against b c A. Applying the mapping (.6 0 id) o A to the identity c = E i (fi, e) e, c c A, yields that

c(i) ,2) c(3 )

Ei

(

LI c)ei,(1)

e,( 2) ®e i ,(3).

(34)

We first pair the third tensor factor of (32) against b. Using the relation (34) and properties of the dual pairing (., •) of the Hopf algebras A° and A we obtain

E

g.E (3)0ei,(2)a(1)0a(2)(fi,b(1))0(1), b (2 )) (g(4), ei,(1))0(2), S-1 (e i,(3)))

= E g(3) O b(2)a(1) 0 a(2) (g(i), b(4))0(4), b(1)) 0(2)1 s -1 (b (3 )))

= E g(2) O b(2)a(1) 0 a(2)(g(i), bms 1 0(3)))0(3), NI)) =

g(i)

b(2)a(i)

a(2) 0(2) ,

)•

Pairing the third tensor factor of (33) against b it follows that

E.E g(00amei0a(2)0(2), b(o)(fi,(2), b (2) ) (f, (3) , a (, )) (f,(1), S -1 (a(3))) = E g( , ) O a(4) (E i. ei(fil S -1 (a(3))b( 2)a( 1 ))) a(2) (g(2), b(,)) = E g(l) 0 a(4)8 -1 (a(3))b(2)a(i) 0 am (g(2), b (, )) =E g(i) O b(2)a(i) O a(2) (gm NO •

256

8. Quasitriangularity and Universal R-Matrices

Thus, we have shown that R.Zi(g a) = Zic°P (g a)R. Since R is invertible, 7 Zi 7 -1 on D(A) D(A). we conclude that 2PP Corollary 10. Any finite-dimensional Hopf algebra A is isomorphic to a Hopf subalgebra of a quasitriangular Hopf algebra.

Proof. A is isomorphic to the Hopf subalgebra 1 0 A of D (A ).

1E)

Remarks: 3. Since the map 0 defined by (26) is an isomorphism of the quantum doubles 13(A,(A°)c°P; Tin ) and D((A°)c 0 P, A; the element

(0 0 0)(7Z) = E i (ej 0 6) 0 (1 0 fi ) is a universal R-matrix of the Hopf algebra D((A°)'°P , A; A). 4. The structure of the universal R-matrix (29) becomes more transparent by the following interpretation. To each element x --= Er a r br E A 0 A° we associate a linear mapping ± - of A° by setting g := (g,ar )b, g G A°. is an injective map of A ® A° into (A° ) . Since Er (g, ei )fi Then, x. g for any g G A°, the so-called canonical element Ei ei 0 fi of A 0 A° corresponds to the identity map of A°. Under the embeddings a E. 0 a of f 0 1 of A° into the double D(A), the canonical element goes A and f into the universal R-matrix R. Therefore, the canonical element of A 0 A° and so the R-matrix (29) are independent of the choice of bases. A

Er

Finally, we return to the general quantum double D (A , B; a). Let A and be (not necessarily finite-dimensional) Hopf algebras equipped with an invertible skew-pairing a such that its convolution inverse is nondegenerate. As discussed at the end of Subsect. 8.2.1, the Hopf algebra D(.4,(A°)C°P; 71 21) is the double D (A , 13; a) with 13 = (A°)e01 , a =- 71 21 and T)-(a, f) = (f, a). That is, for the general quantum double D(.4,13; a) the convolution inverse Tr corresponds to the evaluation bilinear form (., .). Therefore, if we want to translate the assertion of Theorem 9 to the double D (A , 13; a), we have to take dual bases fed and {L} of the vector spaces A and 8, respectively, with respect to the inverse form The element R Ei (1 O ei ) (fi 0 1) is then an infinite sum in general, so it is not a universal R-matrix of D (A , 13; a) in the strict sense. However, when the tensor factors are paired against elements of 13 resp. A, all sums occurring in the proof of Theorem 9 become finite and the whole reasoning remains valid. This means that the sum R is considered as bilinear form on some appropriate dual. An elaboration of this idea leads to the concept of a universal r-form (see Sect. 10.1). Another possibility to remedy this situation is to work with topological completions of the tensor products. The reasoning of the above proof of Theorem 9 also gives the following h-adic version which will be used in Subsect. 8.3.2. Proposition 11. Suppose that A and B are h-adic Hopf algebras equipped with an invertible skew-pairing a of A and B such that its inverse â- is nonbe dual bases of the vector spaces A and B, degenerate. Let led and

8.2 The Quantum Double and Universal R-Matrices

257

respectively, with respect to the form -eY.f If R = Ei (1 ei ) (fi 0 1) be longs to the h-adic completion D (A , B; o- )D(A, B; a), then R. is a universal R-matrix of the h-adic Hopf algebra D(A,B; a). 8.2.3 The Rosso Form of the Quantum Double

Another interesting property of the quantum double is that it possesses a natural adL-invariant bilinear form (see Subsect. 6.3.2 for the latter notion). Let A and B be Hopf algebras and let a be a skew-pairing of A and B. The bilinear form (•, .)R on D (A , B; Cr) x D (A , B; a) defined by

a,

(b 0 a , b' ® a')R = a (S (a) , b' ) (a' , S (b)) ,

G

A, b , b'

E B

(35)

is called the Rosso form of the quantum double D (A , B; a). Proposition 12. The Rosso form (.,.)R on D (A , B; a) is acIL-invariant.

Proof. As noted in Subsect. 6.3.2, it suffices to show that (-c)R satisfies condition (6.91). Since adi, is a homomorphism and the algebra D (A , B; a) is generated by the subalgebras B and A, it is enough to verify (6.91) for elements 1 c and d 0 1, where c E A and d E B. By (21), we have ad i (d 1)(b 0 a) =--

E(d,„

= E d„,bs(d(3) ) adL(S(d)

1)(b

a)(S (d( 2)) 0 1)

a( 2)5- (a(i), S(d( 4)))a(a(3 ), S(d(2))),

1) (b ' 0 a') = E(S(d(2))

1)(b'

d)(512 (d( 1 )) 0 1)

= E s(40b1s2(d (2) )

a'( 2)0(ce( i ), S2 (d(1)))a(d(3)1 ,52 (d(3)))• SO). Using these relations, Recall that Tr(-, -) = o- (S(.), .) and a(.,.) = formula (35) and the fact that a (- .) is a skew-pairing, we compute (adi(d 1)(b0 a), b' ®a')R

E (S (a (2)) , b')a (a' , S((d(ObS(d(3))))0 - (S (am), S (d(4)))cr (a (3) E o- (S (a), S(d(4))b' S (d(2)))6r(a i , S 2 (d(3))S 5 (d(1)))

7 S(d

P)

))

and

=E =E

(I) 0 a, adL (S (d)

1)(b' (S (a) S (d ( 4))b' S 2 (d(2)))cr(d(2) , S Ono' (do) S (d(i)))a (do), S (d (3))) (S (a) , S(d(4))b' S 2 (d(2)))o- (a' , 512 (d(3))51 (b)S(d(1))).

Thus, (adi(d 0 1) (b0 a) , b'

a') R = (b 0 a, adL(S (d) 1)0'

d))R-

In a similar way one verifies that both expressions (adi,(1 6(b0 a) , 0.a') R and (b 0 a, adi,(1 S(c))(b' ®c'))R

are equal to

E a (S 2 (c(4))S(a)S(c( 2)),W)o - (S (c( 3) )a' S(c( i) ), This completes the proof of the adL-invariance of (., • )R.

LI

258

8. Quasitriangularity and Universal R-Matrices

8.2.4 Drinfeld-Jimbo Algebras as Quotients of Quantum Doubles The bilinear forms u(., • ) (.,-) from Propositions 6.34 and 6.38 are dual pairings of the Hopf algebras Uq (b + ) and Uq (b_)°P resp. Uh(b + ) and Oh,(b ) 0P, hence they are skew-pairings of A := Uq (b +) and 13 := Uq (b_) resp. A := Uh(b+) and B := Oh(b_). Therefore, the corresponding quantum doubles D(A, 13; a) are well-defined Hopf algebras by Proposition 8. We denote these doubles by D(Uq (b +),Uq (b_.)) and D(Uh(b+))00-)), respectively. Their products are defined by formula (24). By (20), the convolution inverse of a is just the bilinear form (•, •)' (S(.),-) defined at the end of Subsect. 6.3.1. We shall show that these two quantum doubles are closely related to the Drinfeld-Jimbo algebras Uq (g) and Uh(9)• Proposition 13. As an algebra, D(Uq (b +),Uq (b_)) is generated by 61 elements Ei , Ki , Ki-1 , F1 K Kr 1 . The defining relations are the relations for the generators Ei , Ki, K 1-1 , resp. Kr 1 , of the algebra Uq (b +), resp. Uq (b_), and the following cross relations: ,

,

ICi Ei K7 1 =

K1 Fi K77

qTti

Ki lf; K j1 Ki , EiFi Fi Ei ==6,

F. -1

Proof. As noted above, D(A, 13; a) is the free algebra generated by the algebras A and 13 with cross commutation relations (23). Further, it suffices to require the cross relations (23) for the generators a -= Ei ,K, of A Uq (b +) and b = Fi ,K,;, of B Uq (b_). For example, let a = Ei and b Recall that 4i(E1 ) = E 0 Ki + 1 g Ei and 4A(Fi ) F 0 1 + KT 0 F. Since (Ei , 1) = (Ki , Fi ) --= (Ei , K7 1 ) =--- (1, Fi ) = 0 by (6.82), relation (23) gives K1 .1(Ei , F1 ) +

= F3 E1 (K1 ,1) K7 1 .1(E1 ,Fi ).

Inserting the values (K1,1) =--- (1, K7 1 ) --= 1 and (E1 , F) = bii(C 1 - girl from (6.82), we get [Ei ,Fi l =- 6ij(qi 1 (1C1 - K7 1 ). The other three cross relations are derived in a similar manner. (Note that the equivalent cross relations (22) have already been stated without proof as formula (6.87).) Proposition 14. 73 (Uh(b+)) 0/4-)) is the free h-adic algebra generated by the h-adic algebras Uh(b +) and Uh(b_) subject to the commutation relations (23). For the generators Ei , Hi of Uh(b +) and the elements Pi ,H1 (see Subsect. 6.3.1) of Oh(b_.) these cross relations (23) are [Hi ,

= 0,

[Hi , Pi } == -aii Pi , [Ei 7 Pj]

614 h edi h

-a ii hEi

[E, e -di h

•

8.3 Explicit Form of Universal R-Matrices

259

Proof. The proof is analogous to the preceding proof. We verify (for instance) the relation [Ei , /Pi] = -aijhEi . Setting a = Ei and b = H in (23) and using (6.42) and (6.43), we get

(1,1)Ej ili =

( e dihHi

7

(e d-' hIli fii)Ei. 14) = -ah

= By the formulas in Proposition 6.38, (edj hlij I I) = 1, so the relation follows. and (1,1) = ( ed.; hx; ■

Comparing the doubles D(Uq (b +), Uq (b_)) and D(Uh(b +), Oh(b_)) with the algebras Ug (0) and Uh ( 3), we see that the generators of the DrinfeldJimbo algebras satisfy similar relations, but the number of generators corresponding to the Cartan parts is doubled. Quotienting out a half of these generators we indeed get the Drinfeld-Jimbo algebras. Corollary 15. The two-sided ideal lq (g), resp. Ih(g), of D(Uq (b +),UQ

K' resp. resp. D(Uh(b+), tili(b_)), generated by the elements i 1, 2, • • • , 1, is a Hopf ideal and we have canonical isomorphisms D(Uq (b +),Uq (b_))1;(9)

/IL

Uq (0), D(Uh(b+),0 h(b--))11h(9) U,(9)

of Hop! algebras resp. h-adic Hopf algebras. Proof. The proof follows from Proposition 13, resp. 14, combined with the defining relations of the algebras Uq (g) and Uh(9)•

8.3 Explicit Form of Universal R-Matrices 8.3.1 The Universal R-Matrix for Uh(s12)

Let us recall that Uh(s12) is the h-adic Hopf algebra with generators E, F and H satisfying the relations e hH

[H, =2E, [H, F] = -2F, [E, =

e -hH

eh — e - h •

The comultiplication Zi of Uh(s12) is uniquely determined by the formulas

Zi(E) = Eoe hll +10E, ,i(F) = F 01+C hH OF,

= H01+10 H.

Theorem 16. The h-adic Drinfeld-Jimbo algebra Uh(s12) is quasitriangular

with universal R-matrix given by

7?- = eh(H V I) / 2

E n=0 cc Rn(h)(E n 0 Fa),

(36)

where Rn (h) := qn(n+ 1 )/2 (1 - q -2 )nanbi !) -1 and q = eh . Proof. We begin with the relation (46i id)(7Z) = 1Z 13R.23 . First we remark that

260

8. Quasitriangularity and Universal R-Matrices id)(7?.) =E oe

(A

m,n=0

amn (Eln 0 En 0 Fm+n ),

(37)

where arn, is the element of Uh(t0 3 given by —mn

m {

[miq [Til

: 1

Rm+ ,,(h){(46 0 id )( e h(H0H)/2)}( 1 e rnhH 1 ■) .

(38)

The summand in R131Z23 containing the term Em 0 En 0 Fm+n is equal to

Rm (h)R,(h)e h( " 1") I2 (Ern 01 0 Fra)e h(1"" )12 (1 0 En 0 Fa). (39) By induction on m one shows that {1

H 1/1 Ern

bh H Ee —bhH

0 1 0 F rn i = —2M(Em

H Frn ).

eh e bh E and e bhH F —ebh FF,, ethe latter relation H =e

leads to the equality (1 0 e rnAH e -hooHoH)/ 2 (Evin 01 0 F rn )e h(1®11®11)/2 -=

1)(Em 0 1

0 F m ),

(40)

On the other hand, (A 0 i c e h(11o1')/2) = e h(ZI(H)OH)/2 eh(HO1gH±igHgH)1'2. From formulas (37)-(41) we see that the assertion (A valid provided we know that [rn,

q

(41)

id)(7Z) = R13R23 is

Rm+,(h) R

[Tniq![n]q! This identity is proved by an easy computation. The equation (id 0 A)(R) = R13R12 is proved in a similar manner. Next we turn to the relation A"P(a) = R.ZgarTZ -1 . Since both sides of this equality are algebra homomorphisms from U h (S12) to Uh(S12)(W h(S12), it

is enough to prove that ACOP( a )7Z = R(a)

for a

E, F H.

(42)

,

We carry out the proof for a = E. The cases of F and H are treated similarly. For a =--- E, (42) can be rewritten as Co

(E 0 1

e hH

E Rn (h) eh(HOH)/2(En Fn) n=0 cx)

n h (H OHO ( En 0 = ( E .R.(h)e

Fn))

(E

n=0

The difference between the left and the right hand sides is

e

10

E).

8.3 Explicit Form of Universal R-Matrices

261

CIO

Ee

h(11011)12

( e —hH

Rn (h)(1

e hH ))

n=0

o n e hH _ q -n e -hH )1 (En-I-1 +Rn+i(h)[n + Jig (i q q —i

(43)

Fn) .

(h), the coefficient at En+ 1 Fn in Since [n + lj q R,+ i(h) - qn(q (43) is zero and hence (43) vanishes. Finally, a direct computation shows that the element

e —h(H01 -1)12

E e -hrtRn(h)((e hH E) n

( e hHF)n)

n=0

of Uh(S12) 1 Uh(S12) is the inverse of R..

8.3.2 The Universal R-Matrix for Uh(g) Let g be a finite-dimensional simple complex Lie algebra. The following theorem provides an explicit expression for the universal R-matrix of the DrinfeldJimbo algebra Uh(g). Theorem 17. Uh(g) is a quasitriangular h-adic Hopf algebra. Let Er and For be the root elements of Uh(g) from (6.65) corresponding to the sequence Oil 021 - • 7 On of positive roots of the Lie algebra g. Let (B o ) be the inverse of the matrix (Cii ) (d.T 1 aii ). Then the element n

R.= exp (h

E Bii (Hi

r1, -,r=0

exp (h

-1-rifri-1-1)

E

Hi ))

(

1

—

q 2 ) rj

E

r '

ri

j=1

(44)

(Hi ® Hi)) fi exp q0 ((1- q0-2 )(Eo Fo)) i3E2LF

of Uh(g)Uh(g) is a universal R-matrix of Uh(g), where q o := qi if and the simple root ai lie in the same orbit with respect to the Weyl group W of g, the factors in the product (a) appear in the order On , On - I ,. • • , 01 and the q-exponential function exp q x is defined by expqx = Erw 0 qr(r+1)/ 2 xr [T ]q!

1

Sketch of proof. The key step of the proof is to determine dual bases of the h-adic vector spaces Uh(b +) and C/h(b_) with respect to the bilinear form = (•, .)' from Subsect. 6.3.1. By Theorem 6.24', the set of elements -••

Bri g:,

•

7

m1,-- • 7 Mt 7 7'17

•

NO

(45)

E NO,

(46)

E

constitute a basis of Uh(b+). Similarly, the set of elements

fir 1

Sn

• P-1

;

mi, • • • ,M1 7 S1 7 ' 7

•7

Sn

262

8. Quasitriangularity and Universal R-Matrices

14

is a basis of Oh (b_), where Ek We compute the bilinear form (-, •)' at the basis elements (45) and (46). For this we essentially use the formulas at the end of Subsect. 6.3.1 and the fact that (•, •)' is a dual pairing of the Hopf algebras Oh (b + )°P and Oh(b . Since (Hi , /i Bik(Hi l fik = Akd i i aki =

Ek

Ek

)'

and il (n-1) (H= L-di=0 1 n

( 1/171i

® H1

we obtain

' • • 0 fiir i Y= 606rnini ali!.

= (( zA (ni -1) (H)Yni 7

(47)

A similar slightly longer reasoning shows that

(Ejrn» 7 fr-ini =

mini

1 ni qi

(qi

(ni -1)12

(48)

)

In order to determine the values of (., .)' at arbitrary elements (45) and (46), we need an expression for A(Ef3), 3 E A+ . Clearly, the Hopf subalgebra Uh (9 i ) of Uh (0) generated by Ei , F Hi is isomorphic to Uh(512). By Theorem 16, the universal R-matrix for Uh(gi) is, up to a certain factor, of the form ,

x-, c° (1 - q,7 2 ) m r. ri(m+i)/2 (Er Firn) , - [ml q,! rn.o

where qi = edih . Recall that Ti is the algebra automorphism of Uh(g) defined by the braid group action (6.62)-(6.64) on Uh(g). One can prove that =

Ti)A(a))74

a E Uh(E)-

(49)

By Definition 6.4, the root element Er for a positive root Or G A + is given by Er = Z T12 • • • Ti r _ i (Eir ). We set Or

(Ti1Ti2 • • •

Zr-1

0

Ti1Ti2"

One can derive from (49) that

1'.---1 (f3r)(E07, g eh ar + 1 0 Ex)7(8,),

A(Epr )

where 1?-,(0,.) *8r- 2 • **al and ri > 0. Then we have

Har.

=

di ri Hi if A.

Ei ri ot,

7-?-.-1 ([3,-)(Ep r O ehllor + 1 Ep)nke r ).

L(E)

From the latter formula we derive that A(Erdr ) -

(Ear 0 ehHfir + 1 0 Esi n E Uh(b+)

qv (n+)•

(50)

One verifies that any braid group automorphism Ti satisfies the condition (Ti (a), Ti(b))' = (a, b for a E Uh(n+) and b E Oh(n.4. Therefore, if a positive root and a simple root oti of g belong to the same W-orbit, then )'

8.3 Explicit Form of Universal R-Matrices

(Es,

263

= (E, P')' = h(C 1 — q1) 1 .

= (Er, P)' for n, m c No. Further, from (50) one derives that (Er, Using the above relations, especially (47) and (48), one proves by induction that

(Hr'

Hriz E73:: • • • E

••

(H7ni ii7iY • 6nimMi! i '

PA: • • • fTi y

Tr (Er' 7 Fs [ri ] qp !hr-'

7--r =

7,"

l i=i

6

3

sri a n (1

—

)i

-rifri+1)/2

(51)

go —i2 )rj ql3i

That is, the elements (46) divided by the scaling factor appearing in (51) and the elements (45) form dual bases of Uh(b + ) and Oh,(13_), respectively, with respect to (., .)'. For brevity, let us denote these sets by fe i l resp. {L}. From the particular form of these elements we see that RD := Ei (1® ei )0(f i 01) is an element of the h-adic completion D(Uh(b +), h(b_ ))D(Uh(b +), Oh(b._)). Hence, by Proposition 11, RD is a universal R-matrix of the quantum double D(Uh(b +), Oh,(b_)). Under the usual identification of ei with 1 0 e i and of fi with fi ® 1, we get RD = Ei ei g fi . Let it be the canonical homomorphism of the double D(Uh(b+), Oh,(b_)) to U h (9) given by Corollary 15. Inserting 7)7?,D (TT the elements (45) and (46) into RD, one computes that R, takes the form stated in the theorem. Since RD is a universal R-matrix of the double D(Uh(b +), Oh(b_)) and TF is a Hopf algebra homomorphism, R, is El a universal R-matrix of Uh(g).

Remarks: 5.

Recall from Subsect. 6.2.3 that the sequences {Eoi } and {Foi } of root vectors of Uh(g) depend on the reduced representation wo = wii • • • wit, of the longest element wo of the Weyl group of g. Nevertheless, it can be shown (see [KR21) that the universal R-matrix (44) is independent of the choice of the reduced decomposition of WO. 6. The sum E i,j Bii Hi ®Hi occurring in formula (44) is just the canonical element with respect to the symmetric bilinear form (., .) on given by (Hi , Hi ) = (see Subsect. 6.1.1). Indeed, setting G3 := Ei Bii Hi , we have E Bii Hi g H = Ek Gk g Hk and (Gi

=

Bii

(Hi,

=

E z. Bij (11k Hi) = E

2

C k iBij

6jk

Sometimes it is also convenient to choose other dual bases in b or to take a self-dual basis of b. Then only the first factor exp (hE Bii (Hi Hi )) in (44) has to be correspondingly changed. 7. The h-adic Hopf algebra Uh(gli+i ) is also quasitriangular. A universal R-matrix of it is given by the universal R-matrix (44) for Uh(sli+i ) with exp (hE i B 1 (H HA) replaced by exp (h 0 H)), where H.:, i = 1, 2, • • - , L + 1, is a self-dual basis of the Cartan subalgebra with respect to the bilinear form (., .).

Ei

264

8. Quasitriangularity and Universal R-Matrices

8.3.3 R-Matrices for Representations of Uq (g)

Recall that the h-adic algebra Uh(g) is formally transformed into the complex algebra Uq (g) if we replace edi h by qi and edihili by K. But the universal R-matrix (44) of Uh(g) does not yield a universal R-matrix of Ug (9) in this manner. The first difficulty is that the second factor in (44) is not in Ug (2) 0 Uq (g), because it is an infinite sum. However, it belongs to the completion rig+ ( 9)W q+ ( 9) defined in Subsect. 6.3.3, and it yields a finite sum when a highest (lowest) weight representation acts on the first (second) tensor factor. The second problem is that the first factor in (44) has no counterpart in r./q+(g)calit(g). But the inner algebra automorphism of Uh(g)(Wh(g) generated by.this factor does and we shall work with this algebra automorphism of CI± - q (9) W+q (9) instead. We assume throughout this subsection that q is not a root of unity. One verifies that there is an automorphism 0 of the algebra Uq+(g)Wq±(g) such that

(52) (53) (54) Recall that U-qF (g) and (9)0U-qF (9) are the completions defined in Subsect. 6.3.3 and Co c Uq-- fi(n_)0N3 (n+ ) denotes the canonical element with respect to the pairing (., .) (see (6.82), (6.97) and Proposition 6.37). We define

:=

E q (0,0)(K o O K 0-1 )C0

E

OEQ±

A direct computation shows that 91 coincides with the sum occurring in formula (44) for the universal R-matrix of Uh(g), that is,

=

E ll

( n

r1,•-- 1 r n =0i=1

1 qf-3-j2 [7"

-]

Y 3 g ri

°-1

3 qfii

(ri +1)/2 r Eolj

r3

Theorem 18. The element 9i and ihe above algebra autornorphisrn •P of U± =3:)± q (g)rJ q CO satisfy the following conditions:

a e U: (9),

(55)

(012 0 013)(g123) = 912 3 5

(56)

91A(a) = 4/o("P(a))91, (023 0 013)(9112) = 91121

0 23(9113)9123 = (Zt 0 1)(N), 012(9113)9112 = (1 0 .6)(91). 91 is invertible in the algebra

(g)(7)(1 (9)•

Proof. Our first aim is to prove that 9:1-1

:= E ,v3,0)(1 0 Ko(s 0 1)(C,) pEQ,

(57)

8.3 Explicit Form of Universal R-Matrices

265

is the inverse of 91. Clearly, the product 9191 -1 is a sum over ,3 E Q+ of terms

E c,, . ( K:5-1

0 1) • s 0 iw,5). (

(58)

1, 6 €Q+

1+6=0

For [3 = 0 this expression is obviously equal to 1 0 1. Thus, in order to prove that 9191' = 1 0 1 it suffices to show that all terms in (58) for 0 0 are

zero. The sum (58) belongs to r./q-43 (n_) 0 Uq (g). Since the bilinear form (•, •) is nondegenerate on ue( b +) x U(b_) by Proposition 6.37, it is enough to prove that any term (58) for 0 0 vanishes when the first tensor factor is paired against an arbitrary element a E Ufi,3 (n+ ). We can write

E

,A(a) =

0 (TYq (n+), a6,,y E On+) q

a(1 ®K5 ),

7,6EQ+

1+6=0

E a6 ' 0 a'" '1 b,m

1

CL,5: 6.'Ym

E

Uq5 ( n ± ) ,

a,t6',7„ E U`:(n + ).

m

If a,(73. and

b,f3_ are as in the proof of Lemma 6.40, then pairing (58) in the first

tensor factor against a yields ((a, •) 0 1) (

E

a7..K6-1 8(4) 0 b7..b 6s ) = ;Y .

E

(a, 41‘6-1 ,5(4))Vrf Vs

) b 7, b !

,_ 'Y,f 5,r,s,rn

=

E (E( -Y, 6 ,n1

=

E - Yo5 , 7n

"t1rn

' cO)P) r r

(E(a.56 '1, Ir 1 51(a))b.6, ) s

r

(E(a,(6',77. a7. )1)7) r

s

a1C6 S -1 (41) = S-1 ,fi,rn =-- (51-1 0 M 0 (1 0 S) 0 ,i)

E abas(a,(6,;(K,5 ) 'Y,45, 1n

(a) = 6.(a)1 =0,

where we used various properties of the dual pairing (., .) of the Hopf algebras Uq (b + ) and Uq (b_)°P. Thus, we have proved that 9191• 1 = 1 0 1. Similar reasoning shows that 91-1 91 = 1 0 1. Next we turn to the relation (55). Since zi and 91' • 0 0 ,Ac°P • 91 are both algebra automorphisms, it is sufficient to prove (55) for the generators a = KA, Ei , F. The case a = IC), is easily done. The cases a = Ei and a = Fi follow from (6.99) and (6.100). The relations (56) are proved by straightforward verification. The relations (57) are equivalent to

8. Quasitriangularity and Universal R-Matrices

266

1)(Cp) =

(A

E

q --frf,6)(K-1 ® I ® 1)•(at )13 •(c6) 23 6

7+6=0

and (1 0 46)(c,) = E

q -e7,6)(1 0 1 0 K5).(ay)13.(c5)12.

7,6EQ+

7+6=0

These identities are proved in a similar way as the fact that the terms (58) vanish for O. We omit the details. Suppose that Tv and Tw are two weight representations of Ug (9) on vector VI, spaces V and W, respectively, with weight space decompositions V = Ww . Let B vw denote the linear operator on V 0 W given by and W =

B v w(v

w) := q (4 '4')v w for y E V12 , w E

Now we define the R-matrix Rvw : V 0 W —+ V 0 W and the linear mapping ftvw :V0W--*WOVfor the two representations Tv and Tw by

Rvw :--= Bvw o (Tv 0 Tw)(91) Clearly, ftvw is invertible with inverse

and

itvw := r oR vw .

= (Tv 0 Tw)(91 -1 ) o B

(59)

1

Proposition 19. The mapping itvw intertwines the representations Tv ®Tw and Tw O Tv. The family of mappings itvw has the properties stated in

Proposition 4. Proof. A direct computation shows that on weight vectors we have ((Tv

Tw) o 0) (al 0 a2) = Bvilv (Tv 0 Tw) (al 0 a2)Bvw, al, a2 E Ug g

Therefore, this relation holds on the whole space VOW. Applying the tensor product representation Tv 0 Tw to both sides of (55) we derive

B viv Rvw(Tv

Tw)(A(a)) = ((Tv ØT) 0 45)(A"P (a))Bav Rvw.

Using the previous equation and applying the flip to both sides, we obtain .fivw (Tv 0 Tw)(a) = (Ta, 0 Tv)(a)f?vw. Next we prove (12). It follows from the first relation in (57) that

Bi4-flu ((Tv0Tu)(9113))Bwu(Tw0Tu)(9123)

= (Tv OT w 0Tu)((A0id)(91)).

This implies that B iVuB 1711 RvuRwu = BuRvow,u. Clearly, we have /3/7vlu Bal = w,u . Therefore, applying the flip permuting elements of V 0 W and U, we obtain (12). The relation (13) is derived in a similar manner from the second relation in (57).

8.4 Vector Representations and R-Matrices

267

8.4 Vector Representations and R-Matrices This section is concerned with the R-matrices R 1 , 1 := Rity for the vector representations 711 = Tv of the Drinfeld-Jimbo algebras Uh(g) and Uq (g). These matrices will be the starting point for the definitions of quantized simple Lie groups in Chap. 9. Using the notation qa = e ah 1 a G C, in the h-adic case, we shall obtain expressions for the R-matrices that apply to both algebras Uh(g) and Uq (g) simultaneously. In the case g = 502n+1 we shall use the algebras Uq 1/2 (so 2,2±1 ) and Uh/2(so2n+1)• 8.4.1 Vector Representations of Drinfeld-Jimbo Algebras The "usual" realizations of the Lie algebras gl(N, C), sl(N, C), so(N, C) and sp(N, C) by complex N x N matrices are irreducible representations with highest weight (1, 0, • • • , 0). They are commonly referred to as the first fundamental representations or as the vector representations. We now generalize

these representations to Drinfeld-Jimbo algebras. Definition 5. The vector representation T1 of the Drinfeld-Jimbo algebra Uq (g) or Uh(g) is the irreducible type 1 representation with highest weight A = (1,0, • - • , 0) with respect to the ordered sequence -a i , • • -, -at of simple roots. The simple roots -a i , • • - , -al are chosen only for technical convenience. Note that a representation T of Uq (g) or Uh(g) has the highest weight A with respect to the simple roots -a i , • - • , -ai if and only if the representation To9 has the highest weight A with respect to the simple roots a i , • • • , al (recall that in the first case weights are considered with respect to the elements Ki-l- and in the second case with respect to Ki ). Here 0 denotes the algebra automorphism of Ug CO or Uh (9) from Proposition 6.9 or from formula (6.47),

respectively. Let Eii be the N x N matrix with 1 in the (i, j)-position and 0 elsewhere. The vector representation T1 of Uq (g) is described by the following list: Ug (g1N ): Ti (K) --= q - 1 Eii +

E

Eii,

T1 (E) = Ei+i,i, Ti (Fi) = Ei,i+17

i = 1, 2, • • • , N,

i = 1,2,•••,N-1.

uq (s1N): Ti (Ki) = q —l Eii + q-Ei+1,i+i +

T1 (E) =

E jOi,i+1

TOO = Ei,i+17

i =-- 1,2, • • • ,N -1,

,N -1.

Uq 1/3(soN), N = 2n ± 1: T1 (Ki) = Di-1Di_FiD_i+1D2n-i+21

i =1,2,---,n -1,

268

8. Quasitriangularity and Universal R-Matrices

= Ei,i+1 E2n-i+1,2n-i+27

i= 1,2,• ••n-1,

T1(E1) = Ei+1,i E2n-i+2,2n-i+11

i= 1,2, • • •n-1,

Ti

Ti WTI

C(En,n+1 (1 -112 En+1,n+2), 111(En) - C(En+1,71

_1/2 c rin+2,n+i)

T1(K) = D' D2,

where c=

[21q1/1/22

(q1/ 2 ±q - 1/2 \) 1/2 and Di is the diagonal matrix

Ei q 6i3 Eii .

Uq (sp N ), N = 2n: (Ki ) =

z=1,2,---,n-1,

iD2n --i+

(Fi ) = Ei,i+i - E2n-i,2n-i+13

Ti (E) = Ei+i,i

i=1,2,--•,n-1, i=1,2,•••,n-1,

E2n-i+1,2n-i7

(Fn ) = En , n +i, Ti (En ) =En +L n T1(K) = Dn-2 Dn2 + 1 .

Uq (soN), N = 2n: 1 D2-nl_ i -D271 -i+

Ti (K ) =

- 1,

2-11 (Fi) = E,+ - E2n-i,2n-i-F17

i = 1,2,-• • ,n -1, - 1,

T1 (E) = E • +1,i -

(F7,) = -En , n +2 ± En_ 1, n +1 7 Ti (En) = - En+2,n

En+1,n-17

T1(K) Dn- 1 Dn- i Dn-F1D71+2 •

By straightforward calculations one checks that the preceding formulas define an irreducible weight representation T1 of the algebra Uq (g) on the vector space CN . For the basis vector e l = (1,0, - - , 0) we easily verify that Ti (Uq (g))e i = CN, = elie i and Ti(Fi)ei = O for all i = 1, • • , l. Hence T1 is the type 1 representation with highest weight A = (1,0, • • • , with respect to the simple roots -al, • • • , ai (see also Definition 6.6 and Remark 6.7 in Subsect. 6.2.4). Thus, T1 is indeed the vector representation of Uq (g). If a reduced decomposition of the longest element wo of the Weyl group W of g is fixed, then the action of the corresponding nonsimple root elements can be derived from the above formulas. For example, let g = sl(N,C). We take the reduced decomposition -

wo = (w1w2 • • • wN-1)(w1w2 • • wN-2) • • (w1w2)wi

of the longest element wo of the Weyl group. Here wi is the reflection corresponding to the i-th simple root. Then the roots from Subsect. 6.2.3 are 0121 0131 • • • 01N T-3231* • ' f32N1' ' ,

ON -2,N-17 ON -2,N 13N-1,N

where Oii = ai + ai±i + - • • + ai-i. One easily calculates that

8.4 Vector Representations and R-Matrices

269

T1 (F) =

T1 (E) =

This formula holds also for the root elements of Ug (g1 N ). The vector representation T1 of the h-adic algebra Uh(9) is determined in a similar manner. It acts on the vector space CN [[h]] over the ring C[[h]]. The formulas for the representation operators T1 (E) and T1 (Fi ) are the same as above. The actions of the Cartan generators Hi are given as follows: Uh (g1N ): (H ) —

— 2 ) 3 N•

En )

Uh s1N : (

)

Ti (Hi) = -Eii + Ei+i,i+17 Uh12(S02n+1):

±E2n-i+2,2n-i+27 = 1)

n

T1 (H) = 2( Enn -

Uh(SP2n):

-Eii + Ei+i,i+i - E2n--i E2-j+1,2-i+1, i= 1,2,•••,n— 1,

T1(11 n ) = Enn

E+1,+1.

Uh(S 0 2n):

-Eii + Ei+i,i+i - E2n-i,2n-i E2n-i+1,2n-i-1-11 z - 1 2, • , n — ,

(H) =

Enn E+1,+1 En+2,n+2.

8.4.2 R-Matrices for Vector Representations

In this subsection we compute the R-matrices R 1 , 1 := Rvi v, for the vector representations T1 = Tv, of the Drinfeld—Jimbo algebras Uh(9) and Uq (g). As noted above, for g = so2 n ±1 we shall take the algebras Uq 1/2 (s0 2+1) and Uh12(S02n+1) instead of Uq (502n+1 ) and Uh(S02n+1), respectively. Recall that R1,1 = Rv 1 v1 was defined in Subsects. 8.1.2 and 8.3.3. Let us set R

q l /NR1,1 for g =-- glN ,s1N and R := R1,1 for g =

r? :=

k

SON,

sPN-

As earlier, we put T o R. The notation R and (with sans serif letter R) will be kept throughout the remainder of the book in order to distinguish between these particular R-matrices and general R-matrices. The latter are usually denoted by R and R. We begin with the h-adic algebras Uh (g). Then, by (11), we have R1,1 = (T1 T1 )(1Z), where R. is the universal R-matrix (44) of Uh(g)•

270

8. Quasitriangularity and Universal R-Matrices

First let g = glu , SIN. Using the description of the vector representation T1 from Subsect. 8.4.1 we find that ( T1 0 TO lexp (h Y7 N 1 Bii (Hi e —hIN (eh

V N Eii Eii +

E

For all positive roots 3 we have Ti(E0) 2 = 0. From the expressions for the operators T1 (E) and T1 (F130 ) obtained in Subsect. 8.4.1 we derive that (Ti Ti){expq ((1

e -h, )(Ei,

q-2 )(E0„ F))} =1 +( e h

Eii).

Multiplying both expressions and inserting the convention (la = eah we get R = q E(E2,

En ) + E(E„ 0E33 ) +

q -1 )E(Eij Eii ). (60) 1,>j

ii

The operator (2-11 7-1)(7Z) for Uh/2(so2 n+ 1), Uh(s02,2 ) and Uh(sp2 n ) can be calculated similarly. We state only the outcome of these computations. In order to do so, some more notation is needed. Let (Pi, P21 PN) denote the N-tuple (n -

-

• • • 0,

for g = so2n+i,

• • • , -n +

(n - 1,n - 2,...,1,0,0, -1,

-n + 1) for g = so2n,

(n, n - 1, .. • , 1, -1, • • • , -n) for g = sp2n .

We set Ei := 1, i = 1, 2, • • • , N, for g = soN and Ei, := 1 if i < n, ci -1 if i > n for g = sp N , N = 2m. Put := N +1 i. For Uh/2(so2n-Fi.), Uh(sP2n) and Uh(s02) we then derive that R= q

E(Eii

Eii ) +

E (Eii

Eii ) + q -1

E(Eiti,

Eii )

Joe

E(Eij Eii)

pi—Pi Eie

-(E j i

3

E) (61)

+E(N+1)/2,(N+1)/2 E(N+1)/2,(N+1)/27

where the last summand must be omitted for g =sD 2n,SO2n and the summations are over 1, 2, - • • , N. Now let us turn to the Drinfeld-Jimbo algebras Uq (g). In this case, we have R1,1 = Byv 0 (T1 T1)(91) by (59), where 91 and Byv are defined in Subsect. 8.3.3. Carrying out similar computations as above we find that the matrix R is given also by the formulas (60) for Uq (g1N) and Uq (s1N) and by (61) for Uq 1/2(502n+ 1), Uq (sp2n ) and Uq (s02n ) with q being now a complex 1

8.4 Vector Representations and R-Matrices

271

number rather than an abbreviation for e1 . Strictly speaking, this derivation of R is justified only if q is not a root of unity, because 91 was defined under this assumption. But the expressions (60) and (61) make sense for any q, so we take them as definitions in the root of unity case. By continuity, the intertwining property 1( T1 0 = (T1 711 )f?, the quantum Yang-Baxter equation for R and the braid relation for if'Z hold for all q. We conclude this subsection with an important symmetry property of the matrix R = R1 , 1. for g = 50 2n+1 , 50 2n 5P2n which emerges from the self-duality of the corresponding vector representations T1 . Let T be a representation of a Hopf algebra A on a finite-dimensional vector space V. Then the formula (T' (a)v')(v) := (T(S(a))v),

a C A, V C V,

C

V',

defines a representation T' of A on the dual vector space V' called the contragredient or the dual representation to T. If T is equivalent to T', then T is called a self dual representation. Let us identify the vector spaces V and V' by fixing an isomorphism. For linear mapping A E L(V), we denote by At the transposed map to A defined by (A t v)(w) = v(Aw), y, w E V. One easily checks that a representation T is self-dual if and only if there is an invertible -

linear mapping A E L(T) such that

T (S (a)) = AT (a)t A -1 ,

a E A.

Now we turn to the vector representation T1 of Uq (g) resp. Uh (0) on 171 . We identify V1 with its dual V1' by taking the standard bases of CI' resp. CE[hr as dual bases of V1 and VJ . For g = S 0 2n+11 502n 7 8 P2n we introduce the N x N matrix

C = (C) with C := Note that C2 = / for g = 502n+1l502n and C2

(62) for g =

SD

Proposition 20. Let g be 50 2n+1, 80 2n or sP2n . Then the vector representa-

tions T1 of Uq (g) and Uh(g) are self-dual and we have

111 (9(a)) =- CT1(a)tc-1

a E Uq (9) resp. a E Uh(g),

R = C1(R t1 ) -1- Cr i = C2(R - T2 C2-1 ,

(63) (64)

where t i and t 2 are the transpositions in the first resp. second tensor factors. Proof. Using the formulas from Subsect. 8.4.1 one verifies the relation (63) on the generators. Hence it is valid on the whole of Uq (g) resp. Uh(9)• We prove (64) for Uh(g). Using the first formula of (8) we compute

= (T1 T1)(R -1 ) = (T1 ( Ti 0 S

o Ti)(S

0

icI)(7Z)

TO (R.) = ( CTI C -1 0 Ti)(1?-) = C1RC j4.

272

8. Quasitriangularity

and Universal R-Matrices

This in turn implies that R = C i (Rt i)_ i Cia l . The second equality of (64) follows similarly using the second formula of (8). By construction, the matrices R and C for Uh(g) go into the corresponding eh E C[[h]l is replaced by the complex number q. matrices for Uq (g) if q 0 Therefore, formula (64) holds also for Uq (g). 8.4.3 Spectral Decompositions of R-Matrices for Vector Representations We begin with two general results for the h-adic algebra Uh(g). Proposition 21. Let u be the element from Proposition 5 for the h-adic quasitriangular Hopf algebra Uh(g) and let p be the half-sum of all positive roots of g. Then the element e - hHPu = ue- hHP belongs to the center of Uh(9) • If TA is an irreducible type 1 representation of Uh (g) with highest weight A, then TA(e -hHPu) = exp(-h(A, A + 2p))I. The comultiplication of e -hHPu is given by A(e -hHp u ) _ ( e -hHp u e -hH PU).(R.21R.) -1 . (65) Proof. The proof is given in [Dril ] .

0

Proposition 22. Suppose that TA and TA' are two irreducible type 1 representations of Uh(g) with highest weights A and A', respectively. Let TA be an irreducible representation with highest weight p which occurs in the decomposition of the tensor product TA 0 TA/ into irreducible components. Then the operator (TA OTO(R211Z) acts on the corresponding subspace VA of VA 0 VA' as I (66) where I is the identity map of V. Proof. Since 4(a)/Z2 1 R = R.21 1Z4(a) by Proposition 1, (TA 0 TA(R-21 1Z) commutes with all operators (TA 0 Tv)(a) (TA 0 Tv)(A(a)), a E Uh(9)• Therefore, the restriction of (TA 0 TAI)(R,21 7?) to the carrier space VA of any irreducible component TA of TA 0 TA' is a multiple of the unit operator I. The explicit expression for the operator (TA OTA , )(7Z217Z) on VA can be found by means of Proposition 21. Indeed, by (65), we have (TA 0 TAI)(R21R)(TA TA , )(L(e - hHpu )) TA ( e —hHpu )

TA, ( e —hHp u ) .

.. acts as On the irreducible subspace VA the operator (TA 0 TA,)0(e-hH p u) e-h(A,P+2P)I• Hence (TA 0 TA' )(R-21 7?) takes the value (66) on V. LI

Corollary 23. Let TA be as in Proposition 22. Then all eigenvalues of the operator .hAA := T o (TA 0 TA )( I?) are of the form (67) where p is the highest weight of an irreducible component in the decomposition of TA 0 TA.

8.4 Vector Representations

and R-Matrices

Proof. For the operator IL :-.-=--

T 0 ( TA

0 TA ) ( R.), we have

.TIL = T 0 (TA 0 TA ) (R) •

TO ( TA

0 TA ) (R.) = (TA 0 TA)(R21 1Z).

273

Thus, the squares of eigenvalues of ./4AA are eigenvalues of (TA 0 TA )(R-211z). LI Hence the form (67) follows from (66). Let S(VA 0VA) be the span of eigenspaces for the eigenvalues (67) with sign + for any possible /J. We call S(VA 0 VA) the quantum symmetric subspace of VA 0 VA. The subspace A(VA 0 VA ) generated by the eigenspaces for all possible eigenvalues (67) with sign — is called the quantum antis ymmetric subspace of VA 0 VA. Since TA 0 TA is completely reducible, we have VA 0 VA = S(VA 0 VA ) ED A(VA 0 VA).

(68)

Now we apply these results to the vector representation T1 of Uh(g), g = glu ,s1N,soN, spN • The corresponding R-matrices R1,1 := (T1 0 T 1 ) (7e) have been computed in Subsect. 8.4.2. As noted there, the R-matrix R for the algebra Uq (g) is obtained from the R-matrix R for Uh(g) by replacing the power series q = eh E CP,11 by the complex number q. Under this replacement, the subspaces S(VA 0 VA) and A(VA 0 VA) are also well-defined for Uq (g) and the subsequent results are valid for both algebras Uh (g) and Ug (g). First let g = gl u , slN • By (7.13), the tensor product T1 0 Ti decomposes into two irreducible representations with highest weights (2,0, • • • , 0) and (1,1, 0, • • • , 0). Inserting these highest weights into (67), we find that the matrix .fi 1 , 1 has the eigenvalues ql 1 /1V on the first component S(Vi 0 Vi) and .12, 1 1/N on the second subspace A(Vi 0 V1 ). That is, we have Proposition 24. The matrix k = 7- oR = ql/N it i , i for the algebras Uh(g1N), Uh(s1N), Uq (g1N ) and Uq (s1N) satisfies the quadratic equation

(k — qI)(f? + q -1 I) = 0

or

I=Z2 = (q — C I )I + I.

(69)

The matrix f? has the spectral decomposition

f? = q p + _ 47 -1 p_ 3

(70)

where the projections P+ and P_ are of the form

IF? + 47 -11"

P _. --=

P+ =

The operators

4? + qi

(71)

are quantum analogs of the symmetrization and antisymmetrization operators on CN 0 CN , respectively. Now we consider the cases g = 502,,+1 and g = 50 2n. By (7.14), (71)

Ti 0 Ti =

T2

ED To El) T(1 , 1 ) .

(72)

274

8. Quasitriangularity and Universal R-Matrices

The irreducible representation T( 1 , 1 ) with highest weight (1,1,0, • • • , 0) acts on the subspace A(Vi 171). The subspace S(Vi 171 ) is the direct sum of two invariant spaces. On the first one, denoted by SW1 0171 ), the symmetric irreducible representation T2 with highest weight (2, 0, • • • , 0) is realized. The second subspace is one-dimensional and the corresponding subrepresentation To is trivial. From Corollary 23 we obtain the eigenvalues of R stated in the

following Proposition 25. The operator IEk= roR = 1t 1 , 1 for the algebras Uh/2(--s0271+1); Uh(so2 n ) and U,7 1/2(502n+1); Uq(502n) acts on A(Vi 1 71) as -q -1 1-, on 81 (171 0 VI ) as qI and on the one-dimensional invariant subspace as ql - N I, where N = 2n resp. N =7-- 2n + 1. f? satisfies the cubic equation (r? (bo o'? + trin(I? qi-N 1- ) = 0.

(73)

If N > 2 and (1 q 2)( 1 q 2-N)(1 _ q - N) 0, it follows from (73) that the operator rz for Uq i/ 2 (s027-1+0 and Uq (s02n ) has the spectral decomposition

= qp+ q —i p _ where

IF? 2 P+

q i_N po,

(q i-N _ q -1)És? (q

r? 2

(q

q - 1)(gi

q i—N ) f?

q1.1%r )

(74)

7

q —N+21 (75)

(q + q.-1)(q. - 1 + (11^ N)

Ék 2

(q.

)f?

I (76)

P o = (ql - N q)(q - 1 + q1 - N )•

Here P ± , P_ and Po are the projections onto AM Vi), 4.5'(V1 Vi) and the one-dimensional invariant subspace, respectively. One can easily verify that the matrix f? for Uq l / 2 (so2„± 1) and Uq (502 n ) also satisfies the relations tk--1 = 7 0 k( q — 1 ) 0 ,r,

"1z — iEk -1 =

(q q -1 )(I

K)

(77)

with

K :=

q Pi— Pi EN+1—i,j

Ei,N±i_i

(78)

where the numbers pi have been defined in Subsect. 8.4.2 and f(q -1 ) denotes the matrix f2 with q replaced by q -1 . The matrix K is a complex% multiple of Po and K2

= (q

q l—N

q N-1)( q q -1 ) -1 K

Finally, we turn to the case g --= sp N • Then we also have the tensor product decomposition (72), but now the restriction of 711 0 Ti to 8(171 0 171 ) is irreducible and the space AM 0 VI ) splits into the direct sum of a onedimensional invariant subspace and its irreducible complement /1/(14 14).

8.5 L-Operators and L-Functionals

275

Proposition 26. The operator lk =r oR= Ii , i for the algebras Uh(sp N ) Vi ), as qI on S(V1 0 VI) and as and Uq (sp N ) acts as -q -1 1- on -q -(1 ±N) I on the one-dimensional invariant subspace. We have

_ gnok

q -N-1 _ 0.

q -i nok

If N > 4 and (1 + q 2 )(1 q-N )(1 q -N-2 ) Uq (spN ) admits the spectral decomposition

0, then the matrix ik for

qp+ _ (r i p _ q _N—i po, where

+

( 17 -1

+ q-N- 1 )I

(79)

q - 1)(q q - N - 1) ((if

UNI

CN - 1)

q _N_i ) 7

C1) k

PO =

q - N- 1)(q - 1

(80)

I UN - 1)

Vi) and the one-dimensional are the projections onto S(Vi 0 VI), (VI invariant subspace, respectively. The matrix R for the algebra Uq (sp N ) also satisfies the relations (77) with K

E.1,3 =1 .

Ei,N+1_,)•

(81)

As above, K is a multiple of Po , but now we have K2 = ( 17

1 + q -N- 1

q N-1-1)( q,

q --1)-1K .

8.5 L-Operators and L-Functionals 8.5.1 L-Operators and L-Functionals

Throughout this subsection, let Tv be a representation of the h-adic DrinfeldJimbo algebra Uh(g) on a finite-dimensional vector space V. The elements

= (id 0 Tv)(7?,), L -v = (Tv 0 id)(7Z -1 )

(82)

of Uh(g) 0 .C(V) and L(T) Uh(g), respectively, are called the L-operators of the algebra Uh(g) with respect to the representation T. Proposition 27. The L-operators LiFf and L v satisfy the relations

(4)1(4)2Rvy = Rvv(4)2(4)17

(83)

(L)1(4)2Rvy = Rvv(4)2(4)1,

(84)

276

8. Quasitriangularity and Universal R-Matrices

(2A id)Li4; — W413(4)23, (E

(id 0 llgv= (Lv)12(Lv)137 (id 0 E)./;12 = I.

= /,

(85)

(86)

Proof. Applying the operator id 0 Tv 0 Tv to both sides of the QYBE Ri2R13/Z23 = TZ23/Zi3R.12 we obtain the relation (83) with sign +. The QYBE is equivalent to the identity Re1-31R. 2-31- TZ —12 = Rel2R2-317-C131. Applying Tv 0 Tv 0 id to the latter relation we get formula (83) with sign -. Equation (84) is obtained by applying the operator Tv 0 id 0 Tv to the identity R.1-21 /Z.23 -R.13 = 1Z13 R.231 121 . The formulas (85) and (86) are immediate consequences of (2) and (7), respectively. For example, using the second relation of (2) and the fact that Tv is a representation, we derive that -

-

(id 0 .6)L v- = (Tv 0 id)((id = (Tv 0 id) (R.1-21 R1-31 ) =.(Lv - )13. - )12(Lv

CI

Let us choose a basis fed of V. Then there exist linear functionals tki on Uh(g), called the matrix coefficients of the representation Tv, such that (87)

Tv( . )ei = Ek tki( - )ek-

Further, by (82), there are uniquely determined elements of Uh (g) such that (1 ei ) =E ® e and L v- (ei ®1) = ei These elements it. z3 E Uh(g) are called L-functionals associated with the representation T. This terminology stems from the fact that the elements it are commonly treated as functionals on the corresponding quantum matrix groups (see Subsect. 10.1.3). Let (4) and ( (R -1 )) be the matrices of the transformations Rvv and R v i-T , respectively, with respect to the basis {e i ek} of V 0 V. Since (Tv 0 id)/4 Rvv and (id 0 Tv)L -v = R-v iv by (82), we have tki(r) = Mc" and tia(ii-j ) = (88) In terms of L-functionals the formulas (85) and (86) read as .6(1 i±i ) Ek /Pic t it and

en) = Uij .

(89)

In the remainder of this subsection, Tv, will be vector representation T 1 of Uh(g) on V1 . We fix the basis of V1 used in Subsect. 8.4.1. Let R be the corresponding matrix from Subsect. 8.4.2 and let L+ = (F) and L - = (1,7 ) denote the N x N matrices of L-functionals in the vector representation T1. Then the relations (83) and (84) yield the following matrix equations L-14- L 2-1- R =

L iTLR =

RL4r.

(90)

Properties of L-functionals in the vector representation are contained in

8.5 L-Operators and L-Functionals

277

= 0 for i > j and ll = 177i riFi =-- 1 for

Proposition 28. (j ) l =

i= 1,2,•••,N. 1- N = 1 for g = (ii) /iFi / F2 • • • / N (iii) L±ct (L±)t(c-i)t _ ct (L±)t (c-i)tL± =

I for g = soN, sp N , where C

is the matrix from (62). Proof. (i ): The relations It = Ç = 0, j > j, follow from the particular form (44) of the matrix R, and its inverse R, -1 combined with the fact that T1 (Ei ) and T1 (Fi ) are lowering and raising operators, respectively (see Subsect. 8.4.1). For the same reason, only the Cartan parts of R, and R,-1 contribute to the diagonal L-functionals. From this we derive that ll = ll = 1. (ii) follows from the explicit formulas in Subsect. 8.4.1 and (iii) is obtained from Proposition 20. ,

8.5.2 L-Functionals for Vector Representations The first aim of this subsection is to derive explicit expressions of the Lifunctionals /in terms of the generators of Uh(g). For this we 1, +1 , a+1,1 use the formula (44) for the universal R-matrix R. of Uh(g). In order to treat the first factor in (44), we change the standard basis {111, H2, - • H1} of the Cartan subalgebra I) which appears in the defining relations of Uh(g) to an orthonormal basis {HL 1 1, • • • , Hi} with respect to the symmetric bilinear form (., •) on x 1j (see Subsect. 6.1.1 and Remark 6 in Subsect. 8.3.2). Such a basis will be called self-dual. If {I-I; - • - Hi} is a self-dual basis of 4, then we have EBH 0 H = Ei 0 H. The L-functionals can be calculated by inserting this expression into (44) and using the formulas for the vector representations from Subsect. 8.4.1. We collect the results of these computations and the corresponding self-dual bases in the following list: -

,

Uh (g 1 N):

Hi , tif:i+

1± 71,

1,2,•••, N,

i

(47

— — I i+10, —

(

q q -1 )F2 qH

(q

= (q - q -1)(_1)i-3-1-1 q -H:E 3i31 1- r

1 )F023 q11;.

k

where j > i 1 and the positive roots are as in Subsect. 8.4.1.

Uh(s1N): N-1 k 4--ak=1

=--- 1, 2, - • ---‘1'7-1 Hs,

Hi = .11"i

=

= 1,2, •••,N

lti+ = (q -

j =1,• • - ,N - 2, -

1,

lN =

-(q -

kHkIN

7

8. Quasitriangularity and Universal R-Matrices

278

l. = (q — q -1 )(-1) i-3 + 1 1iti Epii ,

,

= — (q

where j > i + 1 and the positive roots are as in Subsect. 8.4.1.

Uh/2(s02n+1):

g - H: 7

1

= gH:

1, - - n,

+

+ Hi+i + • • • +

=

- 1 7 2 7' ' • 7 n7

i1 = In-4-1,n+1

1-60-1 = (q q -1 )q -14 Ekl

-

1 1c-+1,k = - (q q -1 )Ficek

(q

1 2-n-k+2,2n-k+1

1,

q -1 )q 1141 Ek q -1 ) Ficq -11141

1 < k < n — 1, ,(q ii 2

1+ n ,n+1

q -1/2‘)q -H'n E n

c 1/2 = _ (q

q -1/2)Fn e:,

[2] 1/2

where c

h

q 1/2 q =7" e

1/2 (q 1/2

1 n+1,n+2 + =

,

I n-± 2,n+ 1 =

eq 1/2( 17 1/2

-1/2)En, q -1/2) .Fn

= 2n + 2

and

h (Sp 2n ):

= 1, 2, • • • , 72,

+ Hi+i + • • • +

=

= q-

1 4k+1 = (q

1 ) (1 - 14

1+ 2n k,2n k } 1 = - (q q -1 )q 1141 E1c7 -1‘ -11'1, icq 1- n-k+1,2n- k = (q q -

-

1 ) Fidlik

i k+1,k = - (q -

- -

< k < n 1, Intn+1 = (q 2 Q -2 )q -lini En)

(q2

1 17L+1,n =

2 ) Fnql-Lin

where i' =2n+1 - i.

Uh(so2n): =H +H 1+••-

lin-2 (Ha-1 Hn)/ 2 1

Hn )/2

=

11 : 41;q 11:

44; =

=

(q

1 k-+1,k =

-g (q

-1) g -H;,Ek

(q,

i + 2n-k,2n-k-1-1

g -1 ) Fkg irk

q -1) gHi: +1 Ek

= (q q -1 )Fkq -111:+1 ,

1 < k < n — 1,

1+ n-1,n+1

t,

1 :+1 ,n- 1 =

17 -1) 47 -1-1:i _ i En,

-(

q

i+ n,n+2 -

17-1)(r11:1. En,

(4 - 1 )Fnq H:., ;:+2,n = (q 1

Expressions for the other L-functionals/ti , Ç, i < j, can be derived from the previous formulas combined with the relations (90).

8.5 L Operators and L Functionals -

279

-

Proposition 29. The L-functionals satisfy the recurrence relations

= (/;Ei /i+r - /Pri;Ei )477.,

r< j,

(91)

47,67,),

j
(92)

for g -= glN , slN and, when in addition r

for g = soN ,sp N .

r

Proof. By (90) we have the commutation relations ik ks

E

(93)

/±/± Rik k3 pi mk ts .

We apply this relation in case i=p
r = j' excludes only the functionals 71,71+2 resp. /727+1,n_ 11 1 n-+ 2 ,n for g = sp2n and g = so2, on the left /-7t-1,n+11 i+ hand side of (91) and (92), because in these cases there is no such r. But these functionals for g = 50 2n are already known by the above list. In order to find them for g =-- 5p 2n we use the relations (93). Setting p = n 1, m = r = n, s = n + 1 in (93) with sign + and using the values of the matrix entries Ram , Note that the condition r =

we obtain q -2)/ n4-__

( 47 2

1,n+1

(qintn-1-1 1;+-1-1,n

1:+n--1,n l n+,n-1-1) 1;n•

Inserting the expressions for q_ 1,n , In+ n+ 1 7 I,n TL in terms of En_ i , En , g , we derive

i n+- 1,n+1 =

_

(4' -

1-1:, En q --

En)q H:,

Similarly, we get =

-

FngH4n_ 011;1.-i ),

int n+2 ( q2 CI-2,n

---

' -cin-iti -Hn

(9,2

En _iq - Hi- En ) 7 FnqH4n_i ) .

Corollary 30. Let Uz(g) denote the subalgebra over the ring Z of integers of Uh(g) resp. Uhi 2 (so2n+ 1) generated by the elements Ei , F q 1-1' , . Except for the functionals l, Ç with i = n <j for g = s P2n and with i=n< j ori=n+l< j for g=so2n+1 , we have

li±i E (I? - q -1 )Elz(g) In the excluded cases we have the relations

for

i j.

8. Quasitriangularity and Universal R-Matrices

280

1/2) ( till 2 + C1/2) 1/2uz ( 9 ) ,

1n+i, gn E ( q 1/2 _

el/2( 11 1/2

E

for g

_ 12 -1/2)( q 1/2

q-i/2)1/2 uz

(g)

so2n+i•

Proof. By the formulas of the above list, the assertion holds for all functionals and in addition for int 1,n+17 Intri+2 , 1 41,n-17 In+2,n in the cases g = sp 2n , so2,41 • Therefore, the assertion follows by induction using the recurrence formulas (91) and (92). Finally, let us specialize to the case g = slN. Then, by (60), the relations (90) (or (93)) imply the commutation rules 23 = 1t./±

22. 23 - q -1 1+ 1 ii i 1±1+ + 7 — --lg 1

23 33 7

lt is 1 3ti

-

= /37-Ni if •j

13tA,

<

j s, i S.

Combining the preceding formulas with th recurrence e relations (91) and (92) one derives the following explicit expressions for the L-functionals for m < k: (q Cl) kni - lirri + _1,i, 1j.:

i i1,ii j2-1,i2

L it Jr

4

• - 14+

4

it

(

Xlm — ±i,m+ 112n 7+ 2,272+2 - .1 1- 1)c-1 1 (q ,.. q - 1 )k - m - 1. / -

`km

_ 1+ _ 1+

X ( E(_i)is2iii—i , i ,

1+

+

... / -1,k-1

i ii-2- ,i2 _,..../z , i,_ i ik—A _ Lii-r ,i,_„.. i ..../ 32,3 .7 • 2 -iir . 31,3 1-1

7

12C/

where the summations are over all subsets (2 (including the empty set) of I == {'m + 1, m +2, • • • , k -1}. The numbers in Q are denoted by il, i2, • • • , i. < • • • < 7:3, and the numbers in IVO are denoted by ji 1 2 • '•in il<2 Ji <j2 < • ' • < jr• As usual, if21 denotes the number of elements in Q. ,

,

»

Example 3 (Uh(g12 ) and Uh(s12)). Then the L-operators are of the form ( q

-HI

L± =

(q

qq -H'2

E 7

q H;

L- =

H

(q-1 oFeari,

q

2

O

)

for Uh(g12) and -H/2 11

q

O

(q 1 ) 4r11/2E + H/2

q

HI2

L-

( q -1 oFq 1112,

for U h (S12) Example 4. For Uh(g13 ), the L-functionals 1 -3 and G. are given by 1 -11-3

= _( 7 Cl)C111 (ClEi E2

E2 Ei ) ,

(q

) (F1F2

qF2F0q111-. A

281

8.5 L-Operators and L-Functionals

8.5.3 The Extended Hopf Algebras Ur (g) In the preceding subsections L-functionals for the vector representation of the h-adic Hopf algebra Uh (0) have been investigated. The rest of this section will be concerned with L-functionals for the vector representation in the case of fixed q. The idea to define such L-functionals is to replace in the expressions of /i±i E Uh(g) functions of generators of Uh(g) by their counterparts in Uq (g). Because not all main diagonal functionals 1 counterparts in Uq (g), we need extensions Urt (g) of the Drinfeld-Jimbo algebras Ug (g ) for g = 'N, 5p2 n 7 50 2n. They are constructed by adjoining formally certain products of the elements Ki±l/N , i = 1, 2, • • • , N-1, to Uq (s1N), the elements Ke 2 to 72±1/2 K-n±1/2 7 Kn+1/1 21a1/2 Uq (sp2n ) and IC '1 to Uq (s02n ). Let us give the precise definitions of these extended Hopf algebras. The algebra Urt (slN) is generated by elements Ei , Fi , i = 1, 2, • • - , N -1 7

ki , k1 1 , j = 1, 2, • • • , N, subject to

the defining relations

kiki = ki ki , ki kr 1 = kz- lki = 1, k, k2 - - ki, = 1, =47-1 Ei_ i ,

ki Eik:-1 = qEi , ki Ei ki-1 = Ei , j i,i - 1, (94)

ki Fi ki- 1 = q-1 Fi ,

ki Fi _ 1 ki-1 = qFj_i,

ki Fi ki-1 = Fi , j

Ei Fi - F•E. 3 i = 6t3 kikï+11- k i-lki +1 J. q - q-

i,i -1, (95)

7

and the Serre relations (6.15) and (6.16) for the elements Ei and Fi of Uq (s1N). As in the case of the Drinfeld-Jimbo algebra Uq (s1N) one verifies that the algebra Ugext (sly) becomes a Hopf algebra with comultiplication A, counit E and antipode S such that

A(ki ) =

ki ® ki , .6(ki-1 ) = kr 1 ® ki-1 ,

46(Ei ) = Ei 0 ki ki-±11 + 1 0 Eil A(Fi ) --= Fi 01 + ki-1 ki±i 0 Fi7 e(ki ) =-- 1 7 E(E) = E(Fi ) = 0,

S(k) = ki7 1 , S(E) = -Ei ki-1 ki+i ,

WO =

- kiki-±11 Fi .

Note that the Hopf algebra Urt(s1N) is just the quotient of the Hopf algebra Uq (g1N ) (see Subsect. 6.1.2) by the Hopf ideal (ki k2 • • • kN - 1). The algebra Urt(sP271) is generated by elements Ei7 Fil i = 1 7 2, • • - 7 n, Ki , K77 1 , i = 1, 2, • - • 7 n - 1, and kn , kw, 1 subject to all defining relations of the algebra Uq (sp2n ) involving E, Fi , Ki , Ki-1 , i = 1,2, • • • ,n - 1, and kn ifi = Ki kw

knkrv = kikn =1,

kn Ei k,7 1 = qan11 2 Ei7

EnFn-FnEn =

k,?, —K.2 9,...q

- r,,

7 (96)

282

8. Quasitriangularity and Universal R-Matrices

The Hopf algebra structure for Ut (sP2n) is given by the formulas of Uq(sP2n) for Ei , Fi , Ki , i = 1,2, • • • ,n - 1, and 21(k)

= kn ® kn, 2i(k n7 1 ) = kn7 1 kr,7 1 (E)= En k n2 +1 0 E717 21(Fn ) Fn 1 + k772 Fn7

(98) (99)

= 1, e(En ) =e(Fn ) = 0,

(100)

s(En ) = -En kn-2 , s(Fn ) =

S(k)

The algebra Uqext (so2 n ) is generated by elements Ez , F i = 1,2, • • • , n, K i , i = 1,2, • • • ,n - 2, and ki , j= n n. The defining relations are the relations of the algebra Uq (502n ) containing Ez , F K i , K 1 , as well as the relations ,

,

ki- 1

f i-1 k =- 1, =

kn - kn, = knkn_i, k iKi = Ki ki,

knEikn-1 = q (ani -a

1 )/2

kn Fi k; 1 = q -(aui - a-l'i ) / 2 FiThe Hopf algebra structure for Urt (so2n ) is determined by the formulas of Uq (s02 n ) for the generators i = 1,2, • - , n 2, by the formulas (98) and (100) taken for n and n - 1 instead of n and by -

21(En-i) = E

1 kn-1kn---1 +1 0 En-il

21(Fn _i) = Fn_l 1 + k; l ikn Fn--17

En k n_ i kn + 10 En , 2i(Fn ) = Fn 1 + k77. 1 11‘V S(En _i) = S(En ) = -En kn-1 1 kn-1 ,

.6(En ) =

Fnl

S(Fn _ i ) = - kn--1ff7T 1 Fn-3.7 S(Fn ) = - kn-iknFn• The "usual" Drinfeld-Jimbo algebras Uq (s1N), Uq (sp2n ) and Ug (so2n ) can be considered as Hopf subalgebras of Uq t (s1N), Uq t ( sp2n ) and Ur (so2n), respectively. Indeed, we shall identify Uq (s1N) with its image under the injective Hopf algebra homomorphism : q (s1N ) ---+ urt ( s1N ) such that cp(E) = (PTO ça(K) = kikT+11 . Similarly, the Hopf algebras Uq (sp2,,) and Uq (502n ) become Hopf subalgebras in Uqext (sp 2n ) and Ur (so2n) 7 respecugext (0), g is uniquely tively. The embedding cp : Uq(2) P27 71 502n (p(K) = K, j = 1,2, -7 n 1 for determined by w(E) = E2 , v(Fi ) ugext (sp 2n ) and j = 1,2, • • • , n - 2 for Urt(so2n), ço(K n ) = k72, for Urt (sP2n) = s and (02(1Cn_ i ) = w(Kn) = kn-ikn for Ugext(so2n) The algebra Urt (sp2n ) was defined above by adjoining the square root Kni/ 2 (= kn ) and its inverse to Uq (sp2n ). Similarly, the algebra Urt(s1N) can be obtained by adjoining only the N-th root (KA • - • Kisat1) 1/N (which corresponds to RN -1 in the above notation) and its inverse to the algebra -

8.5 L-Operators and L-Functionals

283

Uq (51N). That is, Urt(s1N ) is generated by the algebra Uq (s1N) with genera- 1, and 1.-CN,14 1 satisfying the relations tors N E 14 1 =

6i

' N

Ei , N Fik = q .5 N _ Fi 7 kN

=

KA

•••

4=1.

In a similar way, Urt (so2n ) can be constructed by adjoining only the square root (Kn _ 1 1C,7 1 ) 1/ 2 (= k; 1 ) and its inverse to Uq (s0 2„). Most constructions and results for the Drinfeld-Jimbo algebras Uq (g) carry over in a straightforward manner to the extended algebras Urt (g). For instance, the vector representation T1 for Urt(s1N) is obtained from that of Uq (s1N) if we leave the operators T1 (E1 ) and T 1 (F1 ) unchanged and set T1 (k1 ) =

T1 (K 1 )_ lIN T1 (K2 ) -2 /N • • • T1 (If1_i) -(2-1) /NTi (K2 ) (N- ') /N x Ti( Ki+ 1 )

, i = 1,2, - • (1v—i-1)IN • • • Ti(Kiv_0 1 / N

,N -1,

where Ti (Ki ) are the representation operators for Uq (s1N ). Similarly, we set T1 (k) = T1 (K) 1 /2 for urt(sp 2n ) and T1 (k- 1 ) = (T1(Kn _1)211(Kn )) 1 /2 7 T1(k) = ( T1 ( K1 ) T1 ( Kn ))1/2 for urt (so2n ). The matrices R, C and K for both algebras Urt (g) and Uq (g) are the same. 8.5.4 L-Functionals for Vector Representations of Uq (g)

A direct translation of the definition of L-functionals for the h-adic Hopf algebra Uh(g) to Uq (g) does not lead to well-defined elements /4 of Uq (g) or Uext(g), because the universal R-matrix does not make sense for fixed complex q. The key for getting L-functionals l for q E C is the algebra homomorphism q from the next proposition. For notational simplicity, we write Urt (g1 N ) := Uq (g1N ) and Urt(so2n+i) := q l/ 2 ( 50 2n+ 1) • Let C[e h , e -h ] be the subring of C[[h]] formed by the Laurent polynomials in eh and let K denote the ring C[eh e -h] for g = glN , slN, so2, 5p2n and C[eh/ 2 , e - h/2] for g =- 50 2n+ 1. Further, we denote by UK(g) the K-subalgebra of Uh(g) resp. Uh/2(s0271+1) generated by the elements E := (qi = (edi h Fi, q 1 = exphli; and c -FL: for i, = 1,2, • • • ,l = rank g. Note that the generators = exp dhH1 of the algebra UK(g) are just the counterparts to the generators K i of Ugext (9) for g = slN,502n7sP2n• Proposition 31. Let q be a nonzero complex number such that qi :La qdi 1, resp. qdj 12 1 if g == 80 2n+1, i = 1,2, • • • ,I. There exists a unique C-algebra Ut ( g) )such that 0q (eh.1) = q.1, atP q (g) homomorphism 1,b q U1163) (qi q (Fi ) = F1 and î/ q (qi 3 ) = K resp. q (V I 1- ) = kj for all j in the case g = glu , slN , 50 2n, 5p2n . For g = 802,2+ 1 the homomorphism 1I5 is given by the same formulas with h replaced by hI2 and qi by q 2 2 . The map V) q intertwines the cornultiplications, counits and antipodes of the Hopf algebras UK(g) and Ugext (9) , respectively.

8. Quasitriangularity and Universal R-Matrices

284

Proof. We carry out the proof in the case g = gl N ,s1N,502n ,sp2n . The case = 2n + is treated similarly. Let E'oi denote the elements of Uh(n+) which are obtained when in definition (6.65) of the root vectors the generators Ei are replaced by E. From Theorem 6.24' it follows that the elements pr i

131

F rn f3n

( 11

1-1; ki

( 11

8: 11;)Ici E/ (3

Ers1 0 17 ri , si

•

E

No ,

ki E Z,

(101)

are linearly independent in Uh(g) over the ring C[[k]. All elements (101) are contained in UK(9). This set is also linearly independent in UK(g) over ) holds in K = C[eh h ] . Since the relation E'Fi - FE = the algebra Uh(g) and the gill',j = 1,2, • • - , 1, can be expressed in terms of q 3 1 the K-linear span of elements (101) coincides with the algebra T-4c(0)• Thus, these elements form a K-vector space basis of uK(g)• Let a E U(g) be a multiple of an element (101) by some coefficient from K = C[e h , e -h ]. We then define IPq (a) by replacing in the corresponding expressions the power series e h E K by the number q E C and the elements Fi, q3-.11 , q 11 of UK(g) by the corresponding generators Ei , F K i , k 1 of urtkg) ‘. From the commutation rules in both algebras Uh(g) and Ut(g) it is clear that q is a C-algebra homomorphism. Comparing the corresponding formulas for the comultiplications, counits and antipodes in UK(g) and ,

td TT

ex g t(

g ) we see that /Pq intertwines these operations.

We now apply Proposition 31 to the definition of L-functionals of Ugext(g). First we set 7(q) = q-q -1 for g = 1 N s ri 50 2n , 7(q) = q2 q2 for u = P272 q -112)\112 q -1)( q 1/ 2 ((q ) for g =-- 502 n+ 1. From Corollary 30 and -y(q) it follows that for some k E N all functionals (eh)kiti belong to the algebra UK(g). We define the L-functionals 1,Pi for the vector representation of U t (g) by -

,

j :=. /.),( 0 -k oqb ( e h)k i ) .

Note that 7(q) - k is only a complex constant. Although this definition may look artificial at first glance, its meaning is clear: any L-functional E Ut (g) is obtained if in the expression for the L-functional /tj E U,1 (g) the elements eah E CM)] and E, F1 , q1 E Uh(9) are replaced by qa E C and the corresponding generators Ei , F1 , Ki , ki E Ugext (g), respectively. Under these replacements the lists in Subsect. 8.5.2 become the lists for Uq (g) resp. uext/_% q kg) • Some main diagonal L-functionals Ç; then take the following form: ( g1 N ) U 1/2

9

1Ti = Ki; U qext (s1N)

(S02n+i):

ur t ( s p 2n Elq (sP2 n )

1; = ki ; Vs1N) •• K-

" • Kn -

< n;

: Ç = K. - • Kn_ i kn ,i n; n; K = (lnn ) 2 ;

1711• i+i,i+11

8.5 L-Operators and L-Functionals

285

i < n - 2, K.."_el

Uq (s0 2n ) : K i = Kn-1 =

Kn =

Some of these formulas express the generators Ki of Uq (g) in terms of L-functionals /± ... Since the elements l 33 are in Uext (g) and not in Uq (g), they 33 cannot be expressed by the generators of U(g) for g = 1N, SP2 n , 502n• Recall that the matrices R and C for Uh(g), resp. Uh/2(so2 n+1), go into the corresponding matrices R and C for Uq (g), resp. Uq 1 / 2(so2n+i ), and Urt (g) if the power series eh is replaced by q. Therefore, because ii),7 is a C-algebra homomorphism, all algebraic properties of the L-functionals /Pi for the vector representation developed above in the h-adic case remain valid for fixed q as well. That is, the formulas (88)-(90) and the assertions of Proposition 28 hold also for the L-functionals l of Ugex t (g). We shall essentially use this fact in the proof of Theorem 33 below. Example 5. The L-operators for the algebra Ur (s12) are of the form

L±

(k -a

(q-q-1)k-lEi)

K'

L-

0Fi k)

(q--1-k

8.5.5 The Hopf Algebras U(R) and U(g) In Subsect. 8.5.1 we have seen that the L-functionals l 1.3 for the vector representation of Uh(g) satisfy the equations (90) and the relations stated in Proposition 28. We now reverse the viewpoint and take these properties as the defining relations of an abstract algebra. We carry out this only for fixed complex q and show that this algebra is isomorphic to the extended DrinfeldJimbo algebra Ur (g). With some technical modifications a similar result holds also in the h-adic case. be an We begin by being a bit more general. Let R arbitrary matrix with complex entries. Let U(R) denote the algebra with N(N ± 1) generators it» Ç , j < j,i,j= 1, 2, • • • , N, and defining relations

R= RJJ,

t z+2, 'et t

/-2,

1- 7

i

R = RL -2' ,

(102)

1,2,• • •,N,

(103)

where we set LI := () and tiFj = Ç = 0, i > j, in (102). Proposition 32. U(R) is a Hopf algebra with comultiplication ZA, counit e and antipode S determined by

'6 (1 ) =

it,

E(itj) = ij

S(L) = (L± ) -1 •

(104)

Proof That Li(R) becomes a bialgebra is easily verified. (The proof of Proposition. 9.1 below contains the details of this reasoning.) The matrices L+ and

286

8. Quasitriangularity and Universal R-Matrices

G — are upper resp. lower triangular and the entries of their main diagonals are invertible in U (R ). Hence both matrices are invertible. The entries it of the inverses (C±) -1 obviously satisfy the opposite commutation relations to (102) and (103). Hence there is an algebra anti-homomorphism S :11(R) 11(R) such that S(tt) We then have the relations S(.&).& --= I which ij mean that the antipode axiom is fulfilled on the generators. Therefore, by Proposition 1.8, U(R) is a Hopf algebra. LI

Now we set R = R, where R is one of the matrices from Subsect. 8.4.2. Let 1(9) be the two-sided ideal in 11(R) generated by the relations

- • • tk N =

fOr

= ct(c±)t(c-i)t

L±ct(L±)t(c-i)t

= MN, = r fror

g =soulsPN-

Since 1(g) is a Hopf ideal of U(R) as easily checked, the quotient 11(R) 11(9) is a Hopf algebra which will be denoted by UqL(g). The following important result says that this Hopf algebra is isomorphic to the extended Drinfeld—Jimbo algebra Urt (g) defined in Subsect. 8.5.3.

q(g)

Theorem 33. Let g be one of the Lie algebras glu , 51N, SON or sp N and set Uqext (g1N ) Ug (g1 N ) and U (s02 11+1) := Uq l/ 2 (S0211,+ 1 ). Then the Hopf algebras (9) and Uqext (g) are isomorphic with a Hopf algebra isomorphism Ut( g) kg) such that 0(t) 0: uql,(0 ) i < j, where lt l and 0(t.;:i ) =

and

are the elements of Ur(9) defined in Subsect. 8.5.4.

Outline of proof. As noted at the end of Subsect. 8.5.4, the elements E Urt (g) satisfy the defining relations of the algebra UL ( g ) . Hence Ut 9 \) such that 0(tti ) = there is an algebra homomorphism 0 :UqL (g) and 0(fi ) = i < j. Comparing the formulas (89) and (104) for the comultiplications and the counits, it is clear that 0 is a bialgebra and hence a Hopf algebra homomorphism. From the formulas listed in Subsects. 8.5.2 and 8.5.4 we check easily that the image of 0 contains all generators of U t ( g) . Therefore, 0 is surjective. It remains to show that 0 is injective. We carry out this proof in the case g =- slN . The proof in the other cases ig similar. First we solve the formulas of / 33 t. 1 t in Subsects. 8.5.2 and 8.5.4 for the generators of U'(s1N) i and set

(q q ' )t1, F: :=

.73

If'71

:= tt. 337

(q

q ' ) ' Ç...1

j = 1,2, • • • , N.

ii

(105) (106)

Using the relations of q(s1N), we next show that the elements of Uql-a(s1N) satisfy the defining relations of the extended Drinfeld—Jimbo algebra Ur t (s1N). In order to do this, we shall write down the relations (102) for Uql-a(s1N) explicitly. The equation rl-L-21- R = RLLt for Ugla(51N) is equivalent to the following relations:

8.5 L-Operators and L-Functionals

(q 'T1)-62+84+31 j < r, j < s, ij ri

tPjt;Es = £rst ij is

,

ii

= -6-3Y

t ij trj

ftcy Li- can be written i > r, j > s,

qt r

> r7 j < .5 1

(113)

> s,

(114)

t n tiz' R = R4LT is equivalent

where i > j and r > s. The matrix equation to the relations

s, Le = q

ri tr+j

q

ltFr

i > r, tZit7t., =

.6 27i ttj =

,

> r,

tTrEs ti::i + (q - q-1 )(tTiEs t; tr±i tL),

tZi tits

(111) (112)

tiitis - qtistiil t it t33

as

>

,

Çjtir-7.9 =

r, j

(110)

,t2:

= t77,-eTi + (q

= tits ti7j , i

(109)

j < s,

s. The relation £jL R =

where i < j and r

(108)

i r, j> s,

= ii

(107)

i < r,

=

287

j < s, < s,

(115)

Ç, i > r, j > s,

-e 33 7-t± sz = -etvi -e7. 33

where i > j and r s. Relation (107) for (i, j) = (i, i 1) and (7-, s) = + 1, + 2) gives

t t-1-1,i+2 tt,i+1 = 4+1,142 and The latter equation expresses tti+2 in terms of t. (by (110)) Inserting this expression into the relation -e-P,i+2-et,i+1 = we obtain 1 tt+1,i+2

t t,i+1 tiF+1,i-F2 t1i-,i+1) tz+1,i-E1 7

t t+1,i-F2 t-P,i+1

=

If we multiply this equation by (47 - q -1 ) -3 q(1 i- i) 2 and substitute the elements E and E4 1 from (105), then a slight simplification yields the Serre relation

E' i2+1 E:

-

(q q - 1 )E:±1 E:E:+1

E:g i2+1 = 0.

The second Serre relation for E follows from (107) and (109). The Serre relations for Fi can be derived similarly from the formulas (111)-(114).

8. Quasitriangularity and Universal R-Matrices

288

1 Equation (108) for i = j 1 and r = j and equation (110) for j =--and s = i lead to the first two defining relations (94). The third relation in (94) follows from (109). The corresponding defining relations for the elements can be obtained from (112)-(114). Relation (95) follows easily from (115). Thus, there exists an algebra homomorphism t9 : Urt (siN) q(s1N)

V(ki ) = K; for i 1, 2, • • • , N - 1, j = such that z9(Ei ) 19(Fi) 1,2, • • - , N. The formulas (105) and (106) imply that 9(E ) ---= E, 0(F21 ) =8WD =-- kJ . Since the sets of elements E F k and g, K.; generate the algebras Urt (s1N) and q(s1N), respectively, 9 is the inverse of 61. ,

,

8.6 An Analog of the Brauer-Schur-Weyl Duality In this section we develop an analog of the Brauer-Schur-Weyl duality for the Drinfeld-Jimbo algebras Uq (g), g =-- slN, SON, sp N • For this we need to extend the Hopf algebras Uq (soN ) by the group algebra CZ2 . Throughout this section we assume that q is not a root of unity. 8.6.1 The Algebras (// (soN) Let CZ2 be the group algebra of the group Z2 consisting of the two elements 1 and x. We denote by Uq (5o2+1 ) the tensor product CZ2 0 Ug li 2 (so2,i+ i) of the Hopf algebras CZ2 and U0/ 2(s02n+1 ). The group Z2 acts as a group of Hopf algebra automorphisms of Uq (50 2n ) such that 10 = id and x ( ) is the Hopf algebra automorphism defined by

x(E) = E(0, x(Fi) = Fx (%), x(Ki) =K,(0, x(i) = i for i = 1,2, • • • ,n - 2, x(n - 1) = n, x(n) = n 1. Since obviously x x = id, this defines indeed an action of the group Z2Let Oq (502n ) = CZ 2 y Uq (s02n ) be the corresponding right crossed product algebra (see Subsect. 10.2.1). That is, aq (so2n ) is the algebra generated by the algebra Uq (so2n) and an additional element x such that x2 1 with the commutation rule x(a) = xax, a E (.1q \so2n). ( One easily checks that Oq (so2r,) is a Hopf algebra with comultiplication defined by the comultiplication of Uq (so2,,) and the relation 21(x) = x x. The finite-dimensional irreducible type 1 representations of the algebra f/q(so2n ) are described by the following proposition. Proposition 34. (1) Let A --= (A 1 , A2 7 Ar,), A i 2(A, ai)/(a i , ai ), be the highest weight of a type 1 irreducible finite-dimensional representation of the algebra Ug

An_1 = A.

There exist exactly two nonequivalent irreducible representations 1;), and i3 of Clq (502,,) such that their restrictions to Uq (502n ) are

Case 1:

8.6 An Analog of the Brauer-Schur-Weyl Duality

289

equivalent to the irreducible representation TA. If v and y0 are highest weight (x)v° = y 0 . vectors for 11), and 7=3; - , respectively, then ii),(x)v = y and -

Case 2: An _i A. Then there exists a unique irreducible representation i"), of Oq (s02n ) such that its restriction to Uq (s02n ) is equivalent to the direct sum representation TA e Two, where x(A) = (Al, - • - 1An--2, An, An-i) if A = (A 17 - - , An _2 ) An -1, An ). (ii) Every type 1 irreducible finite-dimensional representation of fig (so2n) is equivalent to one of the representations from Case 1 or Case 2. Every finite-dimensional representation of Oq (s02n ) is completely reducible.

Proof. The proof of this proposition is similar to the classical case and we 111 omit it. The vector representation Ti of the algebra Oq (soN) is given by the same formulas of ill ' (Ei), ti (Fi) and Ti (Ki ) as for the vector representation of Uq (soN) and by (x) = I if N = 2n + 1, -

ti (X) - En,n+1 En+1,n

— Enn E+1,+1 if N = 2n.

8.6.2 Tensor Products of Vector Representations In this subsection we decompose the r-fold tensor product of the vector representations of Oq (soN) and Uq (g), g = slN,sp N , into irreducible components. This result will be used in the proofs of Theorems 38 and 11.22 below. In order to treat all needed cases at once, we write flq CO instead of Uq (g) for g = slN spN • In what follows, g denotes one of the Lie algebras 51N, SON spN . A set of integers n = (n 1 , n2 , • • • , nk) such that n i > n2 > • • • > nk > 0 is called a partition. Every partition is representable by a Young diagram 0) with ni boxes in the i-th row. consisting of boxes placed in k rows (if nk If n is a partition, then its transpose is the partition n' = (rel ,n'2 , • • • ,nt), where n'i is the number of boxes in the i-th column of the Young diagram of the partition n. In particular, WI is equal to the number of rows in the Young diagram corresponding to n. The number ni := Ei ni is called the length of the partition n. Let P be the set of all partitions (including the partition 0 = (0)). For A n Sin+1 1 B n = 802n+11 Cn sp2n and Dn 802n we define the following sets of partitions (see, for instance, {Wey1): P(A n ) = {n E P

P(B) = In

E P

n+ 1 } , P(C) =

E P 1 nÇ + n'2 < 2n + 1},

P (D n )

= In

I

n},

E P I nÇ + n'2 2n1

and

Pr (An ) = fn E P(An ) I Ln = r} Pr (Xn ) = {n

E

P(X)

I ni

r, In'

r (mod

2 , )}

xn

=

Bn ,CnI Dn.

290

8. Quasitriangularity and Universal R-Matrices

For Ay, and Cn , the sets P(An ) and P(C) coincide with the sets of highest weights (mi, m2, • • .) described in Subsect. 7.1.2 (see (7.2)-(7.4)). Note that different partitions of P(An ) may correspond to equivalent representations of Uq (s1,41 ). If n E P(B72 ) or n E P(D), we set n° := (N n'3 • • •)', where N = 2m + 1 for B7, and N = 2m for D. In fact, n° coincides with n when the first column is appropriately changed. If WI > n in n, then nr < n. Clearly, n" = n. Further, we have n° = n if and only if n E P(D) and nÇ = n. With every partition n E 7"(9) we associate a type 1 irreducible finitedimensional representation f7(n) of tIg (g). It is defined as follows. For g =s1 n+1/ sP2n/ it is just the irreducible representation of Uq (g) with highest weight n. For g = 502 n+1 , the restriction of t(n) to Uq 1/2 (S0271+1) coincides with the irreducible representation with highest weight n if WI < n and with highest weight re if WI > n, and the operator t(n)(x) is given by t(n)(x) =--- (-1) 1 *. In the case g --= s02n we set T(n) := t( n) if < n and (n) i')°,(n) if > n, where i'A and are as in Proposition 34 and A(n)

(ni - n2, • • • , an -1 nn ,nn _i + n r„) A(n) := A(n°)

if i4 < n,

if r4 > n.

Except for the case where g = 502n and nÇ = n, the restriction of the representation T(n) to Uq (0), g = spN , 5027„ resp. Uq l/ 2 (S02n+1) is irreducible. By Proposition 34, if g = s02, and WI = n, then the restriction of i(n) is the direct sum of two nonequivalent irreducible representations of U q (so2n)•

Proposition 35. If g = so2n+1/sP2n/so2n and if n, m E 2(0), n m, then the representations t(n) and T(m) of OW are not equivalent. Proof. Since t-J-q (span ) = Ug (span ) and P(sp 2n ) is the set of all highest weights of type 1 irreducible representations of Uq(sP2n)/ the assertion for 0 =----- sP2n follows. If g = s02n+1, then î'(n)() Pre)(x) and hence i; (n) Pn°). It follows from the definition of t(n) for fig (s02n+1 ) that t(n) and t(m) are not equivalent if < n and m< n or if WI > n and > n. This implies the assertion for g = s02n+1. For g -=-1' 502n, the assertion follows from the results of Subsect. 7.1.2 and Proposition 34. For partitions m, n E P(g) let us write m n if m = n + ei for some i in case g = slN and m = n ei for some i in cases g = soN , spN . Here ei is the vector with 1 in the i-th component and 13 otherwise.

Proposition 36. The tensor product of an irreducible representation i(n), n E 2(g), and the vector representation I-11 of CI q (g) decomposes into irreducible representations as (116) rn,,,n,mEP(g)

8.6 An Analog of the Brauer—Schur—Weyl Duality

291

Proof. For g = SON, one easily verifies that both sides of (116) coincide for the element x. Thus it suffices to prove (116) for elements of the algebras = SiN 7 spN 7 SO2n 7 resp. U47 1/ 2 (SO2n-1- 1 ) • If g = slNI SO2n ) spN or if .9 = SO2n+1 and n'1 n, n + 1, then the decomposition (116) follows from the formulas (7.13)-(7.15) and Propositions 34 and 35. If g -= 50 2n+ 1 and n'i = n (resp. n'i = n+1), then the representation corresponding to the first summand on the right hand side of (7.15) appears when a box in the (n + 1)-th row is El added to (resp. subtracted from) the Young diagram of n. Corollary 37. (i ) The r-fold tensor product ir of the vector representation 1-1. of 0q (9) decomposes into a direct sum of irreducible representations as

ir

(117)

neP,(9)

where m n , m n > 0, is the multiplicity oft(n) in the decomposition. (ii) If g = slN , then ir contains the trivial irreducible representation (that is, the representation with highest weight (0, 0, • • • , 0)) in the decomposition (117) if and only if r = kN , k E No. If g = SON or g --r-- sp N , then Tr contains the trivial representation if and only if r E 2N 0 . 8.6.3 The Brauer-Schur-Weyl Duality for Drinfeld-Jimbo Algebras First we briefly describe the corresponding classical results. Let G be one of the groups SL(N,C), 0(N, C) or Sp(N, C) and let T1 be the vector (first fundamental) representation of G on V1 = CN. It is well-known that the problem of decomposing the tensor product representation Tr of G into irreducible components is closely related to the structure of its centralizer algebra rr (G)l = {A E L (Vi®r ) I AT' (g) -= rr(g)A, g E GI. For G = SL(N, C), a classical result of I. Schur says that the algebra rr(G)' is generated by the flip operators ri,i+i , i =-- 1,2, • • • ,r - 1, of the i-th and (i + 1)-th tensor factors in Vi®r. In the cases G = 0(N, C), SAN, C) the Brauer-Weyl duality theorem asserts that Tr(G)F is generated by ri,i+i and Ki,i+i , i = 1, 2, • - • , r - 1. Here K is the projection of V1 0 Vi onto its one-dimensional r 2-invariant subspace and Ki, i+i denotes the operator K acting in the i-th and (i + 1)-th factors of Vi.°T. We now turn to the quantum algebras. Let Ti be the vector representation of Oq (g) on the vector space 171 , where g = slN , soN , sp N . The image f? r (0 9 (9)) of fig (g) under the r-fold tensor product representation it is a subalgebra of the algebra L (II') of linear operators on K®r. Let R and K be the matrices for Uq (g), resp. Uq 1/3(802n+1), from the formulas (60), (61), (78) and (81). As usual, ki,i+i and Ki, i+i are the operators on 171®r acting as k=T0R and K, respectively, in the i-th and (i ± 1)-th tensor factors and as the identity elsewhere. We denote by B(r) the subalgebra of r(V?r) generated by the operators fli,i+17i = 1,2, - - - , r 1, for g = slN and by ki,i+1 and .

-

292

8. Quasitriangularity

=

and Universal

R-Matrices

—1, for g =7 sopr, spN . Let f?r(C/q (g)) 1 and B(r)' be the sets of operators in £(1/r) commuting with all operators from ir (0q (g)) and B(r), respectively. Recall that a complex number is called transcendental K•, i+i , j

1,2,• • • ,r

if it is not a root of a nontrivial polynomial with integral coefficients. The q—analog of the classical Brauer—Schur—Weyl duality is stated as Theorem 38. Let r E N, r > 2, and let q be transcendental. Then we have

ilr(0q (9)Y = B(r)

and B(r)' =

Moreover, the algebra f?r(tIq (13)) decomposes as a direct sum of algebras

t?r (Oq(0)) = IEB nepr(g) L(V(n)))

(118)

where V(n) is the space of the irreducible representation t(n) of tIg (g). Proof. We carry out the proof for g ----- so 2#7,41 . The other cases are treated in a similar manner. Since the irreducible representations t(n), n C Pr (9)) are mutually inequivalent by Proposition 35, the decomposition (117) of the representation fer r implies (118). Further, since the multiplicities mn in (117) are independent of q, so is m := dim i 1® r(0-q (g))F E nEPr (g) 1 "n2 . On the other hand, let ti be the vector representation of the classical group G = 0(2n+1,C). The r-fold tensor product ti"r of this representation decomposes also into a direct sum of irreducible components according to the formula (117). Therefore, dim inGY =

E

nEpr m2n

= m.

Let us express the dependence of R and K on q by writing R(q) and K(q), respectively. By the Brauer—Weyl duality theorem for the group G =-0(2n 4- 1, C) (see [Bra] [Wey]), there exists a basis vk, k = 1,2, • • • , m, of the vector space TT(G)1 consisting of monomials of the operators Ti , i+1 and j --= 1,2, • • • , r —1. Replacing Ti,i+i by f?(q) i ,i+i and Ki,i+i by K(q)i,i+i in these monomials vk we obtain vectors denoted by vk(q). Consider the matrix of coefficients of the set {vk(q)1 k = 1,2, • • - , m} with respect to the standard basis Eid, 0 • • • 0 Eirir of L(Vi@r). By (61) and (78), the entries of this matrix are Laurent polynomials in q 1/ 2 with integral coefficients. For q = 1, the vectors vk = vk (1), k = 1,2, • • - m, are linearly independent. Hence there is a regular m x m submatrix of the coefficient matrix. For general q, the determinant of this submatrix is a Laurent polynomial, say f, in q 1/2 with integral coefficients. Since f(1) 0, f is nontrivial. Because q O. Therefore, the vectors is transcendental, so is q112 and hence f (q112) {14(4) I k = 1 7 2 7• • .7m} are linearly independent. By Proposition 19, the matrix k(q) and so the polynomial K(q) of k(q) intertwine the representation T1 0 1-'1 . This implies that B(r) Ç T (Uq(g))'.

8.6 An Analog of the Brauer—Schur—Weyl Duality

293

Since the vectors fvk(q) I k = 1, 2, • • • , ml of B(r) are linearly independent and m = dim inffq (9)Y, as noted above, we conclude that B(r) = ( q (g)) 1 . Hence B(r)' = ( q (g))" = (Cfq (9)), where the second equality follows immediately from (118). 8.6.4 Hecke and Birman-Wenzl-Murakami Algebras

In this subsection we introduce Hecke and Birman-Wenzl-Murakami algebras and show that the algebras B(r) appearing in Theorem 38 are images of representations of these algebras. Definition 6. Let q E C, q 0, and r c N, r > 2. The Hecke algebra 11,(q) is the complex unital algebra with generators gi, g, 1 and defining relations (119) gigi+igi = gi+igigi Fil -

gigi =gi g , for Ii -jI>2, = (q 1 ) gi ± 1.

(120) (121)

In the case q = 1 the equations (119)-(121) are the defining relations for the permutation group Pr . Hence Hr (1) is just the group algebra CPI-

0, q2 1, and r c N, r > 2. The Definition 7. Let p,q E CVO}, p,q Birman-Wenzl-Murakami algebra BWMr (p,q) is the complex unital algebra with invertible generators by]. g2, - - • ,gr---i subject to the relations (11 13), (120) and (122) e,g, = p- l e,, —1 (123) eigi_lei = pei, eig i _ l ei = p -1 ei , where ei := 1 - (q - q -1 ) 1 (gi

—1\

gi 1•

(124)

From the preceding relations it follows in particular that ei is a complex multiple of a projection and that g, satisfies a cubic equation. More precisely, we have e=

+ (p

p -1 )(q

q -1 ) -1 )e,,

(g, - q)(g i + q-1 )(g, p -1 ) = 0.

With another set of relations the algebra BWM,(p, q) can also be defined in the cases q = ±1 which have been excluded above. In order to give vector space bases of both algebras, we consider the following sets of monomials: Mn

gn, gngn-11 • " gngn-1 " • gib

M-n = fe1,n+11e2,n+11" •

for n = 1, 2, - r - 1, where

en,n-Fil

294

8. Quasitriangularity and Universal R-Matrices eii := gi_igi__2• • • gi±leigi+1 " •

+ 1 < j, and e,+1 := ei .

Further, we set Mo = {1 } . Proposition 39. (i) The set Br := {s1x2 • • • xr-i j xi G Mil is a basis of the vector space Hr (q). (ii) The set Br := {xni • • • x rir l-r +1 < ni < n2 < • • • < nr < r 1, n i + ni 0 if i, j, sn, C Mn, } is a vector space basis of BWMr(P, q). Proof. See [Bou2j, pp. 54-56, [Wen]] or [HI<W1 for fir (q) and [BW1 or [Wen21 for BWMr (p, q).

The above description of bases is recursive, that is, we have BH-1 = Br 'Mr for the Hecke algebras and Br +i = M_ r •Br Li Br Mr for the Birman-WenzlMurakami algebras. Since the sets Mr and M, consist of r+1 and r elements, respectively, it follows that dim Hr (q) --= r!

and

dim BW1144,q) = (2r - 1)(2r - 3) • - • 3.1.

Both algebras play an important role in knot theory (see, for instance, [HMV], [BW], [Wen2}). The interest in these algebras in the present context stems from the following Proposition 40. Let I-='? and ki ±i be as in Subsects. 8.4.3 and 8.6.3. Set e := 1 for SON and f ::= - 1 for spN . There exists a unique representation 7rr of the Hecke algebra Hr (q) for g = slN and of the Birman-Wenzl-Murakami algebra BWM, (q, EqN - €) for g = soN,spN on the vector space (C N )®r such that Irr (gi ) = j = 1, 2, • , r 1. That is, the algebra B(r) defined in Subsect. 8.6.3 is the image of Hr (q) resp. BWM r (q,eq N- c) under this representation Trr • Proof. We have to show that the matrices --- 1, 2, - • • , r -1, satisfy the defining relations of the algebras _1%40 and BWMr (q, cqN- '), respectively. Since R satisfies the QYBE by Proposition 4, ik fulfills the braid relation (15). This in turn implies (119). Condition (120) is obviously fulfilled. Equation (121) for 13 slN follows from (69). The relations (122) and (124) of the algebra BWMr (q, EqN- ') for g = soN, spN follow from Propositions 25 and 26 and the second equality of (77). Relation (123) can be easily derived (for instance) from (64). D

8.7 Applications The universal R-matrices are the main bridge connecting quantum groups with integrable systems. However, in the theory of integrable systems one needs R-matrices that depend on a spectral parameter. There is a procedure, called Baxterization, for obtaining solutions R(A) of the quantum YangBaxter equation from the R-matrices considered in previous sections. In this section we give a brief review of this method and the relations between Rmatrices and integrable systems.

8.7 Applications

295

8.7.1 Baxterization The parametrized quantum Yang-Baxter equation gives an approach to solvable (integrable) systems of statistical mechanics. A state of such a system is defined by a configuration at the vertices of a square lattice in the plane (for example, by spins). Each edge of the lattice is to be thought of as an interaction and contributes an energy ga, a') to the total energy, where a and a' are the spins at the ends of the edge. For instance, in the Ising model the total energy of a state a is E i (k i crjj cr j+ij + k2crij a,,i+ i). One of the most important characteristics of thé system is given by the partition function

Z

= Ea

exp(-47)1kT)

depending on parameters /e l , k2 , k, T. Usually one collects this dependence of the function Z by a single parameter called the spectral parameter A. Z(A) does not make sense for an infinite sysThe partition function Z tern. For that reason, one considers rectangular approximations of the system with N vertices. If ZN is the corresponding partition function, then the free energy of a site is FPO = ZN VN, where the approximating rectangles have to tend to the whole lattice. One needs to calculate F(A) as a function of A which means the model is solved. The usual technique of finding F(A) is the method of transfer matrices. A transfer matrix is a matrix T(A), depending on the horizontal dimension m of the approximating rectangle, such that Tr (T(A)) is the partition function for the m x n rectangle. The limit of F(A) when n —) oc is given by the largest eigenvalue of T(A). But the size of T(A) grows exponentially with m and its diagonalization is not straightforward. R. Baxter proposed considering systems for which the transfer matrices commute among themselves for different values of A. Then the matrices T(A) have a common eigenvector, say T(A)e = f(A)e. It turns out that one can then determine all possible functions f(A) including the largest one, that is, one can solve the model. In order to find systems for which the transfer matrices T(A) commute it is natural to factorize the matrix T(A) as R,i (A)R,_ 1 (A) • • • R i (A), where Ri corresponds to the contribution obtained by adding energy at the i-th position. Thus, it is necessary to look for local conditions on the matrices R i , which guarantee that the matrices T(A) mutually commute. It can be shown (see [Bax]) that if for each A and A' there exists A" such that

Ri(A)Ri+i (A')Ri (All ) =14+1(A ll )Ri(À i)Ri+i(A), Ri(A)Ri(Y) = Ri(Y)Ri(A),

2,

(125) (126)

then under certain boundary conditions we have T(A)T(A') = T(AW(A). If the matrices Ri are independent of A, then (125) is just the braid relation (15) which is known to be equivalent to the QYBE (6). The method of obtaining solutions Ri (A) of (125) from solutions of the constant(!) QYBE

296

8. Quasitriangularity and Universal R-Matrices

is called Baxterization. Recall from Subsects. 8.1.2 and 8.3.3 that representations of Drinfeld-Jimbo algebras give rise to solutions of the latter equation. In this subsection we carry out the Baxterization procedure for the Rmatrices of the vector representations of the Drinfeld-Jimbo algebras Uq (g), g

= SIN, SON, SPIV

First let R be the R-matrix for the vector representation of Uq (s1N) from 0 R (recall that, as usual, r is the Proposition 24. By Proposition 19, k flip operator) satisfies the braid relation

(127)

k12É‘?23k 12= k23R 12k 231

which is equivalent to the QYBE (6). We shall consider the Baxterization of this R-matrix when the parameters A, A' and A" are related by the condition A" = A'A'. Then the Baxterized R-matrix is a solution of the equation R 12 (x)k23(xY)k12(y) — k25(y) k12 (x023

(x).

(128)

We look for solutions of this equation of the form

k(x) = g (x) (I + f (x)k) , where f(s) and g (x) are functions of the parameter s. Inserting this expression into (128), and using the braid relation (127) and the Hecke condition 12, - 1) És? 2 (q I for the matrix k (see (69)), (128) turns out to be equivalent to the equation + (q - q -1 ) f (x) f (y) = f (xy)

f(s) + f

(129)

for the function f. The function g (x) is arbitrary. Substituting f(s) = (q - q -1 ) -1 (f (x) - 1), we find that the general solution of equation (129) is given by f (s) = (q - q -1 ) -1 (x.)' - 1), where 1/ is an arbitrary parameter. An important particular solution is obtained for -y = -2 and g (x) = (q q -1 )x In this case we get

I(x) = Now we turn to the Drinfeld-Jinito algebras Uq (soN) and Uq (sp N ). In this case we represent the Baxterized matrix k(x) in the form

I(x) = h(s) (I

f (x)k g(x)K)

where k and K are given by (61), (78) and (81), respectively. As in the previous case, inserting this expression into (128) we find that k(X) = h(X)

—

_1

a±x ,

K)

where h(s) is an arbitrary function, -y is a parameter and a* = q -1 )x and "t = -2 we have In the special case when h(s) =

297

8.7 Applications

x -1A -

k(s)

v + -ci-1)(ce±1+ 1) ±x -1 + (q

xr?-1 cf±x - lf

q -1 )(a± + 1))

x - x -1

(x -1 q xq -1 )P + (xq - (xq) -1 )P _

a(x)P o ,

(q, r eg eN ct± ) x2) where a(s) = (x a±x-1 ) -1 (a± x-1 )(eq' N ex± and e, P + , P_, Po are as in Subsect. 8.4.3. The last equation determines the spectral decomposition of kx). Another relation between the values A, A' and A" is obtained under the following change of parameters: x = e (q-1-q)(0-0')/2

y = e (q-1-0072

(130)

Then instead of (128) we have the equation

112(0 - 023012(91 ) = k23(012(6)R23(0 - 0').

(131)

The substitution (130) transforms a matrix k(s) satisfying (128) to a matrix Ék (0) that fulfills (131). 8.7.2 Elliptic Solutions of the Quantum Yang-Baxter Equation The previous solutions of the QYBE are called rational (dependence on x) and trigonometric (dependence on 0). Elliptic solutions are expressed in terms of elliptic functions of the spectral parameter. In this subsection we shall give elliptic solutions which are symmetric with respect to the commutative group Zp Zp p G N, where Zp := ZipZ. Let X and Z be the matrices from Subsect. 7.5.2, where e = exp(27ri/p). We set

71 (a)

71 (al , a2 ) := zaixa2 ,

Cle2 — 0 1 1,2 7 • • •

- 1.

Clearly, this equation defines a projective representation of the group Zp Zp that is, we have T(a)T(13) = ea2 01 T (ct 4_ 0) . Since the algebra generated by X and Z coincides with the matrix algebra Mp (C), any solution of the QYBE on CP CP can be written as R(0) = woMT(a)07(/(3) with wao G C. We are looking for solutions of the following form R(0) = wa (0)7-1 (a) T(a) -1 , w a (0) E C. (132) One immediately checks that such solutions are invariant under the projective representation T of the group zp 0 Zp in the sense that they satisfy the condition

R(0)(T

T)(a) = (71 T)(a)R(0)

for

a E Zp Zp.

298

8. Quasitriangularity

and Universal R-Matrices

In order to obtain elliptic solutions we impose the additional conditions

R(0 + 1) = Z1-1 R(0)Zi = Z2R(0)ZV, R(e + T) = e -i"T

R(0) =

XT 1 R(0)Xi =

0 T',

e -2"i° X2R(0) X 2-1 ,

(133) (134)

where 7- is a complex parameter. These equations are in fact consistent with the quantum Yang-Baxter equation. If we substitute (132) into (133) and (134), we obtain the equations WΠ(9

+ 1) = E Ct2 W a

2i0 7r E-

0) 7 Wa

wa (0 N)

WΠ(0)

= 1.

These equations are satisfied by the elliptic function /D a

m=

&a(11)

where n is an arbitrary parameter and

ea (o)

E

ec:(o)

ce2 ) 2 exp (i7rT (m + —

is the following 0-function:

27ri (m

711=-

c ) (0 + — al )) N N

8.7.3 R-Matrices and Integrable Systems In order to construct integrable systems with periodic boundary conditions corresponding to a given Baxterized R-matrix one uses relation (131) written in the equivalent form Ri 2 (0 0 1 ).R13 (0)R23 (0 1 ) = R23 (01 )R13 (0)R12 (0 — 01 where

R ii := ro î

.

) 7

(135)

We construct a transfer matrix (T(\)) by setting

(0) i 1k22 T1m.km 8 11 - 1 8 m-1

R(0) isik 'l R(0) 81 ,k2 • • • R(0) is1-,7,7' km 82 1 2

Here rn denotes the number of rows in the rectangle. The matrix entries T(0) are linear transformations of L(Vr).Ule partition function Z R(9) is defined by

ZR (9) = Trviem(Tr TOW , where n is the number of columns of the rectangle. If we represent T(0) as a product of R-matrices Ri (0) and use the quantum Yang-Baxter equation, we easily derive the relation

E

s,r

R(0 - 0')T(0) i8 T(0 1 ) 1.

E

8

r

To/yrcT(0):R(0_0/) .;1..

The latter can be expressed by the matrix equation R(0 - 0 1)12T1(0)T2(0 1 ) =-- T2(0')T1(0)R(0 - 0')12

(136)

8.7 Applications

299

in the vector space 171 0 V2 0 Virl, where T1 := R13R14 • • • Ri, m +2 and T2 := R23 R24 • • ' R2,771+2. Equation (136) is called the fundamental relation of the quantum inverse scattering method. Now we multiply both sides of (136) on the right by R(0 — 0') 21- and take the trace Tr in L(V1 0 V1). Then we obtain Trvi ovi T2091 )7'1(9) = Trvi ovi (R(61 — 9')12T1 (0)T2(0')R(0 — 0%7 21 ) =

Tr vi ov,T1(9)T2( 9'),

that is, we have Tr Ti (0)Tr T2 (01 ) = Tr T2(0')Tr Ti (0).

This means that the system is solvable. To every two-dimensional lattice system of statistical mechanics there corresponds an associated one-dimensional quantum chain. This is a quantum system for a particle in the space consisting of a chain of n sites. In the continuum limit n —> oo we obtain a (1 + 1)-dimensional quantum integrable system. This quantum system is the quantization of an integrable system of the equations of motion in 1+1 dimensions. In such a way we also obtain the conformal field theory. As an example, let us consider the elliptic R-matrix

R(0) =

sn (07 + sn T 0

T)

0

0

0

sn (07) sn T 1

sn Or) sn T

ksn (07- )sn (OT + 7)

0

0

ksn (07-)sn(07- +7) \

1

0 ,

0

5n (Or + T) sn T

l

where T is a complex parameter and sn T is the Jacobi elliptic function depending on the parameter k. This R-matrix describes the so-called eight vertex or XYZ model. The term XYZ originates from the existence of anisotropies in the associated quantum model in all three space dimensions. In the limit k —> 0 we have sn T -) sin T and then R(0) describes the trigonometric six vertex or XXZ model. In the limit T -> 0 we obtain the R-matrix for the rational six vertex or XXX model. Equation (135) can be rewritten as 523(0 — 0' )513 (0)Si2 (0') = S12(0')S13 (0)S23(0 — 0'). Under the conditions of unitarity and crossing symmetry

Si2(0)S21(-0)' = I- ,

S12

'---

(S21 Or

— i1\ \t 1

this equation determines factorized S-matrices describing the scattering of particle-like excitations in (1 + 1)-dimensional integrable relativistic models. The term oSvii i2 (0) IS interpreted as the S-matrix for the scattering of two

300

8. Quasitriangularity and Universal R-Matrices

particles with isotopic spins j 1 and i 2 to two particles with isotopic spins j i and 3 2 , and the spectral parameter 0 is the difference of the rapidities of these particles. The many-particle S-matrices in such models factorize into products of two-particle S-matrices.

8.8 Notes The fundamental concepts of a quasitriangular Hopf algebra and of the quantum double and their basic properties were established by V. G. Drinfeld [Dill, Pr2]. Various general approaches to the double have been given by S. Majid (see [Mail] and [Maj], Sect. 10.2). The importance of the quantum double for the construction of universal R-matrices was also observed by Drinfeld who obtained the universal R-matrix for Uh (s12 ) and also Theorem 9. The universal R-matrix for Uh (s1N) was computed in [Ros2]. The formula (44) for the universal R-matrices of arbitrary h-adic Drinfeld-Jimbo algebras was determined in [KR21 and [LS1j. The R-matrices and L-functionals for the vector representations of the Drinfeld-Jimbo algebras and their properties have been derived and studied in [RTF]. The construction of R-matrices for fixed q given in Subsect. 8.3.3 is due to T. Tanisaki [Tan3]. The corresponding approach to L-functionals for fixed q developed in Sect. 8.5 appears here for the first time. The results on the Brauer-Schur-Weyl duality in Subsect. 8.6.3 are taken from {113]. The Birman-Wenzl-Murakami algebra was invented independently in [Mur] and [BW]. The Yang-Baxter equation arose in the work of C. N. Yang [Yam] and R. J. Baxter [Bax] on exactly solvable models in statistical mechanics. Finding solutions of this equation was one of the main reasons for the development of quantum groups. There is now an extensive literature on solutions of the QYBE and on its algebraic aspects (see, for instance, [DV], [Gul ] , [Hi], [Isl.], [Jim5], {LR], [Majl], [Rad3], [Res1], [S1]). The braid relation first came up in E. Artin's theory of braids [Al ].

Part III

Quantized Algebras of Functions

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

With this chapter we take up the detailed study of quantized algebras of functions. The rest of this book will be mainly devoted to this topic, although many results are obtained for general Hopf algebras. The aim of this chapter is to develop basic definitions and facts on the standard quantized simple Lie groups and their quantum vector spaces. Following common terminology in the physics literature, let us adopt the convention from now on to sum over repeated indices belonging to different terms. However, if confusion is possible or if a formula looks too unfamiliar, we shall write the sum sign.

9.1 The Approach of Faddeev—Reshetikhin—Takhtajan 9.1.1 The FRT Bialgebra .4.(R) Let V be a finite-dimensional complex vector space and let R be a linear mapping of V 0 V to itself. Let /4 := T o R, where r is the flip of V 0 V. We fix a basis of V. Then there are complex numbers Ridm kim such that R(x i 0 xi) = kirxn,

xm and .k(xi 0

xm .

=

Let quij ) denote the free algebra with N2 generators uii , j , j = 1, 2, • • N and let 3(R) be the two-sided ideal of quii ) generated by the N4 elements := RVi um k

ult4R 14

j, rn, n

1,2,., N.

Let A(R) denote the quotient algebra C(u.i )/J(R), that is, A(R) is the algebra generated by elements uii , i j = 1, 2, • • • , N, subject to the relations ,

p ji k 1 _ Rik -"klU m Un UlcUl mn)

m, n = 2, • - • , N.

(1)

The N x N matrix (uii ) of the generators uji is called the fundamental matrix of A(R) and is denoted by u. In matrix notation the defining relations (1) of A(R) can be reformulated as

304

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces R111112 = 112111R

(2)

itu 1 u2 = u 1 u2 I

( 3)

or equivalently as ,

I and where u i and u2 are the N2 x N2 matrices defined by u1 = u = I u (Kronecker product of matrices). Then all products in (2) and (3) are just matrix products. The reader should notice that u1u2 u2u1 unless A(R) is commutative. In this matrix notation we shall write the algebra A(R) as (4) A(R) = C(u) I (Juiu2 u1u21). Similar notation as in (2), (3) and (4) will often be used in the following. From the defining relations it is clear that A (r o R o T) is the opposite algebra A(R)°P and that A(ToR-1 o T) = A(R) if R is invertible. Example 1 (R = id and R = T). If R is the identity map, then (1) reads as i44 = uu-1.7, and A(R) is the commutative polynomial algebra in N2 that is, A(R) is the algebra of coordinate functions of theindetrmas, matrix algebra MAC). At the other extreme, if R is the flip T 7 then A(R) is the free algebra C(u) with generators uji . ,

Proposition 1. There is a unique bialgebra structure on the algebra A(R) such that

= uik

IL .I; and e(uii )

6ia ,

j = 1,2, • • , N.

( 5)

In matrix form the equations (5) are commonly written as ,A(u) = uOu and E(u) = I . Proof Since C(u) is the free algebra with generators uji , there are unique algebra homomorphisms A : COLD C such C(u) 0 C(u 32 ) and e : C that (5) holds. Then C(u) becomes a bialgebra with comultiplication A and counit e Indeed, by Proposition 1.8, it suffices to check the axioms (1.15) and (1.16) for the generators uji which is easily done. Since A(I 2 ) = uni k u nt + and e(a) = 0, we get A(j(R)) Ç ,7 (R) C(u) + C(u) J(R) and e(J(R)) = {0 } . That is, ,7(7Z) is a biideal, so the quotient algebra A(R) =-- C(i4)/J(R) is also a I=1 bialgebra. .

Definition 1. The bialgebra A(13) from Proposition .1 is called the coordinate algebra of the quantum matrix space associated with the linear transformation R or, briefly, the FRT bialgebra. Of course, the preceding definition of the bialgebra A(R) depends on the matrix coefficients /Alm of R and so on the basis {xi } of V. The next proposition gives a basis-free characterization of the bialgebra A(R). In particular, this implies that the bialgebra A(R) is uniquely determined up to isomorphism by the linear transformation R which justifies the notation A(R).

9.1 The Approach of Faddeev-Reshetilchin-Talchtajan

Proposition 2. There is a linear map (pv : V

305

V 0 A(R) such that V is a right comodule of A(R) with coaction (pi/ and 14 G Mot (Soy (pv). If A is another bialgebra and "Iv : V -W is a right coaction of A".' on V such that f? E Mor (1,bv 000, then there exists a unique bialgebra homomorphism 0 : A(R) .A such that (id 0 0) 0 (ICI 17 = 11) v -■

Proof. By Proposition 1.13 and (5), the linear map cpv : V

V 0 A(R)

defined by i = 1, 2, - • • , N, (Pv(xi) xi 0 is a right coaction of A(R) on V. Using (1), we obtain (P i/ O

v v )k( xm ® x n ) = = =

(

So y o vv)(14

z 0 xi 0

=

(6)

x k zi)

4î4R

x

=

0

Xj

O Rikii

urn k un /

(14 id)(xk 0 xiu rn k uln ) o

O (pv)(xm 0 xn),

so that 1 E Mot ((py O Soy ). This proves the first part of the proposition. To prove the second assertion, let A and Ov be as above. There are elements t G A , j, j = 1, 2, • • • , N, such that Ov (xi ) = z 0 Pi:. Since Oy is a by Proposition 1.13. right coaction, we have 2k(t) = t O tl and e(t) = From the computation carried out in the first part of this proof it follows that ian for k tni tikqiim E Mor (7Py O 7Pv ) implies that kkiii tni Therefore, the map O: extends to an algebra homomorphism 0 of A(R) to A which obviously satisfies the condition (id 0 0)0 Soy= ftp v . On the other hand, the latter relation implies that 0(u3i-) = t hence the uniqueness assertion follows. ,

By (6), the elements lei are the matrix coefficients of the corepresentation Soy with respect to the basis tx i l. We call the coaction (pv and likewise the matrix u = (14) the fundamental corepresentation of the FRT bialgebra A(R). A similar terminology will be used for the quantum groups GL q (N),S.L q (N),0 q (N),S0 q (N), and Spq (N) which are derived below from FRT bialgebras A(R).

Remark 1. The FRT bialgebra A(R) is a special case of the following more general construction. Let C be an arbitrary coalgebra and let r be a linear functional on C OC. Let M(C, r) be the quotient algebra of the tensor algebra T(C) by the two-sided ideal J generated by the elements /(x,y)

E (r(x( 1 )0y(0)x( 2)0y( 2) - y( l )0x( i )r(x( 2)0y(2))) , x, y E C.

By the universal property of 'T(C), the comultiplication and the counit of C extend uniquely to algebra homomorphisms zl : T(C) T(C) o T (C) and : (C) — ■ C. Then, T(C) becomes a bialgebra. Since e(i(x, y)) = 0 and .6(/(x,y)) = E (/(x ( 1 ) 1y(i ) )

x(2)

Y(2) + Y(1)

1 (1)

0 /(x(2) 7 Y ( 2 ) )) 7

306

9. Coordinate Algebras of Quantum Groups

and Quantum Vector Spaces

j is a biideal of T(C). Hence .A4(C, r) is a bialgebra with structures induced from T(C). Let us consider the special case C =-- MN (C)', see Example 1.4. Let u c C be given by ulic(g) := gici for g = (gm) c AIN(C). As above, let R be a linear transformation of V 0 V with matrix coefficients RiL i and define r E (C 0 C)' by r(uini g := Riditn . Then the bialgebra .A4(C, r) is nothing but the FRT bialgebra A(R). We close this subsection by studying *-structures on FRT bialgebras A(R) and on Hopf algebra quotients of it. Recall that a *-bialgebra (resp. a Hopf *-algebra) is a bialgebra (resp. a Hopf algebra) A which is a *-algebra such that 2i(a*) --= 2(a)* and E(a*) = e(a) for a c A. Suppose that the vector space V is equipped with an involution, that is, with an anti-linear mapping y -+ v* such that (v*)* = y for y E V. Define an involution on V 0 V by (v w)* := v* w* and assume that the basis elements x i of V satisfy xl = x i . Then' the transformation R of V g is called real if R((v g w)*) =-- (R(v w))* y, w E V, and inverse real if R((y Ø w)*) = (R -1 (v w))*, v, w C V. Clearly, R is real if and only if frir = RI 11 for all i, j, m, n (that is, It = R) and R is inverse real if and only if Rrir = (R- l) ir for all i, j, m,n (that is, _II = R-1 ). Since C(uii ) is the free algebra with generators uii , there is a unique antilinear multiplicative mapping 0 of C(u) (that is, 9(Aa +,ub) = A9(a) + FA(b), O(ab) = 0 (a)0 (b) for A, p E C, a, b C C(u)) such that 6+ ( 4) = uji . The next proposition will be used for the treatment of real forms in Subsects. 9.2.4 and 9.3.5 below. Proposition 3. (i) S'appose that R is real. Let I be a subset of C(u) such

that the two-sided ideal J :=(3(R), I) of C(u) is a biideal and the bialgebra A = quii ))1,7 is a Hopf algebra. Assume that 0(I) C j. Let 1,1) be an automorphism of the Hopf algebra A such that 02 = id. Then A is a Hopf *-algebra with involution determined by (uii )* = SOP(u3i:)), i , j = 1,2, • N. (ii) If R is inverse real, then A(R) is a *-bialgebra with involution defined by (uii )* = uii , = 1,2, • • •, N. Proof. (j ): We first prove the assertion in the case V) = id. Define an antilinear anti-multiplicative mapping 19 : C(u) -> A by 19(4) = S ( LI). Let : C( u) * A --= C(t4)/J be the canonical map. Using the assumption that 3 3 R is real and the properties of 5, 0 and /9 we get V(V,.n = 0, .)= so V(J(R)) = {0 } , and 19(x) = S(/(9(x))) for x C quii ). Since 0(I) ç J. by assumption, it follows that 19(J) = 01. Hence 79 passes to an anti-linear antimultiplicative map a -* a* := V(a) of A = quii )1,7 to A. By definition we then have (uii )* = S(u). For A being a Hopf *-algebra we still have to show that a** = a for a E A and that 2i and € are *-preserving. It suffices to do this on the generators uji . The relations 46(( u )*) = .6(u)* and €((uij )* ) = are easily verified. Applying the *-operation to the identity E,n (4)*u.7 = -

9.1 The Approach of Faddeev-Reshetikhin-Takhtajan

Et, s(uni )u.73.

= bij we get

En (uriL)*(u7)**

307

Er, s(u4) (wit)** = 6. This proof in the case 0 = id. =

implies that (4)** = tirii. and completes the Let us turn to the general case. Put a*' := i,b(a)*, a E A. Since 7,b is a Hopf algebra automorphism, V) 0 51 = So 7,/) and hence 0(a*) = 1/0) * , a E A. Using the latter and the assumption 0 2 ,---- id one verifies that (A, *1 ) is also a Hopf *-algebra. (ii): The bialgebra quii) is a *-bialgebra with involution defined by ------ uji . By It z= R -1 , we obtain (Iim )* = - E,,, t ,,, s (R - 1)ik i Irk.19 (R - 1),nrsm , so the involution of quii ) passes to the quotient A(R) --= C(u)/J(R). 0 9.1.2 The Quantum Vector Spaces XL (f; R) and XR(f;R)

We retain the notation from the preceding subsection. Let f = { fi , • - • 7 fn} be a set of polynomials f i in one variable with complex coefficients. We first give a basis-free definition of the algebra XL(f; R). Namely, XL (f; R) is the quotient of the tensor algebra T(V') over the dual V' of the vector space V by the two-sided ideal ,A(f; R) of 11 (V') which is generated by the ranges of the mappings fm (R) t , m = 1, 2, • • • , n, of (V 0 V)'. Here fni (R) t denotes the transpose of the linear transformation fr,-,(i?‘) of V 0 V. Now we describe the algebra el'af; R) in terms of coordinates. Let us identify V and V' and so V 0 V and (V 0 V)' by identifying y E V with the linear functional '6 on V given by i3(w) = E, cii0i, where y = E, a i s i and w = E, 13 ix, E V. Then the tensor algebra T(V') is (isomorphic to) the free associative algebra C(x i ) with generators xi, • • • , xN and Ji,.(f;R) is the two-sided ideal of C(xi) generated by the nN2 elements fni (.1b13/ xkxi, where m = 1, 2, • • •, n and i, j = 1, 2, • • • , N. That is, XL (f; R) is the algebra with generators xi, • - - , xN and defining relations

frn (lbxkx/ = 0,

m = 1,2, • • • , ri; i, j = 1,2, • • • , N.

( 7)

Let x denote the column vector (x1, • • - , xN) t of generators. In matrix form the relations (7) can be written as frn (ft)xi x2

( 8)

Indeed, if x 1 denotes the matrix x0/ and x2 is the matrix /0x, then the expression f„,(ii)xix2 in (8) is just the product of the three matrices fm (Jet), xi and x2. Similarly, the algebra XR(f, R) is defined as the quotient of the tensor algebra T(V) over V by the two-sided ideal generated by the images of the transformations fm (11), m = 1, 2, • • • ,n, of V 0 V. Clearly, XR(f;R) is the algebra with generators xi, - • • , xN subject to the relations

fm (10 121 X k X1

-= 0 1

m=1,2,•-•,n; i,j=1,2,•••,N.

(9)

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

308

Proposition 4. Let A be a bialgebra which is a quotient of the bialge bra A(R). Then there are algebra homomorphisms p : XL(f ; R) —> A0 XL(f R) and (p R XR(f R) —> XR(f; R) 0 A such that

(PL(xi) = î4 0 x i and (pR(x i ) = xi 0 u3i .

(10)

These mappings (p L, and (pR turn the algebras XL (f;R) and XR (f;R) into left and right A-quantum spaces, respectively.

Proof. We carry out the proof for XL,(f ; R). By the universal property of the free algebra C(x i ), there is a unique algebra homomorphism çoL : C(x i ) A® C(xi ) such that çoi,(xi) = u 0 x i . Since PL(frn(MSkXj)

(

frn (ft)Ukr U is SrSs

ki _x r X_ s Tn rs Un k i f (ft) i

by (3), 40L(Si(f; R)) ç A 0 ,71,(f; R). Hence (pi, passes to an algebra homomorphism, still denoted by çoii , of XL(f;R) to A 0 XL(f; R). To prove that ciaL is a left coaction, it suffices to check the conditions (1.52) on the generators si. We omit this simple verification. Definition 2. The algebras XL (f; R) and X R (f; R) are called the coordinate algebras of the left and right quantum vector spaces, respectively, associated with the set of polynomials f and the transformation R. Suppose that the matrix 1 is symmetric, that is, 1 = kj. The matrices

.h for the quantum groups Glig (N), SL q (N), 0 q (N) and Spq (N) considered in Sects. 9.2 and 9.3 will have this property (see formulas (13) and (30) below). Then the defining relations (7) and (9) are the same, so the algebras XL( f; R) and X R( f ; R) coincide and are denoted by X( f ; R) . By Proposition 4, X(f; R) is a left and right quantum space for the bialgebra A.

Example 2 (R = id). Let R be the identity map and f(t) = 1 — t. Then X(f; R) is the commutative polynomial algebra in N indeterminates and the left coaction (pi, is given by the matrix multiplication of u and x. A Next we consider some quantum spaces which are obtained by modifying the preceding construction. For sim - plicity we restrict ourselves to the right handed coordinate versions and to the case of single polynomial relations. For any triple of polynomials f, g h in one variable and any complex number 'y, let X(f,g,h, R) denote the unital algebra with 2N generators X1, ,XN,Y1, n 7 YN and defining relations

f(MVxkxi = g(f)gYkY1 = h(MYfrol Yisi +

Pybki = 0, i j = 1, • • • , N. ,

(11) Proposition 5. Let A be a Hopf algebra which is a quotient of the FRT bialgebra A(R). Then the algebra X E.'. X(f,g,h,ry;R) is a right quantum space for A with respect to the coaction (pR determined by the equations (PR(xi) = xi 0 u-1: and çoR(yi ) = yi S(u).

(12)

9.2 The Quantum Groups GL q (N) and SL(N)

309

Proof. The proof is similar to the proof of Proposition 4. One first defines C (xi, y i ) g A by (12). As usual, an algebra homomorphism çoR : C(xi,y i ) j =1,- • - ,N. Using C(xi, yi) denotes the free algebra with generators (3) one then derives that çoR passes to the quotient algebra X of C(xi , yi ) to CI give a right coaction there. The defining equations (11) may look strange at first glance. However, they become very natural in terms of corepresentation theory (see Sect. 11.1). 1, = g(Nki and Clci = WO. Then the matrices A, B Set At! = f(f?) 4 and C belong to the intertwining spaces Mor (u u), Mor (ue g uc) and Mor (tie u, u uc), respectively, and it is only these properties which are used during the proof of Proposition 5. The term yisi in (11) may also be replaced by Tikil ykx/ with T E Mor (ucou). The second equation of (12) means that çoR acts as the contragredient corepresentation uc on the generators yi . In the following Sects. 9.2 and 9.3 we realize the constructions A(R) and X(f ; R) by taking as R the matrices R which have been derived in Subsect. 8.4.2 for the vector representations of the Drinfeld-Jimbo algebras Uq (g), g -= giN , sip/ 1 502,, spN , and Uq i/ 2 (S027-14-1). Our aim is to develop the coordinate Hopf algebras of the quantum groups Glig (N), SL q (N), °q (N), 80q (N) and SPq (N). Since A(R) is only a bialgebra, this requires additional constructions.

9.2 The Quantum Groups aL q (N) and SL q (N) In this section R is the matrix q l/N R1,1 obtained in Subsect. 8.4.2 for the Drinfeld-Jimbo algebra Uq (s1N). By (8.60), Ê = r o R has the matrix entries

Ran = q6i2 bin 6i, + (q q-1 )Sim,6in 0(j - i),

(13)

where 0 is the Heaviside symbol, that is, 0(k) = 1 if k > 0 and 0(k) = 0 if

k < O. We suppose that A + := q + q-1 0. By (13), the matrix k is symmetric. From Proposition 8.24 we know that k satisfies the quadratic equation

qi)(k + q -11) = 0

(14)

and that

P + := A 1 (k + q -1 /)

and

P._ :=

+ TT)

(15)

are projections (that is, Pi = P±) such that P + P_ = P_ P + -.----- 0 and

= qp+ - q --1 p _ .

(16)

310

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

9.2.1 The Quantum Matrix Space Mq (N) and the Quantum Vector Space CqN We begin by describing the FRT bialgebra A(R) explicitly. Definition 3. The bialgebra A(R), with R = (Kil n ) given by (13), is called the coordinate algebra of the quantum matrix space ltici (N) and is denoted by 0(Mq (N)).

Inserting the matrix entries R;l:Lim from (13) into the relations (1), we see that the algebra 0(Mq (N)) is generated by the elements uii , i, j = 1, 2, • • • , N, with defining relations:

uik tric -z--- quik uik , -';

uik uik = qujk uik , i < j,

(17a)

i < j, k < 1,

(17b)

u-kul,

nik uji - ui ut = (q - q-1 )uik ul,

i < j, k < I.

(17c)

These relations are equivalent to the requirement that for any k, 1, m, n with k 7 b = unk t c = Ulm 1 d = uniof the corresponding k <1, m < n the entries a , um 2 x 2 submatrix of u = (uii ) satisfy the defining relations of Mq (2) (see the formulas (4.1) and (4.2)). From the above defining relations for Mq (N) it easily follows that there are an automorphism 0 and an anti-automorphism79 of the algebra 0(Mq (N)) such that O(u) = u-,! and 79(uij ) =---- 4+1_1. We have Ao0=To (0 0 0) 0 A, A ol9 = (79 019) o A and 0 079 = 79 o O. Moreover, for any nonzero a c C the mapping uij extends to an automorphism of the bialgebra 0(Mq (N)). Definition 4. Let f3 (t) = t - q and fa (t) = t + q-1 . The algebras O(C) := X(f s ; R) and A(C) := X(f a ; R) are called the coordinate algebra and the exterior algebra of the quantum vector space C, respectively.

Note that fs A and fa (I) are just the two factors in equation (14). In order to avoid confusion, we denote the generators of A(C) by yi , • • - , yN and the column vector (yi , • - • , yN) i by y. Since f 8 (Ê) = -AP_ and fa(1) = A + P + by (15), the algebras O(C) and A(C) are defined by the matrix equations P_x1 x2 = 0 and P + yiy2 = 0, respectively. We insert the matrix entries (13) into these equations and obtain the following explicit forms. The algebra O(C) has generators x i , • - • , xN with defining relations xixi = qx i x i ,

i c j.

The quantum exterior algebra A(CqN ) is generated by elements yi l - • - 7 YN subject to the relations

9.2 The Quantum Groups GL q (N) and SL q (N)

311

Obviously, in the "classical" case q = 1 the algebra O(C) is the polynomial algebra in N commuting indeterminates, that is, the symmetric algebra of the vector space C N , and A(CqN ) is the exterior algebra of C N . Proposition 6. The following sets of monomials are bases of the correspond-

ing complex vector spaces: 0(Mq (N)) : 1 (t4) k 11( u D ki2

O(C) qN :

1,.

kN

Xi X2 • • • X N

A(C)

(4)kNN

r1, 1 nd2 ' ' i1,i2l• • •

I

kli, ki2, • • ' )kNN E NO}

kN 6 N 0 1)

iN E { 0 , 1 }}.

Proof. The proof can be found in [PINT], Sects. 3.4 and 3.5. The assertions can also be derived from Lemma 4.8. For the first assertion this is carried out in [NYM], Sect. 1.5. I=1 Proposition 6 implies that for all three algebras the subspaces spanned by the monomials of a fixed degree have the same dimensions as in the classical case q = 1. Let O(C) nand A(C) ndenote the subspaces of O(C) and A(C), respectively, consisting of homogeneous elements of degree n. Since the matrix R is symmetric (see (13)), we know from Proposition 4 that O(C) and A(C) are both left and right quantum spaces for the bialgebra 0(Mq (N)) with coactions (pL and ço R given by (10). Since the coactions are algebra homomorphisms, the subspaces 0(CqN ), and A(C) nare invariant under (pi, and ()OR and the restrictions çoL,n of 40L, and 40R,n of (pR to these subspaces are corepresentations of the bialgebra 0(Mq (N)). We call the mapping clOR,n

0 (C : I )n

0 (C ri inn 0

0(Mq (N))

the symmetric tensor corepresentation of 0(Mq (N)) and the mapping cloR,n

A(C :I )n A(C :I )n

0 (Mq(N))

the alternating tensor corepresentation of 0(Mq (N)). 9.2.2 Quantum Determinants We first introduce some multi-index notation. Let On denote the set of all subsets of {1, 2, • • • , N} consisting of n elements. For subsets I, J E fln we write I and J as I = 1, • • • I in} and J = {ji) • • • , j n } with i
_01( p) uiipi c i)

u iip (n)

(18)

where Pn is the permutation group of the set {1,2, • - • , n } and l(p) is the number of inversions in p. If p E Pn and = p-1 , then

312

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

2i

• • • U4

= U3- V(1)

•i ••U

n

(19)

Indeed, both expressions contain the same factors, but in a different order. Using (17b), one side of (19) can be rearranged to get the other side. Since 1(p') = l(p), (19) yields the alternative expression pEr-n

U 2 p(1)

(20)

3p(r)

Definition 5. The element D JI of 0(Mg (N)) is called the quantum n-minor with respect to the row I and determinant of the matrix u = column J. In the case I = J = {1, 2, • • , NI the element Dij is called the quantum determinant of u and is denoted by Dg . Proposition 7. (PR,n (Yr) = yj 0 V! and (PL,n(Y1) D IJ (ii) = DK I ODIjc and e(D1j ) Sij for I,J C Q.

YJ for

I

E Qn.

(Note that there are summations in (0 and (ii) over J c f2, resp. K G Pa .)

Proof. (i): We carry out the proof for (pR, n . From (10) we obtain (pR,n(Y/) = yi l - • • yi n 0u l ••- uji:. The defining relations for A(0 1 ) imply that yi, • • • y in 0 if jk = ji for some k and that yip(i) - - • yip(n) = (q)t(P)y, • • -yin for= p

Pn and J = fill

' • jrt} E

ço,,,n(y/)= E E

Q. Using these facts we get ip(1)

YiP( 1 ) • • • Y73(n) ®

U il

ip(n) In

•••

=

E Y.70 D I

Jes7r,

JES2npEPn

(ii): By Proposition 6, {YJ I J E On } is a basis of the vector space A(C qN )nHence, by (i), 73-rj are the matrix elements of the corepresentation cpR, n of 0(Mq (N)) and the assertion follows from Proposition 1.13.

We reformulate the above results for the quantum determinant Dg . Let ( be a nonzero element of the one-dimensional space A(C)N. Then, by Proposition 7(i), Dg is the unique element of 0(Mq (N)) which satisfies the equations (10 R,N (() = ® D g and (pL,N(() = Dg ® C. Applying this characterization to ( =-- Yi (_0 1 (1 )( for r c PN, we conclude that

• • •

YN

and

= Yr(1) ' • • Yr(N)

N)) 1 = E (_01(P)--1(r)urx,;...n pIN Dg = E (-01(P)-1(r)U1)4 1 ) ...UP4 N pE PN

pEPN

( 21) for any permutation r E PN. By Proposition 7(ii) applied with I = J {1,2, • • • ,N}, we have

.6(Dg ) =Dg ®Dg and

6' (;)

=1,

(22)

so the quantum determinant is a nonzero group-like element of 0(Mq(N)).

9.2 The Quantum Groups GL q (N) and SL q (N)

313

For j E {1, 2, • • • , N}, let A3i := DI, where J = {1,• - • ,j-1, j+1,• .• ,N} and / = {1, • • • , — 1,i + 1, • • • ,N}. Proposition 8. For i,k E {1, 2, • • • , N} we have

=

E;(- 0j - kuAik

E

= bikl3q)

(23)

3 3 = bikDq•

(24)

Proof. For subsets / and J from {1, 2, - - • , N}, let sgnq (/, J) 0 if / n J 0 and sgnq (/, J) = (-04") if I J = 0, where 1( 1 , J) is the number of pairs (i, j), j E I, j E J, such that j > j. We then have wyj = sgn g (I,J)Y.K1 K = IUJ. Applying (pi? to this equation with appropriate sets I and J, using Proposition 7(0 and comparing the coefficients of yi • • • yN (23) follows. El Using (p L, the identity (24) is derived in a similar way.

n

is i,j = 1, 2, - .- , N. The matrix Set 3 := called the cofactor matrix of u = (u i ,j=1,...,N. The first formulas of (23) and (24) can be rewritten in matrix form as 1-111 = 111-1 = V q • 'N

,

(25)

where IN is the unit matrix. Multiplying (25) by the matrix u on the left and on the right and comparing both equations we obtain Dq • u =uûu = u - Dg which means that Dq uij uij Dq for all j , j = 1, 2, • - • , N. This yields the first assertion of the following proposition. Proposition 9. The quantum determinant Dg belongs to the center of the algebra 0(Mg (N)). If q is not a root of unity, then the center of 0(Mg (N)) is generated by Dg .

The second assertion of Proposition 9 is proved in [NYM], Sect. 1.6. Note that the center of 0(Mq (N)) is much larger than the polynomial algebra in Dq if q is a root of unity. More properties of quantum determinants can be found in [PW] and

[NYM]. 9.2.3 The Quantum Groups GL q (N) and S.Lq (N) The coordinate algebra of the quantum group GL g (N) is obtained by adjoining the inverse Dq-1 of the quantum determinant Dg to the algebra 0(Mg (N)) To be precise, 0(GL q (N)) is the quotient algebra of the algebra 0(Mq (N))[t] of polynomials in t over 0(Mq (N)) by the two-sided ideal (tDg — 1) generated by the element t; — 1. That is, CAGL q (N)) is the algebra with N2 1. generators /L p i j = 1, 2, • - - N, and t and defining relations (17) and

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

314

In the algebra 0(GL g (N)) we have t =

From the very definition

O(GL g (N)) = 0(Mg (N))[t]l(tDg -1) we see that 0(Mq (N)) can be regarded as a subalgebra of 0(GL q (N)) and

00 O(GL q (N))

(26)

m=o The coordinate algebra of the quantum group SL (N) is defined as the quotient

0(SL g (N)) = 0(Mq (N))1(D g - 1) =-- 0(GL q (N))1(Dg - 1) of the algebra 0(Mg (N)) or, equivalently, of 0(GL q (N)) by the two-sided ideal (Dg - 1) generated by the element Dq - 1.

Proposition 10. There are unique Hopf algebra structures on the algebras 0(GL q (N)) and 0(SL q (N)) with cornultiplications L and counits E such that = u,',F and E(u) = bii . The antipodes S of these Hopf algebras are given by S(u) = and S(17,7 1 ) = Dq for 0(GLI g (N)) and by for 0(SL q (N)). Moreover, we have S 2 (u) --= Di7 l uipi for both S(u) = Hopf algebras, where D i := q -2i . and the counit E of the bialgebra 0(Mq (N)) to algebra homomorphism : O(GL q (N)) O(GL q (N)) O(GL q (N)) and E : O(GL q (N)) C by setting ,i(N 1 ) = D-1 0 D-1 and E(D q--1 ) = 1. One easily verifies that then O(GL q (N)) becomes a bialgebra. Recall that by (25), ûu = uû = Dq • IN. Multiplying the defining relations Ru1u 2 = u2u1R for 0(Mq (N)) by ii2 fi 1 D-2 from the right and by 2 R. ii2D q-2 from the left we obtain R(flpq l-) 2 (u,r) ) 1 = ( fiN i) ' ii p-q-1) That is, the entries of the matrix fiDV. satisfy the defining relations of the algebra 0(Mq (N))°P. Hence, there exists an algebra anti-homomorphism S : 0(Mq (N)) 0(M q (N)) such that S(u) = Consider the equation m(id S) = E(a)1 for a E 0(Mq (N)). Clearly, it holds for a product ab if it holds for a and for b. It is satisfied for a = 1 and for a = u, since uii = Dq • IN and E(Dq ) = 1. Thus, the equation is true for all a E (N)). Setting a = Dq , it yields Dq S(Dq ) = 1. Hence, S(Dq ) = Dq-1 and S extends to an anti-homomorphism S : O(GL q (N)) O(GL q (N)) by setting S(D 1 ) = Dq • Since the relations (1.26) hold for the generators a = 1, u 3i.,13V 1 of 0(GL g (N)) as is easily seen, the mapping S is an antipode and O(GL g (N)) is a Hopf algebra by Proposition 1.8. the second formula of (24) yields Since S(uii ) = (-0 2(j - n) S(u! ) S2 (u7) ( —q) 2 (j -72)S(urPuTki = 55 k • Applying S, we get 65k Multiplying this equation by 4 on the left and summing over k, we obtain (-0 2 (i -i) 52 (u) = uii , which gives the formula for S2. Proof. By (26), we extend the comultiplication L

En

.

9.2 The Quantum Groups

G q (N) and S L q (N)

315

Now we turn to the algebra 0(SL q (N)). From the equations 46(7; -1) =-Dq 0Dg -101 = (Dq -1)01+Dq 0(Dq -1), e(Dg -1) = 0 and S(Dg -1) = D-1 - 1 = -D-1 q (D - 1) it follows that Mg - 1) is a Hopf ideal of the Hopf algebra 0(GL q (N)). Hence, the quotient 0(SL q (N)) = 0(GL q (N))1 (Dg -1) is also a Hopf algebra with the structure induced from 0(GL q (N)). Since tl(7.")q ) = Dq 0 Dq and e(Dg ) = 1, the uniqueness assertions are clear. Definition 6. The Hopf algebras C9(GL q (N)) and 0(S L q (N)) defined above are called the coordinate algebras of the quantum groups GL q (N) and SL q (N) respectively.

Recall that the quantum 'vector space O(C) is a left and right quantum space for the bialgebra C9(Mq (N)). Since 0(Mq (N)) is a subbialgebra of 0(GL q (N)) and 0 (S L q (N)) is a quotient of the bialgebra 0(Mq (N)), 0(C qN) is also a left and right quantum space for 0(GL q (N)) and 0(SL q (N)) by Proposition 4. The corresponding coactions (pL and çoR are determined by the formulas ço i, (x i ) -=- u 0 xi and (pR (si ) = z 0 u. The next proposition says that the left (resp. right) quantum space O(C) for 0(G L q (N)) can be considered as a subalgebra of 0(GL q (N)) by identifying x i and ut (resp. x i and ul), where the left (resp. right) coaction on xi the comultiplication of ut (resp. corespndt Proposition 11. Define 0(x i,) = ut and 19(x i ) = u, i = 1,2, • • • , N. Then 0 and 19 can be uniquely extended to injective algebra homomorphisms of O(C) into 0(G L q (N)) such that 0 E Mor (L, tl) and t9 E Mor ((p R , tl).

i < j, by (17a), Proof. We carry out the proof for O. Since uluil the ul satisfy the defining relations of O(C), so there exists an algebra homomorphism 0 : 0(C') 0(GL q (N)) such that 0(x i ) ut. By Proposition 6 and formula (26), {x fp • • kN E No} is a vector space u basis of O(C) and {(u Dk1( /)k2... ( u inkN 1c1, • • • , kN E No} is a subset of a basis of 0(GL q (N)). This implies that 0 is injective. The assertion a E Mor (wL, tl) means that (id 0 0)(p L(a) = ,6(0(a)) for all a E O(C). Since both (id 0 0) 0 (pi, and tl 0 0 are algebra homomorphisms, it suffices to check this equality on the generators x i . But then we have by (10) that (id 0 0)(P L(xi) = u ji 0 0(x i) = uij uji = 46(u1) = 46(0(xi )).

The algebras X( R) from Subsect. 9.1.2 provide another source of quantum spaces for Sid q (N) and also for GL q (N). An important particular case is the algebra Xy, 8 7 where 6 E C, obtained by setting f (t) = g(t)

- q, h(t) = qt 6(1 - q2 ). Inserting the values (13) into (11) we see that the defining relations for the generators xl, • • •, I N, Yi YN of eV7, 6 can be written as follows: isi = qxi x i , yi yi = yixi =

qx i yi ,

< i j,

(27) (28)

316

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces yixi — q2xi yi = + (q2 1)(E

xi yi — 45E

N i= 1

si yi ).

(29)

The algebra X.7 ,6 is a right quantum space for SL q (N) and for GL q (N) with coactions given by (12). We will meet this algebra in the case = 0,7 = 1 as the quantum Weyl algebra in Subsect. 12.3.3. 9.2.4 Real Forms of GL q (N) and SL q (N) and *-Quantum Spaces

Let A = 0(G q.) be the coordinate Hopf algebra of a quantum group G q . If *: A A is an involution such that (A, *) becomes a Hopf *-algebra, then we say that this Hopf *-algebra structure defines a real form of the quantum group G q or briefly it is a real quantum group. A quantum space X for a Hopf *-algebra A is called a *-quantum space if X is a *-algebra and if the coaction ço of A on X satisfies ço(x*) = ço(x)* for all x C X. In order to discuss real forms of GL q (N) and S.L q (N) we consider two cases of values for the parameter q. Case 1: = 1. Then we have R(q) = R(q 1 ) R(q) 1 , that is, the R-matrix R is inverse real. There exist unique involutions of the algebras 0(GL q (N)) and 0(SL q (N)) such that WO* = j = 1, 2, • • • , N. Equipped with these involutions, 0(Gli g (N)) and 0(S.L q (N)) are Hopf *-algebras denoted by 0(GL q (N; r )) and 0(SL q (N;R)), respectively. They are called the coordinate algebras of the real quantum groups GL q (N; r) and Sid q (N; R), respectively. To prove the assertions in the preceding paragraph, we restrict ourselves to the case of 0(GL q (N)). By Proposition 3(ii), A(R) is a *-bialgebra with (u i )* = uii = 1, - - •, N. Since rqr = 1, one verifies that Dq* Dq . Because of (26), there is a unique extension of the involution to the algebra C9(aL q (N)) by setting (Dr)* =Dim , m E N. The algebra 0(C') with the involution defined by (xi)* = x i , i 1, 2, • • • , N, becomes a *-algebra which is denoted by 0(1 61). (It suffices to note that the relations of CqN are invariant under this involution.) One can easily see that O(R) is a (left and right) *-quantum space for the Hopf *-algebras 0(GL q (N;R)) and 0(SL q (N;R)). Case 2: q E R. Then the R-matrix R is real. There is a unique involution on the algebra 0(GL q (N)) such that (uii )* = S(uji ), i j = 1, 2, • • • , N. Equipped with this involution, 0(GL q (N)) becomes a Hopf *-algebra, denoted by O(Uq (N)), which is called the coordinate algebra of the quantum unitary group Uq (N). In the *-algebra 0(Uq (N)) we have Dq*Dq. = ;73: = 1, that is, the quantum determinant Dq is a unitary element. Let u =(ul , • - • , UN) be an N-tuple of numbers zzi E II I —11. The Hopf algebra 0(SL q (N)) becomes a Hopf *-algebra, denoted by 0(SUq (N;u)), with the involution determined by the formulas (u.i )* = vi vjS(u.1), j j = 1, 2, , N. We call this Hopf *-algebra the coordinate algebra of the real ,

,

9.3 The Quantum Groups 09 (N) and Spq (N)

317

quantum group SUq (N;1.). If vi = • • • = UN = 1, then the Hopf *-algebra 0(SUq (N; v)) is called the coordinate algebra of the real quantum group SUq (N) and is denoted simply by 0(SUg (N)). We prove the preceding assertion for 0(SL q (N)) by applying Proposition 3(i). Let //) denote the Hopf algebra automorphism of 0(SL Q (N)) defined by 0(uii) = vi vi u3i-. Let I : C(u) -- A(R) = C(i4)/J(R) be the canonical map. If Ai denotes an element of C(u) such that ADO =Dq , then we have 0(7)0 - Dg E 3(4 Thus Proposition 3(i) applies with I = {Dq - 1} and shows that 0(SL q (N))= C(U i )1(J(R)1Dg 1) is indeed a Hopf *-algebra. It is clear from the definitions of the involutions that for both quantum groups Uq (N) and SUq (N) the fundamental matrix u = (14) is unitary, that is, we have uu* = u*u = I, where (u*) i := (4)* = S(uji ). Because of this property, we call Ug(N) and SUq (N) compact quantum groups. An extensive treatment of such quantum groups will be given in Sect. 11.3. Let 7 and 6 be real. Then the algebra r; 05 defined by the relations (27)-(29) becomes a *-algebra, denoted by X-05(v), with involution determined by (x i )* = vi yi . (In order to prove this, it suffices to check that the relations (27)-(29) are invariant under this involution.) Then we have wR(x7) = vi yi 0S(4) = vi yi 0(u)* = (pR(x i )* by (12). Therefore, the *algebra X-y ,5(1)) is a right *-quantum space for the Hopf *-algebras 0(Uq (N)) and 0(SUq (N; v)). If Py = 0 and v1 --r--- • • • = UN = 1, the *-algebra X(u) for arbitrary 6 E R might be considered as the quantized algebra of polynomials on CN . The elements x l , - • -, xN and Yi = x • • -, yN = x*N are then interpreted as holomorphic and anti-holomorphic generators, respectively.

9.3 The Quantum Groups 0q (N) and Sp q (N) Throughout this section, R denotes the R-matrix for the vector representation of one of the Drinfeld-Jimbo algebras Uq (g), g = s02n , spN , and Uq i/ 2 (S02n+1); see Sect. 8.4 for details. Let us collect some notation and facts (repeated from Subsect. 8.4.3) that will often be used in the rest of the book. We abbreviate i' := N + 1 - i. Let N = 2n if N is even and N = 2n + 1 if N is odd. We define numbers p i , E and Ei by setting .

.1

-• _ N Pi — 7 —1 i f z c i 7

Pt' = — pi. if i < i' ;

c = cl = • • • =

EN

=1

for g = soN and pi El

= 7N + 1 — i,

=-•-=

En

....= 1 1

Pif = -pi if i < i'; f = En+1 = ' • ' = EN = —1

for g = spN . The matrix C --=-- (C) of the metric is defined by Cii := Ei 6if q- Pi. Then we have C -1 = EC and CI, = c(C) 1 . The matrix R has been described by

9. Coordinate Algebras

318

of Quantum Groups and Quantum Vector Spaces

formula (8.61) and investigated in Propositions 8.25 for g = 502n+115027, and 8.26 for g = sp N • A unique formula for the matrix entries of R and k = T o R which applies to all three cases is given by

= kg1,1 = where

6iTn 6in (q q -1 )9(i — m)(6i, i 6in

Kgri :=

Kg7,),

(30)

eCCT and i, j, m, n = 1, 2, • • • , N. In particular we see that and RI, Moreover, R obeys the property

=

= wir.

R = C 1 (Rt 9 - 1 ci-1

c2 (R-1 )t2

1

(31)

where as usual C1 = COI1 C2 = IOC and t1 and t2 mean the transpositions with respect to the first and second tensor factors, respectively. The matrix R satisfies the cubic equation

(IF?

gi)(I +q --1 /0

0.

(32)

Eqe—N) We suppose that (q + q -1 )(q — eq €-N)0,--1 0. This assumption implies in particular that q — q -1 0. Except for Definition 7 below we also assume that N > 2. Then the transformations P + , P_ and P o defined in Subsect. 8.4.3 are mutually orthogonal projections (that is, P i2, --= P, and P u P p = 0 for u p and v, p E {± 7 — 1 0 ) such that }

qp+

_ (r i p _

Eq e_N po•

(33)

Moreover, we have K = (1 + €(q— q -1 ) -1 (q N- f — q f-N ))Po• 9.3.1 The Hopf Algebras 0(0q (N)) and 0(Sp q (N)) Recall that the algebra 0(SL g (N)) is obtained as the quotient of the algebra 0(Mg (N)) by the additional relation Dg = 1. The algebras 0(0q (N)) and 0(Sp g (N)) will be defined as quotients of the algebra A(R), with R = R given by (30), by the metric condition u

cu t c -1 = cut c -1 u

I.

(34)

In the classical case q = 1 the matrix C takes the form

0.

.

\

and 0

1 •••• 0 0 0 •••• 0 0)

\

for Og (N) and Spq (N), respectively, so in this case (34) is just the metric condition which defines the complex Lie groups 0(N, C) and Sp(N,C).

9.3 The Quantum Groups 0q (N) and Spq (N)

319

A heuristic motivation for relation (34) is provided by the following reasoning. Applying the transposition t 1 to the defining relation Ru1u2 = u2 u 1 R for A(R), we derive (ut) iRtiu2 = u 2 Rt1(ut) 1 . Since Rt 1 = CI-1 R -1 C 1 by the first equality of (31), we get (ut) 1 C1-1 R -1 C 1 u2 = u2 C1-1 R -1 C 1 (ut ) 1 . Multiplying this relation by u 1 C 1 on the left and by C -1 1 ul on the right we obtain (uCutC-1)1R-iu2ui = u1ti2R -1( cut c -i. u ) 1. (35) On the other hand, the relation Ru 1 u2 = u2u1 R yields (36)

R -1 u2u1 = u1u2R -1 .

A comparison of (35) and (36) suggests that it is natural to impose the conditions (34). u - I) denote the two-sided ideal of the Let j (uCut C -1 - I, Cut algebra A(R) generated by the entries of the matrices uCu t C -1 - I and CutC - lu - I. The algebras 0(0q (N)) and 0(Sp q (N)) are defined now as quotient algebras A(R)/J, where R is the corresponding matrix given by (30). By Lemma 12 below, the two-sided ideal J has a single generator Qq , 1, so the algebras 0(0q (N)) and 0(Sp q (N)) can also be defined as the quotients A( R)/( Qg 1). This definition closely resembles that of the algebra 0(SL q (N)). Lemma 12. In the algebra A(R) we have the equality .1

.= "-

n

j ‘-n uj

E n fcl

for i,j = 1,2, • • -, N. (37)

The element Qg — 1 generates the two-sided ideal J. Moreover, Qg is grouplike and belongs to the center of the algebra A(R). Proof. Since Po and so K is a polynomial of k, the matrix equation ku1u2 =

u1u2k (by (3)) implies that Ku 1 u2 = u1 u2 K. In terms of matrix entries, the 1 and (4) -1 = latter reads as CIC/u7ur = uin um . Since CI = 0, j' k €C1', this gives the equality (37). Setting k = , 1 -= j' and using the relations €Cnn , = (C -1 ) n' and CI = 0, j' 1, we see that the diagonal entries of the matrices Cut C -1 u - I and uCutC -1 - I are all equal to Qg — 1. Setting k i', 1 = j' and k = 1 j', it follows that the off-diagonal entries of these matrices vanish. Hence the element Qg — 1 generates the ideal J. Using formula (37) it is straightforward to show that A(Qq) = Qq®Qq. Set il := CutC -1 . By the preceding computations, we have ûu = un = Qg • I. As in the case of the quantum determinant (see Proposition 9), the latter implies that Qg is in the center of A(R). Proposition 13. There are' unique Hopf algebra structures on the algebras 0(0,7 (N)) and 0(Sp q (N)) such that A(u.i ) = ul3f and E(u) = For the antipode S of both Hopf algebras we have S(u i ) = C!,4(C-1 ) j: = N. ficiqPi - Piul, and S2 (uii ) =Di-1 14Di with Di := (cf)2 = q2p, ,i,i =

320

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

Proof Since Qg is group-like, J = (Qq -1) is a biideal, so that the quotient A(R)/j is again a bialgebra. Arguing similarly as in the proof of Proposition 10 it can be shown that there is an algebra anti-homomorphism S of A(R)/j

such that S(u) ---- 44(C -1 )i! . By (34), S fulfills the conditions (1.26) of the antipode on the generators 4. Hence A(R)1,7 is a Hopf algebra. The [11 formula for 52 (uji-) follows at once from that for S(u). The matrix D := CC has the entries Dii := 60 D i . In matrix notation the formulas for S and S2 in Proposition 13 can be written as

S(u) = Cu t C -1 and

52 (u) = D -l uD.

For any Hopf algebra structure satisfying A(u) = u 0 u and e(u) = I the matrix 5(u) must be the inverse of u. Thus the formula 5(u) =-- Cu t C -1 is, of course, expected from the metric condition (34). Definition 7. The Hopf algebras 0(0q (N)') and 0(Sp q (N)) are called the coordinate algebras of the quantum groups 0q (N) and Sp q (N), respectively.

By writing down the defining relations for the algebras 0(Sp q (2)) and 0(Side (2)) one observes that they are equivalent, so the mapping u1i -m 3i. defines an isomorphism of the Hopf algebras 0(Sp q (2)) and O(542(2)). We express this fact by saying that the quantum groups Spq (2) and Sido (2) are isomorphic. 9.3.2 The Quantum Vector Space for the Quantum Group 0q (N) Throughout this subsection R denotes the matrix (30) in the orthogonal case. Definition 8. Let f3(t) = t2 (q (11N)t q2—N and fa (t) t The algebras 0(0,7N ) := X(f,s ; R) and A(0) := X(f a ; R) are called the coordinate algebra and the exterior algebra of the quantum Euclidean space 01 , respectively.

By (30), the matrix 1E? is symmetric, so that XL(f;R) = XR(f;R) = X(f;R) for any polynomial f. Hence, by.Proposition 4, both algebras 0(0qN) and A(0) are left and right 0(0q (N))-quantum spaces. 0 by Since fs (k) = A + (q -1 ql - N)P_ by (8.75) and A+(q-1. ql—N ) assumption, the defining relations f3 (I)xix2 = 0 for the algebra 0(0) can also be expressed as P_x1x2 = 0. Clearly, the entries of the N2 x 1 matrix P_x1x2 span im ) t . But im (1p_) t coincides with im P, because the matrix P_ is symmetric. Hence 0(0qN) is aLso the quotient of the tensor algebra T(V) C(xi) by the two-sided ideal generated by im P_ = ker (r? ) , so the algebra We also have fa (k) = = A±P+ + (q-1 ±q 1.-N‘po A(ON) has the defining relations P_Fy1y2 = 0 and PoYiY2 = O. Thus, A(0,7) C(y) by the two-sided ideal is the quotient of the tensor algebra T(V) generated by im P + = ker (rEZ qI) and irn Po = ker (r?

9.3 The Quantum Groups Og (N) and Spg (N)

321

Remark 2. Our exterior algebra A(RIN) differs from the one in [RTF], where A(0 1 1 ) is defined by the relations P_F y 1 y2 = 0 only.

Some algebraic properties of these algebras are collected in the following two propositions. Proposition 14. (i)The algebra 0(0qN) for even N = 2n has the following defining relations: x i xj = qxjx i , i< j, i j', XiI

ii

X iXif

q

-1)\--•

qj-i-1

L-di<j<7-1,

X - X •I

<

(ii) The algebra 0(042,N) for odd N = 2n + 1 has the defining relations xixi = qxi x i , Si t Xi

X i Xi'

(jfni (q112

2

i

)xn2 +1 ±

< i, (q

1

)

q

<

i<j
(iii) The element z := x t Cx Ei xix,= Ei q - Pixix i, belongs to the center of the algebra 0(0qN ) and satisfies (pL(z) =10z and pR(z) = z01. (iv) txki 1 x k2 2 - • -4/Nlic i , • • •, kN E NO is a vector space basis of 0(0,7N ).

Proof. (i) and (ii): Set a :--= (1+q N-2 ) -1 . Since —A + P_ = k—qI-Ect(q—q -1 )K, the defining relations P_x1 x2 = O of O(0) can be stated as kx k x/ — qx i x i + a(q q-1 )Cii x k ask , = 0,

i,j = 1, • • •, N.

(38)

We insert the values k ikii from (30) in this equation. If i < j, i j', then we get xi xi --= qx i x i . First we suppose that N is even. Subtracting the equations (38) for j = n,j = and for i = n+1, j = i' we conclude that xiix/ = xn/xn • Applying (38) with j < i', we obtain x i xi , = qi1.1+1 (17

-1

X 4 Xi i irrP3

cx

p E xkxkl q -h

),

j<21

so that = q(x i, x i — s i x it) + (q — q -1 )x i x i l and — the above relations follow. Now suppose that N is odd. From (38) apwe conclude that plied with j = j = n ± 1 and with j =-- n, j x„ , xn, — X n X n,/ = (q 112 q -1/2 )X n2 + 1. Similarly as for even N, we derive that — xixi, = q(xil_ixi +i xi+ixt, _1) + (q q-1 )xi_F 1xv_i which implies the above relations. On the other hand, the above relations correspond to an N(N — 1)/2dimensional subspace of V017 which is contained in im P_ by the preceding proof. Since dim (imP_) = N(N —1)/2, this subspace coincides with im P_. Hence the above relations define the algebra O (O" ). (iii): That z is in the center of C3(0:1 ) follows from the relations in (i) and (ii). From the definition (10) of (pi, and the metric condition (34) we obtain

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

322

(10/,(Z) =

E

quikuzi g xksi =

k,1

CI • 1 0 skxi = 1 g

Z.

The proof of çoR(z) = zel is similar. (iv): Using the relations from (i) and (ii) the assertion can be derived from the diamond Lemma 4.8.

1=1

Proposition 15. (i) The algebra A(0) for odd N = 2n+1 has the following set of defining relations:

yi =

yeyi yi yi/ = (q— q-1)E ,2

i < j,

yi yi =-

(

n1/2

j-i+1 1<j
YjYi17

n-1./2)V

L-sl<3
-

(ii) The first two lines of equations of (i) are the defining relations of the algebra A(0) for even N = 2n. y y1:2

• • '1iN

E {0,1 1 is a basis of the vector space }

A( O qN Proof. The proof is similar to that of Proposition 14.

El

Note that for even N there is no index i such that i and hence we have y? = 0 for all i = 1,2, • • -, N. However, for odd N = 2n + 1 it follows from Proposition 15 that the element yn2 +1 of A(0) is nonzero in contrast to the classical situation. Definition 9. The quotient of the algebra (.9(0qN ) by the two-sided ideal (z 1) generated by z -1 is called the coordinate algebra of the quantum orthogonal

complex unit sphere 8° ,N-1 . By Proposition 14(iii), we have cp/,((z - 1)) C O(O qN )0(z - 1), so the left coaction (pi, of 0q (N) on Olqv passes to the quotient algebra 0(Se ,N-1 ) = 0(0 N )/(z - 1) and defines a left coaction there. This turns 51° ' N-1 into a left quantum space for the quantum group Gi g (M. In a similar way, 8.70,N-1 becomes a right quantum space for Glq (N). The central element z= x isilq- Pi can be regarded as a quantum analog of the 0(N, C)invariant quadratic form on CN. Finally, we consider the classical limit q 1 of the three algebras 0(04,N ), A(0.7) and 0(Sq" -1 ). Recall that we excluded the classical case q = 1 since 1, the algebra (.9(0qN ) is the we assumed that 1 - q -N O. In the limit q polynomial algebra in N commuting indeterminates (that is, the coordinate algebra of CN), the element z is six, and A(0) is the exterior algebra of the vector space CN . On changing variables to vi defined by Vx = vi + = vi , for j = 1, 2, • n and sn +i = v,2+1 if , N = 2n + 1 is odd, the defining equation z = 1 of 4),N-1 becomes the

Ei

Ei

equation

>:: v2 = 1 of the unit sphere in CN.

9.3 The Quantum Groups 0q (N) and Sp q (N)

323

9.3.3 The Quantum Group SO(N) In order to define the quantum determinant for the quantum group 0q (N) we use the same idea as in the case of SLq (N) (see Subsect. 9.2.2). By Proposition 15(iii), the vector space A(0)N of elements of degree N of A(0) has dimension 1. Let (pR i N denote the restriction of the right coaction PR Of 0 ( 0 q(N)) on A(0,7) to the invariant subspace A(0) N . There exists a unique element Dg of 0(0g (N)) such that çoR,N(C) = (OA, for E A(O N )N. Being the matrix coefficient of the one-dimensional corepresentation cpR ,N, the element Dg satisfies ZI (V q ) = Dg ®Dg and e(Dq ) = 1. The latter facts imply that Dg is invertible in the algebra 0(0q (N)) and S(D g ) = D97 1 . Further, it follows that the two-sided ideal (Dg — 1) of the algebra 0(0g (N)) is a Hopf ideal, so the quotient 0(0q (N))/(Dg — 1) is also a Hopf algebra. It will be denoted by 0(80q (N)). (

Definition 10. The element D g of 0(0g (N)) is called the quantum determifor the quantum group Og (N). The Hopf algebra nant of the matrix u 0(S0 g (N)) is called the coordinate algebra of the quantum group SOq

Let us emphasize that the element Dg defined above is different from the element Dg occurring in Subsect. 9.2.2. They are quantum determinants of different series of quantum groups and belong to different algebras. Example 3. An explicit expression of the quantum determinant for Og (3) is .

23 3 — qu21uu31 Dg = umul 4_ qu3i um qu2 u32ui —q2 u1uu 31— q(q 1/2 _ q--1/2) uum

= utu3u1 + (q + q2 )ululuT. — (1 + q)ulu34

3 qu 1 i u32u2

q2u14/4

Here the first equality is derived from the definition of Dg , while the second one follows from the commutation rules of the A remarkable peculiarity of quantum group theory is that embeddings or homomorphisms of classical Lie groups (for instance, the group inclusions SO(N) c SL(N) and Sp(N) c SL(N) for N > 3) have no quantum analog in general. The technical reason is that quantum groups corresponding to different series of simple Lie groups are derived from different FRT bialgebras A(R). There is a general principle behind this which says that "quantization removes degeneracy". This means that from the quantum group point of view most of the classical group homomorphisms are particular cases. However, at least two relations between classical simple Lie groups carry over to the quantum case. That the quantum groups SL q 2(2) and Spq (2) are isomorphic was already mentioned in Subsect. 9.3.1. The second one is the result SU(2)/{.1,—I } 80 (3). Our next aim is to give a quantum version of this classical fact. The following notions are also used in Sects. 11.5 and 11.6. Definition 11. Let A be a _Hopf algebra. A quantum subgroup of A is a Hopf algebra B together with a surjective _Hopf algebra homomorphism it : A B.

324

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

The elements of the space A/13 := {a E A I (id 0 70,6(a) = a 0 1} are then called right B-invariant. If the Hopf algebras A and B in Definition 11 are coordinate Hopf algebras of quantum groups Gq and Hq , respectively, then we say that Hq is a quantum subgroup of G q . The classical picture for these concepts is clear: Let i : H -> G be the inclusion for a subgroup H of a group G. If f is a function on G, then r(f) is the function on H given by r(f)(h) := AO)), h E H, and the right invariance of f means that Agi(h)) = f (g) for h E H and g E G. Now let A = 0(84(2)) and let 13 = CZ 2 be the group algebra of Z/2Z. The two elements of Z2 are denoted by O and I. Let A, and Z2 A ° be the linear spaces spanned by the basis elements aibick ,di bjck of A with even and odd degrees i + j + k, respectively. Clearly, there is a Hopf algebra homomorphism 7r : A B such that 71- (a,) = e(a)O, a, E Ae , and 7r(a0 ) = e(a0 )I, ao c A ° . Hence B is a quantum subgroup of A. Obviously, we have A/B A. Since A, is also spanned by the matrix coefficients of odd-dimensional irreducible corepresentations T1 of S.Li q (2), Ae is a Hopf subalgebra of A. Let to , j , j = 1, 2, 3, be the elements of 0(84(2)) defined by a2

(to) = ( (1 + c 2 ) 1 / 2 ac --C2

(1 + q -2 ) 1 / 2 ab -b2 1 + (q+ q -1 )bc -(1 + q - 2)1 1 2bd (1 + q-2)1/2cd

.

d2

Then (t o ) is a matrix for the corepresentation T1 of the Hopf algebra 0(574(2)). It is obtained from the matrix described by formula (4.75) when the third basis vector is multiplied by -1. From the explicit form of these matrix entries we see that A, is precisely the unital subalgebra of A generated by the elements to . With some computational work (see [Dij1] for details) it can be shown that the to satisfy all defining relations of SOq 1 / 2(3) and no other relations than those. Hence there is an isomorphism V of the algebras A, and 0(S00 /2(3)) which sends the element t o to the generator zei of SO4,1 / 2(3). Since the tii are matrix coefficients of a corepresentation, it follows that 79 is also an isomorphism of the flopf algebras A, and 0(80q 1/ 2 (3)). Summarizing the preceding, as a quantum anolog of the classical relation SU(2)1{I,-I } e 8 0(3) we have the Hopf algebra isomorphism 0(84(2))/CZ 2 L-./ 0(.90,7 1,2(3)). 9.3.4 The Quantum Vector Space for the Quantum Group Sp(N)

In this subsection let R be the R-matrix from (30) for Uq(sPN)Definition 12. Set f3 (t) = t q and fa (t) = t 2 (q -1 q -1-N )t q -2-N We call O(Sp) := X(f 8 ; R) and A(Sp) := X(f a ; R) the coordinate algebra and the exterior algebra of the quantum symplectic space Spl , respectively.

9.3 The Quantum Groups 0q (N) and Spq (N)

325

We state some properties of these algebras which can be derived in a similar manner as for the corresponding quantum spaces of °q (N). Because the matrix k is symmetric, both algebras 0(84) and A(Sp) are left and right quantum spaces for the Hopf algebra 0(Sp q (N)) by Proposition 4. From the equation fs (k)= -A + P_ (q+q0 we conclude that the algebra 0(Sp qN ) has the defining relations P_x 1 x2 = 0 and P0 x 1 x2 .= 0 C(x i ) by the two-sided ideal and that Q(Sp) is the quotient of T(V) generated by im P_ = ker + q -1 .0 and im Po = ker (Ék + q-1-N /). Since the element Ei xixi,c!, = Ei x ix e ci q- Pi belongs to im Po, it vanishes in the algebra 0(Sp qN ). Since fa (R) = A+ (q+q-1-N) P ±, the defining relations of A(Sp qN ) can be rewritten as P + y 1 y2 = 0 or, equivalently, A(Sp qN) is the quotient of T(V)

C(y) by the two-sided ideal generated by im P + = ker

qI).

Proposition 16. (i) O(Sp) is the algebra with generators xl, • • • ,xN and

defining relations x i xi =qxi x i ,

j7

<

= -(q

silx i (ii)

<

.4N k1, • •, kN E No} is a vector space basis of 0(Sp qN ).

Proposition 17. (i) A(Sp qN ) has the defining relations 2

YiYi

=

q2 yi yii =

= 0, i

j,

j',

1)

(ii) {yll • • •y pi 7 I ii," . 1 1 N E {0, 1}} is a basis of the vector space A(Sp qN ).

Proof. The proofs of both propositions are similar to that of the corresponding assertions for 0(0,7N ) and A(0). 9.3.5 Real Forms of 0q (N) and Spq (N) and *-Quantum Spaces As in Subsect. 9.2.4 we distinguish two cases of parameter values. Case 1: iql = 1. Similarly as in Subsect. 9.2.4, we then have i = R-1 , that is, R is inverse real, and CC' = I. From these facts and Proposition 3(ii) it follows that the Hopf algebras 0(0q (N)) and 0(Sp q (N)) equipped with the involutions determined by (uij )* = uij are Hopf *-algebras. These Hopf *algebras are denoted by 0(0q (n,n)) if N = 2n, 0(0q (n, n + 1)) if N = 2n +1 and 0(Spq (N; R )) , respectively. They are called the coordinate algebras of the real quantum groups 0q (n,n),0q (n,n + 1) and Spq (N; R), respectively.

326

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

The algebras 0(0,7) and O(Sp) equipped with the involutions given by := x i become *-algebras. They are left and right *-quantum spaces for the Hopf *-algebras 0(0g (n, 0(0q (n, n+1)) and 0(Sp g (N; R)), respectively. Case 2: q E R. Then the R-matrix R is real. Let v = (v1, - • vAr) be an Ntuple of numbers vi E {1, -1} such that ye Evi , j = 1, • - N. Moreover, we assume that vri±i = 1 and q>0 in the case of 0q (2n + 1). Then there are involutions on 0(0g (N)) and 0(Sp g (N)) given by (uii )* := vi vi S(u ji ) = Uti „

j j = 1, • • . 7 N, turning these algebras into Hopf *-algebras. ,

(This assertion can be derived from Proposition 3(i) using formula (37) and arguing similarly as in Subsect. 9.2.4, Case 2.) We denote these Hopf *algebras by 0(0g (N;Y)) and 0(Sp q (N; v)) and call the coordinate algebras of the real quantum groups Og (N; v) and Sp g (N; v), respectively. Let v be as above and let b c {1, -1 } . The algebras O(0), 0(Sp qN ), A(0,7) and A(Sp) equipped with the invplutions determined by the formulas := ei viq - sPsxi, and := fvq 5 Pi y1 , , i = 1, , N, become *-algebras, denoted by 0(0qN(6, y)), 0(Sp qN (6, v)), A(O gN(S, v)) and A(Sp qN(S, v)), respectively. (The defining relations listed in Propositions 14-17 are indeed invariant under these involutions.) Then one verifies that Q(0(1, y)), A(O qN (1, v)) and O(SpqN (1, y)), A(Sp qN (1, y)) are right *-quantum spaces for the Hopf *-algebras 0(0q (N; v)) and O(Spq (N; y)), respectively, the coactions being given by (10). Likewise, 0(0,7 (-1, v)), A(O gN (-1, y)) and 0(SpqN (-1, v)), A(Sp qN(-1, y)) are left *-quantum spaces for the Hopf *algebras 0(0q (N ; Y)) and 0(Spq (N; v)), respectively. Recall that the quantum determinant Dg for 0q (N) is defined by the property (pR(‘) =- (0Dg , E A(0)N. For the *-algebras A(O gN (1, y)) and 0(0q (N; v)) we have C*0;* = (pR(C)* = çoR ((*)=C*0; and so Dq* = Dg . Hence the quotient Hopf algebra 0(S0 g (N)) = 0(0g (N))1(D g -1) with the *-structure inherited from 0(0q (N; v)) is a Hopf *-algebra 0(S0 g (N; y)). In the case v1 = • • • = YN = 1 the corresponding Hopf *-algebras defined above are denoted by 0(0g (N;R)),O(S0 g (N;R)) and OW Sp g (N)). (The latter notation is suggested by the fact that the compact form of the classical symplectic group is just the intersection of Sp(N, C) with the unitary group U(N).) For each of these three Hopf *-algebras the fundamental matrix u = (uii ) is unitary, that is, u*u = uu* = I, where (u*) i := (uji )* = S(ujj ). Hence we call Og (N;R), SOq (N;R) and USp g (N) compact quantum groups. Let us return to the *-algebra 0(0qN (6, v)). Its central element z = Ei xixilq - Pi is hermitean (that is, z = z*) and can be written as z = Ei vig(45-1)Pix ixT. Thus, the quotient algebra 0(SqN-1 ) = 0(0qN )/(z -1) be(6, v)), with *-structure induced from comes a *-algebra, denoted by 0(S 1 (-1, Y) are right and left 0(0qN (6,0). By construction, S 1 (1, v) and S *-quantum spaces, respectively, for the Hopf *-algebra 0(0q (N; v)). In the 1 (15,v)) is called the coordinate case vi= • • • =vN = 1 the *-algebra 0(S algebra of the quantum Euclidean real unit sphere SqN,45-1- (R) and is denoted

9.4 Dual Pairings of Drinfeld-Jimbo Algebras and Coordinate Hopf Algebras

327

1 the *-algebra 0(SqN,6-1- (R)) gives by 0(SqN6-1 (R)). Clearly, in the limit q the coordinate algebra of the unit sphere SN ' of R N .

9.4 Dual Pairings of Drinfeld-Jimbo Algebras and Coordinate Hopf Algebras The duality between Drinfeld-Jimbo algebras and coordinate Hopf algebras supports the common viewpoint that both Hopf algebras provide dual approaches to the same "quantum object". It appears to be a powerful tool for the study of quantum groups (see, for instance, Sects. 11.2.3 and 11.5). First let us repeat some notation from Chap. 8. Let T1 be the vector representation for one of the algebras Uq (g), Uqext(g), or Oq (soN) (see Subsects. 8.4.1, 8.5.3, and 8.6.1) and let t o be the matrix elements of T that is, Ti(f)e1 = t(f)e. Let R1 7 1 = (R li l) be the R-matrix (see Subsects. 8.3.3 and 8.4.2) for the vector representation, Rr = ((11 -1-)ii ) its inverse and the L-functionals associated with Ti (see SUbsect. 8.5.4). ,

Theorem 18. There exist unique dual pairings (.,.) of the pairs of Hopf al-

gebras Uq (g1 N ) and 0(GL q (N)), Urt (s1N) and 0(SL q (N)), Uq i/ 2(so2n+i) and 0(0q (2n + 1)), Uq1/2(502n+1) and 0(S0 q (2n + 1)), Urt(so2n ) and 0(0q (2n)), Urt (so27,) and 0(S0 q (2n)), Urt (sp2,) and 0(Sp q (2n)) such that (39) = tki(f), k,1 = 1,2,• • • N. ,

The dual pairing (.,.) satisfies the relations

(l;Ei , u ilc) = 1:116:i and (ii-i ,u 11 ) =

j, k, = 1,2, • • • , N ,

(40)

and it is uniquely determined by these equations. Proof. Let us write Uq and 0q for the corresponding Drinfeld-Jimbo algebra and coordinate Hopf algebra, respectively. Let 13 be the subalgebra of the dual of Uq generated by the functionals t ii . Since t(ab) = tik(a)tki(b), B is a subbialgebra of the dual Hopf algebra Uq°. Clearly, the evaluation of b E B at f E Uq defines a dual pairing (., •) of the bialgebras Uq and B. Since _hvv(Tv 0 Tv) = (Tv 0 Tv)1?), v by Proposition 8.19, the matrix t = (tii ) satisfies the equation ktit2 = tit2. (Recall that R = q l/N R1,1 for g = sIN and R = Ri 1 i for g = soN,sp N .) Consequently, there is an algebra homomorphism : A(R) B such that 1/0:ii ) = t o . Since Zi(uii ) = 14 0 ulc and Igtii) t•k 0 tkl, IP is a bialgebra homomorphism. Therefore, the definition (f , a) := (f ,o(d)) .11)(a)(f), f E Uq , a E A(R), gives a dual pairing of the bialgebras Uq and A(R). Next we show that the pairing (-, .) of Uq and A(R) passes to a dual pairing of Uq and 0q . We carry out the proof in the case where Ug = Urt (s1N) and 0q = 0(SL q (N))). The other cases can be treated similarly. From Subsect. 9.2.2

328

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

we recall the relation wR, N (C) = (07)q for E A(C)N. Setting = Yi • - yN, 4i • • • ttip7 . By (1.64), we obtain we have yoR,N(Y1 • • • YN) = Yii • • • YiN

fic7R,N(Ki)Yi

• • • tlikl yi, • • • yi,

(Ki ,

• YN

= (Ki , uli) • • • (Ki , ur) yi , • • • yi, = (Ki ,Dg ) Yi - • • yN. From the particular form of the representation T1 we see that (K,, ut) tki(K i ) = 0 if k I and (Ki ,u1) • • • (K i ,u) ---- 1. Therefore, by the above equality, (Ki , = 1. Since Dg is group-like, the linear functional (-, DO on Ug is a character. Hence we get (E, D) = (Fi , D) = 0 and so (f, D g -1) = 0 for all f E Uq . Thus, the dual pairing (•, ) of Ug and A(R) induces a dual pairing of the Hopf algebras tfq and 0 q = A(R) I (D - 1). Formula (39) holds by definition and (40) follows immediately from (8.88). By Theorem 8.33, the elements It. generate Ug as an algebra. Except for GL (N), the algebra 0 q is generated by the elements uti . Hence the pairing (., •) is uniquely determined by the values (f (j El rut). ut) 1:essop272. The restrictions in the first variables of the dual pairings for the extended Hopf algebras Ugext (s1N), ugext (sp2 .), r rgext yield corresponding dual pairings for the Hopf subalgebras Uq (s1N), Uq (sp2n ), Uq (s02a ), respectively. Let us collect the values of these dual pairings on the generators /4 of C9(Gq ) and Ei , Fi , Ki of U(g) resp. Uq l/ 2 (S0211+1). From the formulas for the vector representations in Subsect. 8.4.1 we read off the following list:

Ug (g1 N ): (Ki ,t4) (

(K i ,u331) = 1,

,

Ei , uii+1 ) --=

(41) (42)

1;

uq (s 1 N ): = (1‘

=q

(K u) = 1 j i i+1,

=--

U

,

,

uli+ 1 ) =--- 1;

1/2 (S0271+1):

(K

1 ,u)

(Ki ,uV+1) = (K71 7 u(i+1 Y = (K i , (i+1)1

(K, u) =1 for (K7.7 I.

)

i,i+1,

n, n + 2,

= (Fi,u4 1 ) =

(En, U771+1) = (q1/2

2+

tif1/2)1/2 1

n,

+ 1)1 ,

=- (K„,u,71:11) = q, (K, u) = 1 for j

=

=

i n, j

q

(En u n 1:21 ) == .... 17 1/2(q1/2

1, i n, (7112)11 2

(Fa , 41:1) = _ q -3./2( qi/2 (7-1/2)1/2;

9.4 Dual Pairings of Drinfeld-Jimbo Algebras and Coordinate Hopf Algebras

329

Uq(sP2n): (Kr 1 ,14) = (Ki ,u!t1) = (K27 1 ,u i ± 1 :) ,) = (Ki ,u1.)

(Ki7 1 ,u) = 1

i

for

( K; u n = ( K n u

n, j L i i + 1, i', + 1Y, ,

q 2 (Kn,u )i ) = 1 for j

(Ei , /4+1 ) = HE" W(4 1 0 =

(Erl, u

q, i n,

n,n + 1,

(Fi , u(7 1Y ) = 1 i

u:±1 ) =

n,

,

1 ) = (Fn 14+1) = 1;

Uq (so2n ): A1,1 7

=1

(K

urn' (Kn ,

(Ei, u +1- ) =

( re-

(i+1) , )

=

=

for j

= (K 7.7. 1 ti,771) = = 1 for j

U(i+i),

n, j 7 1,

n,

Ui/ )

i i + 1, i', (i + 1 )' , ,

= ( 1 n , Unri)

=q

,

n - 1, n, n + 1, n + 2,

uri+i) ,) = (Fil l4 +1 ) = - (Fi l u!,i+1)1 = 1,

- (En , Unn +2 ) = (En, unnl)

(Fn7 u771+2) = (Fn un+ n - 1) = 1.

The values (Y, un, Y = Ei , Fi , Ki , for all other pairs of generators vanish. In Subsect. 11.2.3 we shall also need a dual pairing of the Hopf algebras tIq(soN) (see Subsect. 8.6.1 for its definition) and 0(0q (N)). Proposition 19. There exists a unique dual pairing (.,.) of the Hopf algebras (1-q (SON) and 0(0 (N)) satisfying the relations (39) for f E Oq (soN ). In particular, we have (x,Dq ) = -1. Moreover, (x, ulic) = -6k1 for N = 2n + 1 and (x,14,1±1) = (x,74+1 ) = 1, (x u) = 1 for i = 1, 2, • • • , n - 1, n + 2, • • • , N and (x, u)) = 0 otherwise for N = 2n. Proof. Since Rt i (x)t 2 (x) = t i (x)t2 (x)R as easily seen from the form of R and ili (x), there is a dual pairing (•,•) of the bialgebras Oq (soN) and A(R) such that (f, = tki(f), f E Oq (soN). An easy computation shows that

Qq) =

ci 'c,71,tni(x)tniii(x) = 1 -

Therefore, since (g, Qg — 1) 0 for g E Uq (502n ) resp. g E Ug li 2 (S°271+1) by Theorem 18, we get (f, Q q 1) 0 for all f E tIq (soN ). Hence the pairing (•, .) of Oq (soN) and A(R) passes to a dual pairing (-, -) of fig (soN) and 0(0q (N)). The values (x, ulic) follow from the particular form of the operator î' (x) (see Subsect. 8.6.1). They easily imply that (x ,73q ) = -1. In case N = 2n one has to use also the relation ynyn±i = -yn±iyn which holds by Proposition 15(ii).

330

9. Coordinate Algebras

of Quantum Groups and Quantum Vector Spaces

Let Uq and Cfq denote one of the dual pairs of Hopf algebras in Theorem 18 or Proposition 19. Then, by Proposition 1.15, any corepresentation (p of Cog yields a representation of Uq . If (pi is the fundamental corepresentation of Cog given by (6), it follows from (39) and (1.64) that çai (f)xi = xi (f,u1) = tii (f)x j , f G Uq . That is, if we take in the carrier space of (p i the same basis as for 711 , then 0 1 is the vector representation T1 of Elq . In fact, the dual pairing (., .) of Uq and 0 q was just constructed such that the latter holds. Clearly, the preceeding is also true for the dual pairs of Uq and A(R). We conclude this section with another duality result. For an arbitrary mawith complex entries, let it(R) denote the algebra trix R = with 2N2 generators i t , = 1,2, • - • , N, subject to the relations R = R4et,

e j-7e -2 R = Re-2 2T,

(43)

where Z± = ( t). Arguing as in Subsect. 8.5.5, it follows that 1,L(R) is a bialgebra with comultiplication and counii given by

=

tt

and e() =

Proposition 20. Suppose that R is an invertible solution of the QYBE. Then

there is a dual pairing (-,-) of the bialgebras 11(R) and A(R) such that (et ,u2 ) = R21

and (C l ,

= R j-21 .

(44)

Outline of proof. First we note that there is a unique dual pairing (•, .) of the free bialgebras C(fti , fi-i ) and C(u) satisfying (44). In order to obtain a well-defined dual pairing of it(R) and A(R) one has to check that the corresponding consistency conditions in both variables hold. As a sample, we shall do this for the relation ZR = Ree i- and compute ZIR1 3 7 U2)= (Z

U 2 ) Z31- U 2 ) R 1 3=

= R13(e 3-E 1 112) (ei-

R 1-21 R23R1 3

= (R13Z 3- ZIT

= R13R23R 1-21 112).

I=1

9.5 Notes The FRT approach and the quantum deformations of the four series of classical simple Lie groups appeared in the fundamental paper [RTF] of L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan. These quantum groups were also constructed by M. Takeuchi [Ta2] and investigated in [H3]. The quantum group SUq (N) was invented and studied by S. L. Woronowicz [Wor2], [Wor3). Another influential early reference is [Mani. The bialgebra M(C, r) from Remark 1 is due to [Doi]. Multiparameter quantum groups appear (for instance) in [AST], [Res31 and [Sud11.

10. Coquasitriangularity and Crossed Product Constructions

This chapter is centered around various fundamental notions (coquasitriangular bialgebras, crossed product and coproducts, braided Hopf algebras) from the advanced theory of Hopf algebras. They are useful tools for the deeper study of quantum groups and for the construction of new quantum groups.

10.1 Coquasitriangular Hopf Algebras Coquasitriangularity is probably the most important general Hopf algebra concept for the second approach to quantum groups. The coordinate Hopf algebras of the quantum groups from the previous chapter and the FRT bialgebra A(R) for any invertible solution R of the QYBE have this property. 10.1.1 Definition and Basic Properties

Recall that a bialgebra B is quasitriangular if it admits a universal R-matrix R.G.808. In order to motivate the subsequent definition of the dual notion, let us consider a bialgebra A which is dually paired with B. Define a linear form r on A 0 A by r(a a') = (a 0 a', R.). Remembering that the role of multiplication and comultiplication is interchanged by the dual pairing, it is not difficult to verify that r is a universal r-form of A according to the following Definition 1. A coquasitriangular bialgebra is a bialgebra A equipped with a linear form r : A0 A -4 C such that the following conditions hold:

r is invertible with respect to the convolution, that is, there exists another linear form 17. : A0 A C such that ri = fr = on A® A, (ii) m A, =r* MA *f on A® A, (iii) r(mA 0 id) = r 13 r23 and r(id Ø m A ) = r13r12 on A0 A 0 A, where mAop(a b) = ba, rri di(a b) = ab, r 12 (a b = r(a r23(a b c) e(a)r(b c) and ri3(a b c) = c(b)r(a a, b, c G A. Recall that the convolution product * is defined by formula (1.21). A linear form r on A 0 A with the properties (i)-(iii) is called a universal r-form of A. A bialgebra, A is called cotriangular if it has a universal r-form r such

332

10. Coquasitriangularity

and Crossed Product Constructions

that r = F. 21 , where 2i (a b) := f(b 0 a). A coquasitriangular Hopf algebra is a Hopf algebra which is a coquasitriangular bialgebra. Remark 1. Coquasitriangular bialgebras appear under a number of different names in the literature. They are called dual quasitriangular bialgebras in [Maj7j, braided bialgebras in [LTo], [Doi] and cobraided bialgebras in [Kas]. A The above name is used (for instance) in [H2], [Si], [Mon].

The above form of Definition 1 is directly dual to Definition 8.1. However, the following reformulations of conditions (i)—(iii) in Sweedler's notation are more convenient to work with. Since linear forms on A 0 A correspond to C as bilinear forms on A x A, we can consider any linear form r : A 0 A a bilinear form on A x A and write r(a, b) := r(a b), a,b E A. Now the equations rt. = Fr =e0e and the conditions (ii) and (iii) read as

E r(a( i ),b(Or(a( 2),b( 2)) = E r(a(i), b(Or(a(2), b( 2)) = e(a)e(b), ba = E r(a(i), b(l))a(2)b(2)f(a(3), b(3))) r(ab,c) = r(a,bc) =

(1) (2) ( 3)

E r(a( i ), 0*( 2), b)

(4)

with a, b,e E A. Multiplying condition (ii) in Definition 1 by r we see that it is equivalent to the requirement

=

r(a(2), b(2))bma(i),

a, b E

A.

( 5 )

Condition (ii) means that the universal r-form r expresses in some sense the deviation of A from being commutative. Clearly, a bialgebra A is commutative if and only if r = e is a universal r-form of A. From the equations (3) and (4) and from Proposition 2(ii) below we see that the map a x b r(a, b) defines a dual pairing of the bialgebras A and .4°P and hence a skew-pairing (see Definition 8.3) of A and A itself. Conversely, a skew-pairing r of A and A is a universal r-form of A if and only if r satisfies (2) or its equivalent version (5). By Proposition 2(iii), the map axb T.(a, b) is a dual pairing of A°P and A. Recall from Proposition 8.4 that the universal R-matrix of a quasitriangular Hopf algebra induces an equivalence of the tensor product representations Tv 0 Tw and Tw 0 Tv. Our next proposition shows that there is a similar result for comodules of a coquasitriangular Hopf algebra. Let A be a coquasitriangular bialgebra with universal r-form r. For right A-comodules V and W we define a linear mapping rvw : V 0 W W 0 V by (6) rvw et; = > r(v(i) W (1)*(0) 0 V(0), V E VI E W.

10.1 Coquasitriangular Hopf Algebras

333

If V = W is the right A-comodule A with respect to the comultiplication, then we obviously have (e e)rvw = r, that is, the universal r-form r can be recovered from the map rvw in this case. Proposition 1. If U, V and W are right A-comodules, then we have:

(1) rvw is an isomorphism of the right A-comodules V OW and W 0 V. (ii) ruo v,w = (ruw Oid)(id0rvw) and ru y ow = (id 0 ruw )(ruv 0 id). (iii) (rvw 0 id)(id ruw)(ruv Old) (id 0 ruv)(ruw id)(id® rvw)• Proof. (i): Define a linear mapping fwv : WO V —> 170 W by wv(w 0 y) = v E V, w e W. Using (1), we obtain

(fw v 0 rvw )( y

w)

=

wv

(E r(V(1) 7 W (i)*(0) 0 V (0))

Er(v (2) , w( 2))*( 1 ), w( i ))y( 0) O w( 0) EE(v( oe(w ( l) )v (0) w (0) = v w and similarly rvw 0 i'wv(w O y) = w v. Hence rvw is invertible. Let cpyow and cpwov denote the coactions of A on the right comodules V W and W 0 V, respectively. By (6), (1.61) and (5), we get

(Pwo v o rv, w ( Y

111)

= (Pwcov (E r(v(i), w(i)*(o) v(0)) Ew (0)

O V (o) O w( i)v( i)r(v ( 2), W(2) )

Ew (o )

0 v(0) 0 v(2)w(2)r(v(i) w(i))

= (rvw 0 id) 0 wv o w (v w). That is, rvw E Mor (wvovv,(Pwov)• Using (3) and (4) the assertions of (ii) are proved in a similar manner.

(iii) follows easily from (II).

111

Of course, the preceding results hold for left A-comodules as well. By Proposition 1(iii), for each right A-comodule V the linear mapping rvv defined by (6) satisfies braid relation, hence T O rVV is a solution of the QYBE. That is, coquasitriangular bialgebras induce solutions of the QYBE on their comodules just as quasitriangular bialgebras do on their modules. A large part of the theory of coquasitriangular bialgebras can be developed by dualizing the corresponding theory for quasitriangular bialgebras. Some basic facts on coquasitriangular bialgebras are collected in Proposition 2. For any coquasitriangular bialgebra A with universal r-form r we have:

(i) r12r13r23 = r23r13r12 on A0 A 0 A. (ii) r(1, a) = r(a, = e(a) for a E A. (iii) i is a universal r-form for the bialgebras A°P and A"P (see Subsect. 1.2.3 for the definitions of these bialgebras).

334

10. Coquasitriangularity and Crossed Product Constructions

:= f(b 0 a). (iv) P21 is also a universal r-form for A, where f21(a (y) If A is a coquasitriangular Hopf algebra, then for a,b E A we have r(S (a), b) = f(a, b), t(a, S(b)) = r(a, b), r(S(a), S(b)) = r(a, b). Proof. All proofs are straightforward verifications. As a sample, we show the first relation of (v). Set ri(a b) = r(S(a)0b). From (3) and (ii) we obtain

E r(a( i ),b(0)r/ (a( 2) 7 b(2))

Er(a(i) ,b(1) )r(S(a(2) )7 b(2)) = = Er(a(i) S(a(2) ),b) = e(a)r(1,b) =

--=

so that re = E 0 E. Similarly, r'r = E 0 E. Hence r' = F.

1=1

The next result describes the square and the inverse of the antipode of a coquasitriangular Hopf algebra. It is dual to the corresponding assertion for quasitriangular Hopf algebras (see Proposition 8.5). Proposition 3. Let A be a coquasitriangular Hopf algebra with universal r-form r. Define linear functionals f and f on A by Pa) =

E r(a( i),S(a( 2)))

and J(a) =

EF(S(a (i) ),a(2) ),

a E

A.

Then we have ff =Jf =e in A'. The antipode S of A is bijective and satisfies S2 = * id* f and S-1 = f * S * 1. That is, for a E A we have

82 (a) =

E f(a(o)a( 2)f(a( 3)),

1. (a) =

Proof. Applying (5) with a 0 b replaced by S(a( 3))a( i)f(a( 2)). This yields

E

E 512 (a(2))f(a(1))

=

E f(a(i))S(a(2))f(a(3)).

E a( 1) S(a( 2))

we get

f (a)1 =

E 8 2 (a(4))S(a(3))a(i)f(a(2)) E S(a(3)S(a( 4)))a( i)f(a( 2)) = E amf(a( 2)).

(7)

Using the first and the third equality of Proposition 2(v), formulas (7) and (4) and Proposition 2(ii) we compute'

f f(a)

E f(S(a(i)), a(2))f(a(3)) = E r(82 (a(1)), a(2)f(a(3))) E r(82 (a(0),8 2 (a(3))f (a(2))) = E r(a(i),a(3))f (a(2)) E r(a( i ), a(4) )r(a( 2), S(a( 3))) = E r(a( i), S(a(2))a( 3)) r(a, 1) =

Similarly, fj=-- e. From the relations ff =

E and

(7) we obtain

E f(a(i))a(2)f (a(3)) = Ef(am)f(a(2))82(a(3)) Since 82 = 1* id*

f as just shown, we

= 82 (a).

have 82 *1= j*id, that is,

10.1 Coquasitriangular Hopf Algebras

E s2 (a(1))f(a(2)) = E f(a(1))a(2)•

335

(8)

Put T := f * S * J. Using (7) and the relation fj-= E. , we get

E T(a(2))a(i.)

E f (a(2))51 (a(3))f (a(4))a(i) = E S(a(3))f (a(4)) 82 (a(2))f(a(1)) = E(a)1.

Similarly, (8) and the equation fl = E imply that E ct( 2)T(a(1)) = This shows that T is an antipode for the bialgebra A°P. By Proposition 1.6, T= Another interesting feature of a coquasitriangular bialgebra A is that its comodules carry (left and right) module structures. Proposition 4. The equation y
Proof. Proposition 2(ii) yields v < 1 = v. By (4), we have (v a a)

b = (E y (0)r (v ( 1 ) , a)) b =

v ( 0)r (y ( 1 ) , b)r (v ( 2) , a)

= E v(o)r(v(i) , ab) = y a ab. Thus V is a right A-module. If V is a right A-comodule algebra, (3) implies that vw a a = E(v i a (1) )(w a a( 2) ), so V is a right A-module algebra. E Similarly, a o v := E v(o)r(a, v( i)) defines a left A-module structure on V. Also, the inverse form i induces left and right A-module structures on V by aov=E f(v( ,) , *(o) and v a := Next we introduce some important bilinear forms. For a coquasitriangular Hopf algebra A with universal r-form r, we define two linear forms q and kR on A 0 A by setting

q(a

b) := r2i r(a

kR(a b) := q(a

S(b))

b)

E r(b( i) , a( 1 ))* (2) , b(2)),

E r(S(b( 2 )), a(o)r(a(2), S(b(1)))•

(9) (10)

As usual we consider q and kR as bilinear forms on Ax A as well. Recall that a linear functional f on a vector space V is called invariant with respect to a right coaction cp of A on V if (fOid)(p = fl. A linear form k on A0 A is called AdR-invariant if k is invariant with respect to the right coaction AdR AdR of A on A0 A or equivalently if

E k(a (2) b(2))

S(a(o)a( 3)S(b( 1))b( 3) = k(a b)1

for a, b E A.

336

10. Coquasitriangularity and Crossed Product Constructions

Proposition 5. The linear form kR on A0 A is Ad R -invariant.

Proof. Applying condition (5) twice, we obtain

E kR(a( 2) b( 2) )8(a(o)a( 3)8(b( 1 ))b(3) E r(S(b(3 ) ), a(2) )S(a( {r(a(3) S(b(2) ))ao) S(b(i) )J 1)(4) E r(S(b( 3)), a(2))8(a( 1 ))[S(b(2))a(3)r(a(4) 7 S(b(1)))]b(4) E S(a(1))[r(S(b(3)),a(2))8(b(2))a(3))]b(4)*(4)1 8 (b(i))) E S(a(0)[a(2)8(b(3))r(S(b(2))1a(3))1b(4)r(a(4)18(b(1))) E S(a( 1 ))a(2)S(b(3))b(4)r(S(b(2)), a(3))*(4), S(b(1))) = kR(a 01.

In a similar way, the linear form kL dn A0 A defined by kL(a

b) = q(S(a)

b)

is AdL-invariant. Further, we note some useful identities: kR(a

8'(b)) = q(a

b) = q(S (0 0 S(a)) = kdb 0 8 (0-

Next we turn to the relationship between *-structures and universal r-forms. Proposition 6. Let r be a universal r-form of a *-bialgebra A. Define linear forms r1 and r2 on A0A by r i (a0b):=r(b*0a*) and r2(a0b):=--f(a*Ob*), a,b C A. Then r1 and r2 are both universal r-forms of the bialgebra A.

Proof. As a sample, we verify condition (3) for r2. By Proposition 2(iv), is a universal r-form of A. Therefore, by (4) applied to i21, we have r2(ab, c) = i;21 (c* b* a*) =

Ef21(c(i) *,a*)

f21

k2i(c(2)*,b*)

E r2 (a, c( i ))r2 (b, C(2)). Definition 2. A universal r-form r of a *-bialgebra A is called real if r = r 1 (that is, r(a b) = r(b* a*)) and inverse real if r = r2 (that is, r(a b) =

?(at b*)). Examples of both cases will be seen in the next subsection. We close this subsection with two examples of coquasitriangular Hopf algebras. Example 1. Let A = CG be the group algebra of an abelian group G. Then the universal r-forms on A are in one-to-one correspondence to the bicharacters C such that r(gh,k) -----r( • , -) of G, that is, to mappings r(., •) : G x G r(g,k)r(h,k) and r(g, hk) = r(g, W (g , k) for g,h,k E G. For instance, let G be the additive group Z of integers. Then, for any nonzero A E C there is a universal r-form of A = CZ given by r),(n, m) = A -nm , n,m E Z.

10.1 Coquasitriangular Hopf Algebras

337

Example 2 (Sweedler's Hopf algebra). Let A be as in Example 1.9. One verifies that the Hopf algebra A is self-dual, that is, there is a dual pairing (., .) of A and A such that (x, x) = - (g g) = 1 and (x, g) = (g, x) = O. If -R(a), a E C, is the universal R-matrix from Example 8.2, then r c,(a b) := (a b,R.(a)), a, b E A, defines a universal r-form r c, on A such that A is cotriangular. A 10.1.2 Coquasitriangularity of FRT Bialgebras A(R) and Coordinate Hopf Algebras 0(G q )

The main source of examples of coquasitriangular bialgebras stems from the following Theorem 7. Let V be a finite-dimensional vector space. Suppose that R is an invertible linear mapping of V 0 V to itself which satisfies the QYBE R 12R 13 R 23 = R231113142. Then the bialgebra A(R) is coquasitriangular. There exists a unique universal r-form r of A(R) such that rvv = R , where V is considered as a right A(R)-comodule by (9.6). Then r is called the canonical universal r-form of ,A(R). For i, j,k,1 = 1, 2, • • , N, we have

r(u 3z. uk1 ) = Ri.31k and

tqui. ukI ) 3

-

(R -1 ) 3i-ki .

(11)

Proof. We first construct two linear forms r and i on C(u) ®C (u). In order

to do so, we define r(u ij ut) and tquii ut) by (11) and r(1 uzi ) = r(uii 0 1) = f(1 074) = 1(uii 01) = 5.

(12)

Since C(u) is the free algebra with generators u and a coalgebra (see the proof of Proposition 9.1), there is a unique extension of r to a linear form r : C(u) C(u) -4 C such that the equations (3) and (4) are satisfied. Similarly, 1 extends uniquely to a linear form 1 : C(u) CM) -4 C such that (3) and (4) hold for the coalgebra The crucial step of this proof is to show that r and i pass to linear forms on A(R) A(R). This is where the QYBE for R comes in. Recall that A(R) is the quotient of the algebra C(u) by the two-sided ideal J(R) generated by the elements _gin, := RAurn um s - uir uis R,,. By (3) and the first equality of (11), we compute

r ( 43,2.

4) =

R-Ar(44, 11.1.is r(ten.

--=

— (u ir i40n ilc)/49,7,"„ ukx )r(14,

.i8 117,1kx R:ifl

-

Rrikx

81

— r(uri

ukx )r(uis

rim

D rip )i D D TIP 13 k = O. kiL12- 14, 13-Lv23) nim k - kii.23111.311.42) nim

a)

Similarly, r(ut = 0 by (4) and (11). By (3) and (4), these equations imply that r(J(R) ut) = r(uil j(R)) = {0}. Further, by (12), we have r(j(R) 0 1) = r(1 3(R)) = {0}. Applying once more (3) and (4) we

338

10. Coquasitriangularity and Crossed Product Constructions

conclude that r(J(R)0C(14)) = r(quii )0J(R)) = {0}. Therefore, r induces a well-defined linear form, again denoted by r, on A(R) A(R) by setting r(a b) := r((a J(R)) (b J(R))), a, b E A(R). Now we repeat the preceding reasoning with r replaced by f and C(u) by quii )°P. Using again the QYBE for R we obtain a well-defined linear form f on .)1(R)0A(R). Next we show that r and t satisfy the conditions of Definition 1 or equivalently the equations (1), (3), (4) and (5). By the construction of r, (3) and (4) are fulfilled. Using (3), (4) and (11), one verifies that both equations (1) and (5) hold for a = a', b = b'c' and for a = alb' b = c' provided that they hold for a = a', b = , for a = a', b = cl and for a = b' , b = cl . Thus it suffices to show (1) and (5) for the elements 1 and uii . If a or b is 1, then (1) and (5) clearly hold by (12). Now let a = u.?., and b = flini . Then, by (11), the equations (1) and (5) are equivalent to the relations RR -1- = R = I and the equation (9.1), respectively. Hence (1) and (5) hold for arbitrary a, b E A(R). This completes the proof that A(R) is coquasitriangular. We prove that ritv = it Since cpv (x i ) = z 0 u3-2,. by (9.6), we have rvv(zkOxi) = r(ut,î4)ziOxi = Rxj0xi =

= ft(xkOxi)

(13)

by (6) and (11). Thus, rify = IL Conversely, if r is another universal rform on A(R) such that fvv = ft, then we conclude from vv(xk 0 xi) = i(ut, ubx j 0 xi by (6) that f( 4 , uji ) = R = r(ut, ul. ). Moreover, r(1, u) = r(uii , 1) --= f(1,uii ) = 1) = 6. Since, and r both satisfy (3) and (4), it follows that r = r on the whole of A(R) A(R). This proves the uniqueness assertion. Remark 2. Theorem 7 can be generalized to the following more abstract result. Let .M(C, r) be the bialgebra which was defined in Remark 1 of Subsect. 9.1.1. Suppose that the linear form r : C C —) C is invertible and satisfies the QYBE r12r13r23 = r23r 1 3r 12, that is, for any X 7 Y 7 Z E C we have r(x( 1 ) 7 y( 1 ))r(x( 2),z( 1 ))r(y( 2),z( 2))=

E r(y( i ), z( i ))r(x( i ), z( 2))r(x( 2 ), y( 2 ))

Then r admits a unique extension to-a universal r -form of the bialgebra M(C,r) which becomes then a coquasitriangular bialgebra. (The proof of this assertion uses the same reasoning as above. Note that the inverse f of r also satisfies the QYBE as r does.) Recall that the FRT bialgebras A(R) are a special case of the bialgebras M(C, r) with C = MN(C)' and r(uni uL) = R. If R is invertible and satisfies the QYBE, one easily checks that the linear form r has the same properties. Thus, Theorem 7 is a special case of the preceding result for the bialgebra .M(C, r). A

As a supplement to Theorem 7 we state another related result which will of be needed later in Subsect. 10.3.1. Let us say that a solution R = the QYBE is regular if Rt2 is invertible (t 2 means the transposition of the

10.1 Coquasitriangular Hopf Algebras

339

second index) or equivalently if there exists another matrix f = (kill) such that RV; flinr-1,1„= 1:4";1:11, 1 = binbkrn

k,l,n,m =

(14)

Proposition 8. Retain the assumptions and the notation of Theorem 7 and suppose in addition that R is regular. Then there exists a unique linear form s A(R) Ø A(R) C such that s(., .) = s(. -) is a dual pairing of the bialgebras A(R)°P and .4(R) and

)— s( ui-Ouk 3

k71

1121 •

(15)

Ar•

Moreover, s is the convolution inverse of r when both are considered as linear forms on A(R)°P .4(R). Proof. The proof is quite similar to the proof of Theorem 7. The crucial steps = s(4 = 0 are again the two consistency conditions s(ini-L which are needed in order to pass from C(u) to A(R). It turns out that both conditions are equivalent to the QYBE. That s is the convolution inverse of 0 r : A(R)°P (S) A(R) C follows immediately from (14) and (15).

Now we apply Theorem 7 in order to inherit coquasitriangular structures on the coordinate Hopf algebras 0(Gq ), where Gq is one of the quantum groups GL q (N), SL q (N), 0 q (N) or Spq (N). Let R = R(q) be the corresponding R-matrix given by (9.13) resp. (9.30). Since R is an invertible solution of the QYBE (in fact, R(q) 1 = R(q -1 )) by Proposition 8.19, Theorem 7 applies to the bialgebra .4(R) with R =-- zR for any nonzero z E C. The following theorem says that for certain values of the parameter z the canonical universal r-form r z of A(R) induces a universal r-form on 0(Gq ).

Theorem 9. O(GL q (N)),O(SL q (N),0(0 q (N)) and 0(Spq (N)) are coquasitriangular Hopf algebras with universal r-forms I., uniquely determined by

rz (uji

=

j, k, I =. 1, 2, • • • , N.

(16)

Here z is a complex number such that z 0 for GL q (N), z N =-- q-1 for SL q (N) and z 2 -= 1 for 0q (N) and Spq (N). The matrix R = (R li l) is given by (9.13) for GL q (N) and SL q (N) and by (9.30) for 0q (N) and Spq (N). Proof. We begin with the quantum groups aLq (N) and SL q (N). Let Z 0, and set R =-- zR. Our first aim is to show that

r,(731 ,uii ) = r z (uij ,Dq ) =

Z N qb 1,3• •

Z

E Cl (

17)

For this proof we use the L-functionals l c U:xt (s1N) defined in Subsect. 8.5.4 and the dual pairing (., .) - of the Hopf algebras Ur (s1N) and 0(SL q (N)) from Theorem 9.18. Since Dg = 1 in 0(SL q (N)) and E(l) = bij , we have = Recall from Subsects. 8.4.2 and 8.5.3 that q-1 /NR is the Rmatrix for the vector representation of Urt (s1N). Using these facts, the first equality of (9.21) and the formulas (8.89) and (9.40), we get

340

10. Coquasitriangularity and Crossed Product Constructions z N q6 ii =

E (_01(p)-i(r) zN \

zN

up(1) r(1)

up(N) \ r(N)

PE 1'N

,p( i ) •..p(N ) i E (-0/(P)-1(r)ZN(R1,N4-1R2,N+1 • " RN,N+1 )7-(1)...r(N)j

(18)

PE PN

for any permutation r E PN. Applying the formulas (3) and (11) a similar computation for the term rz (Dq , uii ) leads to the expression (18) as well. This = z N q8ii . Using the relation R m proves that r z (Dq , ki n = Rink' (which holds Dg ) is by (9.13)) and formula (4) instead of (3), we easily derive that equal to (18) with upper indices p(1) - • p(N)i and lower indices r(1) • - r(N)j interchanged. The latter expression is also obtained if we compute the term zNoii = z N Do by means of the second equality of (9.21). Therefore, we have rz (uii , Dg ) = z N q8ii . This completes the proof of (17). Let An (R) be the linear span of monomials n 31 i.1 . By (3) and (4),

(Ito

equation (17) implies that

r z (Dq a,b) = (z N q)flr z (a,b) and r z (b,Dq a) = (z N q)flr z (b, a)

for a E .A(R) and b E ..47,(R). Therefore, by (9.26), we extend rz uniquely to 0(at g (N)) by setting rz ( D ic a,Dq-i b)

(zNokrn+in-FkiN rz(a, b), a E An(R),b c Am(R),k,1 E N.

For the quantum group S.L q (N) we assume that z N = q-1 . Then, we have that r z (Dg - 104) = r z (uii ,Dg - 1) =--- 0 by formula (17) and hence r z ((Dq -1), A(R)) = r z (A(R),(D g -1)) = {0} by (3) and (4). Therefore, r z passes to the quotient 0(SL q (N)) A(R)/(D g - 1). Now we turn to the quantum groups 0q (N) and Spg (N) and suppose that z = ±1. Using the first (resp. second) expression for Qq in (9.37) and the second (resp. first) formula for R in (9.31), one immediately checks

that rz (uii , Q g ) --= (50 (resp. r z (Q g , uii ) = b ii ). Hence r z ((Q g - 1), A(R)) = rz (A(R), (Qq - 1)) = {0} and r, passes to the quotient Hopf algebra A(R)/ (Q g -1). By Lemma 9.12, the•Iatter is 0(0q (N)) resp. 0(Sp g (N)). The above reasoning can be repeated with R = .zR replaced by R-1 =-z -1 R -1 and r z by 1. (In the proof of (17) for F; one has to use 1 instead of V.) We then obtain a linear form Tz- on 0(G g ) C9(G g ) which is the convolution inverse of rz . Since r z is derived from a universal r-form of A(R), it is clearly a universal r-form of 0(G q ). Finally, we consider the universal r-forms rz from Theorem 9 for the corresponding real forms of the quantum groups (see Subsects. 9.2.4 and 9.3.5). Proposition 10. (1) If the numbers q and z are real, the universal rforms rz of the coordinate Hopf *-algebras of Ug (N),SUg (N,v),O g (N; v) and Spg (N;y) are real.

341

10.1 Coquasitriangular Hopf Algebras

(ii) If Iql = izi = 1, the universal r-forms r z of the coordinate HoPf *algebras of aL q (N;R), SL q (N; R), 0q (n,n),0q (n,n +1), and Sp q (N; R) are inverse real. Proof. We prove the assertion for 0q (N ; u) and Sp q (N; v) and use the corresponding notation from Subsect. 9.3.5, Case 2. The proofs for the other cases are similar. Let rz be as in Theorem 9. Since q E R, the matrix entries R i are real. By (9.30), they satisfy the equations vi vi vkv/R114= R = R. Thus, we obtain (r z

) i

(u 3i. 7 U 1k ) = rz(0/ /1 9 *

= ViVj VkVZ rz(S(4) 49(W1 ))

3 I 1 T z klik

•

) = v'v-v rz ki — zRik — 3 k vii R13

?L k )

That is, the universal r-forms r z and (rz ) i (by Proposition 6) are equal on 111 the generators. Hence they coincide everywhere. We close this subsection with an example concerning Proposition 3. Example 3 (Square of the antipode). Let A be an arbitrary coquasitriangular Hopf algebra with universal r form r and let u = (u i ,i = 1 ,...,d, d e N, be and E(U) = a matrix of elements uii E A such that L\(u) = uik j = 1, 2, • , d. In the notation of Proposition 3, we set -

Dii := f (uti ) r

:= Puii ) r(S 2 (4),u1D.

S(u)),

(19)

((D -1 )ii ) is indeed the inverse of the Since f f = ff = E l the matrix matrix D = (Di). Then Proposition 3 states that s 2 (uii)

= (D -i) niunmD:rin.

(20)

Now let A be one of the Hopf algebras 0(G q ) in Theorem 9 and let u=(u) be the fundamental corepresentation of A. By Propositions 9.10 and 9.13, we then have S2 (u) Set r := r z . By (17) and (18), we get (D -1 )ii = 74) = Ek ZDT 1 IpkWicii . From the latter and the formulas (9.13) and (9.30) one derives = =

= EqN_f Dic5

for G q GL q (N), SL q (N),

=

for G q = 0q (N), Sp q (N).

For the universal r-form r = (r z )21 of A we obtain

Di = Di = zeq 2Pi -N-Efbi

for G q aL q (N), S L q (N), for G q 0q (N), Spq(N).

342

10. Coquasitriangularity and Crossed Product Constructions

10.1.3 L-Functionals of Coquasitriangular Hopf Algebras

In this subsection we investigate some linear functionals on coquasitriangular bialgebras which are important tools for a number of applications. The constructions of bicovariant differential calculi in Sects. 14.5 and 14.6 and of quantum homogeneous spaces in Subsect. 11.6.6 will use these functionals. We suppose that A is a coquasitriangular bialgebra with universal r-form r. For a E A, we introduce linear functionals 1 1 (a), 1(a) and T(a) on A by

1+ (a) := r(., a), 1 - (a) := t(a, -), 1(a) := q(., a), i(a)

q(a, -),

(21)

where q is the bilinear form on A x A defined by (9). If A is a Hopf algebra, it follows easily from Proposition 2(v) that

1(a) = E S(1 - (a(1 )))1±(a(2) ) and r(a) = El+ (a(0 )S(1 - (a(2))).

(22)

Moreover, since q(S (a) S (b)) = q(b , a), the functionals 1(a) and 4a) are related by the formulas [(a) = S(1(S(a))) and 1(a) = Let Li denote the subalgebra of the algebra A' generated by the functionals 1±(a), a E A. If A is a Hopf algebra, then we have S(1±(a)) = 1±(S A-1 (a)) (by Proposition 11(ii) below) and hence 1(a) E Li and r(a) C Li for any a E A by (22). Proposition 11. (0 Li is a subbialgebra of the dual bialgebra A°. The maps 71-± : a —> 1± (a) are bialgebra homomorphisms of A° 13 to U. For a,b,c E A, we have 1 ± (ab)(c) = E 1 ± (b)(c(l))1 ± (a) (c(2)) (23)

A(1 ± (a)) = E

1±(a (1) )

/ ± (a(2))•

(24)

(ii) Suppose that A is a Hopf algebra. Then II is a Hopf subalgebra of A° with antipode determined by the relation S(1±(a)) = 1±(S: 4 1 (a)), a E A. Moreover, for all a,b E A and g E A° we have the identities

A(1(a)) =

/(a(2)) 0 Sr(a(1))) 1± (a(3)))

(25)

46(4a)) =

/±(a (i) )sula(3)))

(26)

4a(2))7

adR (g)(1(a)) = E 1 (a(2) ) g(a(1)S(a( 3))),

(27)

E t(a(2)) g(a(1)S(a(3))))

(28)

adL(g)(i(a)) =

1(a)1(b) = /(amb(2) *(b( i), S(a( 3)))*( 3), a( 2)),

(29)

T(a)l(b) = i(a(2)b(3))*(1), b(2))r(a(3), S(b(1))).

(30)

10.1 Coquasitriangular Hopf Algebras

343

Proof. (i): Since 1 - (a) (.) = f21(-1 a) and 121 is also a universal r-form on A by Proposition 2(iv), it suffices to prove the assertions for 1+ . By (3), we have 1+ (a)(bc) = r(bc, a) =

Er(b, am)r(c,a( 2)) = E 1+(a( i ))(b)1 + (a(2 ))(c).

By the definition of A° (see Subsect. 1.2.8), this implies that 1+ (a) E A° and hence 1,1 C A° . The preceding equation rewrites as (24) using the definition of the comultiplication of A° . By (24), 11 is a subbialgebra of A° . Formula (23) follows from (4). The equations (23) and (24) combined with the relations 1+(1) = e mean that the linear maps r-± are bialgebra homomorphisms of A°13 to U. (ii): The first assertion of (ii) follows at once from (i). Recall that the Hopf algebra A °P has the antipode Si'. Formula (25) is obtained from (22), (24) and the fact that S(1 - (a)) = 1(574 1 (a)). Next we prove (27). Using condition (5) twice, we compute

(1(a)g)(b) =

E q(b(i), a)g(b(2)) E r(a( i ),b(0)r(b( 2 ), a(2))9(1)(b(3)a(3))9(2)(S(a(4))) E r(a( i ), b(1))r(b(3) , a(3))9(1) (a(2)b(2) )9(2) (S(a(4))) Er(a (2) , b(2))r(b(3), a(3))9(1)(bwa(1))9(2)(S(a(4))) E /(a( 2 ) )(b(2))g( 1 ) Om )9(2) (a(i) S(a(3) ))

That is, we have

1(a)g =

E g( 1 )1(a( 2 )) g(2)(a(1)S(a(3)))

(31)

and hence

adR(g)(1(a)) =

E s(g (1) )/(a)g (2) = E S(91))9(2)/(a(2)) g( 3)(a( 1 )S(a (3) )) E1(a(2) ) g(a( 1 )S(a( 3))).

Now we verify equation (29). First we apply the QYBE r12r13r23 = r23r13r12 to x • y for ac, y E A. We then obtain the identity

E S(1 - (s(i)))r(s(2), y ( 1 ) )/+(Y(2) ) = E CE (Y(Or(x(i), Y(2))

(x(2))). (32)

Using formulas (22), (23), (5), (32) and (4) we compute

E

(amb(2))r(b( i ), S(a(3)))r(b(3) a(2)) = Er(b ( l ) , S(a(4)))8(1 - (a(1)))5V - (b(2)))[r(b(4), a(3)))1 + (a(2)b(3))] =

ow 7 5(a(4)))57(1- (a(1)))45(1 (b(2)))r(b(3), a(2)

a(3))

344

10. Coquasitriangularity and Crossed Product Constructions

= E x.0( 1 ), S(a(4)))Sr(a(1)))[S(1 - (b(2)))r(b(3),a(2))/ + (a(3))1/ + (b(4)) = E r(b( i), S(a( 4)))Sr(a( 1 )))/ + (a( 2))r(b( 2 ), a(3))Sr(b(3)))/ +(b(4)) E r(b( i ), a( 2)S(a (3)))/(a( i))/(b( 2)) = 1(a)1(b). The formulas for the functionals ï(a) are derived in a similar manner. Instead of (31) we use the equation gt(a) = E 4a(2))9(2) (S(a(o)a(3))• Equation (30) gives the motivation for the definition of the covariantized product and the transmutation theory of coquasitriangular Hopf algebras, see Proposition 34(ii) below. An immediate and perhaps surprising consequence of formulas (29) and (30) is Corollary 12. If A is a coquasitriangular Hopf algebra, then the sets 1(A) =

S(T(A)) and T(A) = 8 (1(A)) are subalgebras of the Hopf dual A°. In general the algebras 1(A) and i(A) 'are not Hopf subalgebras of A°. Let us recall the notation a = a - E(a)1. From the formulas (25) and (27) we easily obtain the following Corollary 13. Let V be a linear subspace of a coquasitriangular Hopf algebra A such that ,A (2) (V) C AOV A. Then, T(V) MO a E VI is an adR-

invariant linear subspace of A' satisfying X(1) = O and d(X) - e 0 X E T(V) 0 A° for X G T(V). We shall see in Subsect. 14.2.3 that (under some technical assumptions) the subspaces of A° with the above properties are just the quantum Lie algebras of bicovariant first order differential calculi on A. Next we consider *-structures. Recall from Corollary 1.11 that the algebra A° for a Hopf *-algebra A becomes a *-algebra with respect to the involution f f* (a) := f(S(a)*), a E A. Proposition 14. Suppose that A is a Hopf *-algebra.

(i) If the universal r-form r is real, then = /T(S -2 (a*))

and

1(a)* =-1(8(a)*).

(33)

(ii) If the universal r-form r is inverse real, then 1± (a)* = 1± (S -2 (a*)) and 1(a)* =

r(a*( 1 ),S (a* (3)))!(a*(a*(4))1(a* ( 2)) , (34)

where J is the linear functional from Proposition 3. Proof. As samples, we prove the formulas for 1+(a)* in (33) and for 1(a)* in (34). Using the assumption that r is real and Proposition 2(v), we get l+(a)* (b) = 1+(a) (S

= r(8 (b)* , a) = r(at, S(b))

= r(8-2 (a*), S-1 (b)) = 1(S-2 (a*), b) =

10.1 Coquasitriangular Hopf Algebras

345

The second proof will be only sketched. Let b, C E A. Using the first formula of (34) and the relation * o S -1 = S o* (see (1.39)), we compute

S(1 - (b))* = (S -1 (b))* = (8 -2 (8 -1 (b)*)) = 118 -2 (8(b*))) =

(b*))

and hence

1(b)* = (E 8(1 - 0(1)))1 + 0(2))) *

/±(s-2(b* (2) ))sq--(b* (1) )).

(35)

Farther, by Propositions 3 and 2(v), we have

b = E Ab (i) )s2(b(2))f(b(3)) =

r(b(i) 51-2 0(2) ) 15-2 (b(3) )f ( b(4) ) • (36)

Using the formulas (22), (36), (32), (35) and (7) in this order, we get

1(c)

=

= E

(c(i) ))*(2) , 8 - 2(c(3) ))/+(s - 2(c (4) ))f(c (5))

E 1±(s-2(c (3) ))r(c (1) , 8-2(c(4) ))sr (c(2)))f(c(5))

= = E r(, ) , coof(c(3))1((c(2))*)*• Inserting the latter formula into the expression E r (c(1), S(c(3)))/(c(4))/(c(2) ), it simplifies to 1(e*)*. Setting e = a*, this gives the assertion. Now let u=(ziii )i,j-1,...4 be a fixed matrix corepresentation of A. Setting = r(uii ,unm )

(37)

i ) given by we obtain a matrix R = (R) with inverse R -1 = ((R -1 ) A = t(uji , uT,'„ ).

(R -1

Comparing (37) and (6) we recognize .k = TOR as the matrix of the map rvv, when V is the right A-comodule Cd with coaction determined by the matrix corepresentation u = (uji ). Therefore, by Proposition 1(iii), it satisfies the braid relation and so R fulfills the QYBE. From Proposition 1(i) we obtain ut " klu'n rn

We define dxd matrices LI = A by = l(u), 1 = The linear functionals

15i on A

pplk 4 -nm 7 - 1

i,j=1,•••,d.

[= (4)

of functionals on

ti = 1(u)), i. =

(39)

L = (11i ) and

1- 4), (

(38)

are called L-functionals associated with the matrix corepresentation u. Since A(uii ) =utOull and 6(14) = 6ij , (24) and

Proposition 2(ü) imply that

346

10. Coquasitriangularity and Crossed Product Constructions

Li (1 ii ) = IV 0 1 ±k and

(40)

c(iti ) = 6ii .

The latter relations mean that the mappings LI(.) : A > Md(C) are algebra homomorphisms. Further, if A is a Hopf algebra, then the matrices L and E are just the matrix products L = S(L)L+ and E = L -Es(L - ). From the corresponding definitions we see that the evaluation of the functionals /ti 3 at the matrix entries unm, are given by —

ii.(un ) = (142)rm. iti (li n ) = Rni . / -:i ( Un ) =-(R -1 ) in / i(m nti ) = (fer i im) 3 my 3‘m ‘ 3m m3 7 3 m .

In matrix notation these formulas can be expressed as L(u2 ) = R21,

1-1- (I-12) = Ri-21 1 Li(u2) ----- (M)21, E1(u2) =

(j )12. (41) 42

Proposition 15. The matrices LI satisfy the relations LiF 14R =- RL -21- LiF, LT L2- R = R1_ 2- LT, L i- L-2F R = RL-2F L 1- .

(42)

Proof. Using the formulas (21), (4) and (38) we conclude that

(/ +j/+ kl k I i )( a)R nm

=E =

nkini

E r(a( i ), ujk )r(a( 2),W)Rnkim

= r(a,Wujk )Rtim = kit *, unk ulm ) . Rikil l:.il le (a), that is, LiEL -2FR = RL -2F Lt. Replacing r by the universal r-form 1 21, the same reasoning shows that LT L2- 17 = RL2- LT . The QYBE for r (by Proposition 2(i)) implies that f 12 r23 r 13 = r13 r23f- 12 • Applying this equation to uni 0a0u-In and using formula (37) we obtain

E 17 (4, a( 1 ))r(a( 2),uji )Rnklm = E Rr(a( i ),uim )i(u nk ,a( 2)). This means that rkii+iiRnkim = R i i+mi rnk . Thus, L i- L -2F R = RL-2I- L i- .

0

Proposition 16. Suppose that A is a coquasitriangular Hopf algebra and is a matrix corepresentation of A. (0 The matrices L = S(L)L+ and I = L+ S(L- ) defined above satisfy the reflection equations (43) L2R 1 21_1R21 = R12 1-1 R2 1 1-2)

[ 2R21 1--1R12 = R21r-1R12 1 2-

(44)

(ii) The elements Tr Lk D-1 , k E N, belong to the center of A°, where D -1 is the matrix given by (19) and L = S(L1L+. Proof. (1): By the third equality of (42), we have LiER1 2,5(Ln .= S(Ln.R1214. Using this relation and the first equality of (42) we get

10.1 Coquasitriangular Hopf Algebras

347

L2142L 1 = S(LnI4R12S(LT)LiE = S(L 2-- )S(LT).R12L -21- Lii= S(Li- L 2- )1414R 1 2.

(45)

Interchanging the lower indices yields L 1 R21 L2 =

)L -2E 1-jR21•

(46)

Multiplying (46) by R12 on the left and using the first two relations of (42) we obtain R12 Ll R21 I-2 := S( LT L2- Lt 14Ri2R21. By (45), the right hand side is equal to L2R12L1 R21. Thus we obtain formula (43). The proof of the second reflection equation (44) is similar. (ii): From formula (31) we derive by induction on k the identity 1(a 1 ) • • • 1(ak )g = E g(1) 1(a (' 2) ) • . • /(alt2) )g( 2) (ati) S(at3) ) • • • a/4 ) S(4) ))

for arbitrary elements al, • • , ak E A and g E A°. Using this equation and -1 )ii = (D -1 )r-n "g-1 (uji ) by (20), we obtain the relation S(urri)(D 3 i (L kp -iyig

1 Zi2 l i 32 ..

- 1) 3i

E g( 07,12 4,-23 • • . l g(2) (uin S(u in:)uS(u in33 ) .. = E g(1) l i 2 iz . • • l g(2)(uS(uT)(D -1 )3i. ) Eg(i)(L k )rn , g(2)( ,_, (u3i s(u)))(D-1) g

j (LIC

1 )i3

El

We now assume that A is a Hopf *-algebra. We shall express the conditions in Proposition 14 in terms of the functionals / ±ii and l. First suppose that r is real and (7.L i )* = S uii ) for i,j = 1,2, • • • d. Since then S-2 ((4.)*) = S -1 (trii ) and S(74)* =74, the equations (33) yield (

(1±11)* = S(i i')

and

(13i.)*

(47)

Next we assume that r is inverse real and (uii )* = uji for i, j = 1,2, • • • , d. By un m (D-1 7 and (D -1 ) 3i. (20), we have S-2 04) Du(D1 )T , where the numbers are given by (19). Inserting this into (34) and setting fliknin := r(4, Sei4.0), we get (48 ) (111) * = D inl± „?(D7 1 )T and = fek7n(D-1)jmink It might be necessary to emphasize that the matrices L±, L and of functionals depend on the choices of the universal r-form r and of the matrix corepresentation u of A. If we take the universal r-form t 21 (by Proposition

348

10. Coquasitriangularity and Crossed Product Constructions

2(iv)) instead of r, then 1+1 and l j interchange. Further, if A is a coquasitriangular Hopf algebra and u is replaced by y = (vi) with v := then the representations LI of A go into the contragredient representations L, where (LIfl i := S(/-1 ). Let us specialize the preceding to coquasitriangular FRT bialgebras. Suppose that R is an invertible solution of the QYBE. Then, by Theorem 7, the FRT bialgebra A(R) is coquasitriangular and its canonical r-form r of A(R) ) denotes the fundamental corepresentasatisfies equation (37), when u = tion of A(R). Thus, all the above facts are valid in this case. Let U(R) denote the bialgebra U defined above for A = A(R). That is, U(R) is the subalgebra of A(R)° generated by the functionals l +3-i and 1 7 , j , j = 1, 2, - • • , N. Now let us suppose that Gq is one of the quantum groups GL q (N), SL q (N), 0 q (N) or Spq (N) and that A is the coquasitriangular Hopf algebra 0(G q ) with universal r-form r, (see Theorem 9) and fundamental corepresentation u = (uii ). Let U(G q ) denote the bialgebra U in the case A = 0(Gq ). By Proposition 11 (ii), U(G q ) is a Hopf algebra. All the above formulas remain valid with R = zR(q), where z and R(q) are as in Theorem 9. In addition, the L-functionals / ±ii associated with the fundamental corepresentation u of 0(G q ) have the following properties.

Proposition 17. (î) l = = 0, i > j, and l+kk l -: = E, k = 1, 2,• • - N. 00 1 +11 1 +22 .. .i+NN = E for Gq = SLq (N). ( iii) L+ ct (L+)t (c-i)t = = el for Gq = 0q (N), Sp q (N). Proof. (i ): Suppose that i > j. The matrices R given by (9.13) and (9.30) are lower triangular. Thus, by (41), 1(unm ) = zR -= 0 for all n, m = 1, 2, • - - , N. Since 1 ±ii (ab) = l(a)l(b) and l±ii (1) = 6ii by (40), we conclude that 1 +1 = 0 on the whole of A for Gq = SLq (N),0 q (N), Sp q (N). In the case of GL q (N) this is also true, since LI- (DV) is the inverse of the upper triangular matrix L±(Dq ). The proof of l7 = 0 is similar. Since el = 1T= as just shown, we have A(irii ) = irii 0/rii again by (40). Hence is a character of the algebra A. By (41), the characters 1ri i and E are equal on the generators and hence on A. (ii) follows from the fact that 1+11 • • • / 4-1 and E are both characters which coincide on the generators u m n by (41) and the assumption z N =q 1 . (One may also verify that E = 1 1 (;) 1111 • • • 1 ±:). (Hi) is obtained from (3) and the relations (9.34). 1=1

Let A be A(R) or 0(Gq ). Then the L-functionals l for the fundamental corepreseqation u of A are uniquely determined by the equations (40) and (41) or equivalently by the requirement that the matrices LI = (1 ±1) are algebra homomorphisms of A to MN(C) satisfying (41). For most applications this characterization of the functionals i is sufficient to work with. The theory of L-functionaLs for 0(Gq) developed above is closely related to that for Uh(g) and Uq (g) in Sect. 8.5. Both L-functionals have similar

10.2 Crossed Product Constructions of Hopf Algebras

349

properties (see, Propositions 17 and 8.28 and formulas (42), (40), (41) and (8.83), (8.84), (8.89), (9.40), respectively) and even some proofs are analogous (compare, for instance, the proofs of Propositions 15 and 17 with those of 1,i(G q ) are Propositions 8.27 and 8.28). However, the L-functionals l defined as linear functionals on 0(G q ), while the L-functionals iti and iti in 8.5 belong to the abstract algebras Uh(g), Ut( g ) and UqL (g), respectively. By Propositions 15 and 17 and formulas (40), there is a surjective = Hopf algebra homomorphism 9 : UqL (2) ti(G q ) such that VW; and V(e7) = l , i < j, where g --= glN , slN, soN , sp N and G q = 3 GL q (N),SL q (N),0q (N),Sp q (N), respectively. Likewise, there is a Hopf algebra homomorphism 0 of the algebra Urt (g) generated by l , Ç onto U(G q ) which maps /ti to rEji and /7,i, to l , i < j. In general, both homomorphisms V and 0 are not injective. (For instance, if g = s12 and q is a root of unity, one can verify that (/ ±i l ) = E in U(SL q (2)) for some n c N.) Obviously, if we identify Ugext (g) and U(g) by the isomorphism from Theorem 8.33, V and 0 coincide.

10.2 Crossed Product Constructions of Hopf Algebras Crossed products or semidirect products are useful tools for the study of covariance problems and for the construction of new objects from old ones. Most of such constructions occurring in the literature can be thought of as generalizations or combinations of two fundamental concepts developed in this section: crossed product algebras or coalgebras and twisted cocycle algebras or coalgebras. In the course of this the quantum double will be treated as a guiding example. If not specified otherwise, A and B are bialgebras in this section. 10.2.1 Crossed Product Algebras Recall from Subsect. 1.3.3 that an algebra X is a left A-module algebra if X is a left A-module (that is, there is a bilinear map a : Ax X 3 (a, z) at>s c X satisfying al> (bc. x) (ab)i> x and 1 i> x = x for a, b c A, s c X) such that

a (xy) =

(a(1) x)(a( 2)› y), a >1 = E(a)1.

(49)

Proposition 18. Let X be a left A-module algebra. Then the vector space X OA is a (unital associative) algebra, called the left crossed product algebra and denoted by X >l a A or simply X >1 A, with multiplication defined by

(ac a)(y b) =

E s(a( i ) t> y) 0 a( 2)b,

c, y c X, a, b E A.

(50)

Proof. It suffices to show that this product is associative. Using the formulas (49) and (50) and the left A-module property of X we obtain

350

10. Coquasitriangularity and Crossed Product Constructions

(x ®a)((y ®b)(z c)) =

Es(a() t (Y(b (i) z))) a( 2)b( 2)c x(a(i) r> Y)((a(2)b(1)) z)

a(3)b(2)c

> y) a( 2)b) (z e) ((x

a)(y

b))(z

111

c).

Clearly,themapsX s-4s01 EXmAandADa-4 10aEXA A are injective algebra homomorphisms. Therefore, by identifying x with x 0 1 and a with 1 0 a, we can consider X and A as subalgebras of the algebra X >i A. Then definition (50) of the product yields the commutation relations ax =

(a(1) x)a( 2) ,

a

E

A, s E X.

(51)

The algebra X )41 A may be thought of as the universal algebra generated by the algebras X and A with respect to the commutation relations (51). If cp is a representation of the algelSra X >1 A on a vector space V, it follows at once from (51) that the restrictions (p x :=-- (prey and (to A := [ A are representations of X and A on V such that the compatibility condition (pA(a)c,ox(x) =

E ,(a (1) x)c,oA(a( 2)),

a E A, x E X,

holds. Conversely, if cpx and cpA are representations of X and A on the same vector space V satisfying the latter condition, then the equation (p(ax) = cpA(a)9o x (x) defines unambiguously a representation of X >1 A on V. For instance, by (49), the representations cp x (x)y = sy and A(a)y = a > y of X and A on V = X have this property. In this case the corresponding representation cp of X xi A on X acts as (p(xa)y = x(a> y), a E A, s, y E X. It is called the Heisenberg representation of the crossed product algebra X xi A. Before we discuss a number of examples, we briefly mention the corresponding right handed version A x X. Let X be a right A-module algebra, that is, X is an algebra equipped with a bilinear map a : XxA (x, a) —3 xaa E X such that (x a a) ab=sa (ab), s <1 =s, (xy) a a= E (s a( 1 ))(y zi a( 2)) and 1 a a = e(a)1 for a, b E A and x, y E X. Then the vector space A X is an algebra with product defined by the fofinula (a 0 x)(b y) =

ab( i ) (s b( 2 ))y, a, b c A, x,y E X.

This algebra is the right crossed product algebra and denoted by A, x X or simply A x X.

Remark 3. In the Hopf algebra literature (see, for instance, [Swl or [Mon], 4.1.1 and 10.6.1) crossed product algebras and coalgebras are usually called smash products and coproducts, respectively.

Example 4. Any algebra X is a left A-module algebra with respect to the trivial action a t s = E(a)s. In this case X >i A is the "ordinary" tensor product algebra X 0 A with product (x

a) (y 0 b) = xy 0 ab.

A

10.2 Crossed Product Constructions of Hopf Algebras

351

Example 5. Each Hopf algebra A is a left A-module algebra with respect to the left adjoint action a b adL(a)b = E ambS(a(2)). Then the corresponding left crossed product algebra A xi ad L A has the product (a

b)(c 0 d) = E ab( i)eS (b( 2)) b( 3)d,

a, b, e, d c A.

If A is quasitriangular, then the algebra A x ad i, A is even a Hopf algebra for d some appropriate coproduct, see Example 25 in Subsect. 10.3.6 below.

Example 6. Let A be the universal enveloping algebra U(g) of a Lie algebra g, see Example 1.6. Suppose that 7r is a homomorphism of the Lie algebra g to the Lie algebra of smooth vector fields on a, C'-manifold M. (An important special case is when M is a Lie group G with Lie algebra g and the elements of g act as left invariant vector fields on G.) Then 7r extends uniquely to an algebra homomorphism of U (g). For simplicity let us assume that this extension is injective and identify A = U(g) with its image. Then the algebra X = Ccic (M) (or any other subalgebra which is invariant under the action of g) is a left A-module algebra. Indeed, it suffices to check (49) for elements a of g. Since Z1(a) = a 1 +1 Oa, the first equation of (49) reduces to the Leibniz rule which holds because a c g acts as a vector field by assumption. In this case the elements of the crossed product algebra X A act as differential operators on M with coefficients in X. Let us specialize to the case where g = 111n, M = R X = C[Xl, ' ' • xn] and 7r(e) = 0/a, i = 1, 2, - • • , n, for some basis fed of g. Then X >4 A is just the Weyl algebra and its action on Rn is the realization as the algebra ,

of differential operators with polynomial coefficients.

Example 7. Let G and H be groups such that H acts on G by group automorphisms, that is, h c:;. (gg') = (h g)(h g') and 1 > g = g. The semidirect product G >i H of G and H is the set Gx H with group operation (g, h)(g' , h') := (9 (to. g'), hh'). Extending the actions of H and G by linearity to the group algebras X = CG and A = CH, X becomes a left A-module algebra and the crossed product algebra X xi A is just the group algebra C(G >1 H) of the semidirect product G >4 H. Important crossed product algebras are the Heisenberg doubles. Let us suppose that (•, •) is a dual pairing of two bialgebras B and A. Define

b>a := E(b,a(2) )a(i) ,

a E A, b c B.

(52)

Using the properties of the pairing (1.31) and (1.32) it is easy to check that the map (b, a) b > a defines an action of B on A such that A is a left Bmodule algebra. The corresponding crossed product algebra A >1 B is called the Heisenberg double of the pair A, B and denoted by H (A, 8). When the bialgebra B is the dual A° of A, then H(A) := B) is said to be the Heisenberg double of A. Inserting the definition (52) of the action into (50), we get the following expression for the product of the algebra .1/(A, B):

352

10. Coquasitriangularity and Crossed Product Constructions

(a 0 b)(a'

(b(i) , a'(2) )ad 1)

b')

b( 2)b',

a, a' E A, b, b' E B.

We consider this general construction in two examples. Example 8. Let G be a Lie group with Lie algebra g, B = U(g) and let A be a bialgebra of C°°-functions on G. It is well-known that any b E B acts as a left invariant differential operator b on G. The evaluation of the function 6(a), a E A, at the unit element of G defines a dual pairing of the bialgebras B and A (see Example 1.6). The corresponding Heisenberg double 11(A, 13) is an abstract algebra of differential operators Ei ab, ai E A, b, E B, and the Heisenberg representation of H(A, 13) ---- A >a B describes the action Ei ab of these operators on functions of A. That is, the elements of B and A act as differential operators and multiplication operators on G, respectively. This picture is the reason for the name "Heisenberg double" . It also motivates the interpretation of (52) for a general Heisenberg double as the action of the A "generalized left derivation" b E B on the "function" a E A. Example 9. Let R be an invertible solution of the QYBE and let A(R) and it(R) be the corresponding FRT bialgebras defined in Sects. 9.1 and 9.4. By Proposition 9.20, these bialgebras are dually paired with pairing described by (9.44). Suppose that A is a quotient bialgebra of A(R) and 13 is a quotient bialgebra of .11(R) such that the pairing of it(R) and A(R) passes to the quotients. Then, by (9.44), the actions of the generators tti of B on the generators unn, of A are written in matrix form as ziF u2 = u2 R21 and ° u2 = 112Ri21 . Hence the commutation relations (51) between both sets of generators in the algebra II (A, B) are = u2R21eil- and

,C i- u2 = u2 R1-21 ,C IT.

(53)

That is, the Heisenberg double I/(A, B) has 3N2 generators ej, u j j, n, m = 1, 2, • • • , N. The defining relations are those of A and B and the cross relations (53). A ,

,

10.2.2 Crossed Coproduct Coalgekoras

This subsection is concerned with the dual notion to crossed product algebras. A coalgebra X is called a right A-cornodule coalgebra if X is a right Acomodule such that the comultiplication A x and the counit Ex are comodule maps. The latter means that

E(x (0) )(1)

(x(0))(2)

s( i ) _ E(x(1)) (0)

(x(2))( 0)

( x( 1) )(1) ( x(2) ) (1)

= ex (x)i.

Here and in the following we use the Sweedler notation O(x) = E so) ® s(1), Ax(x) = s(i) x(2) and 46.4(a) = E a(1) a(2), where : X X A is the coaction of A on X.

10.2 Crossed Product Constructions of Hopf Algebras

353

Proposition 19. Let X be a right A-comodule coalgebra. Then the vector space .4 0 X becomes a coalgebra, called the right crossed coproduct coalgebra and denoted by Af3 D< X or A X X, with comultiplication and counit given by ZA(a 0 x) = E a(1) 0 (x(1)) (°) 0 a(2)(x(0) (1) 0 x(2), E(a 0 x) = EA(a)Ex(x),

(54)

a E A, x E X.

Proof. The proof of the coassociativity of ZA is an advanced exercise in the D use of the Sweedler notation. We omit the details, see [Mon], 10.6.3. Let us turn to the corresponding left handed notion. A left A-comodule coalgebra is a coalgebra X which is a left A-comodule such that Zl x and Ex are comodule maps, that is,

® (x(0))(1) ® (x( 0) )(2) _ E(x(i)) " )(x(2))( _1 ) 0 (x(1))( 0) 0 (x(2))( 0) , = Ex(x)i l where we write fi(x) = E x (-1) 0x (°) for the coaction ,(3 : X —* A® X. For such a left A-comodule coalgebra X, the vector space X 0 A becomes a coalgebra with comultiplication and counit defined by Z1(x 0 a) ---=

Ex°, 0 (x( 2)) (-1) a( 1) 0 (x (2)) (°) 0

E(x g a) =-- Ex (x)E-A(a), x E X, a E A.

This coalgebra is called the left crossed coproduct coalgebra. It is denoted by X >03 A or X >1 A. Example 10. Each coalgebra X is a right A-comodule coalgebra with respect to the trivial coaction f3(x) = x 0 1. Then A)3 Ix X is just the tensor product coalgebra A ® X with coproduct d(a 0 x) --= E a( i) ®X (l) øa(2) ø x(2 ). A Example 11. The right adjoint coaction AdR(a) = E a( 2) 0 S(a( 1 ))a( 3) turns each Hopf algebra A into a right A-comodule coalgebra. Then the coproduct of AAd R ix A is given by 46(a 0 x) = E a(i) o x (2) 0 a(2)S(x( 1))x(3) 0 For a coquasitriangular Hopf algebra A, we shall show in Subsect. 10.3.6 (see Example 24) that the coalgebra A AdRix A equipped with a suitable product becomes a Hopf algebra. A

10. Coquasitriangularity and Crossed Product Constructions

354

10.2.3 Twisting of Algebra Structures by 2-Cocycles and Quantum Doubles In this subsection we construct new bialgebras from old ones by twisting the product by 2-cocycles and keeping the coproduct unchanged. Definition 3. A bilinear mapping y : Ax A C is said to be a left 2 cocycle on A (with values in C) if it satisfies the condition -

E 7(a( 1 ) b( 1))7(a( 2)b(2),e) = E ry(b(i) , c( 1 ))7(a,b(2)c( 2))

(55)

for a,b,c e A. It is called a right 2 cocycle on A if it satisfies the equation -

E 7(amb( i),c)7(a( 2), b( 2)) =

y (a, b( i)c( i))1 /0( 2) , c( 2))

(56)

for a,b,c E A. The mapping y is said to be unital if y(a,l) = y(1, a) --= a e A.

: A x C is a bilinear mapping which is convolution invertible, then (55) for 7 is equivalent to (56) for its inverse -7. That is, 7 is a left 2-cocycle on A if and only if -7 is a right 2-cocycle on A. If 7

Example 12. Any universal r-form r of a bialgebra A is a imitai left 2-cocycle on A and its inverse 1 is a unital right 2-cocycle on A. Let us verify (55) for r. Indeed, if we apply (3) to the term r(a(2)b(2), c) and (4) to r(a,b(2)c(2))) then (55) reduces to the QYBE which holds by Proposition 2(i). Thus r is a left 2-cocycle. By the preceding remark, 1 is a right 2-cocycle. A Proposition 20. (i) Let y be a unital left 2-cocycle on a bialgebra A which

is convolution invertible. Let y be its inverse. Define a new product on A by

a...y b := E y(a(1) ,b(1) )a(2) b(2)7(a(3) ,b(3) ),

a, b E A.

(57)

Then the vector space A equipped with the product . 1, and with the coalgebra structure of A becomes a bialgebra. It is denoted A(7) and called the twist of A by the cocycle 7. (ii) If A is coquasitriangular with universal r-form r, then ri.), := 721r7 and j; are universal r-forms of the bialgebra AN. In particular, A(y) is also coquasitriangular. (iii) If A is a Hopf algebra, then so is .A(7) with antipode given by SA(.. y )(a) = S(a( 2))) and f (a(OSA(a(2))7(a(3)), where f(a) := J(a) := E-7(S(a( 1)),a( 2) ), a E A. Moreover, rf- = Jf = e in A'.

Proof. All assertions follow by direct verifications of the corresponding axioms. As an example, we prove the associativity of the product Inserting the definition (57) of the product ••y we obtain

10.2 Crossed Product Constructions of Hopf Algebras

E

355

b( 1 ))-0( 2)b( 2), c( o )a( 3)b(3)c(2) x Py(a(4)b(4), c(3))e7(a(5), b(5) ) 7

Eey (b(1) , C(1))7(a(1) b(2)c(2) )a(2)b(3)c(3) x 77(a(3) 7 b(4)c(4) )70(5) 1 c(5)).

Since -y is a left 2-cocycle,'7 is a right 2-cocycle. Using (55) for -y and (56) for it follows that the right hand sides of the preceding relations coincide. Hence the product • is associative. For the last assertion, see also the proof of Proposition 3. CI

Example 12 (Continued). If the left 2-cocycle ey in Proposition 20 is the universal r-form of a coquasitriangular bialgebra, then, comparing (57) and (2), A we see that A(7) is just the opposite bialgebra A°I3 Let us retain the assumptions and the notation of Proposition 20. Then the twisting procedure can be extended to A-comodule algebras and to Acomodule coalgebras as well. Let X be a left A-comodule algebra. Define a new product • by x-7 .y =

-Y(X ( -1 ) 7 Y(-1))X(0)Y(0),

XI

Y E

X.

The product • makes the vector space X into an algebra / X which is a left comodule algebra for the twisted bialgebra A(-y). (To prove the associativity of the product we note that the scalar factors appearing in the expressions (x l .y).y .z and x-y -(y l -z) are just the two sides of left cocycle condition (55). Since 7 is unital, the unit of the algebra X is also a unit for the product „y •.) Similarly, a right A-comodule algebra X is a right A(7)-comodu1e algebra with respect to the new product defined by X2J = Es (o) yor-y(s( 1)1y( 1) ),

z, y E X.

If X is a right A-comodule coalgebra, then

,6 -y(x) = E(x( )(0) ® (x(2 (0)7( ( x (1) )

))

)

(I)

(x(2))(1)),

X E

X,

gives a new coproduct 461 on the vector space X such that X becomes a right A(-y)-comodule coalgebra. Next let us consider cohomologous cocycles. Let u be an invertible element of the dual bialgebra A° with inverse 77,. If 7 is a left 2-cocycle on A, then one verifies that -y' :=ri,)-y(u), that is, (a,b) = Eri(a (o )ri(b(1) )-y(a (2) , b(2))u(a( 3)b(3)),

a, b E

A,

is also a left 2-cocycle on A. Then we say that 7 and 7' are cohomologous. A left 2-cocycle -y is called a coboundary if 7 is cohomologous to the trivial

356

10. Coquasitriangularity and Crossed Product Constructions

cocycle 70(a, b) = c(a)c(b), that is, if 7 is of the form (77/ 0 Ti)70,6(u) for some convolution invertible element u E .

Proposition 21. If 'y and 1, are two cohomologous invertible unital left 2cocycles on A with 0 17)74(u), then the map 0 -= u * id * i (that is, 0(a) = u(a(1))a(2)%a(3))) is an isomorphism of the twisted bialgebras A(7) and A(Y). In particular, if if is a coboundary, then the twisted bialgebra A(7) is isomorphic to A.

E

Proof. Using the relations

(u0u)ey' -=

7d(u) and 7' ( 7, 0 'FL) = A(Tiry, we get

0(a). 0(b) = (Eu(a(1 ))a(2)%a(3)))

(Eu(b(o)b( 2)u(b(3)))

E u(a ( ou(b( 1 ) ),,f(a ( 2) , b(2))a(3)b(3)7i (a(4), xri(a( 5) )7EL(b( 5) ) =

h )11(

h

1

h

h 57(

,, (1),. ‘ a(2)-(2),a(3)-(3).,a(4)-(4),

E*7(a ( i ) , b(1)) 0 (a(2)b(2))Va(3) b (3)

17-v( 7 -(5), h ,a(5)

0 ((a'7b))•

It is easy to check that 0 preserves the coalgebra structure as well. We conclude this subsection by showing how the quantum double (see Subsect. 8.2.1) can be obtained by the twisting procedure. Proposition 22. Let A and B be bialgebras which admit a skew-pairing : AX B C. Suppose that a has a convolution inverse a. We define bilinear mappings : B A x 13 0 A —+ C and 5, :A0BxA0B —+ C by

(b 0 a, b'

a') = (b)Tr (a, b')E(a' ) and '5'(a

b, a' 0 b') = E (a)a (a' , b)e

.

Then 7 and '5/ are invertible unital left 2-cocycles on the tensor product bialgebras BOA and A0B, respectively. The corresponding twisted bialgebras are the quantum doubles D(A,B; a) and 1 3(13,A; 721), respectively. That is,

B; a) = (B A)(y)

and

1)(13, A;

=

(A 0 1 ) ( 1).

Proof. We carry out the proof for ey. The assertion concerning follows then by interchanging the role of A and B and of a and 7721. The left 2-cocycle condition (55) for 7 is equivalent to the equality E (a( i), bi )Tr(a( 2 )d, b)

b(1))5(al Vb(2))-

Therefore, since 5-21 is an invertible skew-pairing of B and A, -y is an invertible unital left 2-cocycle on BOA. Comparing the formulas (8.21) and (57) we see that the' products of the algebras D(A, B; a) and (B A)(7) on the vector space B 0 A coincide. By definition, the coalgebra structures are the same. Corollary 23. If A and B are coquasitriangular bialgebras with universal r-forms rA and r 8 , then the quantum double 13(A, B; a) is coquasitriangular with universal r-form i = (741(r13)13(rA)24(723, that is,

10.2

Crossed Product Constructions of Hopf Algebras

357

f(boa,bioal).E er- (a (1), b(o)r8(b(2)7/1(1))rA(a(1), a/(2) )17 (a(2) N2)). Proof Obviously, r = (1.8) 13(rA)24 is a universal r-form for the bialgebra B A. Since D(A, 8; cr) = (B A)(Py), i = ty = 7201', = d41(r8)13(rA)244 723 I=1 is a universal r-form of D (A , B; cr) by Proposition 20(4 Example 13. Let A be either the FRT bialgebra A(R) for some invertible solution of the QYBE or the coordinate Hopf algebra 0(G q ) for Gq = GL q (N), SL q (N),0q (N), Spq (N). By Theorems 7 and 9, A is coquasitriangular with universal r-form r such that r(ui,u2) = R. Therefore, by Corollary 23, the quantum double D(A, A; r) with respect to the skew-pairing r of A and A admits a universal r-form I such that T(111 0 112 1 113 0 114) = R4-11 R13R24R23.

(58)

Since 17.2 1 is a universal r-form of A as well, we also get three other universal r-forms of D (A , A; r) in this manner. They are obtained if the term R13R24 in (58) is replaced by R3-11- R 24 R13R4-21 and R3-11 R4-21 , respectively. Remark 4. There is a more general concept of crossed product algebras that contains the constructions from Propositions 17 and 19 as special cases. Suppressing technical details, it is obtained if the action of the Hopf algebra A on X. the A-module algebra X is twisted by an X-valued cocycle a : A A

Such crossed products have been introduced and studied in [DTI.] and [B CM], see also [Mon], Chap. 7. 10.2.4 Twisting of Coalgebra Structures by 2-Cocycles and Quantum Codoubles

In this subsection we dualize the main considerations of the preceding subsection. That is, we construct new bialgebras by twisting the comultiplication and preserving the multiplication. Definition 4. An element yEAOA is called a 2-cocycle for A if "Y12 • ( 2i 0 id)(PY) = PY23 •

The element y is called counital if (E it has an inverse, denoted by Py -1 , in

(id 0 2i) ("Y)•

(59)

(id 0 E)Fy = 1 and invertible if the algebra A&A. id) -y =--

Example 14. The universal R-matrix R, of a quasitriangular bialgebra A is an invertible counital 2-cocycle for A. (Formula (8.7) means that R., is counital and (8.2) and (8.6) imply (59).)

The next proposition is the counterpart to Proposition 20. Proposition 24. (i ) Let y be an invertible counital 2-cocycle for a bialgebra A. The algebra A equipped with the comultiplication ZI- f (a) := Fy.ZIA(a)-7-1,

10. Coquasitriangularity and Crossed Product Constructions

358

a E A, and the counit EA becomes a bialgebra, denoted by A(-y) and called the twist of A by the cocycle (ii) If A is quasitriangular with universal R-matrix R, then Ri : 721R-7 -1 is a universal R-matrix of A(7) and so A(7) is also quasitriangular.

(Hi) If A is a Hopf algebra, then the bialgebra A(7) is also a Hopf algebra with antipode given by Sy (a)=v5A(a)v' , where v := mA(id SA )( )) E A and v -1 := mA(SA id)(y -1 ) E A is the inverse of v in the algebra A. Proof. We only sketch the proof of the coassociativity of A-y . From the defi-

nition of

2i-r

we obtain

id),y (a)

(id 0

)

(a) =

VY12(ZA 0 id)(y)] •

id)f(a)] -

id)(y -1 )yi-21 ],

[y23(id A)(-y)] • [(id 0 21)2i(a)1 • [(id 0 2i)(y -1 ) -yÇ31 ].

The corresponding expressions in the squared brackets of both equations coincide, the first and the third ones by the cocycle condition (59) and the middle ones by the coassociativity of = AA. Remark 5. If yE A0A does not fulfill condition (59), then the comultiplication Ay = y • ZAA • y-1 is no longer coassociative in general. However, P := y12 • ( 2i id)(7) • (id 0 20(7 -1 ) 72-31

1

is an invertible element of A 0 A 0 A such that (id 0 2i-y )A,r (a) =--- 0 -1 • ((Ay id)tly (a)) • 0. That is, Li.1 is coassociative up to conjugation by 0. Such a weakening of the coassociativity axiom leads to the notion of a quasi-Hopf algebra, see [Dr51. Note that (59) implies that 0 = 1 and hence that Zly is coassociative, but the latter property is weaker than (59) in general. Our next aim is to define a dual notion to the quantum double. In order to do so, we first dualize the notion of a skew-pairing (see Definition 8.3). Definition 5. A skew-copairing of tWo bialgebras A and B is an element e A0B such that ( 6

A 0 id)cr

17

(id 0 ez3 )o- =1,

(60)

(61) (AA 0 id)(u) = 0130237 (id 0 218)(0) = 0'130'12. Note that if a e A 0 B is invertible, then the two equations (61) already imply (60 ) (see the proof of formula (8.7) in Subsect. 8.1.1).

Example 15. The universal R-matrix R. of a quasitriangular bialgebra A is a skew-copairing of A and A itself (see (8.7) and (8.2)).

A

Proposition 25. Let a E A0 B be an invertible skew-copairing of two bialE B A0 B® A is an invertible gebras A and 13. Then the element -y :=

10.2 Crossed Product Constructions of Hopf Algebras

359

counital 2-cocycle for the tensor product bialgebra B 0 A. The corresponding twisted bialgebra (13 0 A)(y) has the algebra structure of the tensor product 13 0 A, the counit e(b 0 a) = e(b)EA(a) and the comultiplication .60 0 a) =

E b(i) O cr -1 (a( i) 0 bp) )o Øa( 2).

(62)

If A and 13 are Hopf algebras, then so is (13 0 A)(y) with antipode SO 0 a) = cr21 (S8(b)

SA(a))0-2-11 .

(63)

Proof. From (61) one easily derives that 11 is indeed a counital 2-cocycle for B 0 A. Thus, Proposition 24 applies. The formula tl,), = 'y • .6 130.4 11- 1 for the twisted bialgebra (13 0 A)(1) now reads as (62). To prove the formula (63), it suffices to show that the element y in Proposition 24 is C121 • Let = . Then we have

Ei

i (1 a')(913(bi) 0 1)

-= m/3 0,A(id 0 id 0 SsigA)(-y) =

= (8 B id)(cr2) = an, where the last equality follows similarly as the second formula in (8.8).

ID

Definition 6. The bialgebra (13 0 A)(y) in Proposition 25 is called the (generalized) quantum codouble of the bialgebras A and 13 with respect to the skew-copairing a- and denoted by D"(A,13;cr).

10.2.5 Double Crossed Product Bialgebras and Quantum Doubles In this subsection we are concerned with a generalization of the crossed product algebra construction by considering two bialgebras which act on each other. This situation is precisely described by the following definition. Definition 7. A pair t13, A} of two bialgebras is called matched if A is a right 13-module and B is a left A-module with actions a(a b) = a < b of 13 on A and )3(a b) = a> b of A on B satisfying the following conditions: 46A (a 46/3(a c

E a( 1 ) < b(i) O a( 2) b( 2), b) = E a( i) > b( i ) O a( 2) > b( 2), b)

(ad)
a > (bb') =

EA(a
(64)

E B (a c• b) = A(a)E (b) ,

(65)

lab = ej3(b),

(66)

a > 1 = EA(a),

(67)

(a < (d(1) r b(1)))(a(2) < b(2))1

E(a(1) t b(1))((a(2) < b(2)) D

E a( l )
a( 2) >1)( 2) =

Ea(2) 4 b(2)

a( i) >

(68)

Equations (64) and (65) mean that the actions a and )3 are coalgebra homomorphisms. They can be expressed by saying that A is a right 13-module

360

10. Coquasitriangularity and Crossed Product Constructions

coalge bra and B is a left A-module coalge bra. Equations (66)—(68) are compatibility conditions of the two actions. Note that (68) is always fulfilled if A and B are both cocommutative. Proposition 26. (i ) Let {B, A} be a matched pair of bialgebras. Then the vector space B 0 A becomes a bialgebra with the tensor product coalge bra structure of B 0 A and with the product (b a)(b'

a') =

E b(a (i) > Wo.) )

(a (2) V(2) )ai .

(69)

This bialgebra is called the double crossed product bialgebra of the pair {B, A}. It is denoted by B a k' 0A or simply by B m A. (ii) If A and B are Hopf algebras, then B ki A is also a Hopf algebra with antipode S (b 0 a) = (1 0 SA(a))(SB(b) 0 1), a E A, b G B. Proof. The proof follows by direct verification using the conditions in Definition 7. As in the case of similar constructions (see Subsects. 8.2.1 or 10.2.1) 7 we can consider B and A as subbialgebras of the double crossed product bialgebra B 0.1 A by identifying b E B with b01EBN A and a E A with 10aEBN A. We then have the following cross commutation relations between elements a E A and b E B in the algebra B txi A: ab =

E a( i) >

a( 2) 1)( )

E a( 2) 1)( 2)

a( 1) b( 1) .

We now show that the quantum double D(A, B; a) can be considered as an example of double crossed product bialgebras. Let us assume that A and B are bialgebras equipped with an invertible skew-pairing a (see Subsect. 8.2.1). For a E A and b E B, we set aab=E a( 2) a(a( 1)7 b( 1 ))47(a( 3) ,b( 2) ),

(70)

E b(2) cr(a( 1)7 b(o)a(a (2) , b(3) ),

(71)

a> b =

where 6; denotes the inverse of a. With some work one verifies that (70) and (71) define actions of B on A and of A on B such that { 5, A} is a matched pair of bialgebras. Let B txi 0. A denote the corresponding double crossed product bialgebra. By (70) and (71), we have

E a( i ) t> V( 1 )

a(2) <1 //(2)

El/(2) 5-(a ( i ) , b1(1) )a(a(2), V(3) )

a(4) ii- (a( 3),1/(4) )a- (a( 5), 1/(5) )

E b/(2) a( 2) r7(a( i ),N0a(a( 3),N3) ). of the algebra B>ci crA on the vector space 13 A is just the product (8.21) of the algebra D(A, B; a). Recall that both This shows that the product

(69)

10.2 Crossed Product Constructions of Hopf Algebras

361

B 1>ci A and D(A, B; a) have the tensor product coalgebra structures. Thus we have proved the following equality of bialgebras:

D(A, B; o-) = B

o. A.

(72)

Let us suppose in addition that A and B are Hopf algebras such that the antipode SB of B is invertible. Then the two actions < and c defined above can be nicely expressed as duals to the right adjoint coactions of the Hopf algebras A and /3°P. First a few preliminaries are needed. Using the formulas (8.20) and (8.19), we rewrite the equations (70) and (71) as (73)

aab=E a c> b =

b(2)a(a,b( 3 )85-1 (b( 1 ))).

(74)

Let (•, •) be a dual pairing of two Hopf algebras g and 71. By Proposition 1.15, any right 7-t-comodule V becomes a left G-module with action given by g> V :--= E V(0) (g, v( i)) , g E G, V E V. Likewise a right G-module is a left N-module with action h> y = E v(0) (v(1) , h). For the right adjoint coactions AdR (a) = E a (2) S(a (i ))a( 3) of g and 71 on itself, the associated left actions of G and 7-t on each other read as

h>g = g

h=

E g(2) (8 (g( 1) )g (3) , h),

(75)

h( 2 )(g S (h (o )h(3 )).

(76)

-) of A and /3°P. We now apply the preceding to the dual pairing (-, -) Then the left action (75) of B°P on A yields the right action (73) of B on A. In terms of the product and antipode of B the right adjoint coaction of the Hopf algebra /3°P can be written as AdR(b) = E b(2) ®b (3) s,3-1 (b ( ,) ), b E B°P. Hence the left action (76) of A on B°P gives just the left action (74). We close our study of the quantum double by treating an instructive example. It is based on the FRT bialgebra A(R) which has always been our guiding example.

Example 16 (The quantum double D(A(R), A(R); r)). Let R be an invertible linear transformation on C N satisfying the QYBE. Since the canonical universal r-form r of the FRT bialgebra A(R) (see Theorem 7) is a skewpairing of A(R) and A(R) itself, the quantum double D(A(R), A(R); r) is a well-defined bialgebra. Let u and y denote the fundamental matrices of two copies of A(R). By (11), we have r(ui,v2) = R12 and 1(U1, v2 ) = .R 1-21 . Inserting these expressions into the formulas (70) and (71) we see that in matrix notation the actions i and > of A(R) on A(R) are written as ul
and 1.1. 1 > v2 = Ri-21v2R12.

362

10. Coquasitriangularity and Crossed Product Constructions

Similarly, the commutation relations (8.22) of the entries of u and v in the algebra D(A(R),A(R); r) are u1v2 = R-1 v2u1ft. Hence, by the discussion after Definition 8.4, the algebra 73(.A(R), A(R); r) is generated by the entries of the matrices u and v subject to the defining relations RU1 112=----- U2 111R ) RV1V2 = V2V1R 1 U1V2 = R-1 V2V1R.

On the other hand, the algebra it(R-1 ) has the defining relations (see (9.43))

RC-IF 4 = .4Z-2E R, RZre = £jeR , RCi",4 = Therefore, the assignments v v 0 1 bialgebra isomorphism

R.

Z+ and u 1 0 u -0 Z - define a

D(A(R), A(R); r) = A(R) 3r A(R) 11(R -1 ) = it(R)°".

(77)

We will give two other interpretations of the latter result. Let /4(4 =-- ±, -, be the algebra with matrix generators ef and defining relations = It is clear that the map £C --0 u is an algebra isomorphism of it, (R) to A(R -1 ), so it(R) is a bialgebra with comultiplication LC) =-..CC 0 ,C€ and counit E(2,€) = I. Under the isomorphisms it_ (R) A(R 1 ) and it+ (R) A(R 1 ) the universal r-form r of A(R 1 ) goes into a skewpairing i of the bialgebras it_(R) and 14(R) determined by the equation , = Ri-21 . Thus, (77) yields the bialgebra isomorphism

(R) , it+ (R) ; ) = U+ (R) m it_ (R) 11(R)

(78)

This representation of the bialgebra it(R) as the quantum double of it_ (R) and .LWR) explains the source of the third matrix relation ZIT ;FR = . It is just the cross relation of the quantum double. Likewise, we could have expressed it(R) also as the quantum double of it+ (R) and it_ (R) built over the vector space it_ (R) 0 it+ (R) rather than it + (R) LL (R). To give still another view of the isomorphism (77), we recall from Proposition 9.20 that there is a dual pairing (•, •) of the bialgebras A(R) and it+ (R) such that (ui, = R12. The corresponding skew-pairing cy of A(R) and U+ (R)°" is just the canonical universal r-form r of the bialgebra A(R) under the isomorphism U+ (R)°P = £t+ (R 1 ) A(R), so (77) means also that

D(A(R), it+ (R)°P o-) it.(R - 1 ) . 10.2.6 Double Crossed Coproduct Bialgebras and Quantum Codoubles In this very short subsection we only state dual versions to the main results of Subsect. 10.2.5. We consider two bialgebras which coact on each other. Definition 8. A pair {A, B} of bialgebras is said to be comatched if ,8 is a A and A is a left right A-comodule algebra with right coaction a : 8

10.2 Crossed Product Constructions of Hopf Algebras

363

B-comodule algebra with left coaction 0:A --BOA such that for a E A and b G B the following compatibility conditions hold: (Z1 0 id) o c(b) =

((id 0) 0 ex(b (1) ))(1

(id 0 Z1) o ,3(a) =

(Nam)

a(b(2)) 7

1)((a 0 id) o

a(b)0(a) = 3(a)a(b). Proposition 27, Suppose that {A, B1 is a comatched pair of bialgebras. Then

the tensor product algebra A013 becomes a bialgebra, called the double crossed coproduct bialgebra and denoted by Aa 3 B or A M B, with counit e = EA 0 es and comultiplication Z1(a 0 b) =

E a( 1 )

Proposition 28. Let a-

oe(b (0) )3(a ( 2)) 0 b ( 2) ,

a

C A, b C B.

c A® B be an invertible skew-copairing of the bialge-

bras A and B. Then the pair {B. A} is comatched with respect to the coactions a(a) = a-1 (a 0 1» and OW = a-1 (1 0 b)a and the corresponding double crossed coproduct bialgebra B 14 A is just the generalized quantum codouble vco (A , B ; 10.2.7 Realifications of Quantum Groups

Let A be the coordinate Hopf algebra of a quantum group G q . Roughly speaking, a realification of Gq should be a Hopf *-algebra B which contains A as a Hopf subalgebra and which is generated as an algebra by A and A*. Here A* denotes the set of adjoint elements a* for a E A with respect to the involution of the larger *-algebra B. The elements of A are then interpreted as "holomorphic polynomials" and the elements of A* as "anti-holomorphic polynomials" on Gq . The purpose of this subsection is to show that the quantum double D(A, A; r) = A m i. A provides a general procedure for the construction of a realification of a coquasitriangular Hopf *-algebra A equipped with a real universal r-form r. The involution of A will be used in order to inherit an involution on the algebra D(A, A; r). Definition 9. A Hopf *-algebra B is called a realification of a Hopf algebra A if A is a Hopf subalgebra of B and if the linear map tz : A0 A* B defined by 1.1,(a O b*) = ab* is bijective, where A* is the set {a* I a E A} in the *-algebra B.

Remark 6. Some authors call the passage from A to B a complexification and B a complex quantum group., A Suppose that A is a coquasitriangular Hopf algebra with universal r-form r. By Proposition 8.8 and Corollary 23, the quantum double 1) (A, A; r) is also a coquasitriangular Hopf algebra. Because this Hopf algebra is essentially needed in the following, we briefly repeat its structure (see Sects. 8.2,

364

10. Coquasitriangularity and Crossed Product Constructions

10.2.3, and 10.2.5 for details). As a coalgebra, D(A, A; r) is the tensor product coalgebra A 0 A. As an algebra, D(A, A; r) is the vector space A 0 A with product (a

b)(c 0 d) = E a c( 2)

b(2)d Ob( i ), c( i ))10( 3), c( 3)).

(79)

Further, D (A , A; r) is the free algebra generated by two copies, denoted by A and À, of the algebra A with cross commutations relations =

Eiim aoo r(a (2) , 6- ( 2)),

Here a E A is identified with a®1 and 6- E

a E A,

A with 1®

E

A.

(80)

.

Proposition 29. Suppose that A is a Hopf *-algebra and r is a real universal r-form of A. Then the quantum double 13 := D (A , A; r) is a Hopf *-algebra with involution defined by (a 0 b)*

b* 0 a* ,

a, b E A.

(81)

B is a realification of the Hopf algebra A. There is a homornorphisrn 1 of the Hopf *-algebra B to the Hopf *-algebra A such that 0(a b) = ab, a, b e A. Moreover, r' := N1r13r24r23 is a real universal r-form and r" := f'41k31r24r23 is an inverse real universal r-form of 13. Proof. Since r is a real universal r-form of A, r is a real universal r-form of A°P . Using these facts and equations (79) and (81), we obtain ((a g b)(c 0 d))* = (b(2)d)*

= d* b* (2)

, c(o)r(b( 3 ), c(3)) (ac(2))* c* ( 2)a* r(c* ( 1 ) b* ( 1 ))r(c* (3) 7 b* (3 )) at) = (c d)* (a b)* =(d*c)b

The other requirements for 13 = A; r) being a Hopf *-algebra are clear. Obviously, A becomes a Hopf subalgebra of B if we identify a E A with a 0 1 E B. Then, because of (79) and (81), it follows that E i a 0 b = Ei (ai 1)(bi 0 1)* = ab. Hence the mapping p, : A 0 At B is bijective and B is a realification of A. By Corollary 23, r' and r" are universal r-forms of B. The reality of r', the inverse reality of r" and the fact that 0 is a homomorphism of Hopf *-algebras are easily verified. CI

Ei

Let us emphasize that A is not a *-subalgebra of B because the adjoint a* in B of an element aag1E AC B is the element 1 g a*. We now apply Proposition 29 in order to construct realifications of the coordinate Hopf algebras 0(G q ), G q GL q (N), SL q (N), 0q (N), Spq (N), for real q. In doing so we use the Hopf *-algebras A = O(G) for the corresponding real quantum groups G eirte = Uq (N), SUq (N;u), 0q (N;u), Spq (N;u) and the real universal r-forms r z from Proposition 10 (i) If we consider G q as

10.3 Braided Hopf Algebras

365

a complex quantum group (which means that we "forget" the involution of 0(GRe q )), then Proposition 29 yields a realification of G q . By the discussion before Proposition 29 the structure of this realification B = D(A, A; rz ) can be explicitly described as follows (see also Example 16 in Subsect. 10.2.5). Let u = (ui.3 ) and y = (vi.) be the fundamental matrices of two copies of the Hopf algebra A = 0(G q ). Then, the algebra B = D (A , A; rz ) admits the generators j = 1, 2, • • • , N. The defining relations of B are the defining relations of 0(G q ) for each of the sets fu jil and {v } together with n k / = Rnklm or in matrix form Ru 1 v2 = v2 u 1 R. the cross relations R uvm Note that the latter are just the relations (80) for the generators of the two copies of A. The involution of B is determined by the equations (74)* = v. As noted above, we consider the uii as holomorphic generators and the vni as anti-holomorphic generators (or as complex conjugates to uii ) of the algebra B. The coalgebra structure of B is that of the tensor product AoA. It is given by the formulas di and e(u) = = 60 . = u ® 74 AO) = Clearly, one can also verify directly that the preceding description gives a realification B of A. However, an advantage of the above approach is the explanation of the cross commutation relations Ru 1 v2 = v2 14 R as the defining relations for the quantum double. Further, the general theory of the quantum double provides the universal r-forms r' and r" of B.

gut

10.3 Braided Hopf Algebras A braided Hopf algebra X is in general not a Hopf algebra according to Definition 1.4. The main new feature is the requirement that the comultiplication is an algebra homomorphism of X to X 0 X, where the latter carries a particular product, the so-called braided product. 10.3.1 Covariantized Products for Coquasitriangular Bialgebras

The right adjoint coaction AdR(a) = E a(2 ) S(a(o)a( 3) of a commutative Hopf algebra A is obviously an algebra homomorphism of A to A 0 A. In the general case this is no longer true, so the algebra A is not necessarily a comodule algebra for the Hopf algebra A with respect to the coaction AdR• For coquasitriangular Hopf algebras one can remedy this defect by introducing a new "covariantized" product on A such that AdR is a homomorphism of the corresponding algebra B(A) := (A,) to B(A) B(A). Further, there is another "natural" product • on B(A) I3(A) such that the comultiplicadon 44 is an algebra homoniorphism of B(A) to (B(A) B(A),$). In this manner B(A) becomes a braided Hopf algebra. In this subsection we consider a similar passage from A to B(A) for bialgebras. This requires some additional technical preliminaries. The Hopf algebra case will be treated in the next subsection.

10. Coquasitriangularity and Crossed Product Constructions

366

Definition 10. A universal r-form r of a coquasitriangular bialgebra A is called regular if there is a linear form s : A0 A C such that for a,b E A,

E r(a( 1 )0b(2))s(a( 2)01)( 1)) = E s(a( i ) b(2) Wa(2) 0b(0) =-- e(a)e(b).

(82)

Recall that A°P (resp. A"P, A°P'"P) is the bialgebra built on the vector space A with opposite product (resp. coproduct, product and coproduct) of A. Then equation (82) says that s is the convolution inverse of r when r and s are considered as linear forms on A"P 0 A. The universal r-form r of a coquasitriangular Hopf algebra A is always regular. Indeed, one easily checks that the linear form s(a b) r(a 0 8(b)) satisfies (82). In the bialgebra considerations below the form s is used, roughly speaking, as a substitute for the possibly missing antipode. Example 1 7 ( The FRT bialgebra A(R)). Let R be an invertible solution of the QYBE. Then the canonical universal r-form (see Theorem 7) of the FRT bialgebra A(R) is regular if and only if R is regular, that is, if Rt2 is invertible. (The sufficiency was stated in Proposition 8. The necessity follows from (82) applied to a = 4 b = u lic and combined with (14) The R-matrices for the quantum matrix groups SL q (N),0q (N) and Sp q (N) are all regular, because they lead to coquasitriangular Hopf algebras by Theorem 9. The flip R = T is a solution of the QYBE which is not regular. A ,

Throughout this subsection we assume that A is a coquasitriangular bialgebra and r is a regular universal r-form of A. Since r is a dual pairing of the bialgebras A and A°13 (see Subsea. 10.1.1), it follows easily that s is a dual pairing of A°P and A and of A and A"P. Hence s is a skew-pairing of the bialgebras A and A0P ' 013 7 so the quantum double D(A, A°P ' "P ; s) is well-defined by Proposition 8.8. By (72), it coincides with the double crossed product bialgebra A°P'"P N. A from 10.2.5. In order to avoid notational confusion, let us denote the identity map from A to A°P'"P by i and assume that the Sweedler notation always refers to the comultiplication of A. That is, i(ab) = i(b)i(a) and ,AAop,cop (0)) = ® i(b( 1 )) for a, b E A. Then, by Proposition 8.8, the bialgebra

o(2))

D(A)

A'P' cop ;

s) _

Aop,cop

s

A

(83)

is built on the vector space A°P ,c°P 0 A with product

(i(b)

a)(i(b')

a') =

E i(1/(2) b) g a(2)a( r(a( 1 ), V(3) )s(a( 3),b1(1) )

(84)

and coproduct ,

AD(A)(i(b) 0 a) -= E i(b(2)) g a(i) O i(b(1))

a(2).

Proposition 30. Let A be a bialgebra equipped with a regular universal rform r. The vector space A equipped with the new product

10.3 Braided Hopf Algebras

a•b :=

367

E a( 2) b(3)r(a( 1) , b(2) )s(a(3)7b(i) = E b(2) a(i) r(a(2) ) boos(a(3) , boo (85)

is an algebra, denoted B(A). The algebra B(A) is a right comodule algebra for the bialgebra D(A) with respect to the coaction : B(A) —) B(A)0D(A) given by (P(a) : => a(2) i(a(1)) a(3), a E A. The product of A can be recovered from that of B(A) by the formula -

ab =

a( 2) J0( 3) f(a(i) 7b(i))r(a(3), b ( 2 )),

a , b G A.

(86)

Proof. We begin with the associativity of the new product. Using first the

definition (85) of : , then formula (5) and finally the facts that r(-, -) and s(-, • ) are dual pairings of A and A°P resp. A°P and A, we compute (a=b) :e =

E a(3)b(4)c(3)r(a(2)b(3), C(2) )s(a(4)b( 5 ) , c (l) Ma (l ), b(2))s(a(5), b(i) E a( 3)b( 4)c(3)r(b( 2) a( 1 ) , C(2) )s(a(4)b(5) c (i) Ma(2)7 b(3) )s(a(5)7 b(i) E a( 3 )b( 4)c( 5 )r(b( 2 ), coOr(a(i), C(4))s(a(4)7C(2))s(b(5)7c(i)) X r(a( 2) , b(3) )s(a(5), b(i) )7

E a(2)b(4)c(5)*(1) b( 3)c( 4 ) )s(a(3)7b(2)c(a) ) 10(1)7c(2))s(b(5)1c(i) = E a( 2)b(4)c(5 )r(a( 1 ) 7b(3)e(4) )s(a(3) , c(2)b(i) )r(b(2)7C(3) )s(b(5)7 C(i))

a•(b•c) =

=

a(3)b(4)c(5)*(1),c(4)*(a(2), b( 3))s(a( 4 ), c(2))s(a(5), No) x r(b( 2 ), c(3))s(b(5), c(i)).

Proposition 2 (u i) implies that 1 :a = a:1 = a, a E A. Thus, we have shown that B(A)=(A, :) is indeed an associative unital algebra. It is clear that (i) is a coaction of D(A) on B(A). Using first the definitions (85) and (84) of the products and then the identity (82) we get

(p(a):(p(b) =E a( 2) lb(2)

(i(a(1))

a(3))(0(i))

a( 3)b(6 ) r(a( 2), b(5 ))s(a( 4) b(4) )

b(3)) i(b( 2) a( 1) )

a( 6)b(7)

r(a( 5 ), b(3))s(a(7)7b(i))

E a( 3)b(4)

i(b(2)a(1)) 0 a(4)b(5) r(a(2)7b(3))s(a(5)7b(i))7

where : = :0• . On the other hand, applying (85) and (5) we obtain

Ea(3) b(4) = E a( 3)b(4)

p(a•b) =

i(a(2)b(3)) 0 a( 4 )b(5 ) r(a(i), b(2))s(a(5) NO i(b(2)a(1)) 0 a(4)b(5) r(a(2)7b(3))s(a(5)7/9(i)).

Moreover, (p(1) = 1 0 1 0 1. This proves that (p is an algebra homomorphism of B(A) to B(A)®D(A).

368

10. Coquasitriangularity and Crossed Product Constructions

Inserting (85) into the right hand side of (86) and using (82) one obtains I=1 the equality (86). The second equality in (85) follows from (5). The product is called the covariantized product and the algebra B(A) the covariantized algebra of the coquasitriangular bialgebra A, because they turn the right D(A)-comodule A with coaction cp into a right D(A)-comodule algebra. Note that the algebra A itself does not have this property in general. Let us describe the covariantized algebra for the FRT bialgebra A(R). Example 18 (The FRT bialgebra A(R) - continued). Let R be a regular invertible solution of the QYBE. By Proposition 8, the canonical universal r-form r of A(R) is regular. Recall that r(u, u ilc) = kill' and s(ujj , = ) -1 . Hence formula (85) for the new prodby (11) and (15), where k 2 n u(Rnirs . In matrix n utfirrk kr, and so uiilulicki.rk = um uct yields u i=tegic --= u m notation the latter reads as U1R12112. = R12 11 1112.

(87)

Each of the three last formulas expresses the products of B(A(R)) and of A(R) by each other. From (87) we obtain u2:R21 u 1 R 12 - R21 u2 u 1 R 12 and

R21 u 1LR 12 u2 = R21 R 12 u 1 u2 . (88)

Therefore, since R 1 2u 1 u2 = u2u1R12 by (9.2), the matrix u satisfies the equation u2•12 1 u 1 R1 2 '= R211-11.:R12u21 (89) which is called the reflection equation. Recall that we have met this equation already as formula (44); see also (43). An equivalent form of (89) is u2:fli2u2.fii2 = iii2u2j112u2-

Since R is invertible, it follows from (88) that the reflection equation for the new product is equivalent to the equation R 12 u 1 u2 = u2 u 1 R 12 for the old one. Thus, the covariantized algebra B(A(R)) for the FRT bialgebra A(R) has generators uji , i , j = 1, N, with defining relations (89). This algebra is denoted 13(R) and called the braided matrix algebra associated with R. Z\ Example 19. Let R be the matrix (3.40). Using the notation a u , b c --= ui, d --= ui for the matrix entries of u, the defining equation (89) of the braided matrix algebra I3(R) is equivalent to the following six relations:

= (q- 2

-

as- c:a (1- q-2 )ds, b:c-c:b = (q-2 -1)Ca- d), a:4 = da b•d = q -2 0, c:d = q2 d:c.

A

The comultiplication ZAA is not necessarily an algebra homomorphism of B(A) to B(4)0B(A) if the latter carries the "ordinary" tensor product multiplication. However, there is another product • on the vector space

10.3 Braided Hopf Algebras

B (A) such that ZAA is a homomorphism of B (A) to (B (A)

B (A)

369

13 (A) , •)

This product is derived from the following more general lemma. Lemma 31. Let a:Ax.B C be a skew-pairing of two bialgebras A and B

and let X and y be right comodule algebras for A and B, respectively. Then the vector space X ® y is an algebra with product defined by

(x0y)•(x' Oy i ) := E xx/(0) 0y(0)y' o- (y( i), x4 ) ), x, x'

X, y,y 1 G

y.

(90)

If A = B and a is a universal r-form of A, then this algebra is a right A-comodule algebra. Proof. Using the assumption that X and Y are comodule algebras and the properties of the skew-pairing a we get

((x

y) • (x'

V)) • (x"

y") = E(xxi(o) Ø y(o)v) • (x"

y") cr(y(i), X1(1))

E xx/( ,) x ) y(0)y6y" cf(y ( i)yi, x") (y( 2), x /(1) ) E xxlm x (0) y(o)v(0) y" a(y(i), 41) ) cr(V 7 6)) cr(y( 2), x4 ) ). The same expression is obtained for (x0y) • ((x' y')• (x" y")). Hence, the product • is associative. Clearly, (x y) • (1 0 1) = (1 0 1) • (x 0 y) = x 0 y. We prove the last assertion of the lemma. By property (5) of the universal r-form o• , the expressions

ciox

cloY((x

Y) • (x i 0 YI ))

= E x (0) x/(0) (PAX 0 OPX 0

(19

y(0)y o)

x(i)x /(1) y(04 ) 0-(Y(2), X1(2)),

(PAX / 0 y l )

E x (0 ) x/(0)

y(0)yo,

xo,y(2)x/(2 ) y/( 1 ) a(y(i) 7 xl(1))

coincide, so tpx cpy is indeed an algebra homomorphism. Proposition 32. Let A and B(A) be as in Proposition O. The equation

a- 0. (b )0a, i(b1 )0d) :=

ET (b (* d(i) )r(b( i ), W(2) )r(a (i) , d(2) )s(a (2) , b4 ) ) (91)

defines a skew-pairing of the bialgebra D(A) with itself. With the product (a 0 b) • (CO d) = E a,c (2) b (2):dc,(0 (1) )

b (3) , i(c(i) )

C(3) )

(92)

the vector space B(A)®13(A) _becomes an algebra, denoted B(A)0B(A), such that 2IA is an algebra hornomorphism of B(A) to B(A)0B(A). Proof. We omit the lengthy verification that a is a skew-paring. By Proposition 30, B(A) is a right D(A)-comodule algebra with respect to the coaction w(a) = a(2) ®i(a( 1 )) a(3), a E A. The product (90) specializes to (92) in

E

370

10. Coquasitriangularity and Crossed Product Constructions

the case when A and B are the bialgebra D(A) and X and Y are the comodule algebra B(A). Therefore, by Lemma 31, (B(A)0B(A),•) is an algebra. It remains to prove that AA is an algebra homomorphism of B(A) to BA0B(A). Applying first the definitions (92) (combined with (91)) and (85) of the products of B(A)0B(A) resp. B(A) and finally (82), we get 46.4(a) • AA(b) =

E(a,(1) a( 2)) • (b(i) 0 bp))

E a( 2)b(5) 0 a( 7)b( lo ) *(l) 7 b(4))S(a(3)

b (3)) r ( a (6) 7 b(9) ) S ( a (8)

b(8))

xiqa,( 5),b(6 ))r(a( 9 ), b(7))*(4)7b(2))s(a(10)1b(1))

= E a(2)b(3)

I=1

a(3)b(4) r(a(i),b(2))s(a(4),b(1)) = AA(aJ)).

Note that the skew-pairing o- defined by (91) is not a universal r-form of the bialgebra

MA).

Example 20 (The FRT bialgebra A(R) - continued). Recall that the covariantized algebra B(A(R)) of A(R) is the algebra B(R) with product Then the product • on the vector space B(R) 0 B(R) is given by

(a 0 ti i ) • (ulic

b) = afulr'n, tirs:b (1?-1 ):in' RpiY7i R zs f

,

a, b E B(R).

In matrix notation this may be rewritten as

(a cs) R i-21 u 1 ) • (R12u2 0,b) = a=u2R 1-21 0 ui.LbRi2-

A

10.3.2 Braided Hopf Algebras Associated with Coquasitriangular Hopf Algebras We now assume that A is a coquasitriangular Hopf algebra and show that then the above results take a much simpler form. Some necessary technical facts for that are collected in the next proposition. The first three assertions give the motivation for the use of the forms s and a and of the bialgebra D(A) in the preceding subsection. Proposition 33. For any coquasitriangular .Hopf algebra A with universal r-form r, we have: (i) r is regular and s(a,b) = r(a, S(b)) for a,b E A. (ii) o- (i(b)0a,i(b')0d) --= r(S(b)a, SOW) for a, b, a' ,b' E A. (Hi) There is a Hopf algebra homomorphism 0 : D(A) A such that 0(0) 0 a) = S(b)a. (iv) Define 57(a) := E S(a( 2))r(S2 (a(3))S(a( i )),a( 4)) for a E A. Then the linear mapping S : A A intertwines the right adjoint coaction AdR of A (that is, S E Mor (AdR)) and satisfies the condition

E awl i, (a(2)) = E Na(i))La(2) = e(a)1,

aE

A.

(93)

10.3 Braided Hopf Algebras

371

Proof (0 and (ii) are verified in a straightforward manner using the properties of r. In the case of (ii) we apply (1) and the identity r(S(a), = Y(a, b). (iii): Using the formulas (3) and (5) and the assertion (i), we obtain

0((i(b)

a)(i(W)

d))

E 0(i0/(2) b) a(2)d) r(a(i),N 3) )s(a(3), b'(1) ) .= E S (b ) (S(11(2) )a( 2))a' r(a( i),N3) )*( 3), S( 1/(1))) = E S(b)a( 3)5(N 1) )ai r(a(i), W 3 ))r(a(2), S(1/( 2))) = E S(b)a( 2), (11(1) )al r(a( i), S(b'(2) )N3) ) = S(b)aS(V)ai = ONO 0 a) 0(0) a').

It is easy to check that (0 0 0) o ,6 D(A) = ,6A o 0. Thus, 0 is a Hopf algebra homomorphism. (iv): Using the functionals f and f from Proposition 3 we compute

E a( l) : ,i(a(2)) E a(2) S(a(5) ) r(a( i ), S(a( 6)))r(a( 3), S2 (a( 7)))r( 52 (a( 8)), a( 9)) x r(S(a(4)), a(io))

— E a( 2)S(a(5)) r(a( i ), S(0,(6) ))*(3), S 2 (a(7)) f (a(8)))r(S(a(4)), a(9)) E a( 2)S(a(5)) r(a( i), S(a( 6)))*( 3), S2 (a( 7))) xr(S(a(4)),S 2 (a(8)))7 (a(9))

E a( 2)S(a( 4)) r(a( i) , S(a(5) ))r(a (3) , S(a( 7))52 (a(6)))f (a( 8)) E a( 2)5(a( 3)) r(a(i), S(a(4)))1 (a(5) E r(a( 1 ), S(a(2)))7 (a(3))1 = E f(a(1) )7(a (2) )1 = E(a)1. Here the first equality follows from the definitions of and S and assertion (i ). The second and third equalities are obtained by inserting and applying formula (8), respectively. Similar (slightly simpler) reasoning yields the second equality of (93). In order to prove that S E Mor (AdR), we compute

7

E S(a(3)) 0 5 2 (a(4))5(a(2)) r(S 2 (a(5)) ,*(1)), a(6)) = E S(a(3)) 52 (a(4))51 (a(2)) f (a(5))r(S(a(1)), a(6)) = E S(a)) O a( 5)S(a( 2)) f (a(4))r(S(a(i)) a(6)) = E s(a (3) ) O S(a(1))a(6) f (a(4))r(S(a(2)) a(5))

AdR(S(a)) =

= E S(a(3) ) O S(a(1) )amr(S2 (a(4))S(a( 2) ), a(5) ) = E 8 (a(2))

S(a(1))a(3) = (ii 0 id)AdR(a).

372

10. Coquasitriangularity and Crossed Product Constructions

Here we used the formulas (8), (4) and (5).

I=1

The next two propositions restate some key results and formulas from the preceding subsection in the case of a coquasitriangular Hopf algebra A with universal r-form r. Proposition 34. (i) The vector space A equipped with the product

a•b =

E a( 2)b( 2)r(8(a( o )a(3), S(b( l ))) = E b( 2)amr(a( 2), S(b(o)b( 3))

is a right A-comodule algebra, denoted by B(A), with respect to the right adjoint coaction MR. (ii) Recall that r(a) is the linear functional on A defined by (21). Then the map a —> r(a) is a homomorphism of the algebra B(A) to the subalgebra f(A) of the dual Hopf algebra A°. Proof. (i ): From Propositions 32(i) and 2(v) and formula (3) we get r(S(a)al, S(b)) =

E r(S(a), S(b( 2)))r(a l ,S(b( 1 ))) = E r(a, b( 2))s(a 1 ,

and r(a, S(b)b') = E r(a (1 ), bi)s(a (2) , b), so the product (85) takes the above form. Since obviously AdR = (id ® 8) o, B (A) is a right A-comodule algebra with respect to the coaction AdR by Propositions 29 and 32(iii). (ii): Since s(a, b) = r(a, (0) by Proposition 33 (i), a comparison of the formulas (30) and (85) shows that i(a:b) = i(a)T(b) for a, b E A. Let us say that a coquasitriangular Hopf algebra A (or more precisely a pair (A, r) of a coquasitriangular Hopf algebra A and a universal r-form r on A) is factorizable if the bilinear form q given by (9) is nondegenerate. Since q(S (a) , (b)) = q(b, a) and the antipode of a coquasitriangular Hopf algebra is bijective, it is clear that A is factorizable if and only if 1(a) = 0 implies that a = 0 or equivalently if [(a) = 0 is only possible for a = O. Therefore, by Proposition 34(ii), for a factorizable Hopf algebra A the map a t(a) is an algebra isomorphism of the covariantized algebra B (A) to the subalgebra i(A) of A°. Note that cotriangular Hopf algebras are those coquasitriangular Hopf algebras for which the form q is trivial, that is, q = E 0 E. The factorizability of a coqua.sitriangular Hopf algebra is just the other extreme where the form q is far from being trivial. The following proposition is only a reformulation of Lemma 31. Proposition 35. Let X and y be right A-comodule algebras. Then the vec-

tor space X®Y becomes a right A-comodule algebra, denoted by respect to the tensor product coaction and the product

(x y) • (x' y') := (x 0 1)r y x (y 0 s')(1. 0 y'),

E

xoy,

X, y y'

E

with

y, (94)

where the product on the right hand side is the tensor multiplication and

10.3 Braided Hopf Algebras

ryx (y

s') := Ex/(0) y(o) r(Y(1), si(1))•

373

(95)

The product (94) on X 0 y is called the braided product and the mapping ryx : Y X X 0 Y is called the braiding of the A-comodule algebras y and X. Recall that ryx is just the mapping defined by formula (6). Some properties of the braiding maps can be found in Proposition 1. In particular, it was shown therein that each map rxx satisfies the braid relation. Applied to the case when X = y is the right A-comodule algebra B(A) with coaction AdR, the product (94) of the algebra B(A)0B(A) writes as

(a

b) • (c 0 d) =

as(2)

b(2) :d r(S (b ( 1 ))b ( 3) , S (c(o)c( 3)).

(96)

By Proposition 33(ii), it coincides with the product (92) of the algebra B(A)0B(A). Now let us summarize the main facts established above. By Proposition 34, B(A) is a right A-comodule algebra. Obviously, B(A) is a right A-comodule coalgebra with comultiplication ,AA and counit EA inherited from A. By Proposition 32, A4 is an algebra homomorphism of B(A) to C is an algebra homomorB(A)0B(A). It is obvious that EA : B(A) phism. Proposition 33(iv) says that the mapping S : B(A) B(A) defined therein intertwines the right adjoint coaction Ad R and satisfies the antipode axiom. Thus, the algebra B(A) and the mappings 4,4, EA and S fulfill all Hopf algebra axioms except for one modification: the algebra B(A)0B(A) carries the braided product (94) rather than the usual tensor multiplication. We shall express all that by saying that B(A) is a braided Hopi' algebra associated with the coquasitriangular Hopf algebra A. The precise definition of this concept is Definition 11. Let A be a coquasitriangular bialgebra equipped with a fixed universal r-form r. A right A-comodule algebra and right A-comodule coal-

gebra X is called a braided bialgebra associated with A (briefly, a braided bialgebra) if Ex : X C and ,Ax : X X0X are algebra homomorphisms, where X0X carries the braided product (94). A braided bialgebra X is a braided Hopf algebra associated with A (briefly, a braided Hopf algebra) if X admits an antipode that intertwines the coaction of A on X. In other words, a braided bialgebra X associated with A is a right Acomodule X which admits a product and a coproduct such that:

(i) X satisfies the bialgebra axioms with the modification that the vector space X0X is equipped with the braided product (94); (ii) the product and the coproduct of X intertwine the corresponding coactions of A, that is, X is a A-comodule algebra and coalgebra. 'It should be emphasized that the definitions of the braided product (94) and of a braided bialgebra depend on the choice of the universal r-form of A.

374

10. Coquasitriangularity and Crossed Product Constructions Recall that the antipode of a Hopf algebra is an anti-homomorphism of

the algebra and of the coalgebra. For a braided Hopf algebra X there is a similar result which says that Sx 0 mx ---= mx 0 rxx 0 (Sx 0 Sx), 2ix 0 Sx = (Sx0Sx ) 0 rxx 0 .6x1

Sx(i) =--- 1 7

(97)

Ex 0 Sx =- Ex-

(98)

Let us turn to examples of braided Hopf algebras. Each "ordinary" Hopf algebra X is, of course, a braided Hopf algebra associated with any coquasitriangular bialgebra with respect to the trivial coaction (3(x) = x 01, x E X. A nontrivial simple but still instructive example is the following Example 21 (The braided line). Let A = CZ be the group Hopf algebra (see Example 1.7) of the additive group Z of integers. That is, A is the Hopf algebra of all complex polynomials in a single group-like generator, say x, and its inverse x'. As noted in Example 1 above, for any nonzero complex number q there exists a universal r-form r = rq on A such that rq (xn, km) = q - ", m,n E Z.

(99)

The algebra X = C[zj of all polynomials in one variable is a right A-comodule algebra with coaction 0 given by f3(z) = zn 0 xn , n E Z. By (95) and (99), the corresponding braiding map r xx acts on the vector space X 0 X as rxx(zn 0 en) = CT' el 0 zn , m,rt E No .

Therefore, the braided product (94) of the algebra X 0 X is given by (z 2 0 zi) • (zn 0 el) = q -inz i+n 0 zi+m In particular, we see that the two generators x := z 0 1 and y := 1 0 z of the algebra X 0 X satisfy the relation yx -= q -l sy. Hence the algebras X 0 X and O(C) are isomorphic. Now we define algebra homomorphisms d x : X --- X0X and ex : X —p C such that 2ix(z) =---- z 0 1 + 1 0 z and Ex (z) = O. From Proposition 2.2 we obtain a n-k 0Z k l nE No. 2,x(zn) = En [72] k =0 k q1 Then the vector space X becomes a right A-comodule coalgebra (see Subsect. 10.2.2) with comultiplication Zix, counit Ex and coaction )3 as given above. (Indeed, since all structure maps are algebra homomorphisms, it suffices to check the axioms on the generator z which is easily done.) Rather, one verifies that the linear map Sx : X -- X defined by n E No, S x( z n ) : = (_ 1)Tz q - (n-1)7q2 z n , „

satisfies the antipode axiom. Obviously, we have 0 o Sx = (Sx 0 id) o 0. Thus, we have shown that X is a braided Hopf algebra associated with the coquasitriangular Hopf algebra A. It is called the braided line algebra.

10.3 Braided Hopf Algebras

375

In this example the first two formulas of (97) and (98) read as

Sx(Zin+n )

A x (Sx(zn)) =

Z—dic=0

7n q- infSx( Z)Sx(Zn), [n] -k(n-k) s x (z n-k) k q-.., n

sx ( z k) .

A

Obviously, Example 21 remains valid if the Hopf algebra A = CZ is replaced by the subbialgebra C[x] of polynomials in x. If we interpret this bialgebra C[x] as the FRT bialgebra A(R) (with R being the 1 x 1 matrix (q-1 )), then the braided line algebra becomes a special case of an important class of examples which will be constructed in Subsect. 10.3.5. Nevertheless, the braided Hopf algebras B(A) developed above are still a main source of interesting examples. In fact, the same arguments of proof as in the case of B(A) yield the following more general result. Proposition 36. Let p : 1-t —> A be a Hopf algebra homomorphism from an arbitrary Hop! algebra N to a coquasitriangular Hop! algebra A. Then the vector space 7-t becomes a braided Hopf algebra associated with A, denoted by B(7-(; A, p), with structures given as follows. The product of the algebra B(H; A,p) is defined by

hIg =

E h( 2)g (2) r(p(S (h( i))h( 3)) ,

(g ( 1)))),

h, g c

The coalgebra structure and the unit element of B ( H; A, p) coincide with those of N and the antipode S of 13(7i; A, p) acts as

(h)=

E S(h( 2)) r(p(52 (h( 3 ))S(h( i))), p(h( 4))).

Further, B (fl ; A, p) is a right A-cornodule algebra with respect to the coaction

O(h) :=

h(2) p(S(h( 1))h(3 )),

h E H.

Clearly, if H = A and p is the identity map, then B (Ii; A, p) is just the braided Hopf algebra B(A) defined above. At the other extreme, if p is the trivial Hopf algebra homomorphism (that is, p(h) = e(h)1 for h E H), then obviously B('1-t; A, p) = H. In this case the coaction of A on H is trivial, so 7-t is a braided Hopf algebra with respect to the trivial coaction. Following S. Majid, the passage from the Hopf algebra A resp. the Hopf algebra homomorphism p : H A to the braided Hopf algebras B(A) resp. B(Ii; A, p) will be called transmutation.

376

10. Coquasitriangularity and Crossed Product Constructions

10.3.3 Braided Hopf Algebras Associated with Quasitriangular Hopf Algebras

In this subsection let B be a quasitriangular Hopf algebra with universal Rmatrix R. -= Ei x i ® yi . We briefly state (without giving proofs) the dual versions to the main results of the preceding subsection. Proposition 37. Let a> b := adL(a)b = E ambS(a( 2 )), a, b C B, denote the left adjoint action of B on itself. With the new comultiplication 46 defined by

41(b) = E.

2

Eb(l) S(y ) ® xi › bp),

b E B,

the vector space B becomes a coalgebra, denoted by B(B), such that B(13) is a left 13-module coalgebra. Moreover, 4 (1) = 1 01.

and the counit

68 1

Proposition 38. If X and Y are left B-module algebras, then the vector space X®Y is a left B-module algebra, denoted by xoy, with the tensor product action of B and the braided product .'defined by

(x0y)•(x'00 = E(x01)7,,,x( y 0x1)(100, x, xi E X, y, y' E y, (100) where Ry,x(y 0 x') :=

E i yi › x' 0 x i > y.

The algebra B itself is also a left B-module algebra with respect to the left adjoint action. Therefore, the braided product (100) makes the vector space BOB into an algebra denoted by BOB. It can be shown that 4 : B B0B is an algebra homomorphism. Further, the mapping S : B —> B defined by S(b) := E i y i Ss(x i > b),

b E B,

intertwines the left adjoint action (that is, a › S (b) = S (a > b)) and satisfies the equation E b(1) ,5(b (2) ) = > (b(1) )b(2) = 6 (b )1, b E B. Here 4 (b ) = E b( ,) ® b(2) is the Swee-dler notation for the new comultiplication A. The left B-module coalgebra 8(B) is also a left B-module algebra with the algebra structure inherited from B. We call B(Bj equipped with these structures a braided Hopf algebra associated with the quasitriangular Hopf algebra B. The general definition is easily guessed from Definition 11 above. A braided bialgebra associated with 13 is a left B-module algebra and a left B-module coalgebra X such that ex : X —> C and 4&v : X —› X0X are algebra homomorphisms, where X0X is equipped with the braided product (100). If X has an antipode which intertwines the action of B on X, then X is called a braided Hopf algebra. Finally,, we want to make more precise the sense in which the preceding constructions of B(A) and B(B) are dual to each other. In order to do so, we consider in additiOn to the quasitriangular Hopf algebra B a coquasitriangular Hopf algebra A with universal r-form r. We suppose that there is a dual pairing (•, .) of the Hopf algebras A and 5 such that r(a 0 a') = (a 0 a', R.)

10.3 Braided Hopf Algebras

377

for a, a' E A. The next proposition shows that there is a close link between the two new structures (the product • of B(A) and the coproduct 4 of B(8)) and the two old ones (the product of B and the coproduct LtA of A). Proposition 39. Retain the preceding assumptions on A and B and define a linear mapping Q : A —* B by Q(a) = E ir (a, ai )bi , a E A, where 1Z211Z•12 = ai bi E BOB. Then, Q is an algebra hornornorphism, of B(A) to 8 (that is, Q(a:b) =-- Q(a) • Q(b)) and a coalgebra homomorphism of A to B(B) (that 4Q(a) (Q CA(a)).

Ei

Idea of proof. Some computations show that Q(a:b) and Q(a).Q(b) are both equal to (a0b, 112R21 7Z13R31) and that the expressions ,Q(a) and (Q0Q)zA(a) coincide with (a, 7Z137Z317Z23R.32)• See the proof of Proposition 7.4.3 in [Mai] for the details. LI 10.3.4 Braided Tensor Categories and Braided Hopf Algebras

In this subsection we shall see how both braided products (94) and (100) and both braided Hopf algebras B(A) and B(13) can be considered from a unique point of view. The common notion behind these concepts is that of a braided tensor category. In the following treatment we shall concentrate on the essential points and suppress mathematical subtleties. A category C is a collection of objects X, Y Z • • • and of sets Mor (X, Y) of morphisms between two objects X, Y such that a composition of morphisrns is defined which has similar properties to the composition of maps. That is, given morphisms f E Mor (X, Y) and g E Mor (Y, Z) there always exists a morphism gof E Mor (X, Z) and the composition of three morphisms has to satisfy the associativity law. Moreover, each set Mor (X, X) has to contain a morphism idx such that f o idx = f and id x o g = g for any f E Mor (X, Y) and g E Mor (Y, X). A mapping f E Mor (X, Y) is called an isomorphism if there is another morphism g E Mor (Y, X) such that g o f = idx and f 0 g = idy. An obvious example of a category is obtained by taking the vector spaces as objects and the set of all linear mappings f : X Y as Mor (X, Y). For our purpose the categories MA and MA of a bialgebra A are most important. The objects of MA and MA are the right A-comodules resp. the right A-modules and the morphisms are the linear intertwiners of the corresponding coactions resp. actions. In what follows we consider tensor (or rnonoidal) categories. These are categories C that have a product, denoted 0 and called the tensor product, which admits several "natural" properties such as associativity and existence of a unit object denoted 1. The associativity of the tensor product requires that there is an isomorphism qS X ,Y,Z (X 01 7 )0 Z X0(YOZ) for any triple X ,Y , Z of objects in C. We will not list the axioms of a tensor category here (see, for instance, [Kas], Sect. XI.2). For our subsequent discussion it suffices to know that the categories MA and MA for any bialgebra A become such tensor categories if we take the tensor product of comodules resp. modules as ,

378

10. Coquasitriangularity and Crossed Product Constructions

the tensor product 0 and the trivial one-dimensional comodule resp. module as the unit object 1 in the category. The categories we are interested in are tensor categories which are equipped with some kind of transposition of tensor products. A tensor category (C, 0) is called a braided tensor (or monoidal) category if for any pair 17 0X such that X, Y of objects in C there is an isomorphism Wx ,y : XOY (g f) 0 WX,Y = WX 1 ,171 0 f. g) for arbitrary morphisms f E Mor (X, X') and g E Mor (Y, Y') and the hexagon axiom holds. For simplicity we will suppress writing the isomorphisms Ox,y,z which govern the associativity of the tensor product. Then the hexagon axiom is the validity of the two conditions (

WX,Z ° WY,Z = WX0Y,Z7 WX,Y ° WX,Z = WX,YOZ

for all objects X, Y, Z in C. A family W of mappings Wxy having the above properties is called a braiding (or a quasisymmetry) in the tensor category (C, 0). Obviously, the flip gives a braiding in the category of vector spaces. But in contrast to this example, the -mapping Wy, x need not be the inverse of Wxy in general. The two hexagon identities imply the relation (PY,zOidx) 0 (idyetPx,z)0(Wx,Y0idz) = (idz0Wx,y)0(Wx,zOidy)0(idx®Wy,z)•

Indeed, there are two ways to go from (X0Y)0Z to ZO(Y0X) by applying W, one over Z0(X0Y) and another one over (Y0X)0Z. The hexagon conditions for these ways give the above 'relation. In the case X=Y-Z it is just the braid relation for Wx ,x and this explains why W is called a braiding. From now on let A be a coquasitriangular bialgebra with universal rform r and B a quasitriangular bialgebra with universal R-matrix R. Then, by Propositions 1 and 8.4, the families of mappings Wx,y := rxy and WX,Y := Rxy are braidings in the tensor categories MA and MB, respectively. Thus, MA and MB are braiding tensor categories. Let C be a braided tensor category. By an algebra in the category C we mean an object X in C together with morphisms mx : X0X X and 77 : 1 X from C (!) satisfying the usual associaivity and unit properties. From the corresponding definitions we conclude that the algebras in the categories MA and MB are precisely the right A-comodule algebras and the right 8-module algebras, respectively. Now let X and Y be two algebras in the category C. Then it can be shown that the object X®Y in C becomes an algebra in C, denoted by X®Y and called the braided product algebra of X and Y, with product morphism mxoy := (mx 0my ) o (idx 0Wy,x0idy)

and the tensor product unit morphism. If Wy,x is the flip, then X031 is just the tensor product algebra X 0 Y. In the case of the category MA with braiding Wx,y = rxy, the algebra X®Y in the category MA is nothing

10.3 Braided Hopf Algebras

379

but the right A-comodule algebra X®Y from Proposition 35 with braided product (94). Likewise, for the category MB the algebra X®Y in MB is the right 13-module algebra X®Y from Proposition 38 with product (100). In a similar way, a number of other concepts from "ordinary" Hopf algebra theory make sense in the category C. In addition to the usual axioms one has to require that the corresponding object and all structure maps belong to the category C. In particular, a Hopf algebra X in the category C is defined in this manner, whereby the comultiplication LX x has to be an algebra homomorphism of X to the braided product algebra X0X. It is clear that the Hopf algebras in the braided tensor categories MA and MB are precisely the braided Hopf algebras associated with A and B, respectively. Let us illustrate the preceding by a simple example. It can be generalized to any abelian group rather than Z2 equipped with a fixed bicharacter. Example 22 (A = CZ2 ). Let A be the coquasitriangular group Hopf algebra with universal r-form r(,3) CZ2 of the group Z2 = Z/2Z {0, j G {0, 1 } . Then any right A-comodule X with coaction ço splits into a direct sum of an even part XT, :=--- Ix G X I (}0 ( x) = x0-61 and an odd part XT = {X E X 1 (P(x) = x0I}. Therefore, the objects of the category M A are the Z 2-graded (or super) vector spaces and the morphisms are the Z 2-graded linear mappings. The braiding obtained from r is determined by the formula Wx,y(x7 yy) = (-1) ijyy 0 x7 ,

E

y.7 E

j E {0,1}. (101)

As noted above, the algebras in MA are the right A-comodule algebras. If X is an algebra equipped with a coaction ço of A on X, then ‘,61:X--X0A is an algebra homomorphism if and only if X6 X6 C Xo, X6X1 C I , XIX() C X

and Xi-Xi C X.

Thus, the algebras in MA are precisely the Z 2-graded algebras. Similarly, the Hopf algebras in MA or equivalently the braided Hopf algebras associated A with A are just the super Hopf algebras (see Subsect. 1.2.9). We close this subsection with some comments on the physical interpretation of the braided product algebras. For this purpose we assume that the right A-comodule (or 13-module) algebras X and y are algebras of observables of two physical systems. Let us consider the braided product algebra xoy as an algebra of observables of a joint system which is built from the two systems. First let A = C. Then the braiding of the category MA is the flip and the algebra X®Y carries the product (x0y) • (xi0V) = xxiOyy', so the subalgebras X and y of xoy mutually commute. This means that the two systems are independent • in the joint system. Next we set A = CZ2 . As usual in "super physics", we consider Xr) and Y6 as algebras of observables of bosonic systems and Xi: and y, as algebras of observables of fermionic systems. From the formula (101) of the braiding it is clear that bosons and bosons as well as bosons and fermions of the two systems commute, while

10. Coquasitriangularity and Crossed Product Constructions

380

fermions and fermions of the two systems a.nticommute. That is, in this case the braiding contains the statistics between the two systems. In the general case the braiding W describes the commutation relations between observables of the two systems. Having the above example A = CZ2 in mind, one says that the systems obey the braid statistics W. This picture opens an important conceptual view of the braiding and the braided Hopf algebras and their possible future role in physics. 10.3.5 Braided Vector Algebras

From Proposition 9.4 we know that the algebra X R( f ; R) is a right quantum space (that is, a right comodule algebra) for any quotient bialgebra A of the FRT bialgebra A(R). Recall that X R(f R) is the unital algebra with generators x 1 , x2 , . xN and relations f(h) iii x k x i = 0, i j = 1,2, • - • , N, and that the coaction çart of A on XR(f;R) is given by the equations (pR(x) = xi 0 u,ji , see (9.10). The purpose of this subsection is to show that under suitable assumptions the algebra X R ( f ; R) becomes a braided Hopf algebra. ,

Proposition 40. Let R be an invertible solution of the QYBE and let f be

a complex polynomial such that + I) f(ii) = 0.

(102)

Suppose that A is a quotient bialgebra of A(R) such that the canonical universal r-form r of A(R) passes to a , universal r-form of A. Then the right A-comodule algebra X := XR (f;R) is a braided Hopf algebra associated with the coquasitriangular bialgebra A with comultiplication 2i x , counit e x , antipode Sx and braiding rxx determined by the formulas

/-X(Xi) = x i 01+10 x i ,

Ex(x i )

rxx (xi 0 xi ) = Xk 0

0, Xi

S(x) = —xi ,

(103) (104)

.

Proof. As in similar earlier proofs (for instance, of Proposition 9.1), we first show the corresponding result for the free algebra C(xi ) and then we pass to the quotient algebra X of C(xi ). Because C(xi ) is a right A-comodule algebra with coaction PR (xi) = Xi 0 tr i: 1 the braided product algebra C(x i )0C(x i ) is well defined by Proposition 35. Since C(xi ) is the free algebra with generators xi , there are unique algebra homomorphisms : C(x i ) —> C(xi )0C(xi ) and E : C(Xi) —> C such that 2i(xi) = xi 0 1 + 1 0 x i and E(xi ) = 0. The set {1, xii • • • xi n } is a vector space basis of C(xi ). Hence there is a unique linear map S of C(xi ) such that 5(1) = 1, 5(x) = —x and -

S(Xi• - -x n ) =

(-1) 1 x..

• *Rln R23' "R2n• • •Rn-1,n

7

n? 2, (105)

in the usual matrix notation. That is, we define 8 (1) = 1, S(xi) = xi and extend S to a linear map of C(xi) according to the formula (97). One easily —

10.3 Braided Hopf Algebras

381

verifies that C(xi ) equipped with these mappings becomes a braided Hopf algebra associated with A. In order to complete the proof we have to show that these structures pass to the quotient algebra XR(f;R)=C(x i )1J, where J denotes the twof(11)i xkx/. For sided ideal of C(xi) generated by the ./V2 elements the coaction ço R this holds by Proposition 9.4. Formula (104) for the braiding rxx is obtained by repeating the reasoning of (13). By the definition of E we have E(J) = {0). Using the formulas (103), (94), (104) and finally (102), we obtain

= f (14) ikjiA(xkxi) = f(ki kj(xk ® 1 + 1 ® x k ) • (xi 01 + 10 x i ) f( fo iki xm xr, = f(ft)ikjxkxi el + f (Ncjxk ® x l + +10 f(14) ikjxkx1

® + ((k+ 1- )f (M)Ti.r; n Trol Xn +10

1+ 1

and hence .A(a/ii b) = 2i(a) •

01+10

• 2i (b) E J C(x i ) + C(xi) 0

for a, b C C(xi). From equation (105) it follows that

S(I) = f(1ViciS(xkx1)= f(k) icjx,x n Rircir = RjIkl C J. By (97), this implies that S(aIti b) E J for a, b E C(x2 ). Thus we have shown that J is a Hopf ideal of the braided Hopf algebra C(i). Hence the quotient X=C(xi )/J is also a braided Hopf algebra. Note that (105) also hold for the antipode Sx of the braided Hopf algebra XR(f; R) and allows us to compute Sx on products of generators. In order to discuss the applicability of Proposition 40, let us assume in addition that the transformation it admits a spectral decomposition le = Ei A,Pi , where Pi are nonzero projections such that Ei P = I and Pi Pi = 0 and A i Ai if i j. It is well-known from linear algebra that the projections Pi are then polynomials in ft, say, Pi = Mk). If an eigenvalue A i is —1, then fi satisfies (102) and hence, by Proposition 40, XR( fi ; R) is a braided Hopf algebra associated with A. In the case A = A(R) the transformation R can be replaced by any nonzero complex multiple, so each eigenspace of it gives rise to a braided Hopf algebra. However, if A is one of the coquasitriangular Hopf algebras 0(G q ), Gq = SL q (N),O p (N),Sp q (N), and R = zR as in Theorem 9, such a scaling is impossible and k cannot have the eigenvalue —1 unless q is a root of unity. Our next proposition shows how to overcome this difficulty: XR(f;R) becomes a braided Hopf algebra associated with the coquasitriangular bialgebra A. which is obtained from A by adjoining an additional central invertible group-like generator. Recall that the group algebra CZ of the group Z of integers is the algebra of Laurent polynomials in a single generator x. It is a Hopf algebra with

382

10. Coquasitriangularity and Crossed Product Constructions

comultiplication Zi(xn) = xn xn, n E Z. For each nonzero A E C, there exists a universal r-form r), on CZ such that r),(xn g xm) = A -nm (see Example 1 in Subsect. 10.1.1). One easily verifies that the tensor product bialgebra A := A g CZ is coquasitriangular with universal r-form i defined by î(a xn

b xm) = r(a g b)A - nm,

a, b E A,

n, m E Z.

(106)

Clearly, each algebra XR(f; R) is also a right kcornodule algebra with coaction given by c,oR (xi ) = xi 0 u3i: g x. Proposition 41. Let R, A and r be as in Proposition 40. Suppose that A E C, A L 0, and f is a polynomial such that

(il + All f

--= 0.

(107)

Then X :--= XR (f R) is a braided Hopf algebra associated with the coquasitriangular bialgebraA defined above. The Hopf algebra structure of X is determined by (103) and the braiding map kxx is given by

x(s i 0 x i ) ==

(108)

Proof. Clearly, (106) implies (108). The term + ilf(k) of the expression Zi(iii ) in the proof of Proposition 40 now becomes (A -1 k + .00) which vanishes by (107). The rest of the proof follows by the same reasoning. The braided Hopf algebras XR(f ; R) in Propositions 40 and 41 are called braided vector algebras. Note that Propositions 40 and 41 are also valid for the coquasitriangular Hopf algebra 0(Gid q (N)) with the same proofs verbatim. Finally, let us consider the case when the number A in (107) is a root of unity, say, Ak = 1 for k E N. Then, rx gives a universal r-form of the group algebra C(Z/kZ) of the group Z/kZ. Hence the assertion of Proposition 41 holds also for the coquasitriangular bialgebra A := A 0 C(Z I kZ). In particular, in the case A = 1 we can take k = 1 and recover Proposition 40. 10.3.6 Bosonization of Braided Hopf Algebras In this subsection we show that the crossed product of a braided Hopf algebra X by the coquasitriangular (or quasitriangular) Hopf algebra A which X is associated with becomes an ordinary Hopf algebra. In order to avoid possible confusion with the comultiplications of X and A, let us write the Sweedler notation for the coaction @ of A on X as a(s) = E so) 0 so). This notation will be kept in the rest of this section. To begin with, suppose that X is a braided bialgebra associated with a coquasitriangular bialgebra A. Then, by Definition 11, X is in particular a right A-comodule coalgebra with respect to the coaction0 of A on X. Therefore, by Proposition 19, we have the crossed coproduct coalgebra A x X built

10.3 Braided Hopf Algebras

383

on the vector space A0 X. On the other hand, X is also a right A-comodule algebra by Definition 11. Hence, by Proposition 4, the right coaction @ induces a right action 0' of A on X by xaa=-E x (°)r(x (1 ), a), a E A, x E X, such that X becomes a right A-module algebra, where 0(x) = E x ) ® x('). Thus the vector space A® X carries also the structure of the crossed product algebra Ap, tx X (see Subsect. 10.2.1). The next proposition says that these two structures (algebra and coalgebra) on the vector space A0 X indeed fit nicely together.

(°

Proposition 42. Let X be a braided bialgebra associated with a coquasitriX is a bialgebra, denoted angular bialgebra A. Then the vector space A by AS x X or Ax X , with the coalgebra structure of A 8 x X and the algebra structure of Aoi tx X . That is, the bialgebra A 3 tx X has the coproduct (54) and the product (a 0 x)(b y) =

E ab( l)

x (°) y r(s (I) , b ( 2 )),

a, b E A, x, y E X. (109)

The subalgebra1®X of AD< X is a right Ax X -comodule algebra with coaction PR(10X) := 2■ Apc x (1®x) =--

E 1® (x (1) )(0)®(s (1) )(1)0x (2) , X E X. (110)

If X is a braided Hopf algebra associated with the coquasitriangular Hopf algebra A, then the bialgebra A Lx X is a Hopf algebra with antipode S (a 0 x) = E(1 0 S x (x (1) ))(S A(ax (°) ) ® 1) ,

a

E

A, z E X.

(111)

Proof. Being the restriction of the comultiplication to an invariant subspace, (p R is a coaction. One has to show that (10 R 1 ZA and E are algebra homomorphisms and that S satisfies the antipode axiom. These technical verifications are carried out in [Wei] and in the dual situation of quasitriangular Hopf algebras in [Maj10]. Formula (110) follows from (54). The construction of the "ordinary" Hopf algebra A x X from the braided Hopf algebra X is a process called bosonization by S. Majid. It is in a sense dual to the process of transmutation which converts the Hopf algebra A or more generally any Hopf algebra homomorphism p : A into the braided Hopf algebras .13(A) resp. B(Ii; A; p), see Subsect. 10.3.2. Let us see how A and X are related to A tx X. From the formulas (54), (109) and (111) it is clear that the map a a01 is an injective Hopf algebra homomorphism of A to A x X, so we may consider A as a Hopf subalgebra of A x X by identifying a with a 0 1. Also, the map x 1 tg x is an algebra isomorphism of X to the subalgebra 1 tg X of A x X, but as seen from (110) and (111) it does not preserve the comultiplication and the antipode. Being a crossed product algebra (see Subsect. 10.2.1), Aix X is the universal algebra generated by the algebras A and X with cross commutation relations xa =

E a( l )x ° )r(x (1) , a(2)),

a E A,

X

E X.

(112)

384

10. Coquasitriangularity and Crossed Product Constructions

We now illustrate the bosonization procedure by two examples. The Hopf algebras A y X appearing in these examples are also of interest in themselves without referring to braided Hopf algebras. Example 23 (The braided line - continued). Let us retain the notation of Example 21. By Proposition 42, A(3 ix X is a Hopf algebra. It is generated by the elements x x01 and z 10z. Since 0(z) = zox and rq (x0x) = the cross relation (112) with x := z and a := x reads as zx xzq -1 . Since AA A, (z) = z 1 1 z, formula (54) for the coproduct of A y X yields (

X = X X7 igz) )

z

x 1

(113)

z.

From these formulas (or from (97)) we easily derive e(x) = 1, e(z) = 0, S (x) = x -1

(z) =

-

zx -1

(114)

That is, the Hopf algebra A ix X has two generators x and z with defining relation xz = qzx and structure maps determined by (113) and (114). For q =1 the Hopf algebra (A Ix X)"P is just the coordinate algebra of the group of affine transformations of the real line (the so-called ax + b-group). This suggests the use of the bosonization for the construction of inhomogeneous quantum groups. This idea will be realized in the next subsection. A Example 24 (The Hopf algebra A Ad R y13(A)). Let X be the braided Hopf algebra B(A) associated with a coquasitriangular Hopf algebra A. Recall from Subsect. 10.3.2 that B(A) has the same coalgebra structure as A and that the product of B(A) is given by (85). The coaction of A on B(A) is

the right adjoint coaction Ad R (a) = E a( 2) S(a( i) )a( 3 ). Inserting these data into the formulas (109) and (54), the product and coproduct of A D.ti 13(A) read as (a 0 s)(b y) =

E ab(i ) 0 x(20 r(S(x(1))x(3), b(2))

= E ab( i ) O 5 (3)Y(2) r(S(s(2))x (4)1 S (Y (1))*(S (x (Os (5) (a Ø s) = E a( 1 ) X(2) a(2)51 (5(i))5(3)

11 (2) )

X(4)-

The following interesting result emphasizes once more the fundamental importance of Drinfeld's double construction. Statement: The linear map 0 defined by 0(a 0 E ax (1) Ox (2) is a HoPf algebra isomorphism of the quantum double D(A, A; r) = A ix, A (see Subsects. 8.2.1 and 10.2.5) to the Hopf algebra AAd R x B(A). Proof. Using the definitions (115) and (8.21) of the corresponding products

and some properties of the universal r-form r we compute

10.3 Braided Hopf Algebras _

0(a0x)0(bOY)

385

Ek ax( 1) 0 x(2))(bY(i) ® Y(2))

E ax( l )b( l )y( l ) 0 x()y(4) r(S(x(3))x(5)

S(Y(3)))

xr(S(x(2))/(6)7b(2)Y(2))

E ax( i )boo( i )

x(5 )y( 4 ) s(S(x(4) ) x (6) 7 Y(3) )

xr(S(x( 3 ) )x (7) Y(2)*(S(x(2))x( 8),b(2)) =

ax(i)b(i)Y(1)

x(3)Y(2) r(S(x(2))x(4), b(2))

n rr h --(1)-(1)y(1)

X(3) Y(2)

au(2)x(2)y(1)

x(3)y(2) r(x(i) 7b(o)r(x(4), bo

r(x(2), b(2))*(4), b(3))

E e(ab (2)

x(2 ) 0 r(x(i)lb(1))r(x(3)

0 ((a 0 x)(b

y)).

o

b (3)

Here the third equality follows from Proposition 33(0, the fourth from (82) and the sixth from (5). Recall that the quantum double has the tensor product coalgebra structure. Hence, by (116), we obtain

61 (a 0 X) =

=

E ,i(ax( i ) 0 x(2)) E a (l )x (l ) .T(4 )

a(2)x(2)S(x(3))x( 5 )

x(6)

a( l )x( l) 0 x( 2 ) 0 a( 2 )x( 3 ) 0 x( 4 ) = (0 0 0) o ,i(a

x).

Since obviously 9(1 0 1) = 1 0 1 and E o8 = e 0 is an algebra and a coalgebra homomorphism. It is easy to check that 8 -1-(a x) := E aS(x( i )) 0 x( 2 ) is the inverse of 0. Therefore, 0 is a Hopf algebra isomorphism. El A ,

We briefly turn to the dual situation of a quasitriangular Hopf algebra B and state the counterpart to Proposition 42 as Proposition 43. Suppose that X is a braided Hopf algebra associated with a quasitriangular Hop! algebra B. Then, by definition, X is a left B-module algebra. Let a : (b o x) bi> x denote the corresponding left action of B on X and let R. = x 0 yi be the universal R-matrix of B. Then X is a left B-comodule coalgebra with respect to the coaction a'(x) := yi 0 x t x. The vector space X 0 B equipped with the left crossed product algebra structure of X m a B (see Subsect. 10.2.1) and with the left crossed coproduct coalgebra structure of X Na' B is a Hopf algebra, denoted by X >1„B or simply X >1 B. The product, the coproduct and the antipode of this Hopf algebra are given by

Ei

Ei

(x 0 ar)(y Zi(x 0 a) =

Ei

=

E x(a(i) L> y) 0

x ( i ) Yia(i) 0 xi f> x(2) 0 a(2) 7

386

10. Coquasitriangularity and Crossed Product Constructions

sB(Y ) a (2) )xi (x i t> Sx(x)) 0

S(x 0 a) =

where LA(a) = am a( 2) and Ax(x) = notations for the comultiplications of A and X.

S13 (Yia(i) )1

x(i) ® x (2) are

the Sweedler

Example 25 (The Hopf algebra B(8)>4 ad,8). Let B and R, be as in Proposition 43 and let X be the braided Hopf algebra B(8) from Subsect. 10.3.3. In this case a is the left adjoint action adi, (Os = E b ( i ) xst3(b ( 2) ). The product and the coproduct of the Hopf algebra B(8) >ad,, B are determined by (s

,A(x ® a) -=

b) = E xamySB(a( 2))

a) (y

E. E x ( i ) yia(i)

a( 3 )b,

ximx(2)SB(Xi(2))

a(2).

10.3.7 *-Structures on Bosonized Hopf Algebras

The definition of * - structures on quantum spaces of a coquasitriangular *bialgebra depends essentially on the reality type of the universal r-form. In this subsection we treat only the inverse real case which is much simpler than the real case. Proposition 44. Suppose that r is an inverse real universal r-forrn of a *bialgebra A. Let X and y be right A-comodule *-algebras. Then the vector space xoy equipped with the braided product (94) and the involution

(x

y)*

x(0)* ® y (°) *r(y (1) *, x (1) *),

becomes a right A-comodule *-algebra, denoted by

s E X, y E

(117)

3),

x(gy.

Proof. By Proposition 35, xoy is a right A-comodule algebra, so only the assertions concerning the involution have to be proved. Using the assumption that r is inverse real, we compute ((x0y)(z0w))* =

E

z (0)* s (0)* •

(z w)* and

(

* * _

Y )

w (o)*0)* r(w (i)* , x (1)* ) r (w (2)* , z (1)*)

xr(y (1) *, x (2) *)r(y (2) *, z (2) *)f(y (3) *, z (2) *) w (0)* )(x (0)* y (0)* ) r(w (i)* z (1)*) r ( y (1)* 7 s (1)*)

= E(z(0)*

(s

yMw r(y (1) , z (1) ))*

z(0)*x(0)* • w (0)*0)* r (w (1)* y (1)* 7 z (i)* x (i)*) r (y (2) , z (2))

—

(

E(xz(0)

®

y)

*

y (o) ,. (y (1 ) , x (1 )) ,.. (y(2) , y(2) )

y.

)

Hence we have an involution of the algebra xo_y. Next we show that the coaction cpR of A on ev(Iy is *-preserving. Indeed, by (117) and (5), we get

387

10.3 Braided Hopf Algebras

(pR((s

y)t) =

x( o) *

y( o ) *

x (ot y(i)*

so )*

yo) *

y (2)* x(2)* r (y (1)t , x (1)* )

E (s (o)

y(0) ) *

r(y (2)* , x (2)* )

(x (l)0.) ) * = ipR(x 0 y)t.

I=1

The next proposition shows that the involution (117) turns the bosonized Hopf algebra A x X from Proposition 42 into a Hopf *-algebra. Proposition 45. Let r be an inverse real universal r -form of a *-bialgebra A. Let X be a braided bialgebra associated with A such that X is a A-comodule

*-algebra and ,(x*) = ,ix(x)* for x e X, where X0X carries the involution (1 1 7). Then the bialgebra A Ix X from Proposition 42 equipped with the involution (a

x )t = E(a (i) )*

x (°) * r(x (1) *, (a( 2))*),

becomes a *-bialgebra, where dA(a) =

E a( i)

a E

A, x

E X,

(118)

a(2).

Proof. Since the involution (118) is a special case of (117) (by considering A as a right A-comodule *-algebra with respect to the comultiplication), A x X is a *-algebra by Proposition 44. It only remains to show that the counit and the comultiplication of the bialgebra A x X are *-preserving. For the counit this is obvious. For the comultiplication this requires some technical computations which will be omitted here. They are carried out in [Weil. 1:1 As usual, let us identify x E X with x 0 1 E X03) and y c y with 1 o y E xoy. Then it is clear from the definition (117) of the involution that X and y are *-subalgebras of the *-algebra X®Y from Proposition 44. Conversely, formula (117) already follows from that property, since

(x

y)* =

((x 0 1)(1 0 y))* = (1 0 y)*(x 0 1)* = (1 0 y*)(x* ® 1)

E x(0)*

y (°) * r(y(1) *, x (1) *).

In other words, there exists a unique involution of the algebra X®Y which extends the involutions of its subalgebras X and Y. The last assertion is no longer true if the universal r-form r is real rather than inverse real. More precisely, if A is a Hopf *-algebra with real universal rform r and X is as in Proposition 45, then there is no well-defined(!) involution of the algebra A x X which coincides with the involutions on its subalgebras A and X. An example will be seen in Subsect. 10.3.9 below. One possibility of handling this difficulty is to work with the realification 13 = A; r) of A (see Subsect. 10.2.7) rather. than with A itself. Recall that, by Proposition 29, 13 always admits an inverse real universal r-form r", so that Propositions 43 and 44 apply to 13 and r". Details can be found in [Wei].

388

10. Coquasitriangularity and Crossed Product Constructions

10.3.8 Inhomogeneous Quantum Groups

In this subsection we combine Propositions 41 and 42 in order to construct coordinate Hopf algebras of inhomogeneous quantum groups. In what follows let Gq be one of the quantum groups GL q (N), Sid q (N), 0q (N) or Spq (N). By Theorem 9, the Hopf algebra A = 0(Gq ) possesses a universal r-form rz such that rz (u1, u2) = R12 with R := zR, where the parameter z and the matrix R are as stated there. Let us suppose that 0 for q2 + 1 0 for Gq = GLq (N),SId q (N) and (q2 + 1)(€ + Gq = 0q (N),Sp q (N). Then the spectral projector P_ of the matrix I to the eigenvalue -q -1 is of the form f(k) for some polynomial f and the right quantum space X := X(f,R) of A has the defining relations P_x1x2 = O. Note that X = O(C) if Gq = Glaq (N),SL q (N) and X = 0(0,2N) if Gq = 0q (N).

+ zq-1 )P_ = -zq -1 P_ + zq -1 P_ = 0, Since (it + zq-1 )f(ft) = Proposition 41 applies to X with A = zq -1 . Hence X is a braided Hopf algebra associated with the coquasitriangular Hopf algebra ./1- = A 0 CZ. If (zq -1 ) k = 1 for some k G N, it is possible to take the quotient group Z/kZ instead of Z (see the remarks at the end of Subsect. 10.3.5). By Proposition 42, there exists the (ordinary) Hopf algebra .A y X. Definition 12. The Hopf algebra A D< X is called the coordinate algebra of the inhomogeneous quantum group IGq and denoted by O(IGq ).

We are now going to desribe the structure of the Hopf algebra 0(1.GO explicitly in terms of generators. We shall do this 'first in the case where = A 0 CZ and recall briefly the structure of the Hopf algebra A- Y X. As a vector space, .4 y X is A 0 CZ 0 X. We can consider A as a Hopf subalgebra ofAv X by identifying a c A with a 0 1 0 1 E Av X. Likewise, CZ becomes a Hopf subalgebra of .,21 x X and X a subalgebra of „.1 y X by identifying g E CZ with 1 0 g 0 1 and x E X with 1 0 1 0 x. As an algebra, A- ix X is a crossed product algebra of .A and X. Therefore, as noted in Subsect. 10.3.6, it is the algebra generated by the algebras A (and so by A and CZ) and X satisfying the cross relations (112). The coalgebra structure of A y X is that of a crossed coproduct coalgebra of .4 and X and described by Proposition 19. Finally, the coaction (pR and the antipode of A ix X are given by (110) and (111), respectively. Inserting the data of the coaction (0(s) = x i Oulx), of the universal r-form (i.(u i , u2) = zR12, x, x) = r(uji , 1)A -1 = 5ijqz -1 ) and of the braided Hopf algebra (2ix (xi ) = xi 0 1 + 1 0 xi , Ex (x i ) = 0, Sx (x i ) = -xi ) in all these formulas, we obtain the following description of the Hopf algebra 0(IG q ) for the quantum groups Gq SL q (N),0 q (N),Sp q (N). Algebra structure of 0(IGq ):

0(IGq ) has N2 + N + 2 generators uii , xil x, x---1 ) j)./ = 1,2, - • , N. The defining relations are the defining relations of C9(Gq ) for the generators u3i-

(see Sects. 9.2 and 9.3), the defining relations of X for the generators x i (that is, (P_)ijxkx/ = 0 for i, j = 1,2, • • - N) and the following relations:

10.3 Braided Hopf Algebras

61',niklUkX1 i X n Uii = ZuN 7 XUji -- Uji X,

XiX = qZ -1 XXil

XX

-1

389

= X 1X = 1.

Coalgebra structure of 0(/G q ): ,A(uii ) =

4 ®i4, ,i(xi ) = 1 ® xi + xi 0 ti3, x , ,A( x ) = x 0 x , E(u) = Su, E(xi ) = 0,

E(x) = 1.

Antipode of 0(IG q ):

8(u) = 51 0 ( G ,)(uii ) , 51(xi ) = -xiSo( G )(u)x - , S() = x -1 .

Coaction 'PR of 0(1G q ): X is a-right 0(/G q )-comodule algebra with coaction (pR(xi) = Let us emphasize that the preceding formulas already give a complete description of the structures of the Hopf algebra 0(/G q ). One may also work alone with these formulas combined with the fact that they define a Hopf algebra and ignore the braided Hopf algebra approach presented above. For the quantum group G q = GL q (N) it suffices to add a generator D-1 and to require that it is the inverse of the quantum determinant Dq . The corresponding formulas for Dq-1 then follow from the above structures. For example, the commutation relations for x n and uii imply that xn Dq, z NqDq xii , so we obtain Dq-1- x n = z N qxnD q-1 . The new phenomenon for the quantum inhomogenous groups IG q is the appearance of the additional element x which is not present in the classical case. It can be interpreted as a scaling (or dilation) generator which is adjoined as a central element to the homogeneous part. The generator x cannot be avoided if the parameter A = zq -1 is different from 1. Let us give two ar1. First, tracing back the above approach guments for its necessity when A without x or more precisely with x = 1, it would require that X is a braided Hopf algebra associated with A. However, the proof of Proposition 41 is not therein valid for A 1, because the term (R 1 )1(R) =-- (1 - ) )P_ of does not vanish. Secondly, having the above set of formulas, one can set = 1 and try to verify directly that they define a Hopf algebra. Computing (recall that (P_) ilci(xkx/) = 0 by using the formulas ,A(xi ) = 1®x +xi = 1) and xnuii = zk lj uik x / leads to the relations (A - 1)(P_ )flurk'x i = 0 for 1. all j, j, n = 1, 2, • • • , N. This is not reasonable when A Next let us discuss the case when A k Z k q -k = 1 for some k E Z. Then, in our above approach, the group Z can be replaced by ZikZ. In terms of generators this means that we have to add the equation )( k = 1 to the defining relations. For the quantum groups SL q (N), 0q (N) and Spq (N) an equation Ak = 1 with k E Z is only possible if q is a root of unity. However, in the case of GL q (N) the parameter z 0 is arbitrary, so it may be choosen as q. Then we have A = 1 and X is a braided Hopf algebra associated with A 0(GL q (N)) by Proposition 40. Thus, in this case the generator x can indeed be avoided.

390

10. Coquasitriangularity and Crossed Product Constructions

That is, for A = 0(GLq (N)) there exists a Hopf algebra with N 2 + N +1

generators uji , x i , Dq-1 and the defining relations and structure equations as listed above with x := 1. This Hopf algebra can be taken as the coordinate algebra of the inhomogeneous quantum group IGL q (N). The Hopf algebra structure of 0(/Gq ) can be nicely kept in mind if we use the matrix = The above formulas for the comultiplication are obtained from the entries of the product of û by itself and for the antipode from the entries of the inverse matrix S(u) 0

-xS(u)x -I 1) • 10.3.9 *-Structures for Inhomogeneous Quantum Groups As in earlier subsections, we shall distinguish between two cases of parameter values and reality types. First we assume that 11 = izl =--- 1. Let Gq be one of the real forms GL q (N;IR), SL q (N;R), 0q (n,n), 0q (n,n + 1) or Sp q (N; R) from Case 1 in Subsects. 9.2.4 or 9.3.5. Then, by Proposition 10(i), the universal r-form of the Hopf *-algebra A = 0(Gq ) is inverse real. It is easily checked that the Hopf algebra À = A 0 CZ becomes a Hopf *-algebra with the involution defined by (a0xn)* := a* 0 xn, a E A, n E Z, and that X is a right _A1, the universal r-form r), of A is comodule *-algebra. Since IA = also inverse real. Therefore, by Proposition 45 and the subsequent remarks, the Hopf algebra 0(IGq ) = A D.< X is a Hopf *-algebra with respect to the involution determined by the equations (14)* = i4 (x i )* = x i and x* = x. From now on we suppose that q and z are real and we turn to the real forms Uq (N), SUq (N;v), 0 q (N;v) and Sp q (N; v) treated in Case 2 of Subsects. 9.2.4 and 9.3.5. Then the universal r-form r, of A is real (by Proposition 10(ii)) and Proposition 45 does not apply in general. In order to point out the difficulties that occur in this case; let us consider the Hopf *-algebra A = 0(0q (N ; 0) with vi = • = UN = 1 and its right comodule *algebra X = O(0(1; v)); see Subsect. 9.3.5, Case 2. As in the inverse real case above, we define the involution on A by (a xn)* := a* 0 xn. Then A is a Hopf *-algebra and X is a right sa-comodule *-algebra. The universal r-form r), of A is real, because A = zq -1 is real. However, there is no involution of the algebra A y X which extends the involutions of and X. Assume on the contrary that such an involution exists. Recall that by .definition ()* = 8(u-ii ) and (xn )* = q - P-sni. Applying the in,

volution to the equation xnuii = A nkli ulxi, then multiplying by tir on the left and by t4 on the right and finally summing over j and i, we get sniq-Pnuran = Atturxilq -P1 - Comparing the latter with the relation

10.3 Braided Hopf Algebras

xretesn = A 37.41,18 urx1i (note that the elements urx/t

391

ur 0 1 0 xi/ E A 0 X

kt8li en- =

for all j,1, n, s. This are linearly independent), we obtain contradicts formula (9.30). Thus, the involutions of the *-algebras A and X cannot be extended to an involution of the algebra A y X. One way to circumvent this difficulty and to still get a Hopf *-algebra is to "double" the generators of the translation part. This procedure will be carried out in the rest of this subsection. Let XR, denote the algebra X( f , g, h, 7; R) from Subsect. 9.1.2 with f (t) g(t) = t - q, h(t) = qt, = 0. That is, XRe has 2N generators xi, • • •/ yN and the defining relations (9.11) of XRe read as y17 • • • 7

P lk

Irki ii XkXi = (1XiX j

qyiyi,

Pi k

xkyi = q

-1

YiX j •

(119)

Let X and Y denote the algebras with generators • • , xN resP• yi, • • • , Y N and defining relations klxkx/ = qx jx j resp. Wykyt = qyiyi• Clearly, X and 3) are right A-comodule algebras with coactions given by the formulas çoR (xi ) xj x and ço R (yi ) = yi S(uii ) 0x -1 . Then the algebra XR, is nothing but the braided product algebra X®Y with product defined by (94) with respect to the universal r-form (i.) 21 of „4, where î is given by (106) and is the convolution inverse of f. Indeed, the cross commutation relations of the braided product algebra are Yij

= xicYi (021(S(t4) x -1 , ui; x) = qz -l xkY/

uit ) = qkixkY/ •

This is just the third group of relations in (119). The algebra XRe becomes a *-algebra with involution determined by x i* := viyi, where vi are the numbers in the definitions of St/q (Ar, v), 0q (N; v) and Spq (N; v) and vi := 1 in the case of Uq (N). To prove this assertion, it suffices to check that the relations (119) are invariant under this involution. Using the facts that ki = this is easily done. In the special case = • • • = vN = 1 the elements x i and yi are interpreted as holomorphic and anti-holomorphic generators, respectively, and the *-algebra XRe may be considered as a realification of X (see Subsect. 10.2.7). Obviously, the algebra A is also a *-algebra with involution (a a* 0 a E A, n E Z. Recall that, by Proposition 9.5, XRe is a right quantum space of A. Hence XRe is also a right quantum space of A with coaction çoR given by

ki

(PR(xi) =

0

u3i

X/ 'PR(Yi) = yj 0 S(u) 0 x -1 .

From the relations x = viyi ,.(q)* = vi vj S(uii ) and x* --= x' we conclude that çoR(x) = (p R (si )*. Therefore, XRe is a right À-comodule *-algebra. Next we turn XRe into a right À-module algebra. Since i and (ii )21 are universal r-forms of it follows from Proposition 4 that X and y are right À-module algebras with actions defined by

392

10. Coquasitriangularity and Crossed Product Constructions

s
à E .;49

)

(120)

respectively. Hence the vector space X 3) is a right À-module with respect to the action (121)

X®Y even becomes a right We shall show below that the algebra XR, À-module algebra in this manner. Now we consider the right crossed product algebra A < XRe of the Hopf algebra .;4. with its right module algebra XRe . Recall from Subsect. 10.2.1 that A y XRe is generated by the algebras  and XRe subject to the cross commutation relations xa= E a(i) (x i 0 (2) ). From (120) we get Sn Z1 t

yn

4 = xii(Uin x l u 0 1) -= x1 r(u/r„ uji ) =

= yi

= yi z

0 1, S(u) 0 x -1 ) = yi

xn x

xn A =

yri x =-- yriqz -1 .

Hence A ix XRe is generated by the algebras .A and XRe with cross relations

0" ki xn u_;. = zrxukxt,

yn uii =zkt4yj , x n x = qz - 1 xx„,

gnx --= qzxy

(122) for the generators. The involutions of ./4 and XRe extend uniquely to an involution of ./zi- ix XRe such that A ix XRe becomes a *-algebra. Indeed, it is not difficult to show that the cross relations (122) are invariant under the involutions. Proposition 46. The *-algebra A XRe defined above is a Hopf *-algebra

with comultiplication, counit and antipode determined by 2i(u3i.) = ut ul; , =-- 1

A(X) = X Xl

yi + yiS(u i )x -1 , a6(x i ) = 1

xi + xi ® u3i x,

E(u) = ii, E(x i ) = e(yi ) = 0, E(x) = 1, S(uii ) =

So(Go (tei ),

S(y) = -yi S,)(G,) (u))x,

S(x) =

S(x i ) = -xi So(Go (u3i )x -1 .

Proof. The proof follows by similar reasoning as Proposition 9.1. We do not carry out the details. D The Hopf *-algebra A y XR, from Proposition 46 is called the coordinate Hopf *-algebra C9(/G q ) of the real inhornogeneous quantum group IGq , where Gq is one of the real quantum groups Uq (N), SUq (N; 0,0q (N; v) and Spq (N; z'). The structures of the Hopf *-algebra CA/G q ) can be summarized as follows:

10.3 Braided Hopf Algebras

393

As an algebra, 0(.1Gq ) admits the generators u), x i , yi ,x,x -1 . The defining relations are the defining relations of 0(Gq ) for uii , the relations (119) and (122) for x i and yi and xx -1 = =1. The involution is given by the equations (u,i )* = 11i vi So (Go(u1), x = vi yi and x* = x'. The coalgebra structure and the antipode are determined by the formulas in Proposition 46. Knowing that .;4 y XRe = 0(IG q ) is a Hopf *-algebra (by Proposition 46), some of these formulas can be derived from others by means of axioms or properties of a Hopf *-algebra. For instance, by applying the involution the second relation of (119) leads to the first one, the first and the third relations of (122) yield the second and the fourth one, respectively, and the formulas of 2i(xi), E(xi) and S(z) imply the corresponding formulas for yi . The preceding constructions become clearer if they are reviewed in terms of the Hopf *-algebra A with respect to of the realification L = DO, .,4; the real universal r-form i (see Subsect. 10.2.7). This will be sketched below. First recall that, by Proposition 29, 8 admits an inverse real universal r-form r" = - 41- 31- 24- 23. Obviously, X and 3) are right 8-comodule algebras = E y(0) 0 y(1) 0 1, x (°) 1 Ox(1) and with coactions given by 0(x) = where ço(x) = E xo) x(') and ço(y) = E y (° ) 0 y (1 ) denote the coactions of sa on X and y, respectively. Since r" (Et 0 1, 1 0 6) = 21 (â, 6) for FL, b E the braided product algebra XRe = x0y with respect to ,21- and 1-21 coincides with the braided product algebra X03) with respect to 8 and r". Therefore, by Propositions 34 and 4, XRe is a right 8-comodule algebra with coaction 0(x 0 y) =

x(0)

(123)

y(°) 0 (1 0 x (1) )(y (1) 0 1)

and so a right 8-module algebra with action (x

y) a

(a ® b) := E ,(0)

y (o)e ((i

x(i) )(y (i) 0 1 ) , 0 g) .

Hence XRe is a right Â-module algebra with action (x0y)a c- i := (x0y)<(ã01). For elements x x 0 1 E X and y10yEY this is just the action of ;1 on X and 3) given by (121). Therefore, it follows that this action of À on XRe coincides with the action of di' on X ® y defined by (122). In particular, as promised above, we have proved that XRe is a right À-module algebra. As we have seen above the involutions of ,2( and X do not extend to a well-defined involution of the algebra ;ix X, while the involutions of A and XRe becomes a *-algebra lies XRe do for ;iv XRe . The deeper reason why in the facts that XRe is a 8-comodule *-algebra with coaction //) and

(a,

w)* = E(a (1) )*0w(0)*rii(w(1)*, (a (2) )* ® 1), à c À

,

wc

XR, , (124)

where 1P(w) E wo) w(1) is the Sweedler notation for the coaction //) of on XRe given by (123). Formula (124) shows that the involution of ;iv XRe can be obtained in a similar manner as in Proposition 44 by means the inverse real universal r-form r" of the larger Hopf *-algebra 8. In order to prove

394

10. Coquasitriangularity and Crossed Product Constructions

ale

if it holds for a' formula (124), one first checks that it is true for a = and for a" and that it is valid for z = z'z" if it holds for z' and for z". Then it suffices to show (124) for the generators of A" and XRe We omit these verifications. In the cases where à = uji 0 1 and z = x n, 0 1 resp. z --= 1 0 y n, equation (124) turns out to be equivalent to the first two defining relations (122). .

10.4 Notes Coquasitriangular Hopf algebras appeared implicitly in the announcement [Lyn and more explicitly in [Maj6]. They were first studied in detail by R. G. Larson and J. Towber in [I,To] and by T. Hayashi in [H21. Theorem 7 can be found therein. Other early papers on coquasitriangular Hopf algebras are [Maj7], [Si], [Doi]. The theory of L-functionals for the quantized simple Lie groups was developed in [RTF]. Our Subsect. 10.1.3 deals with the general case of coquasitriangular Hopf algebras and contains a large amount of new material. Crossed products are standard constructions in the Hopf algebra literature (see [Mon] and the references therein). The twisting of Hopf algebras was studied by V.G. Drinfeld {Dr5]. The approach to the quantum double via twisting is taken from [DT2]. Double crossed products of Hopf algebras were introduced in [Maj2]. Realifications of quantum groups have been investigated in [Pod4], [Zak], [DSWZ], [Maj8], [WA and others. Most of the constructions and results of Sect. 10.3 are due to S. Majid {Maj5], {Maj8-10]. The results on braided Hopf algebras and on transmutation of Hopf algebras can be found in [Maj9j, {Maj8], while the bosonization theory VMS developed in [Maj10]. Hopf algebras in categories occurred much earlier (see [MM], [Pa]). Inhomogeneous quantum groups were introduced and studied in [SWW1], [SWW2], [Maj8], [OSWZ], [Dt2]. Our main reference for Subsect. 10.3.9 is [Wei].

11. Corepresentation Theory

and Compact Quantum Groups

The material in this chapter is related, directly or indirectly, to the corepresentation theory of Hopf algebras. In Sects. 11.2 and 11.3 we investigate two classes of Hopf algebras resp. Hopf *-algebras, called cosemisimple Hopf algebras resp. compact quantum group algebras, which are determined by their irreducible resp. unitary corepresentations. The second class leads at once to Woronowicz' compact quantum group C*-algebras studied in Sect. 11.4. In Sect. 11.5 the irreducible representations of the quantum group GL q (N) when q is not a root of unity are classified. The final Sect. 11.6 gives an introduction to quantum homogeneous spaces. In this chapter we retain the summation convention to sum over repeated indices.

11.1 Corepresentations of Hopf Algebras In this section the basics of general corepresentation theory are developed. For the convenience of the reader, this section is self-contained and some notions which have already appeared in Chaps. 1 and 4 are repeated. If not stated explicitly otherwise, A denotes a coalgebra.

11.1.1 Corepresentat ions First we recall the following definition. Definition 1. A corepresentation of A on a complex vector space V is a linear map ço : V V 0 A such that (id 0 .6) o ço = ((p 0 id) o (p and (id 0 e) o = id. In other words, that (p is a corepresentation of A on V means that V is a right A-comodule with right coaction (p. We want to give some reformulations of the conditions in Definition 1 and assume that the vector space V is finite dimensional. First we introduce some more notation. Recall that L(V) and L(V,V 0 A) denote the vector spaces of linear mappings of V to V and of V to V 0 A, respectively. For an element = E .C(V) 0 A with Ti E L(V) and ai E A, we define -

Ei

396

11. Corepresentation Theory and Compact Quantum Groups

= Ei Ti (.) 0a

V O A). Since V is finite-dimensional, the mapping (;-(3 ça is an isomorphism of the vector spaces E(V) ® A and E(V,V A). Let (p E £(17,V 0 A) and let ei , j = 1, 2, • • • , n, be a basis of V. Then there exist uniquely determined elements vii E A, j j = 1, 2, • • • , n, such E L(V,

,

that ço(ei) --= e 0 vii, j = 1, 2, • • • , n. The elements yii are called the matrix coefficients or the matrix elements of ço with respect to the basis {e}. For f C A', we define a linear mapping ça(f) C L(V) by setting 43(f) = (id f)(p. Recall that the dual A' of A is an algebra with product fg = (f O g),A and unit element E. Proposition 1. Let V be a finite-dimensional vector space and let ça E E(V,V 0 A). Then the following conditions are equivalent: (0 (id 0 4) o ça (cp 0 id) o ça. ° = 012013 • (ii) (id 0 (iii) g.fg) = gfYgg) for f, g C A'. = Vik O Vkj for j, j = 1,2, • • • , n. (iv)

Proof. The proof follows easily from the corresponding definitions combined with the coassociativity axiom. Proposition 2. If V is a finite-dimensional vector space and ça C E(V,V OA) satisfies the conditions in Proposition 1, then the following are equivalent:

e) 0 = id. (ii) ker ça = (iii) c,O(E) = id. (iv) e(v ii ) = 60 for i, j = 1, 2, • • • , n. If A is a Hopf algebra, then these conditions are also equivalent to: (y) (T) admits a left inverse in the algebra .C(V) 0 A. (vi) (:6 is invertible in the algebra L(V) 0 A and 0-1 = (id 0 S)0 . (i) (id

Proof. Since fo(E)(*) = AO by Proposition 1(iii), fo(E) is a projection. If x C ker fo(E), then ‘,O(f)(x) = gfE)(x) = ‘,S(f)(7)(e)(x) = 0 for any f E A', hence ça(x) = O. Thus, ker fo(E) = {O} and 'OW = id. (iii)—(vi) for a Hopf algebra A: Let Ø = Ei T, 0 a" with Ti c .C(V) and a' G A. The equation (id 0 0 ç = 012013 by Proposition 1(u) yields the identity Ei E T 0 ai(1) O ai(2) = Ei TTi 0 a 0 Applying id 0 m(id 0 S) to both sides, we obtain OW 0 1 = Ei TT Since ç(E) = id by (iii), this means that (id 0 S)0 is a right 'inverse of Ø in the algebra (V) ® A. Replacing the mapping id® m(id O S) by id 0 m(S 0 id) in the preceding, it follows that (id 0 S)0 is also a left inverse of 0. (ii) are easily verified. The (ii) and (IT) (iv) The implications (i ) equivalence (i) 4-+ (iii) is just the definition of AO. (vi) ---0 (v) is trivial. A matrix y =

of n2 elements vii E

A satisfying

= vik vki and e(vii ) = 1, i j = 1,2, • • • , n, ,

11.1 Corepresentations

of Hopf Algebras

397

is called a matrix corepresentation of A. By Propositions 1 and 2, such a matrix y defines a corepresentation w on an n-dimensional vector space V with b asis ei , j = 1, 2, • - • , n, by (p(ei ) = ei vij . Sometimes we abuse language and shall not distinguish between corepresentations and corresponding matrix corepresentations. 11.1.2 Intertwiners

Let w and be corepresentations of A on finite-dimensional vector spaces V and W, respectively, and let y (vii ) and w = (wii ) be the corresponding matrix corepresentations of A with respect to bases e i , i = 1, 2, • • • , n, and fk k = 1, 2, - • - , m, of V and W, respectively. Let T : V —› W be a linear mapping. We write T(e) = Ek Tkifk• Proposition 3. In the above notation, the following are equivalent:

(T id) o = 0 T (ii) (T 0 id) o =?:/.) o (T 0 id). (iii) T43(f) = 7:b(f)T for f E A'. (iv)Ei TkiVij = Ei WkiTii for j = 1, 2, • - • , n and k = 1, 2, • • , m. Proof The proof is straightforward and is omitted. Definition 2. A linear mapping T : W satisfying the conditions in Proposition 3 is called an intertwiner of (p and O. The vector space of all such operators T is denoted by Mor ((p,0). We set Mor ((p) Mor (ça, p).

The space of matrices satisfying condition (iv) is denoted by Mor (y, w). We write Mor (y) := Mor (y,w) if V = W, ei = fi and yii = wii . Note that Mor ((p) and Mor (y) are subalgebras of .C(V) and Ma (C), respectively. Two corepresentations w and ?,/) of A are said to be equivalent if there exists an invertible intertwiner of w and 7,b. In this case we write ça 7,b Remark 1. Those parts of Propositions 1-3 which do not contain the mapping ço` remain valid for infinite-dimensional vector spaces. The corresponding matrices (yii ), (wki) and (Tk i ) have the property that for a fixed second index all but finitely many terms vanish, so all occurring sums are still finite. ,L 11.1.3 Constructions of New Corepresentations

The direct sum ED, w, of a family { w,} of corepresentations of A is defined in the obvious manner as the direct sum of the linear mappings wv . a) Tensor product of corepresentations: Let (p and ?,/) be corepresentations of a bialgebra A on vector spaces V and W, respectively. Then the mapping w00:=(idOid0m)(id0T0id)(w00): VOW--V0W0A defines a corepresentation of A on V 0 W. It is called the tensor product of w and ?,/). In Sweedler's notation, we have

398

11. Corepresentation Theory and Compact Quantum Groups ((p //))(x 0

y) =

Es (o) y(o) 0 x(i)Y(i),

xEVI yEW.

Retaining the notation of Subsect. 11.1.2, the matrix corepresentation of the tensor product 4,o 0 V) with respect to the basis { ei fk } of V 0 W is given by y w ((y 0 w) ik,i i) = (viiwki). Clearly, the tensor product of corepresentations is associative. If A is noncommutative, it is no longer commutative in the usual sense that the flip provides an equivalence of ça 0 and 00 cp. (This may be seen by comparing the corresponding matrix elements.) However, if A is a coquasitriangular bialgebra, Proposition 10.1 (i ) states that (p and V) (p are equivalent. b) Conjugate corepresentation: Let ça be a corepresentation of a *coalgebra A (that is, A is a coalgebra and a *-algebra such that 41 and E are *-preserving) on a vector space V. Let V be the complex conjugate vector space of V, that is, V consists of elements 1-3 where x c V, and the vector space structure of V is determined by 7 --Ey := x + y and AT :=7\x, x l yE V, A E C. Put 7 := (— *)(p, where — and * denote the mappings x and V 0 A is a corepresentation of A on V a -4 a*, respectively. Then rp : V and the corresponding matrix corepresentation is f) (Tii ) =((yini )*). c) Contragredient corepresentation: Let cp be a finite-dimensional corepresentation of a Hopf algebra A on V. For T E L(V), let T t E L(V') denote the transposed mapping of T, that is, (rx')(x) = x' (Tx) for x' E V', x E V. If (0 --= Ei Ti 0 ai with Ti E L(V) and ai E A, define (Pjc := Ei Tit S(a i ) E L(V') 0 A. The corresponding map (pc E 0 A) is uniquely determined by the requirement (pc(x')(x 0 id) = (x' ids)(p(x), x' c V', x E V. If ça is represented by the matrix y = (yii ) with respect to a basis of V, then (pc is obviously given by the matrix yc --a- ((vc) ij ) = (51 (v)) with respect to the dual basis of V'. Then (pc is a corepresentation of A on V'. (The easiest way to see this might be to check the conditions (iv) in Propositions 1 and 2.) We call cpc the contragredient corepresentation of ço. 11.1.4 Irreducible Corepresent at ions

Let ça be a corepresentation of A on a vector space V. A linear subspace W of V is called cp-invariant if ça(W) C W A or, equivalently, if (33(f)(W) C W for all functionals f E A'. Then the restriction of (p to W is obviously a corepresentation of A on W, called a subcorepresentation of ça. An element xE V is said to be (p-invariant if (p(x) = x 0 1 or, equivalently, if 45(f)(x) = f(1)x for all f E A'. Definition 3. A corepresentation ço of A on a (not necessarily finitedimensional) vector space V is said to be irreducible if (p has only the trivial invariant subspaces W = lol and W = V . A coalgebra A is called if it has no nonzero proper subcoalgebra.

cosimple

Remark 2. In the Hopf algebra literature, the irreducible corepresentations are the simple right comodules and a cosimple coalgebra is usually called a

11.1 Corepresentat ions of Hopf Algebras

399

simple coalgebra. However, the name "cosimple coalgebra" seems to be more appropriate for the latter, because it can be shown that a coalgebra A is cosimple if and only if A' is a finite-dimensional simple algebra. A Next we prove two finiteness results. Proposition 4. (1) If (p is a corepresentatian of A on V, then any element E V is contained in a finite-dimensional (p-invariant subspace. (ii) Any element a E A belongs to a finite-dimensional subcoalgebra of A. Proof. (i): Let {ci } be a basis of the vector space A. Write (p(x) = Ei x i 0 ci and ,6(ci ) = >j , k 'YijkCj Ck with xi E V and -yij k E C. (All sums in this proof are finite.) Comparing the coefficients of ck in the identity

E

(p(xk) ck = ((p id)(p(x) = (id 0 d)(p(x) =

E

xi 0 -yiikCi

Ck

we conclude that (p(xk) = E i xi 0 -yiikci. Thus, W := Lin {x, xi} is a finite-dimensional ça-invariant subspace. ' (ii): By (0 applied to V = A, = 4 there is a finite-dimensional subspace W of A such that a E W and d(W) C W0A. Let feif be a basis of W and let ,6(ei ) e i 0wii . Since d(wii ) = wik owki by Proposition 1, 8 = Lin {6,, wii } satisfies ,6(13) C B B and a E B. Corollary 5. (i) Each irreducible corepresentation is finite-dimensional. (ii) Any cosirnple coalgebra is finite-dimensional.

The following very simple but very useful result, called Schur's lemma, will often be used in the following. Proposition 6. Let (p and V) be irreducible corepresentations of A. Then:

(i) If (p and 1,I) are not equivalent, then Mor ((p,i,b) = (ii) If (p and lib are equivalent, then dim Mor (ço, 7,1)) = 1 and any T E Mor ((p, lib) is either 0 or invertible. In particular, Mor ((p) = C • I. Proof. The proof follows immediately from the facts that ker T is ça-invariant and im T is 1,1)-invariant for any T E Mor ((p , . Proposition 7. Any irreducible corepresentation of A is equivalent to a subcorepresentation of the corepresentation d on A.

Proof. Take a functional f E V', f O. One easily verifies that T :=-- (f Oid)ço belongs to Mor ((p, ,6). Since eaT=f, we have T O. Since ker T is a (lainvariant subspace, T is injective and provides an equivalence of (p and the restriction of ,6 to its invariant subspace im T.

Let (p be a corepresentation of A on an n-dimensional vector space V and let du = (vij ) be the corresponding matrix corepresentation with respect to some basis of V. Then we have Lin {(x'

id)(p(x) I x' E

V i ,x

I f E L(V) 1 } Lin {vii I i , j = 1,2, • • • , n}.

e V} = Lin {(f =

id)

400

11. Corepresentation Theory and Compact Quantum Groups

From the latter and Proposition 1(iv) we see that this vector space is a subcoalgebra of A. It will be denoted by C(w) or likewise by C(v). Definition 4. 409) (resp. C(v)) is called the coefficient coalgebra of the corepresentation p (resp. of the matrix corepresentation v). If w and ti) are equivalent corepresentations, then C(99) --= C(0). Further, we have = C(cP) + C(0), C(V 0) C(4 0)C(0), COP e

C ( ,7) = C( o)* , C(wc) = S(C((p)). For the second formula we have to assume that A is a bialgebra, for the third that A is *-coalgebra and for the fourth that A is a Hopf algebra. The next proposition shows that the cosimple subcoalgebras of A are precisely the coefficient coalgebras of irreducible corepresentations of A. Proposition 8. (i) A corepresentation ri9 of A is irreducible if and only if the coalgebra C(p) is cosimple. Any cosimple subcoalgebra of A is of the form C((p) for some irreducible corepresentation of A. (ii) Let (09 be an irreducible corepresentation of A on an n-dimensional vector space V. Then the corepresentation ZIC ( p ) : C((p) —> C((p) 0 A of A on C(p) is equivalent to a direct sum of n corepresentations (p. In particular, dim e(cp) = n2 and the matrix elements of (09 with respect to any basis of V form a vector space basis of C((p).

Proof (0: If

is the restriction of (,o to a nontrivial, invariant subspace of w, then is a nontrivial subcoalgebra of C(w). So C(w) is not cosimple if (p is not irreducible. The converse will follow from the proof of (ii) given below. Let 13 be a cosimple subcoalgebra of A. Since B is finite-dimensional by Corollary 5, the corepresentation LIB : B BOA of A on B admits a nonzero invariant subspace E which does not contain any proper invariant subspace. Then the restriction o of Zi8 to E is an irreducible corepresentation of A and C(w) is a nonzero subcoalgebra of B. Since B is cosimple, we have Op) = B. (ii): The proof consists of showing that V' ® V considered as a twosided (that is, right and left) comodule Qf A is irreducible. Let e l , • • • , en and ell , • • • , en be bases of V and V', respectively, and let w(e i ) = e i vii. Define right and left A-comodule structures on V' 0 V by wR (e'i ei ) = e: ek Vkj and (pi, (ei ej ) = vi k ® 4 ® ej , respectively. Since LIcep) (vii) = vik Vki, the C(v) given by T (e"i ej ) = vii intertwines (PR linear mapping T : V' 0 V (resp. cpL) and the comultiplication Zle(,p). Hence k := ker T is an invariant subspace for (pR and wL. 101. Take a We prove that k = {0}. Assume on the contrary that k nonzero wR-invariant subspace ?Co of IC such that the restriction of wR to k o is irredutible. Define linear mappings Ai : V' 0 V —> V by Ai(ei x) = 6iix, x E V. Then the restriction Bi of Ai to /C0 belongs to Mor (îP, cp), so Bi is either zero or bijective by Schur's lemma. Since ko {0}, Bk 4 0 for some k. Then BiBk-1 E Mor (ç9), so that BiB k-1 = Ai/ with A i E C again by

op)

11.1 Corepresentations of Hopf Algebras

401

Schur's lemma. Hence any nonzero y E Ko is of the form y = Ei Bi (y) O. Since IC is invariant (Ei Ai e'i ) Bk(y) =: x with x' O and x under (p R and (pL , it follows that IC D x f V and iC D VI x 7 so /C =-V' V and T = O. The latter implies that = 0, which is a contradiction. Thus, we have proved that IC = ker T = 101. Hence T is bijective and provides an equivalence of a direct sum of n corepresentations ço and the corepresentation 2i e ( 1 ). Moreover, the elements vii constitute a basis of C(cp). Therefore, the dual algebra C(cp)' is isomorphic to Ma (C) (in fact, the dual basis to {yii } corresponds to the matrix units). Thus, C((p)' is simple and hence C(cp) is cosimple. Corollary 9. If c,o and V) are irreducible corepresentations of A, then either C(cp) = C(tP) and ço and 1,b are equivalent or C(v) n C(p) = {0} and cp and V)

are not equivalent. {0}. Since C(cp) n c p) is a subcoalgeProof. Assume that C(o) n bra of the cosimple coalgebras C(c,o) and C(P), it follows that C(v) = Thus, 2ic ( ip) and 2ic(p) are equivalent corepresentations, hence cp and 1,b are equivalent by Proposition 8(ii). (

Corollary 10. Let {çoa

Ic

E I} be a family of pairwise nonequiva-

lent irreducible corepresentations of A with matrix corepresentations {v = E I}. Then the set {v i j = 1, 2, • • • , dcr , a E I} of matrix elements is linearly independent. ,

Proof. The proof follows from Proposition 8(ii) and Corollary 9. 11.115 Unitary Corepresentations In this subsection we suppose that A is a Hopf *-algebra.

Let V be a finite-dimensional Hilbert space with scalar product (•, •) conjugate linear in the first variable and linear in the second one. Then L(V) and L(V) 0 A are *-algebras. Let cp : V V 0 A be a corepresentation of A on V and let y = (vii ) be the corresponding matrix corepresentation with respect to an orthonormal basis fed of V. Set y* ((e) ii ) = ((v3i)*). Proposition 11. The following conditions are equivalent: (x(i)) * x, y E V. (i) E(x, NON') = E(x(o), (ii) E(x ( 0 ) , y(0))x(i ) * y( i) = x, y E V. (iii) * = * I in the *-algebra L(V) A. (iv) `0*(,O = I (resp. cPc,a* = I) in the *-algebra _C(V) 0 A. (v) v*v = vv * = I (that is, S(v ii ) = evii r for all i, j). (vi) v*y = I (resp. vy* ).

Proof. If eki E L(V) are the matrix units defined by eki(ei) = bikei, then (P can be written as = E i eii vii. From this we immediately obtain that (Hi) (v) . and (iv) (vi). Setting x = ei ,y = e• into (i) yields vij = so v*v = vv* = I. Conversely, the relations v u = S(vii)* imply (i). Thus,

402

11. Corepresentation Theory and Compact Quantum Groups

(v). Similarly, (ii) is equivalent to the equality vv = I. Multiplying (i) v*v = I (resp. vv* = I) by 8(v) on the right (resp. on the left), we obtain v* = 8(v) and so vv = vv* = I. This proves (N) 4-+ (vi). Definition 5. A corrpresentation cp of A on a finite-dimensional Hilbert space V is called unitary if it satisfies the conditions in Proposition 11. A corepresentation cp on a finite-dimensional vector space V is said to be unitarizable if there exists a scalar product on V such that (p is unitary.

Let v =(vii ) i,j , i ,...,n, be the matrix corepresentation of a corepresentation (p of A on a vector space V. Then y is unitarizable if and only if there exists an invertible matrix A E Mn (C) such that w := AvA. -1 satisfies the equality S(w) = w* or, equivalently, if BS(v)B -1 = v* with B = A t A. It is clear from Proposition 11 that direct sums, tensor products and subcorepresentations of unitary (resp. unitarizable) corepresentations are again unitary (resp. unitarizable). However, contragredient and conjugate corepresentations of unitary corepresentations are not necessarily unitary. Proposition 12. Any unitarizable (in particular, any unitary) corepresentation is completely reducible, that is, it is a direct sum of irreducible corepresentations.

Proof. It suffices to show that for a unitary corepresentation y the orthogonal complement W I of a y-invariant subspace W in V is also y-invariant. To prove this, let x E W and y E WI. Write (p(y) = yi ai such that fail is a linearly independent set. Since W is y-invariant and y E , the right hand side of the equation in Proposition 11(i) vanishes, so E(x, yi )ai = O. Hence, (x, yi ) = 0 for arbitrary x E W which implies yi E

11.2 Cosemisimple Hopf Algebras This section is concerned with Hopf algebras which are spanned by the matrix elements of their irreducible corepresentations. 11.2.1 Definition and Characterizations

First we put the subsequent considerations in the proper algebraic context and give Definition 6. A coalgebra is called cosemisimple if it is the sum of its cosimple subcoalgebras (that is, of coalgebras which have no nonzero proper subcoalg eb ras

Example 1. Any group algebra CG is obviously cosemisimple.

A

Example 2. The universal enveloping algebra U(9) of a Lie algebra g {0} is not cosemisimple, because its only cosimple subcoalgebra is C•1.

A

11.2 Cosemisimple Hopf Algebras

403

Example 3. For the Hopf algebra of polynomial functions on an algebraic group G cosemisimplicity means that G is linearly reductive, that is, the finite-dimensional polynomial representations of G are completely reducible (see [Ho], Sect. V.3). A large number of classical groups such as SL(N, C), SO(N, C), Sp(N, C) has this property. A Example 4. If q is transcendental, the Hopf algebras 0 (G L q (N)) , 0 (S L q (N)) 0 (0q ( N)) , 0 (Sp q (N)) are cosemisimple (see Subsects. 11.2.3 and 11.5.3). A

A be a coalgebra. Denote by Â. the set of equivalence classes of irreducible corepresentations of A. For any a E A , we fix a corepresentation (,oa Let

from the class a and a basis of the underlying space. Then (pa is given by a matrix corepresentation ua =-- (UZ)i,j,_1,...,d a,. Recall from Subsect. 11.1.4 that irreducible corepresentations are always finite-dimensional and that the coefficient coalgebra C(cp') = C(P) = Lin fuZ I j, j = 1, 2, • • • , c/} depends only on the class a, but not on the choice of the representative (pa E a and of the basis. Definition 7. A linear functional h on A is called left-invariant (resp. rightinvariant) if for all a e A,

(id 0 h)A(a) = h(a)1 (resp.

(h

id)A(a) = h(a)1).

(1)

Equation (1) means that h intertwines the left (resp. right) comodule A with respect to the comultiplication and the trivial comodule C - 1. Clearly, a functional h on A is left-invariant (resp. right-invariant) if and only if f h = f (i.)h (resp. h f = f (1)h) for all functionals f e A'. In the Hopf algebra literature such functionals h are called left (resp. right) integrals on -

A

(see [Mon], Definition 2.4.4). A number of characterizations and properties of cosemisimple Hopf algebras are collected in the following theorem. Theorem 13. For any Hopf algebra A the following statements are equivalent:

(i) A is cosemisimple. (ii) Every corepresentation of A is a direct sum of irreducible corepresentations. (Hi) There exists a left-invariant linear functional h on A such that h(1)=1. (iv) There exists a unique left- and right-invariant linear functional h on A satisfying 141) =1. (y) A is equal to the sum of all C(u), a E A. (vi) The set {uZ I a G A, j=1,2, • -•,d 4,} is a vector space basis of A. If A is a cosemisimple Hopf algebra, then the antipode S of A is bijective and we have S 2 (C(u")) = qua) for any a G Â.

Proof.

The equivalence of (i) (iv) is Theorem 14.0.3 in [aw]. By Proposition 8(1), the cosimple subcoalgebras of A are precisely the coefficient spaces —

404

ii. Corepresentation Theory and Compact Quantum Groups

C(0), c E Â. Hence we have (i)4-qv). (v)—(vi) follows from Corollary 10. (vi)— ( v) is obvious. The last assertion is proved in [La] , Theorem 3.3. The crucial implications of the theorem will be proved in the next subsection. ID Let A be a cosemisimple Hopf algebra. The unique left- and right-invariant linear functional h such that 1 ( 1) = 1 from Theorem 13(iv) is called the Haar functional of A. By (v) and (vi), we have

A =

e

aE..4

(2)

C ( ua).

The decomposition (2) is referred to as the Peter-Weyl decomposition of A.

11.2.2 The Haar Functional of a Cosemisimple Hopf Algebra The existence and uniqueness of the Haar functional is a crucial property of a cosemisimple Hopf algebra. Because of its importance we carry out the proof of the relevant implication (vi) --+ (iv) of Theorem 13 and obtain additional information in this way. (In the above proof we referred only to [Sw].) Suppose that (vi) is fulfilled. Then A is the direct sum of the coefficient coalgebras C ( P). Let a = 1 denote the trivial one-dimensional corepresentation of A with matrix element 1. We define a linear functional h on A

by h(1) = 1

and

h(a) = 0 for a E C(0), a E

:Al a

I•

(3)

Since ,I(C(u)) C C(0) 0 C(0) for each a E Â, h is obviously left- and right-invariant. This proves the existence assertion of (iv). For the uniqueness, let h denote a left-invariant linear functional on A such that h(1) = 1. Then we have E awit(a(2)) = h(a)1 for a E C(0). Since LX(a) E C(0) g C(ua) and 1 C(u) if a 1, it follows that iii(a) := 0 for a E OM, a 1. That is, h is of the form stated in (3). The same reasoning proves the uniqueness of a right-invariant linear functional h on A satisfying /41) = 1. This finishes the proof of the implication (vi) (iv) in Theorem 13. Since the antipode S of A is injective, one easily verifies that h(S) := WO) is also a left-invariant functional on A. Since h(1) = 1, the uniqueness of h yields h = h. That is, we have

h(S(a)) = h(a),

a C A.

(4)

Note that the condition 141) = 1 in Theorem 13(iv) is essential. Indeed, for any finite-dimensional Hopf algebra there exists a unique (up to complex

multiples) left-invariant linear functional hi (see [Mon], 2.1.3), but hi is not right-invariant in general (see Example 5). But if kW 0, then the Hopf algebra is cosemisimple by Theorem 13(iii) and hi is right-invariant as well. Example 5 (Sweedler's Hopf algebra). Let us retain the notation of Examples 1.9 and 10.2. One easily verifies that the functionals 140 := (x + gx,.) and (x — gx, -) are left- resp. right-invariant on A. Note that h1(1) = 14)

11.2 Cosemisimple Hopf Algebras

405

hr (1) = 0. If A would possess a Peter-Weyl decomposition (2), then we have hi(a) = h r (a) for a E C(0) by the above reasoning and hence hi = hr . Therefore, A is not cosemisimple, because hi hr . In the rest of this subsection we assume that A is a cosemisimple Hopf algebra. Let h denote the Haar functional of A. Our next aim are the Schur type orthogonality relations (7) and (8) below. A key step for this is Lemma 14. If v = (NJ) and w = (wii ) are matrix corepresentations of A, then

E h(vii s(w,n)),,, =

En

( 5)

Vinh(VnjS(Wki)),

E h(s(wkov) % = E wkn h(S(wra)Yii)).

(6)

Setting iLl 3n''k) := h(viiS(wk n )) and k l 'i) h(S(wkOvi n ), the equations (5) and (6) can be written as A ( I'k) w = vA 5,k ) and B( 1,i)v = wB( 1,i) , respectively. That is, A ( i , k ) E Mor (w, y) and B (1,0 E Mor (y, w). Proof. By the left invariance (1) of h, we have

h(viiS(wkn ))1 = (id 0 h)(vii8(wkn)) =

h(yri S(wks ))yir S(wsn ).

Inserting this into the left hand side of (5) and using the identity S(wsn)wni = 6 .9/ 1 , we obtain (5). Applying the right invariance (1) of h with a = S(wki)vin) (6) is derived in a similar manner. Proposition 15. Let v = (vii ) and w = (wki) be irreducible matrix corepresentations of A. Then we have:

(i) If v and w are not equivalent, then for any j, j,k,l, h(v ii S(wki)) = h(51 (wki)v ii ) = 0.

( 7)

w is equivalent to its bicontragredient corepresentation w". If F = (Fin ) is an invertible intertwiner of w and w", then Tr F 0, Tr F -1 0

and h(wk/S(wii)) =

Tr

F

and

h(S(wki)w 2,3 ) = (5k3

(F-1)i1

Tr F -1-

(8)

,Proof. (i ): Since y and w are not equivalent, their intertwiners AO' k) and are zero by Schur's lemma (Proposition 6(0). This gives (7). (ii): First we apply Lemma 14 in the case y = w. By Schur's lemma (Proposition 6(ii)), the intertwiners A ( i , k ) and B (1,i) are scalar multiples of the identity, so there are complex numbers aik and Oti such that A U' k) = h(wiiS(wki)) = aik6ii

Setting j

and B k(13:4) = h(S(wki)wii) =

1, j = k and summing over j and j, respectively, we obtain

(9)

406

11. Corepresentation

Theory and Compact Quantum Groups E aii = E 022=1.

z

(10)

Next we apply Lemma 14 with y = wcc. Then yii = S2 (wii ). Combining (5) and (6) with (4) yields E h(wkns(wi i ))wrii=E 82(wirt)mwo(wni )),

‘--dn

h(s(w inwk1 )s2(wni ) =

wk n h(S(w ii )w ni).

(11)

(12)

With Cle ) := h(WknS(Wij)) and D k( in'i) := h(S(win)wki) the equations (11) and (12) mean that C (i' k) E Mor (w,wcc) and D (l 'i) E Mor (w", w). Moreover, by (9), we have Ci(ni'k) = an ibkj and .13)c( in'i) = finkbli which gives (8) for k j. From (10) together with the preceding expressions we obtain Tr C (i,j ) = Tr = 1. Since the antipode S is bijective by Theorem 13 and w is irreducible, wcc is irreducible as well. Because CU'i) E Mor (w, tucc) is nonzero, w and wcc are equivalent by Schur's lemma. Let F be an invertible element of Mor (w, wcc). Since such an F is unique up to a complex factor (again by Schur's lemma), there are complex numbers ai and Oi such that ai F = C(i,i ) and f3i F -1 = D(i , i). Thus, ai Tr F = Tr CUM = 1 and 3Tr F' =Tr B(i ,i) = 1, and (8) for k = j follows.

Let us restate formula (8) in the case where S2 = id. Recall that, generally speaking, all Hopf algebras coming from groups have this property. Then we have wii = ( w "), so (8) applies with F = I. If d, is the dimension of the representation space of w, then (8) reads as h(wk/S(w ii )) = h(S(wki)wii) = bki bi/c/V.

Next we apply the Schur type orthogonality relations (7) and (8) to the study of characters. Definition 8. Let cp be a corepresentation of A on a finite-dimensional vector space. The element x,p := (Tr 0 id» of A is called the character of cp. If (vii ) is a matrix corepresentation of cio , then x‘p = Ei vii . For any f E A', we have f•x v, = x(p .f,, where f .x,i; and x,p .f are given by (1.65). Note that T O 46(X(p) = 46 (X49) x(Pelb = 740 ± )(lb X(p00 = Xy0X10 and Xwe = Moreover, x,p = xv, if y; 0. Of course, Definition 8 and the preceding simple facts are valid for any Hopf algebra A. Let x,„ a E A, denote the character of 0. An immediate consequence of the formulas (7) and (8) is Corollary 16.

h(x„S(xs)) = h(8(x4 ) xs) = b as for a, E A.

Proposition 17. Let c,o be a corepresentation of A on a finite-dimensional vector space V. For a E Â, we set na := h(S(xa)x
11.2 Cosemisimple Hopf Algebras

exEÂ

n

407

a

Pa is a projection of V onto the (p-invariant subspace Vc := Pa (V) such that the restriction of cp to Va is equivalent to the direct sum of n a copies of tta . Moreover, vp = E a naxa and h(S(vp )x,p ) = Ea n2c, . Proof. From the definition of ha we immediately obtain that f h a = ha f for f E A' and ha ho = baoha . Hence (f)(h) = ((ha )(gf) and ((h a)(13(ho) = bas((h a ) by Proposition 1(iii). Therefore, by Proposition 3(iii), P0 = gha ) E Mor ((p) and PPo = 6a0Pa , ot, E Â. Thus, {Pa I c E AI is a family of projections such that E ct P1 = I on V and Pa Po = 0 if a Pa (V) )3. Let (p a denote the restriction of cp to Va (74,) be the matrix corepresentation of (pa with respect to and let va a basis {4} of Va . For f E A', we have En 4tI f(v m ) = Afkro(h a )(x)) = 4(fh a)(v 7citni ), so f(v 7c,t,i ) = f ha (ert,i ). Since f ha vanishes on C ( P), a )3, this implies that v7c,, E qua), so there are complex numbers A'X i (depending also on a) such that v'T"= E i Arrini,j uZ. Applying the comultiplication to both sides and comparing the coefficients of =-- ArXi S„. Using the latter we compute that for u .9 , we obtain E k Arkiri A fixed m and j the elements yi := A771 i x,;.ti satisfy cp(yi ) = yi ucli . Thus, Lin fyi = E 71 ATi x(41 is {0} or the restriction of (00 to Wami either Wami is equivalent to uct. Since obviously Va = Ent , i Warn, it follows that (p a is equivalent to a direct sum of copies of uct. Since cp = ED 0 (p a , the remaining assertions follows at once from Corollary 16. 111 ,

Proposition 17 implies in particular that each finite-dimensional corepresentation of A is completely reducible. Now let (p be an arbitrary corepresentation of A on V. By Proposition 4(i), any element of V belongs to some finite-dimensional (p-invariant subspace of V. Thus, by Proposition 17, V is the sum and hence the direct sum of (p-invariant subspaces such that the corresponding restrictions of (p are irreducible. This proves that any corepresentation of A is completely reducible. Corollary 18. Two finite-dimensional corepresentations of A are equivalent if and only if their characters coincide. Corollary 19. For any finite-dimensional corepresentation cp of A on V, (Ph == (id 0 h)cp is a projection of V to the subspace of ça -invariant elements.

Proof. The proof follows from Proposition 17 applied to a = 1, since then ha = h. , LI Of course, the assertion of Corollary 19 can also be verified directly without using the orthogonality relations (7) and (8). It is then obtained for an arbitrary (not necessarily finite-dimensional) corepresentation of A.

408

11. Corepresentation Theory and Compact Quantum Groups

11.2.3 Peter—Weyl Decomposition of Coordinate Hopf Algebras Throughout this subsection we suppose that Gq is one of the quantum groups SL q (N), 0 q (N), S0 q (N), Sp q (N) and that the complex number q is transcendental. Our aim is to prove that then the Hopf algebra 0(Gq ) is cosemisimple and to determine its Peter—Weyl decomposition. The main tools for that are the Brauer—Schur—Weyl duality (see Subsect. 8.6.3) and the representation theory of the corresponding Drinfeld—Jimbo algebras Ug (a). First let us collect some notation and facts needed later. Recall that Oq (so N ) is the Hopf algebra defined in Subsect. 8.6.1. For notational simplicity, we also set al (s1N) Uq (s1N) and ùq (sp N ) := Uq (spN ). Let denote the vector representation of 04 (g) with matrix coefficients t ii (see (8.87)). By Theorem 9.18 and Proposition 9.19, there exists a dual pairing (•, .) of the Hopf algebras r/q (g), g = slN , soN, spN , and 0(Gq ), Gq q (N),0 q (N),Sp q (N), of Uq (s02,i ) and 0(S0 q (2n)) and of Uq l/ 2 (S02 n +1) Sid and 0(S0 q (2n +1)) such that (.,u) =--- t ii (.). Let A(R) be the FRT bialgebra corresponding to 0(Gq ), where R is given by (9.13) and (9.30), respectively. Since 0(Gq ) is a quotient of A(R), the pairing of Cfq (g) and 0(Gq ) induces a dual pairing of Clq (g) and A(R). Therefore, by Proposition 1.15, any corepresentation (,o of 0(Gq ) or A(R) yields a representation (03 of Cfq (g). Let (p i denote the fundamental corepresentation of 0(Gq ) or A(R) defined by (9.6) and let V1 be its carrier space. As noted in Sect. 9.4, we have 01 -= 1. Proposition 20. Suppose that r E N. The assignment fo gives a oneto-one correspondence between subcorepresentations of the coreprusentation (pr : Vi®r Vi®r A(R) of the bialgebra A(R) and subrepresentations of the representation f'?r of the algebra q (g). Two subcorepresentations and V) are equivalent if and only if the subrepresentations("03 and are. A subcorepresentation (09 is irreducible if and only if (To is. The corepresentation ()or of A(R) is completely reducible.

((pr).

®i Proof. Since (;9" 1 we have and any subcorepresentation (09 of (pr leads to a subrepresentation (r9 of fl?r . Conversely, let T be a subrepresentation of ir realized on a subspace V of the carrier space Vi® r of f'?r. Since the representation irr is completely reducible by Theorem 7.8, there exists an invariant subspace W for f E®r such that V?r = V e W. Obviously, the projection P of Vi®r onto V is in the commutant inCig (g)Y. Therefore, by Theorem 8.38, P belongs to the algebra B(r) generated by the transformations for g = slN and by k, # i+i and K i,i+i for g soN7sPN• Since '1?i,i+j ui ui+i = ujui±i k i±i and K +l ui ui±i = ui ui±i Ki, i+i , we have P E Mor (Or) and so P E Mor ((pr) by Proposition 3. Hence PK®r = V is also an iiivariant subspace for (pr. If (p denotes the restriction of (pr . to V, we clearly have (''09 = T. It is obvious that any equivalence of subcorepresentations (09 and yields an equivalence of (03 and ?i) and that any invariant subspace for (09 is invariant

11.2 Cosemisimple Hopf Algebras

409

for fo. The opposite direction, as well as the last assertion of the proposition, 0 can be verified as in the preceding paragraph using Theorem 8.38.

By Corollary 8.37, any representation Î'(n), n E Pr (13), of 0q (0) is an irreducible subrepresentation of Hence, by Proposition 20, there exists a unique irreducible subcorepresentation, denoted ça„,,, of the corepresentation cpr of A(R) such that (i)„, r t(n).

'tr.

Theorem 21. The bialgebra A(R) decomposes as a direct sum of coefficient coalgebras C(v„,r), n E 1),(0), r E No, that is,

A(R) = 401) 1 C(Vn,r)w r>0 EDnE, P(9)

(13)

Any irreducible core presentation of A(R) is equivalent to one of the corepresentations w n,r . Proof. Let r E No. We denote by A r (R) the homogeneous part of A( R) of homogeneity degree r. Clearly, A r (R) is just the coefficient coalgebra of the corepresentation cor of A(R). By Proposition 20, the decomposition (8.117) of the representation fr leads to the direct sum decomposition or (14,

- qp.

inn (Pn,r 7

of the corepresentation cor. The coefficient coalgebras of both sides must coincide, so we get

Ar (R) =

E nEP.r(g) c(vn,r).

(14)

n', n, If n E Pr(s), then the representations T(n) and Pn!) and so the corepresentations çan,r and çani,r (by Proposition 20) are not equivalent. Ar (R), this Hence the sum (14) is direct by Corollary 9. Since A(R) = yields the decomposition (13). Let ça be an irreducible corepresentation of A(R). By Proposition 8, C(ço) is a cosimple subcoalgebra of A(R). Therefore, by (13), C(ça) = C(ço n,r ) for some n E Pr(g) and r E No. Hence w ça n,r by Corollary 9.

Our next step is to formulate the Peter—Weyl decomposition theorem for the Hopf algebras 0(G q ). Recall that these Hopf algebras are defined as

0(S L g (N)) = A(R)/ (D —1), 0(Sp g (N)) = A(R)I (Q g — 1), 0(0g (N)) = A(R)/(Q g —1), 0(S0g (N)) = ATV (Q g — 1,Dg — 1), where A(R) is the corresponding FRT bialgebra, Qq , is the element of A(R) defined by (9.37) and Dg is the unique element of A(R) such that Dg for ( E A(G)N. That is, Dg is the quantum determinant VR,N(() = from Definition 9.5 for Gg = SLg (N). In the case G g = O(N) the quantum determinant from Definition 9.10 belongs to the algebra 0(0q (N)), but here rog denotes the corresponding element from the FRT bialgebra A(R).

410

11. Corepresentation Theory and Compact Quantum Groups

In the proof of Theorem 22 it will be shown that the irreducible corepresentations of 0(G q ), Gq S L q (N), S0 q (N), Sp q (N), are characterized by the highest weights of those irreducible representations of the corresponding Drinfeld-Jimbo algebra Uq (g), g = 51N , soN , spN , which are contained in tensor products of some number of vector representations. These are precisely the weights of the set qa(g) from Proposition 7.20. We write P(Gq ) := q3(9) in this case. Further, it will be proved that the irreducible corepresentations of 0(0g (N)) are parametrized by the set P(soN) described in Subsect. 8.6.2. We set P(0q (N)) := P(soN ). In both cases the corresponding irreducible corepresentation (ion , n E P(Gq ), of 0(Gq ) has the property that (,:à„ is just of Uq (g) resp. Og (g). the irreducible representation Tn resp. T(n) _=

4,

Theorem 22. Let Gq be one of the quantum groups SL q (N), 0 q (N), SO q (N) or Spq (N). Suppose that q is transcendental. Then the Hopf algebra C9(G q ) is cosemisimple and it decomposes as a direct sum

(15)

0(Gq ) r-'EDnep(Gq) C((ion )

of coefficient coalgebras C(çon ) of irreducible corepresentations (ion , n E P(Gq ), of 0(Gq ). Every irreducible corepresentation of 0(Gq ) is equivalent to one of the corepresentations n E P(Gq ). Formula (15) is the Peter-Weyl decomposition of the cosemisimple Hopf algebra 0(Gq ). Before we begin with the proof of Theorem 22, let us derive •) of the Hopf algebras an important corollary concerning the dual pairing tIg (g), resp. Uq (g), and O(Gq ) defined in Sect. 9.4. Corollary 23. The dual pairings (•,.) of the pairs of Hopf algebras Oq (p), g = sl N , soN, spN , and 0(G q ), Gq = SL q (N),0 q (N),Sp q (N), of Uq (s02n,) and 0(S0 q (2n)) and of Uq i i 2(so2n+i) and 0(S0 q (2n+1)) are nondegenerate.

Proof We give the proof for the Hopf algebras Og (g) and 0(Gq ), Gq q (N), 0 q (N), Sp q (N). The proof in the other cases is completely similar. SL Let N. be the set of a E 0 (G q ) such that ( f , a) = 0 for all f E Oq (0). Let (,o = L be the corepresentation of 0(Gq ) on itself. By Theorem 22 and Proposition 8, we have ço ED. dn çon , where dn = dim ço n . Since (1b 1 = ED. cintn . If a E N. and f E tig (0), by the definition of (Jon , this yields ço (g, a( 1 ) ) (f, a(2)) = (gf, a) = 0 for all then, by (1.65), we get (g,43(f)a) = g E t7q(g) , so that ço(f)a c N. That is, AT is 0-invariant. Assume that .Ar

{0}. Since fo

ED. dn tn - , there exists a c,O-invariant subspace go {0} of Ar

rN-0

such that No c C(çon ) for some n and • Hence we can find a basis fed of No such that 0(f)e = Ei (f, qi )ei , where (NI) is the matrix of the corepresentation rpn for this basis. We write the vector ei as ei = with czj E C. By (1.65), we have (g, fi .a.f2) = (f2gh,a) = 0 for a E and 11 ,12 ,9 E Oq (9). Therefore, fi .a.f2 E Af for arbitrary 11 ,12 E U(9). The matrix coefficients (., vri ) of the irreducible representation fin Of

r.Aro

11.2 Cosemisinnple Hopf Algebras

411

f/q (g) are linearly independent. Hence, for any k, 1, r, s = 1, 2, • • • , dim.Vo, we can find elements f , f2 E q (9) such that (fi , vri ) = 6k 1 60 and (f2, vfli ) = Since k,l,r,s are arbitrary, 5rib'sj . Then, we obtain f1 .e 1 .f2 = ariusnk E any matrix element vrj belongs to N , that is, (f, v) = 0 for all f E tIg (g). This in turn yields that 0(f)a = 0 for any f E f/q (g) and a E M. Since {0 } by construction, this is a contradiction. Thus, N. = {0}. This means that (f, a) =- 0 for all f E C4(9) implies that a = O. Now suppose that (f, a) = 0 for all a E 0(G q ) . Since ( f , a) = (a)( f ) by the definition of the pairing (•, .) (see the proof of Theorem 9.18), it follows that t ii (f) = 0 for all matrix coefficients t o of subrepresentations T of an r-fold tensor product fr. of the vector representation ti of tiq g If g = SIN or i = spN , then Ûq (g) = Uq (g) and so f = 0 by Proposition 7.21. Combining the results of Subsect. 8.6.1 and Proposition 7.21 one can easily verify that if an element b E tig (soN) is annihilated by all irreducible representations 7' contained in some r-fold tensor product i?r, r E N, then b O. •

Corollary 23 implies that the maps a -) (-, a) and f (f -) are injective Hopf algebra homomorphisms of 0(G q ) to (1.q ( 9) ° and of Oq (9) to 0(G q )° The injections (.9(G q ) -> aq (g)° are not surjective, because not every irreducible representation of Clq (9) is of type 1. Recall from Sect. 7.1 that each finite-dimensional representation of Uq (9) is a direct sum of (irreducible) representations of the form Twn , where n E P+ and w E {1, -1 } n, n = rank 9. Since RE is precisely the set P(G q ) for G q = SL q (n + 1), Sp q (2n), it follows from the preceding and Theorem 22 that there are Hopf algebra isomorphisms

Uq (sln_E i)° = 0(S L q (n + 1)) >1 CZ3, Uq(sP2n) ° 0(Sp q (2n)) >1 CZ. Let us briefly recall the structures of the Hopf algebras on the right hand sides. As a coalgebra, (.9(G q ) >1 CZ is the tensor product coalgebra (.9(G q ) 0 CZ. As an algebra, it is a left crossed product algebra (see Subsect. 10.2.1). The generators xk, k = 1, 2, ... n, of CZT2i act as Hopf algebra automorphisms xk of 0(G q ) which are determined by the following formulas: xk (74) = -74 if i k j or j < k i for Gq = SL q (n + 1), and if i < k < < lc' or k j < k' < i or j < k < i lc' or k i < k' < j for G q Sp q (2n) and Xk(u ii ) = u 3i- otherwise. Thus the algebra (.9(G q ) >1 CZ r2i is generated by the algebra 0(G q ) and n additional pairwise commuting generators xk such that = 1 and xk(a) = xkaxk-1 for a E 0(G q ) and k = 1,2, • • • , n. In the case g = SON there is still an injective Hopf algebra homomorphism of 0(50q (N)) >I CZT2' into Uq (so2n )° resp. Uq1/2 (502n+1)° where as usual N 2n or N = 2n +1. However, this homomorphism is not surjective, because its image does not contain the matrix coefficients of the spinor representations

of Uq (s02n) resp. Uq1/2 (S02n+1)The rest of this subsection is devoted to an outline of the proof of Theorem

22. The decomposition (15) will be derived from (13). In order to do so, we

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11. Corepresentation Theory and Compact Quantum Groups

have to find the parts of the decomposition (13) that cancel when we pass to the quotient 0(G q ). We begin with some notation. The right A(R)-comodule corresponding to the corepresentation r is denoted by M(n, r). Let M be a right comodule of A(R) or 0(G g ). Then' , by Proposition 1.15, M is also a left module of the corresponding algebra Oq (g) resp. Uq (g), denoted by Mo resp. M. Since A(Dg ) = Dg 0 Dg and A(Qq ) = Q47 0 Q q , the one-dimensional spaces CQg and CDg are right A(R)-comodules. They are described in Lemma 24. CDg M((NY 7 N) for g = slN , CDg f_•2 M(0°,N) ) CQ g M(0 1 2) for 9

(16) 50 N,

(17)

CDg M(O I N), CQ g -^2 M(0,2) for g = spN ,

(18)

Dq2 =

for g = soN and 73q = Qqn for g = sp2„

Proof. We carry out the proof of (17). Clearly, (CQq )o- and (CDOÙ are onedimensional Oq (soN)-modules. But there are precisely two one-dimensional type 1 representations of tIq (soN). They correspond to the partitions 0 and 0 0 . Since (f,Q q ) = (f, 1) for f E g (SON) and (x,Dq ) = —1 by Proposition 9.19, it follows that (CD00 = (M(0°, no and (CQq )0 = (M(0, 2))0. By Proposition 20, the latter implies that C; M (0° N) and CQq M(0, 2). The formulas (16) and (18) are verified similarly. Let us prove the equation Dq2 = Q g = SONS. The equality 73,7 = Qqn for g =sP2n is proved similarly. We have Dq2 E A2N (R) and Q .A2N(R)• Both Dq2 and Q qN are group-like elements belonging to the one-dimensional coalgebra C(v)0, 2N ) in the decomposition (13). Hence we get 73 q2 = Q. (IN

Lemma 25. There are the following A(R)-comodule isomorphisms: M(n,r) O C;

M(n +

(N)' 7 r

M(n,r) O CAI M(n,r M(n,r) M(n,r)

O CDq M(n°, r• CQq M(n,r + 2)

N)

for

g -= slN ,

(19)

N)

for g = spN ,

(20)

N)

for

(21)

for

g =

SON,

g = spN ,so N .

(22)

Proof. The relations (19)—(22) are derived from the formulas (16)—(18). In order to prove the relation (19), we first note that both sides of (19) correspond Thus , by Proposition 20, it is to AT + N (R)-subcorepresentations of ço ®fr+N) i sufficient to show that (M(n,r) O CD00 M(n + (N)' , r N) t-j- . But the latter follows at once from (16). Let us, prove also the relations (21) and (22) for g = so2n . Using again the dual pairing of C/q (so2,i ) and A(R) we find that x E 0(so2n ) acts on 131q and Qq, as x(Dq ) = —731q and x(Q q ) = Qq. Therefore, by the definition of the irreducible representations t(n) and t(n°) (see Subsect. 8.6.2), we have .

11.2 Cosemisimple Hopf Algebras (M(n,r)

CDOti M(n°,r +N)0,

413

(M(n,r) CQ q )0 M(n,r + 2)0,

which proves the assertion for g = so2n . Now we turn to the corresponding 0(Gq )-comodules. Clearly, any A(R)comodule M gives rise to an 0(Gq )-comodu1e by composing the coaction with the canonical mapping of A(R) to its quotient 0(G q ). It will be denoted by Mo if G q SL q (N), SO q (N), Sp q (N) and by Mc-5 if Gq = 0q (N). Recall that the irreducible representations of U(g) are described by the A(n) = partitions n from Subsect. 8.6.2 and also by the highest weights A (A i , A2, • • • I An ), where A i = 2(A, ai )/(a i ,ai ), as in Subsect. 7.1.2. Lemma 26. (0 There are the following carnodule isomorphisms: M(n, 0 c-5 Agn,$) c5, M(n,r)0

M (n, r) M(n,

M(n, s)0 if n e Pr (g) n Ps (g),

M (n° , s)0 if n E Pr (soN),

E Ps (soN),

Agn + (N)' , r + N)0 if n E Pr (s1N).

(ii) If n E Pr (g), m E Ps (g) and A(n) = A(m), then Agn,r)0 M(m, s)o. n, where n' = (n'a , n', • - • ,n's ) is (iii) Let n E P(so2 n ) be such that the transpose of the partition n. Then the comodule Af(n, r)0 is the direct M(A(n)) sum of two irreducible comodules M' and M" such that MI M(o- (A(n))) are irreducible Uq (so2n )-modules with highest weights and Mg A(n) and u(A(n)), respectively (see Proposition 8.34). = n (considered in (iii)) the comodule (iv) Except for the case g = s02n , M(n, 00 is irreducible for any partition n E Pr (9).

Proof. (i): We prove the first two lines of relations for g = 50 2n • The cases g = 'N SO2n+11 sp N are treated similarly. By Proposition 8.36, if n E Pr (s0 2n) n ,

P8 (so2 n ), then r — s E 2Z. Since the comodules (CQ q )6- and (CQ q )0 are trivial, the first line of relations follows from (22). By the definition of n°, the assumptions n E Pr (5o2n ) and n° E P3(so2n) imply that r — s E 2nZ. M (n° s)0 Since (CD00 is the trivial comodule, the relation M (n, follows from (21). Similarly, the last relation in (i) follows from (19). (ii): By the definition of A(n), the equality A (n) = A(m) yields that n = m for g = sp N , n = m or n m k(N)' k E No , for g = slN and n = m or . n = m° for g = SON • Hence the assertion follows from (i). (ill): Since n = n° in the present case, it follows from (21) and (22) that there is an A r+2n (R)-comodule isomorphism

P: Agn, r) CQqn M(n, r) CDq . Let y be a nonzero highest weight vector of M(n,r)u. Since 0 is also a Ûq (so 2n )-module isomorphism, we may suppose that sP(v Qqn) = y 0 Dq . Since x(Qqn) = Q and x(Dq ) = —Dq , we have 0(0.0 Qqn) = —x(y) Dq . On the other hand, since 0 (S 0 q,(2n)) = A(R) I (Q q — 1, D — 1), the linear

11. Corepresentation Theory and Compact Quantum Groups

414

map V defined by :u u Dg , u E M(n,r), gives a comodule Q qn isomorphism (M(n,r)0C(4)0 (M(n,r)0CD00. We define 0(S0 q (20comodule isomorphisms 40± := ± Then we have Tm 40+ = 40 ±(Uq (so2n)(v 0 QT:) Uq(s 0 2n)(X(v) Q T:)) = Uq (s0271)4$-F(v QT4i, ) Uq(s0 2n) 4 $F(X(v) gc;) = Uq (502n)(V

Im

DO,

Ug(so2n)(X(v) D g ).

Thus, Tm 40+ and Tm 40_ are distinct irreducible components of the Uq (so2n )module (M(n, r) CD0u. This implies the assertion. (iv): The 0(G q )-comodule M(n,r)0 is irreducible, because the corresponding Uq (g)-module M(n, r)u is so. 0 By Lemma 26(1), the irreducible corepresentations of 0(0q (N)) which are derived from irreducible corepresentations of the bialgebra A(R) are parametrized by the set of partitions 'P(SON) from Subsect. 8.6.2. Now let G g = S L g (N), SO g (N), Sp g (N). We define an irreducible 0(G g )comodule Mo(A) for any dominant integral weight A = (Al l A2, , ) n ) E q3(g), A i = 2(A, cri )/(cri , cri ), of the corresponding Drinfeld—Jimbo algebra Uq (g). First we consider the case where g = 502n and An —i A,. Then there exists a partition n E P(so 2n ) such that A = A(n) and nÇ = n. (Recall that (74, n'2 , • - - , n's ) is the transpose of the partition n and A(n) is given by the formulas in Subsect. 8.6.2). In this case we set Mo(A) := M' and Mo(a(A)) := M", where M' and M" are the irreducible 0(80g (20comodules from Lemma 26(iii). In all other cases we set Mo(A) := M(n, r)o, where n E Pr (g) is such that A(n) = A. Thus we obtain irreducible 0(G q )comodules Mo(A) or equivalently irreducible corepresentations of 0(G q ) parametrized by the elements A of the set q3(9) from Proposition 7.20. If A jt then these corepresentations of 0(G q ) are not equivalent, because the corresponding representations of the Drinfeld—Jimbo algebras are not. ,

Proof of Theorem 22. Taking the coefficient coalgebras of both sides of the equivalent A(R)-comodules in (19)—(22) we obtain the relations C(V ii ±(N)/ ,r+N) = C(CiOn,r)

C( (pn,r+N) = C(Vn,r) C(Ci'n°,r+N) = C(Pn,r) C(Vn,r+2) =

g = slN, .Dg for g = sp N , - Dg for g = soN, for g = sp N , soN • Qq Dg

for

The direct sum decomposition (15) will be derived from the decomposition (13) and the preceding formulas. Let us carry out the proof in the case G g = Spq (N). Since C(,(Pn,r+2) = C(IOn,r)• Qq , the direct sum (13) is now A(R) =

sED nEP(sp N ),InjE2Z

C(,on,o) ®

ED nEr)(813N),IniE2Z+1

c(pn , i ) ® (Qq

1)A(R).

11.3 Compact Quantum Group Algebras

415

Since the union of In E P(spAlinj E 2Z} and In E P(spAlinj E 2Z + 1} is just the set P(Sp q (N)) and 0(Sp q (N)) = A(R)/(Q q —1) by the definition, we get 0(Spq (N)) AEP(Sp q (N))

which proves (15). Using the above results the other cases are treated simiLI larly. The last assertion is proved as in the case of Theorem 21. Remarks 3. That q is transcendental was only used in order to apply Theorem 8.38. If the latter theorem could be shown for q not a root of unity, then all results of this subsection remain valid under this assumption. 4. The representations Tr, and T(n) of Uq (g) resp. 0,7 (g) and the corepresentations (pi, of 0(Gq ) are parametrized by highest weights with respect to the simple roots ai, • • • , at. Likewise we could have used highest weights al. Then we obtain the same corwith respect to the simple roots —a 1 , • • respondence of irreducible corepresentations with Young frames as for the classical groups. For G q = GL q (N) this is carried out in Subsect. 11.5.3. —

11.3 Compact Quantum Group Algebras This section is devoted to a detailed study of the class of Hopf *-algebras which are linear spans of the matrix coefficients of their finite-dimensional unitary corepresentations.

11.3.1 Definitions and Characterizations of CQG Algebras Let G be a compact topological group and let A = Rep (G) be the linear span of matrix elements vii of unitary finite-dimensional continuous representations of G (see Example 1.2). Then A is a (commutative) Hopf *-algebra with *-structure and comultiplication given by (vii)* = t7i (the bar means complex conjugation) and A(v ii ) = Vik®Vki . Further, the algebra A is finitely generated if and only if G is a compact Lie group or, equivalently, if G is isomorphic to a closed subgroup of the group U(N) of N x N unitary matrices. Such groups are called compact matrix groups. These facts provide the motivation for the following two definitions. Definition 9. A Hopf *-algebra A is called a compact quantum group algebra (abbreviated, a CQG algebra) if A is the linear span of all matrix elements of finite-dimensional unitary corepresentations of A. Definition 10. A compact .matrix quantum group algebra (abbreviated, a CMQG algebra) is a CQG algebra which is generated as an algebra by finitely many elements.

We give first two slight reformulations of Definition 9. Let A be a Hopf *-algebra. Since finite-dimensional unitary corepresentations are completely

416

11. Corepresentation Theory and Compact Quantum Groups

reducible by Proposition 12, A is a CQG algebra if and only if it is spanned by the matrix elements of irreducible (hence finite-dimensional) unitary corepresentations. Combining the latter characterization with Theorem 13(v) it follows that A is a CQG algebra if and only if A is cosemisimple and each corepresentation v a , a E Â, is unitarizable. Since CQG algebras are cosemisimple, the theory developed in Subsects. 11.2.1 and 11.2.2 applies in particular to any CQG algebra. Further characterizations and basic properties of these algebras are contained in the following two results.

Theorem 27. For any Hop! *-algebra A the following statements are equivalent: (0 A is a CQG algebra. (ii) A is generated as an algebra by the matrix elements of all finitedimensional unitarizable corepresentations of A. (iii) Any finite-dimensional corepresentation of A is unitarizable (and hence a direct sum of irreducible corepresentations). (iv) There is a linear functional h on A such that (h

id)Z(a) = h(a)1 and Ma'ra) > 0 for all a C A, a

O.

Proof. (i) (ii) is trivial. (ii) —> (i): Let fuil be a set of finite-dimensional unitarizable corepresentations such that their matrix elements generate A as an algebra. Let R. be the smallest family of corepresentations of A which contains all u i and is closed under direct sums and tensor products. These operations preserve unitarizability, so all corepresentations in R. are unitarizable. Since c ((,0 zp) = ( w ) + c (' p) and C(ço zP) = C(v)C(0), the union of C(v), is an algebra which contains all C(u). By assumption (ii), this algebra is A. For any (p C R. there is a unitary corepresentation ço such that CM = C(ç). Hence A is a CQG algebra. (iv) —> (Hi): The proof is an adaption of the proof for the corresponding result on representations of compact groups. Let (p be a corepresentation of A on a finite-dimensional vector space'V. Let (., .) be an arbitrary scalar product on V. Define another Hermitian sesquilinear form (., •) on V x V by (x, y) :=

(x(0) , y(0))h(x( 1 )*y( l )),

x, y G V.

Let v be the matrix elements of (it) with respect to an orthonormal basis {xi } of (V, (a, -)). Then, we have (xi , x i ) = SO

E k,I (xk ,x0h((vki)*vii ) = E

h((Vkir Vki),

11.3 Compact Quantum Group Algebras

417

0, the left hand side vanishes only for any ai E C. Since h(a*a) > 0 for a when ajVki = 0 for all k. But then ak = e(E i ctivki) = 0 for all k. Hence (-, .) is a scalar product on V. Using the right invariance of the functional h, we conclude that

Ei

E(x (0 ) ,y(0 ) )X 0) *y ( 1 ) =E(x (0) ,(0 ) )h(x ( i ) * N ox (2 ) *y (2 ) ___,_,x (o) ,y (o) )(hoz id)A( (1) * y(0 )=E(x (o) , y(0) )h(x (i) * y( 1 ) )1 .E(x

Therefore, by Proposition 11, the corepresentation c,o is unitary with respect to the scalar product (. 7 .). (iii) 4 (i ): Let a E A. By Proposition 4, there is a finite - dimensional subcoalgebra C containing a. The restriction c,o of z to C is a finite-dimensional corepresentation of A and hence is unitary with respect to a certain scalar product on C. Since (e id)(p(a) =-- a, we have a E C(cp). The proof of the implication (i ) -4 (iv) is given in Subsect. 11.3.2 along with the proof of Proposition 29 below. -

Proposition 28. For a Hopf *-algebra A, the following are equivalent:

(0 A is a CMQG algebra. (ii) Them is a finite-dimensional unitary compresentation 'a of A whose matrix elements generate A as an algebra. (iii) There is a finite-dimensional corepresentation u of A such that u and uc are both unitarizable and the matrix elements of u and uc generate A as an algebra. Proof. (i)--)(ii): Take a finite set of generators of the algebra A. Since A is a CQG algebra, each generator is a linear combination of matrix elements of finite-dimensional unitary corepresentations of A. The direct sum of these corepresentations has the desired property. (ii)--(i) follows by the proof of implication (ii)-40 of Theorem 27 applied to a singleton ful. (ii)-4(iii): By the implication (ii)-4(i), A is a CQG algebra, so uc is unitarizable by Theorem 27(iii). (Here the more involved part of Theorem 27, which contains the implication (i) --> (iii), is needed. This can be avoided by an alternative proof based on Proposition 12.) (iii) —(i) is obvious.

Let A be a CMQG algebra and let u be a corepresentation of A as in Proposition 28(iii). By choosing an appropriate basis, we can assume that u is given by a unitary matrix corepresentation u = Then we have u*u = uu* = / and hence uc --= j Since uc is unitarizable by Proposition 28(iii), there is an invertible matrix A E MN(C) such that v := Auc.A.' is a unitary matrix over A. Setting E := A* A, one immediately checks that the relations vv = vv* = I are equivalent to the matrix equations ut EfLE -1

L

(23)

That is, any GANG algebra A is generated as a *-algebra (I) by the entries of a unitary matrix u = (uii)i,j=1,...N which satisfies the equations (23). Further,

418

11. Corepresentation Theory and Compact Quantum Groups

if V E MN(C) is a unitary matrix, then ii := Vt ul7 is another unitary matrix corepresentation of A such that the entries of el generate A as a *-algebra and 71 satisfies (23) with E replaced by .t := VEV -1 and u by ft. By the spectral theorem, the matrix È is diagonal with positive numbers on the diagonal for some suitable choice of V.

We close this subsection by considering some examples.

Example 6 (Universal CMQG algebras). Let E E MN(C) be an invertible positive Hermitian matrix. Let A(E) denote the free *-algebra with generators j = 1, 2, • • , N, subject to the 4N 2 relations u*u = uu* = I and (23), where u (u ji), u (Fiji ) := ((u ii )*) and u* := fit . Then A(E) is a

CMQG algebra with comultiplication given by Z1(u ii ) = uik 0 uki •

Proof. The free algebra qu ii , vii ) with 2N 2 generators u ii and vii is a bialgebra with comultiplication determined by L(u) = u i k 0 uki and ZI(vii ) = vik (gvki. Let J be the two-sided ideal of quii , Yii ) generated by the elements corresponding to the relations vu = uvt = utEvE -1 = EvE - lut = I, where u = (uii ) and y = (vii ). As in the proof of Proposition 9.1 we verify that J is v ii ) I J is a bialgebra. One easily checks that the a biideal, so that A := elements vii, v := (E -l ut E) ii of A satisfy the defining relations of the opposite algebra A°P. Hence there exists an algebra anti-homomorphism S : A A such that S(u) = and S(vii ) = v. Since S(u)u = uS(u) = I and S(v)v = vS(v) = I by construction, it follows from Proposition 1.8 that S is an antipode for the bialgebra A. Thus, A is a Hopf algebra. Clearly, C(u ii , vii ) is a *-bialgebra with respect to the involution given by u;ri = vii . Since the matrix E is Hermitian, .7 is a *-ideal and hence A is a Hopf *-algebra. Obviously, as a *-algebra, A is just the *-algebra A(E) defined above. From the defining relations u*u = uu* = I and (23) E1/2,Et-E1/2 it follows that u and w := Ei/2 ucE -112 are unitary matrix corepresentations. Since w uc, the corepresentation uc is unitarizable. Since the matrix entries of u and uc = ft generate A as an algebra, the Hopf algebra A is a CMQG algebra by Proposition 28(iii). CI From the remarks preceding this example it follows that any CMQG algebra is a quotient of some CMQG algebra-A,,,,(E) for a certain diagonal matrix E with positive diagonal entries. That is, the CMQG algebras A(E) are universal CMQG algebras. Note that the CMQG algebras A(E) are very large in general, because no relations among the entries u ii are required. A Example 7. Each of the Hopf *-algebras 0 (Uq (N)), O(SU q (N)), 0(0, 7 (N ; R)), 0(S0q (N; R)) and OW Spq (N)) (see Subsects. 9.2.4 and 9.3.5 for their definitions) for q E R is a CMQG algebra. In these cases condition (ii) in Proposition 28 applies, since the fundamental corepresentations u are unitary (recall that uu*, = I by the definitions of the involutions). Except for Uq (N), the matrix elements u generate the corresponding algebras. The algebra 0(Uq (N)) is generated by the matrix elements of the unitary corepresentation u631(;7 1 ). A Recall that Dq-1 is a unitary element of the *-algebra 0(Uq (N)).

11.3 Compact Quantum Group Algebras

419

11.3.2 The Haar State of a CQG Algebra Because any CQG algebra is a cosemisimple Hopf algebra, it possesses a unique Haar functional. In this and the next subsections we are mainly concerned with the Haar functional h of a CQG algebra A. Recall that a linear functional f on a *-algebra A with unit is called a state if f(1) = 1 and f (a* a) > 0 for a E A. A state f on A is called faithful if f (a* a) > 0 for all nonzero a E A. Proposition 29. The Haar functional h of a CQG algebra A is a faithful state on the *-algebra A. The key step for the proof of this proposition is the following Lemma 30. Let w be an irreducible unitary corepresentation of A on a (finite-dimensional) Hilbert space W. Then there exists a unique invertible intertwiner F E Mor (w, w") such that Tr F = Tr F-1 > O. This operator F is positive definite, that is, (Fx,x) > 0 for all x c W, s O.

Proof. We fix an orthonormal basis of W and consider w and w" as matrix corepresentations. Since wc is also irreducible, there exist an irreducible unitary corepresentation y of A and an invertible complex matrix A such that = A - lyA. Let  denote the matrix with entries A. Since y and w are unitary, we have (y9 ii = S(vii ) = vii*, (wc) ii wii * and (wcc) ii = S((wc) ii ). Therefore, we conclude from yA = Awc that ycA = Aw and Atyc = wCCAt and so AtAw = wccA t il. That is, At A E Mor (w, w"). Since A is invertible, AtA is positive definite. Since dim Mor (w, wcc) = 1, F E Mor (w, wcc) is uniquely determined by requiring that Tr F = Tr F-1 > O. Let A be a CQG algebra. By the second reformulation of Definition 9 given in Subsect. 11.3.1, we can choose a unitary corepresentation ua in each equivalence class a E A. Moreover, let us fix Fa, G Mor (ua ) ) such that Ma := Tr Fa = Tr F;1 > O. Since (4)* == S(4,3) and (4)* = S(4) by the unitarity of u)3 and ua, (7) and (8) can be rewritten as

h(uck'i (u)*) = 5,035kiMaT i (Fa)ii,

h((urk )* 14) = 84,06kimc-g-1 (Fa— I)

(24) (25)

These two equations are referred to as the Schur orthogonality relations for the CQG algebra A.

Proof of Proposition 29. Since A is a CQG algebra, any element a G A is a finite sum Acrij uZ with Actii E C. By (25), we have

h(a* a) =N--`

/ki 4—s a,,,

AcitikAalkM (F ct—i )il.

Since F;' is positive definite as well, h(a*a) > 0 and h(a* a) -= 0 implies that all coefficients Accik are zero, so a = O.

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11. Corepresentation

Theory and Compact Quantum Groups

Because of Proposition 29, the Haar functional of a CQG algebra will be called the Haar state. The Haar state is a fundamental tool for the study of CQG algebras. It was essentially used in the previous subsection in order to prove the important result that any finite-dimensional corepresentation of a CQG algebra is unitarizable (and hence completely reducible). Also, the character theory in Subsect. 11.2.2 was based on the Haar functional. Other applications will follow below. Since the Haar state h is a faithful state on A, the equation

(a, b := h(a* b) ,

a, b E A,

defines a scalar product (., •) on A. Let 1-1(A) denote the Hilbert space completion of the pre-Hilbert space (A, (., .)). The left and right invariance of h are expressed in terms of the scalar product (., •) as (a, b) =

E 0, (1) *b(1) (a(2) b (2 ))

=

(a( i ), b ( o)a(2)*b (2)

a, b E A.

Equation (25) shows that for a [3, a, @ E Â, the coefficient coalgebras C(L') and C(u0 ) are orthogonal subspaces and the characters x,, and x0 are orthogonal elements of li(A). Therefore, for any CQG algebra A the Peter— Weyl decomposition (2) is an orthogonal sum in the Hilbert space 11(A).

Example 8. Let G be a compact topological group, j.t the Haar measure of G and A the CQG algebra Rep (G), see Example 1.2. Then the Haar state h on A is given by h(a) = f a(g)dp(g), a E A, and MA) is just the Hilbert space L 2 (G,p) of square integrable functions on G with 'respect to /I. The latter follows from the Peter—Weyl theorem which asserts that Rep (G) is dense in L 2 (G, 11.3.3 C*-Algebra Completions of CQG Algebras

In this subsection we show that each CQG algebra A admits a completion to a C*-algebra. In order to do so, we extend the considerations of the first two paragraphs of Subsect. 4.3.4 to general CQG algebras. First let us recall some standard terffiinology (see, for instance, [BR] or [Mu]). A C*-seminorm on a *-algebra A is a seminorm p which has the so-called C*-property p(a* a) = p(a) 2 , a E A. It can be shown that any C*seminorm p satisfies p(a*) = p(a)* and p(ab) < p(a)p(b) for a, b E A and *1) = 1 if p O. A C*-norm is a C*-seminorm which is a norm. A C*algebra is a *-algebra A equipped with a C*-norm 11'11 such that the normed linear space (AA II) is complete. A *-representation of a *-algebra A on a Hilbert space 7-1 is a *-homomorphism of A into the *-algebra of bounded linear operators on 1-1. We née-d a simple technical lemma. Lemma 31. Let V be a dense linear subspace of a Hilbert space with scalar product (.,.). Suppose that 7r is an algebra homomorphism of a CQG algebra

11.3 Compact Quantum Group Algebras

421

A into the algebra of (not necessarily bounded) linear operators on V such that (71- (a)x,y) = (x,r(aly) for all a E A and x, y E V. (i) Let y = (vii ) be a unitary matrix corepresentation of A. Then each linear operator 7r(vii ) on V is bounded and 117r(vii )11 5_ 1. (ii) 7r extends uniquely to a *-representation of A on Proof. (ii) follows immediately from (i). To prove (i ), recall that vv r= so E k Vki * Viej = 1 by Proposition 11(v). Hence, for x E V we have

I and

117r(vii )x11 2 <E k (7r(vk i )x,7r(vki )x) = E k (7r(vk j *vki )x,x) = 1411 2 .

El

We now apply Lemma 31 to the case V = A, H = H(A), 7r = 7rh, where rh is defined by 7rh(a)b = ab, a, b E A. Since (7rh(a)b, c) = h((ab)* c) h(b* a* c) = (b, h (a* )c) for a, b E A, 7rh satisfies the assumptions of Lemma 31. Hence there exists a unique extention of 7rh to a *-representation, denoted again by 7rh, of the CQG algebra A on the Hilbert space Ii(A). The *representation 7rh of A on H(A) is called the GNS representation of the state h (see [BR] or [Mu]). From the definition it is clear that rh(A)1 is dense in the Hilbert space IAA) and that h(a) = (7h(a)1, 1) for a E A. These two properties characterize the GNS representation up to unitary equivalence. Moreover, T-h(a) = O implies that a = O. (Indeed, rh(a) = 0 yields h(a*a) = 1lrh(a) 1 11 2 = 0, so a = 0 by the faithfulness of h.) Hence 117rh•)11 is a C*-norm on A. Proposition 32. Let A be a CQG algebra. For a E A, let Majj„,3 denote the supremum of p(a) for all C* -seminorms p on the *-algebra A. Then, 11 • II 00 is a C* -norm on A and the completion of A with respect to this norm is a C* -algebra A with unit element. The counit c and the Haar state h of A have unique continuous extensions, denoted by the same symbols, to the C* -algebra A. The comultiplication of A can be uniquely extended to a *-hornomorphism A : A -4 MM. Here A(gdzi denotes the C* -algebra obtained by completing the algebraic tensor product A g A in the least C* -norm on A 0 A. These extensions satisfy the relations (A 0 id) o A = (id 0 A) o zl on AAA, 0 id) o L = (id 0 6) o = id and (h id),6 = (id h )Z1 = hi on A. (c

Proof.

First we have to show that 011,0 is finite for all a E A. Since A is a CQG algebra, it suffices to do so for a = vii , where vii is a matrix element of some unitary matrix corepresentation of A. Let p be a C*-seminorm on A and let ip denote the canonical mapping of A to the quotient *-algebra A/ker p. Then p(ip (-)) := p • ) defines a C*-norm on A/ker p. Let .Ap be the C*-algebra which is the completion of A/Icer p with respect to p. The C*algebra flp has a faithful norm preserving *-representation (see [ 1311], 2.3.4), say frp , on some Hilbert space. lip . Then rp = ïrp o ip is a *-representation of A on lip . By Lemma 31(i), 117rp (vij )11 1 and hence p(vii ) =- 75(ip (vii )) = II Trp (ip (vii ) )11= II7rp (vii ) if _Ç 1. Thus, II vii 11,, 5_ 1 and II • II co is finite on A. Since if wh011 _is a C*-norm on A as noted above, • 11,,„ is a norm, hence a C*norm, on A. Clearly, the completion of (A, II 1100) becomes a unital

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11. Corepresentation Theory and Compact Quantum Groups

C*-algebra A. Let II • denote the least C*-norm of A 0 A (see [MO. Then 114A(•)iimin is a C*-seminorm on A. Also, WI is a C*-seminorm on A and Ih(•)1 1lrh (-)11. Therefore, II • Hoe) le(*)1 It • 1100 and I h()I < P • 1100 by the definition of II li c,Q . Since 4 e and h are continuous with respect to the norm • he) they have unique continuous extensions to A. The algebraic relations are preserved by continuity. Remarks: 5. The antipode of A is not necessarily bounded with respect to the norm • li„„, so it cannot be extended to the C*-algebra A in general. 6. Since • II 00 is the largest C*-seminorm on A, each *-representation of A on a Hilbert space has a continuous extension to a *-representation of the C*-algebra A. This property characterizes the C*-algebra A in Proposition 32. However, the largest C*-seminorm is not the only possible choice of a C*-norm on A for getting a C*-algebra completion . Â of A such that the comultiplication of A extends continuously to di (see also Subsect. 11.4.1). Further, it should be mentioned that in general there are several inequivalent C*-norms on the algebraic tensor product A 0 A (see [MO.

11.3.4 Modular Properties of the Haar State Throughout this subsection let A be a CQG algebra with Haar state h. A linear functional f on A is called central if f (ab) = f (b a) for a, b E A. It is called a character if f(1) = 1 and f (ab) = f (a) f (b) for a, b E A. We shall see below (Corollary 35) that the Haar staté h on A is not central if the square of the antipode of A is not the identity. In this subsection we investigate a family of characters and of related automorphisms of A which provide qualitative measures of the extent to which both properties fail. First we state a preliminary result on holomorphic functions. Let us say that an entire function g is of exponential growth on the right half-plane if there exist constants M > 0 and u E R such that Ig(z) l < MeARe Z for any z E C, Re z > O. Lemma 33. If g i and g2 are entire functions of exponential growth on the right half-plane such that gi (z) = g 2 (z) for z E N, then gi(z) = g 2 (z) for all z E C. Proof. This is in fact Carlson's theorem (see, for instance, [Ta]).

0

Proposition 34. There exists a family If zlzEc of characters on the CQG algebra A such that for all z, z' E C and a,b E A the following conditions are fulfilled: (i) The function z f(a) is an entire function of exponential growth on the right half-plane. (ii) fzfzi = fz+z/ and fo = E. (iii) fz (1) = 1. (iv) fz (S(a)) = f_(a) and fz (a*) = f_2-(a).

11.3 Compact Quantum Group Algebras

423

(v) S2 (a) = where is defined by (1.65). (vi) h((fz-i.a.fzi---1)0= h(b(fz.a.fzI)). In particular, (26)

h(ab) = h(b(fi.a.fi))-

The linear functionals h, z G C, are uniquely determined by the (0, (II) and equation (26).

conditions

Proof. Recall that by Lemma 30 the intertwiners Fa G Mor (ua, (u1") defined in Subsect. 11.3.2 are positive definite. Hence Fa has positive eigenvalues, say A 1 , • - • , Ada . For z G C, let Daz denote the diagonal matrix with diagonal entries Al , • • • , . By the finite-dimensional spectral theorem, there is a unitary matrix Uct such that Fa = (Ia ./Jai-tic:. Define the complex power z G C, of Fa by Tic', := Ua Daz Uc*,. We write klz, as the matrix (F ) ii with respect to the fixed basis of the representation space of ua. Now we define the linear functional fz , z E C, on the elements of the vector space basis fu'icrit la G A , i , j = 1,2, • • • of A (see Theorem 13(vi)) by setting f(u) := (Faz)ii. Then the properties (i)-(iii) are clear. For (ii) we use the

relation Plz,F,z: . By (ii), f_ z is the inverse of fz in the algebra A'. This implies the first equality of (iv). To verify the second one, it suffices to take a = u. Using the facts that Tjt- z = (FV)* and u is unitary, we compute fz((uZi r) = fz(S(ucli)) = f—zett7 i ) = (Fct— z) j i =

(1 1—Z \ ._ = f_2(u'icrti).

Thus, (iv) is proved. Since Fa G Mor (u", (0 ) ), we have S2 (u) = (F,t )ik4i (F; 1 )ii = (fi id 0 f_i)(L\ 0 id) 04(u) = f_i.uZ.fi. This proves (v) for a = uZ and hence for all a G A. It suffices to prove (vi) for elements a = u and b = (41 )*. If a )3, then both sides of (vi) vanish by (24) and (25). If a = 0, then using once more the Schur orthogonality relations (24) and (25) we compute that both sides F ta ik ( F zi) ii. of the equation in (vi) are equal to /UV- Lz-i) Next we show that the functionals fz are characters. Put p(a) := By (26), we have h(cp(ab)) = h(abc) = h(bcp(a)) = h(cp(a)p(b)) and hence h(e(p(ab) p(a)p(b))) = 0 for any c E A. Setting c = (p(ab) - p(a)p(b))* and using the faithfulness of h (by Proposition 29), we conclude that p(ab) p(a) p(b), so that h.ab. f = (h.a.h)(fi.b.fi). Applying e to this equality and using the relation fifi = f27 we get f2 (ab) = f2 (a) f2 (b). Since the convolution product of characters is again a character as is easily seen, the latter implies that fn(ab) = f(a)f(b) for n = 2,4, • • From Lemma 33 (M we derive applied to the functions g1 (z) = f22 (ab) and g2 (z)f \ay., 2z = J(2z)f that fz (ab) = Ma) f z (0 for all z G C. To prove the uniqueness assertion, let triatec be an arbitrary family of linear functionals satisfying (i), (ii) and (26). By a similar argument to that used in the preceding paragraph, (26) implies that fl.a.fi =

424

11. Corepresentation Theory and Compact

Quantum Groups

Therefore f2 (a) = /2 (a) and hence f(a) = J(a) by (ii) for all n = 2,4, • • •. Finally, Lemma 33 yields that h(a) = fz (a) for any z E C.

Corollary 35. The Haar state h of A is central if and only if 5 2 = id. This holds if and only if f, = E for all z E C. Proof. If S2 = id, then u" = Oil' and hence Fa = I for a E À. Therefore, h = E for all z E C. By (26), the latter implies that h is central. Conversely, if h is central, then fi.a.fi = a by (26) and so fn = E for n = 2,4, • • by the reasoning used in the above proof. Hence f„ = e for all z E C by Proposition El 34(i) and Lemma 33.

In terms of the functionals h the Schur orthogonality relations (24) and (25) can be expressed as h((4)*43.) =45,06ki M; l f_ i (uici).

h(4/ (4)*)

Since the functionals h, z E C, are characters, they belong to the Hopf dual A° of A. Thus, the first equality in (iv) can be rewritten as S(f) — f—z. Further, for any z, E C the equation

(27)

a E A,

Pz,zi(a) =

defines an automorphism pz , z , of the algebra A. From Proposition 34 we immediately obtain the following formulas for these automorphisms: Pz,z' °

Pw,w' = Pz-Fw,z' -Fw'

and

poi = id,

(28)

(p,, w ® p_ w , z ,) o Z = X o pz,z ,, Pz,z'

(e) =

P—

(29) (30)

(a) * 7

p,/(S(a)) = S(p_z , ,_z(a)),

(31)

h(p„,„,(a)) = 144

(32)

Set Oz piz ,i, and piz ,_ iz , Z E C, where i = Then Ot ItER and OthER are commuting one-parameter groups of *-automorphisms of A by (30). They are important tools for the study of the CQG algebra A. By condition (vi) in Proposition 34, for any a, b E A we have Nab) = h(b0_ 1 (a)) and h(O z (a)b) = h(b0,_ 1 (a)),

z E C.

(33)

That is, {Oz } measures the extent to which h fails to be central. Recall that, by the proof of Corollary 35, h is central if and only if 0, = id for all z E C. Further, by Proposition 34(i), the function z h(0,(a)b) is entire and bounded on any strip {z I a < Tm z < 0 } , a, 8 G R. Thus, by (33), for any a, b E A there exists a bounded function f on the strip {z 0 < Tm z < 1} which is holomorphic in its interior and satisfies the boundary conditions f(t) = h(O t (a)b) and f(t + i) = h(bO t (a)), t G R. The latter is expressed by saying that {Ot } tE R satisfies the KMS condition relative to the Haar state

11.3 Compact Quantum Group Algebras

425

h. The KMS condition plays an important role in the theory of operator algebras and in statistical mechanics (see [BR]). It characterizes the modular automorphism group of a faithful normal state on a von Neumann algebra. We call {Ot } tE R the modular automorphism group of the CQG algebra A. The second automorphism group {0,} is related to the behavior of the antipode S of A. First we note that each V, commutes with S by (31). Iterating condition (v) in Proposition 34 we obtain 82n

Set U(a) :=

p_n,n oin

n E Z.

19_0(a)), a c A. From this definition it is clear that =U o19112 =1900U.

(34)

The map U is a *-preserving anti-automorphism of A such that U oU = id. It is called the unitary antipode of the CQG algebra A. Note that U0/9, =19,0U, U 0 Oz = 0_ z oU, US = SU and h(U (a)) = h(a) for z E C and a E A. We close this subsection by considering once more the coordinate Hopf *-algebras of compact quantum matrix groups.

Example 9. Let Gq denote one of the compact quantum groups Uq (N), SUq (N), 0 q (N; R) or USpq (N), q E R. As noted in Example 7, the Hopf *-algebra A := 0(Gq ) is a CQMG algebra. In order to treat all four quantum groups Gq at once, we set i := 2i

-

N

-

1 for Uq (N), SUq (N),

-

2pi for 0q (N; R), USpq (N),

where pi are the numbers defined in Subsect. 8.4.2. Then the value of the character h at the entry u i of the fundamental matrix u = (u i ) of Gq is given by f2 (u) = for j = 1, 2, • • • , N and z E C. (35) Let us prove this formula. By the definition of the involution in (.9(Gq ), the fundamental corepresentation u is unitary. As noted in Sect. 9.4, the vector representation Ti of Uq (g) is the associated representation with u by Proposition 1.15. Therefore, since T1 is irreducible, so is u and we can assume that u is one of the corepresentations u', c c Â. Set F = (Fii ) := (q1 60 ). By Propositions 9.10 and 9.13, S2 (u) = qiu.;47 -3 . Hence F E Mor (u, u"). Since = -P with i' := N + 1 i by the definition of we have Tr F = Tr F -1 . Thus, F is the intertwiner from Lemma 30 and formula (35) follows from the definition of the functionals h given in the proof of Proposition 34. Further, let (., .) denote the dual pairing of the Hopf algebras Uq (g), g = glN ,s1N, 502n,5pN, resp. U0 / 2 (502 n+ 1) and 0(Gq ) from Theorem 9.18. If p is the half-sum of positive rots of g and K2 p the element of Uq (g) resp. Uq 1/2(5027, + 1) defined by (6.26), then fi (a) = (K2,,, a), a E 0(Gq ).

(36)

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11. Corepresentation Theory and Compact Quantum Groups

Because fi is a character and K2p is group-like, it is sufficient to prove (36) for a By (35) and (9.39), formula (36) is then equivalent to the equaThe latter can be verified by using the explicit forms tion ti(K2) = of the vector representations T1 (see Subsect. 8.4.1). Under the additional assumption that Uq (g) resp. Uq 1/2(so2n+1) separates the elements of C9(Gq ) (see Corollaries 23 and 54), there is a more instructive proof of (36) given as follows. Since S2 (x) = K2px1(2-pi for z in U(g) resp. Uq 1 / 2(s02n+1 ) by Proposition 6.6, we have S 2 (U n i )) (K2p, Ujn ) = ( 82 (X)K2p, Ui)= (K2 091 U n )(X,

and hence S2 (uni )(K2p , (K2p , uni )u.7. Therefore, the linear transformation F = (Fii ) := ((K 2 , u)) is the intertwiner of u and ucc from Lemma 30. By the definition of fi , (36) follows for a = u. Finally, combining the formulas (27) and (35) we obtain the action of the algebra automorphism p,,,, of 0(Gq ) on the generators u as Pz,zi(ILD = qz 14 z 3 uii . 11.3.5 Polar Decomposition of the Antipode

With formula (34) a decomposition of the antipode as an anti-automorphism of A was given. The following proposition describes the polar decomposition of the closure of the antipode in the Hilbert space 7-t(A). Proposition 36. Let A be a CQG algebra. The antipode S of A, considered as a linear operator in the Hilbert space 'NA), is closable. Let S = Uolgi be the polar decomposition of the closure S' of the operator S. Then U0 is a unitary self-adjoint operator on 71(A) and we have

Uo(a) = fi.S(a)

and II(a) = a.fl,

a E A.

Proof. Using the formulas from Subsect. 11.3.4 and formula (4), we derive that (b, 51 (a)) = (p l , i (S(b)), a), a, b E A.,Therefore, the domain of the adjoint operator of S contains A, hence it is dense and S is closable. Define Uo (a) = .h.S(a) and So(a) = a.f1 , a c A. Then, by (31), we have U0S0(a) = p i ,o(S(po, i (a))) = S(p0,-1 0 po,i(a)) = 8(4

a E A.

Since S 2 = p_i,i by Proposition 34(v) and (27) and So* = * S' by (1.39), we obtain p i ,o (S(a))* --= po,_ 1 (S(a*)) by (31). Using Proposition 34(vi) and the formulas (32) and (4) we get (U0(a ) ,U0(b)) =

h(Pi,o(S(a)) * Pi,o(S(b))) = h(Po,-1(S(a * ))pi,o(S(b)))

h(f)1,0(S(b))Pi,o(S(e))) = h(Pi,o(S(a * b))) = h(S(a*b)) = h(a* b) = (a,

11.3 Compact Quantum Group Algebras

427

Therefore, since U0(.4) = A, U0 extends uniquely to a unitary operator, denoted again by U0, on 7-1(A). Further, by (32) and (30), we have (a, So (a))

h(a*p0 , i (a)) = 4,90,1/2(Po,-1/2(e)Po,i/2(a))) ()= h(P0,112(a) * k,112(a))

That is, S0 is a positive operator on A. Thus, by the uniqueness of the polar decomposition (see [RS], Sect. VIII.9), U0 and the closure of S0 are the two factors of the polar decomposition of S. From Proposition 34(v) and (31) it follows that Ug = I. Hence U0 is unitary and self-adjoint.

11.3.6 Multiplicative Unitaries of CQG Algebras First let us motivate the definitions given below. Let G be a unimodular locally compact group and let 7-1 = L 2 (G) be the Hilbert space of square integrable functions on G with respect to the Haar measure of G. The unitary operator V on the tensor product Hilbert space 7-107-1 = L 2 (G x G) defined by (V f)(s,t) -= f (s, st), s,t E G,fE L 2 (G x G), plays a fundamental role in the duality theory of locally compact groups. This operator, originally invented by W. F. Stinespring, is usually called multiplicative unitary or the KacTakesaki operator. In fact, V allows one to introduce a Hopf algebra structure on the von Neumann algebra generated by the left regular representation of G from which the group G itself can be recovered (see [ES] and [Tk] for treatments of this topic). The unitary operator W on 7-1 7-1 given by (Wf)(s, t) = f (st,t) is used in a similar manner. It is easy to check that V and W satisfy the following equations on 7-1 0 7-1 0 7-1: V12 1723 = V23 1713 1/121

(37)

W23W12 = W'12 W'13 W23.

(38)

Equations (37) and (38) are called pentagon equations. Obviously, V satisfies (37) if and only if V -1 satisfies (38). Now let A be a Hopf algebra with invertible antipode. We define linear mappings V, W:A0A-4 A0A by

V (a 0 b) = 46(b) • (a 0 1)

and

W(a 0 b) = .6(

• (1 ®b).

(39)

Clearly, these are generalizations of the mappings V and W mentioned above, but because of the commutativity of the function algebras we could have also taken qa 0 b) = (a 0 1)-(b) and 1-47 (a b) = (1 0 b). 46(a) . Proposition 37. (0 V and W are bijections of A0 A. Their inverses act as V - I (a(g)b) = (G9-1 0i46(b))•(a01) and W -1 (a0b) = ((id0S)41(a)).(10b). (it) V and W satisfy the pentagon equations (37) and (38), respectively. (iii) The comultiplication, the counit and the antipode of A can be recovered from the mappings V and W by

428

11. Corepresentation Theory and Compact Quantum Groups

(a)=V(1 0a)V - ', E(a)1=m0T0V -1 (1 0a), S -1 (a)=m(id0E)V -1 (1 0a), Zi(a) = W(a 0 1)W -1 ,

(a)1 = mW -1 (a 0 1) , 8(a) =--- m(e

id)W -1 (a 1),

where .A(a) is considered as a multiplication operator on A 0 A. Proof All assertions follow by straightforward algebraic manipulations. We verify (for instance) the pentagon equation (38) for W. In the Sweedler notation, we have W(a b) = a(1 ) a(2)b and hence

E

(E a 0 b(1) b(2)c) 1V12 (E a (1 ) 0 b(i) 0 a(2)b(2)c) = E a(1) 0 a(2)b(i) 0 a( 3 )b( 2 )c,

W12W13 W23 (a 01)0

W2314712 (a

b

= Wi2W13

= W23 (

= E a( i )

a( 2 )b( 1 )

a (i ) a( 2)b

c)

a( 3)b( 2 )c.

A be a CQG algebra. Then the mappings V and W defined by (39) have unique extensions to unitary operators of the Hilbert

Proposition 38. Let

space 1-1(A) 1-1(A). Proof By Proposition 37(i), V and W are bijective mappings of A 0 A. Therefore, since A 0 A is dense in 7i(A) 0 'H(A), it suffices to show that V and W preserve the scalar product of A0 A. We prove this in the case of W. Let a, b, c, d E A. Using the definitions of the scalar product on A 0 A and A and the right invariance of the Haar state h, we, compute (W (a 0 b)

=

W(C

0 d)) =

(a (i) O a(2)b,c( i) O c(2)d)

(a (i) , c( i ))(a( 2) b, c(2) d) =

E h(a(1) *c(o )h((a(2) b)*c(2) d)

= h (b* (E h((a* c)( 1 ))(a* c) (2 )) d) = h(b* h(a* c)l)d) = (a, c) (b d) = (a b, c d).

Let us denote the continuous extensions of V and W to 7-0A) 0 H (A ) again by V and W, respectively. These extensions (which also satisfy the pentagon equations (37) and (38)) are called multiplicative unitaries of the CQG algebra A. The unitary operators V and W are linked by the relation V* = T(U0 O 1)W(U0 0 1)T,

(40)

where T is the flip and U0 is the unitary self-adjoint operator from the polar decomposition of the closure of the antipode (see Proposition 36). Indeed, (40) is equivalent to the equation V-1 T(U0 01) = 0 1)W, which can be verified by inserting the corresponding formulas for V', U0 and W. Multiplicative unitaries play a crucial role in the C*-algebra approach to noncompact quantum groups (see [BS], [B], [Wor81).

11.4 Compact Quantum Group C . -Algebras

429

11.4 Compact Quantum Group C*-Algebras In Proposition 32 we constructed a C*-algebra completion of a CQ.G algebra. In this section we take the C*-algebra of a compact quantum group as the starting point and show that it always contains a unique dense CQG subalgebra.

11.4.1 CQG C*-Algebras and Their CQG Algebras Definition 11. A compact quantum group C*-algebra (briefly, a CQG C - algebra) is a C* -algebra A with unit element such that the following two conditions hold:

AA which satisfies the (i) There is a unital *-harnomorphism z : A id),6 = (id 0446 on AAA. relation is an index set and Iv E 11 (ii) There is a family fuv = d, E N) of matrices with entries in A such that: 7 dv • (ii.1) d(u) = uZ i for v E I, i,j =1,2, (ii.2) uv and its transpose (uv)t, v E I, are invertible matrices over A. (ii.3) The *-subalgebra A of A generated by the entries 'al) ." is dense in A. If the family fuv I v E 11 consists only of a singleton, then A is called a compact quantum matrix group C*-algebra (a CQMG C*-algebra).

As in Subsect. 11.3.3, A07;)A denotes the completion of the algebraic tensor product A 0 A in the least C*-norm (see [Mu], p.190). The invertibility of a matrix y = (vij ) over A means that there is another matrix w = (w ii ) with entries in A such that Ek VikWkj = Ek Wikkj = kl for all i, j. Let A be a CQG C*-algebra. Note that the *-homomorphism 46 maps A to the completion AA rather than to the algebraic tensor product A 0 A. Let us still call 46 the comultiplication of A. However, 46 maps the *-algebra A to the algebraic tensor product A® A by (ii.1) and satisfies the "ordinary" coassociativity axiom (46 0 id) o46 = (id o on A 0 A 0 A. By Theorem 39 stated below, A is even a Hopf *-algebra, but at the moment we do not yet have a counit and an antipode on A. Next we define finite-dimensional corepresentations of A. Definition 12. Let V be a finite-dimensional vector space. A linear mapping ço : V 7 A is called a corepresentation of the CQG C* -algebra A on V if (id 2i) = (y) id) ço and ker ço={0}.

By Propositions 1 and 2, the two conditions in Definition 12 characterize a corepresentation of a coalgebra. Let us emphasize here once more that a CQG C*-algebra is in general not a coalgebra according to Definition 1.2. The concepts and results on finite-dimensional corepresentations of coalgebras developed in Sect. 11.1 go over almost verbatim to corepresentations of CQG C*-algebra.s (except for the few statements where the counit occurs). In particular, the first condition (id 0 Z1) o ço =(cp 0 id) o cif) in Definition 12

430

11. Corepresentation Theory and Compact Quantum Groups

is equivalent to the relation (id .6) o (`i6 012013 , where (;6 is the associated element to ço of L(V) A (see Subsect. 11.1.1). Let us recall the notion of a unitary corepresentation which plays a crucial role in the following. A corepresentation ço of a CQG C*-algebra A on a finite-dimensional Hilbert space V is called unitary if ço is a unitary element of the *-algebra L(V) 0A, that is, (P*0 -== Iv 01A. For a CQG C*-algebra A it is more important to treat finite-dimensional corepresentations of A as elements E £(17) 0 A rather than as mappings (p : V V 0 A. The technical reason is that L(V) A is itself a C*-algebra, so the C*-algebra theory applies to O. On the conceptual side, this view paves the way for the definition of infinite-dimensional corepresentations of CQG C*-algebras (see Subsect. 11.4.3 below). There is a rather close link between CQG C*-algebras and CQG algebras which we will now discuss. On the one hand, for each CQG algebra Ao the C*-algebra completion A of Ao from Proposition 32 obviously satisfies the conditions of Definition 11 (with A = A o and 1 = dio ), so A is a CQG C'algebra and ,6.40 = 2iAr.ito. Thus, each example of a CQG algebra Ao gives rise to an example of a CQG C*-algebra A in this way. On the other hand, for each CQG C*-algebra A there exists a unique CQG algebra Ao such that Ao is a dense *-subalgebra of A and 41A0 = 2■ ArAo. The latter assertion is the main content of the following theorem. Theorem 39. Let A be a CQG C* -algebra and let A be the *-algebra from Definition 11 (11.3) . Then A is the linear span of matrix elements of all finitedimensional unitary corepresentations of A and A is a CQG algebra with comultiplication 2■ A := LA[ A. It is the 'unique 'CQG algebra which is a dense *-subalgebra of A and whose comultiplication is the restriction of the comultiplication of A.

The proof of Theorem 39 will be given in Subsect. 11.4.3. It is essentially based on the existence of a Haar state on a CQG C*-algebra which will be proved in Subsect. 11.4.2. Let us continue the above discussion concerning the connections between the CQG C*-algebra A and the associated CQG algebra A. By Lemma 45 below and Theorem 27(iii), finite-dimensignal corepresentations of A and A are unitarizable. Therefore, the first assertion of Theorem 39 implies that there is a one-to-one correspondence between finite-dimensional corepresentations of A and finite-dimensional corepresentations of A. Clearly, this correspon dence preserves all standard operations and concepts from corepresentation theory such as direct sums, tensor products, equivalence, irreducibility and unitarity. That is, the CQG algebra A carries the full information on the finite-dimensional corepresentation theory of the CQG C*-algebra A. It might be necessary to emphasize that despite the close relations explained .above the correspondence between CQG C*-algebras A and their associated CQG" algebras A is not one-to-one. As already mentioned in Subsect. 11.3.3, for a given CQG algebra A there is in general no unique C*-norm on A such that the corresponding completion is a CQG C*-algebra.

11.4 Compact Quantum Group C*-Algebras

431

11.4.2 Existence of the Haar State of a CQG C*-Algebra

The main aim of this subsection is to prove the following

Theorem 40. Let A be a CQC C* -algebra. Then there exists a unique state h of A such that (id 0 h).6(a) = (h id)d(a) = h(a)1 for all a G A. The state h in Theorem 40 is called the Haar state of the CQG C*-algebra A. The proof of Theorem 40 is divided into several steps stated as lemmas.

Lemma 41. If A is a CQC C* -algebra, then the sets 46(A)(A 0 1) Lin 1,6(a)(1 b) I a,b E AI Lin 1,6(a) (b 0 1) I a,b E A} and ,6(A)(1 0 A) are dense in AA.

Proof. Let y = (yii ) i ,i =1,...4 be a matrix with entries in A such that 46(vii ) = Vik OVki • Let V be a d-dimensional vector space with basis {e j = 1, 2, • • , dI and define (,0 G (V, V A) by cp(e i ) = ei 0v ii. If vt is invertible, then ker = {0} and hence c,o is a corepresentation and y is a matrix corepresentation of A. If vt is invertible, then 7/ is also invertible and hence 3 is also a matrix corepresentation. From these remarks and conditions (ii.1) and (ii.2) it follows that and ---7 u are matrix corepresentations of A. Now let R be the smallest set of matrix corepresentations of A that contains all ?iv and uv, u E I, and is closed under direct sums and tensor products. Then A is the linear span of matrix coefficients of corepresentations in R. Let u = (u ii ) be an arbitrary matrix corepresentation in R. By induction one can easily verify that the matrices u and fi are invertible. Hence ut is also invertible. For y := u -1- and w (ut) -1 , we have Ei ,6(uni)(1 ® vim) = unj ® uj ivi„,, = u„,„, ® 1 and Ei 46(u,„)(wim ® 1) = 1 ® umn . Therefore, A 0 A. Since A is dense in A by 460)(1 0 A) A 0 A and 46 (A)(A 0 1) condition (ii.3), the assertion follows. El For the following three lemmas we assume that A is a unital C*-algebra such that condition (i ) of Definition 11 is satisfied and ,i(A)(A 0 1) in dense in AA.

Lemma 42. For any state f on A there is a state g such that gt=fg=g.

Proof. Define a state jrn on A by the Cesaro sum fn := (f + f 2 + • • ' fn), n E N. Since the state space of a unital C*-algebra is weakly compact (see [Mu] or [BR]), there exists a weak accumulation point g of the sequence Ifn l. From the inequality II jen f frLII = — fil we obtain g f = g in the limit. Similarly, fg = g. Lemma 43. Let f and g be .states on A such that fg = g. If (p is another positive linear functional on A such that (p < f, then (pg = (p(1)g.

Proof. Let a E A and put b := (id

g),6(a). Then we have

(id 0 ALI(b) = (id 0 fg),d(a) --= (id 0 g).(a) = b

11. Corepresentation Theory and Compact Quantum Groups

432

and so (id 0 f)((b) — b 1)*((b) — b 01) = (id 0 f)(b*b) b*b — (id 0 f)(b*)(b 0 1) — (id 0 f)(b* 1)(b) = (id 0 f)LX(b*b) — b*b. Now let be a state on A such that Of=7,b. Such a state always exists by Lemma 42. Applying to the preceding equation, we obtain b 1) = O.

1)*((b)

(0 0 f)(z(b) —

By the Cauchy—Schwarz inequality, this yields p® f)((cod)((b)—b01)) = for all c, d E A. Inserting the definition of b, we get (

g)((c

(0 0 f

d 1)(LX

id)46(a)) = f(d)(0 g)((c

1)(a)).

We have 1)W 0 idW(a) =

(cØ d

(c ®dØ 1)(id ,4(a)

= (1

d

1)(id

,)((c 0 1)(a))

Applying the involution to ,i(A)(A 0 1) we see that (A 0 1)z(A) is dense in AA. Therefore, we can replace (c 1)d(a) by 1 0 x, z E A, on the right hand side of the preceding equalities and obtain OP

f

g)((1

d

1)(id ,i)(1

x)) = f(d)(cp 0 01 0 x).

Setting yx := (id 0 g)46(x) and using the equality OM = 1, this reads as f (dy s ) = f (d)g(x), d, x G A. Now let 7rf denote the GNS representation of f on the Hilbert space 7-tf (see [Mu] or {BR] for this concept). Since 1(a) = (Cf, f(a)Cf), a E A, by the definition of 7rf , we have f(dy s ) = (71-f(d*)( f ,ir f (yx )(f ) = g(x)(1-f(d*)( 1 ,(1 ) = g(x)f(d), d,x E A.

Because 7rf (A)(f is dense in Hy, the latter implies (71,7rf(Y s )(f) = g(x)(77, (f) for all n E Nf . From the inequality (it) < f it follows that there exists a vector nv, E Nf such that ço(a) = (7,rf (a)(f ) for all a E A. Hence (10 (Yx) = g(s)(p(1). But ço(Ys) = ((pg)(x) by the definition of yx . Thus, 1=1 (pg(x) = g(x)c,o(1) for x E A.

Lemma 44. There exists a state hi on A such that (id 0 hi)(a) = h i (a)1 for all a E A. Proof For a positive linear functional f on A, let K 1 be the set of all states g on A satisfying fg = f(1)g. Clearly, Kf is a weakly compact subset of the dual space of the C*-algebra A. By Lemma 42, Kf is not empty. By Lemma 43 we have K f C K p if ço < f. Hence K1 1 +12 C K1 1 n K1 2 for arbitrary positive linear functionals fi and f2 on A. Since the state space

11.4 Compact Quantum Group C*-Algebra.s

433

of A is weakly compact, the intersection of all ICf is nonempty. If hi is an element of this intersection, we have fh 1 = f (1)hi for all positive and hence El for all continous linear functionals f on A. This implies the assertion. Proof of Theorem 40. By Lemma 41, the sets d(A)(A 0 1) and 2(A)(1 0 A) are dense in A(, so the preceding lemmas apply to the CQG C*-algebra A. Let hi be as in Lemma 44. By similar arguments (or by applying Lemma 44 to A equipped with the opposite comultiplication 2icIP = T 0 2i) there is a state h, on A such that (h, id)(a) = h r (a)1. Applying h, to the formula for hi and hi to the formula for hr., we get kJ-4(a) = hi(a) and hr hi(a) = hr (a) h and the existence of h is proved. The latter for a E A. Thus, hi --- h, 1=1 reasoning proves also the uniqueness assertion. C • 1. Remark 7. Let A be a C*-algebra with unit element such that A Define a linear mapping LX : A -4 JUM by 2i(a) = 1 g a. Obviously, 2i satisfies the coassociativity axiom and 2i(A)(A 0 1) is dense in AA. In this case, each state h on A is left-invariant, but there is no right-invariant state on A. Hence A is not a CQG C*-algebra. (If A is the algebra of the continuous functions on a compact topological Hausdorff space, the above comultiplication L corresponds to the semigroup structure on X defined by x.y = y for all x, y E X.)

11.4.3 Proof of Theorem 39 The following lemma is used in the proof of Theorem 39. It is the counterpart to assertion (iii) of Theorem 27 for CQG C*-algebras.

Lemma 45. Let ço be a corepresentation of the CQG C* -algebra A on a finite-dimensional vector space V. `(,6 is an invertible element of L(V) 0 A, then w is unitarizable. Proof. Let (•, -) be a scalar product on V. Then L(V) 0 A becomes a CY - algebra. Since (,`(,' is invertible, (0*(,6 > 0 1) for some 8 > O. Put E := (id 0 1)((;6* (0), where h is the Haar state of A. Then E is a self-adjoint operator on the Hilbert space V such that E > 61. Since (p is a corepresentation, we have (id 0 2 ) ((,6*(P) =-- 013 012 012013 (see Proposition gii)). Applying (id 0 h 0 id) to both sides of this identity and using the right invariance of h we obtain (0* (E 0 1)0 = E 0 1 and hence (,O(E -1 01)0* = `0(E -1 1)[0*(E 1)0145 -1 (E -1 0 1) = E -1 0 1. Let .iPj := (E 1 / 2 01)ç(E -1 / 2 01). Then is a corepresentation of A which is equivalent to (p (via E" 2 c Mor ((p, 0). By the two equalities at the end of the preceding paragraph we have IN = 17)11/)* /4 = I® 1, that is, V) is unitary. Therefore, (p is unitarizable.

Now we begin the proof of Theorem 39. Suppose that A is a CQG C*algebra. Let '14_ be the class of all finite-dimensional corepresentations of A which have a matrix u such that .1) and i3 are invertible. Since this

434

11. Corepresentation Theory and Compact Quantum Groups

property is preserved under direct sums, tensor products and conjugations, the linear span Ao of the entries of all such matrices y is a *-algebra. Any corepresentation from R o is unitary with respect to some scalar product by Lemma 45 and hence a direct sum of irreducible unitary corepresentations by Proposition 12. Since unitary corepresentations of A belong to Ro , Ao is spanned by the matrix coefficients of all irreducible unitary corepresentations of A. Let A denote the equivalence classes of such corepresentations. For each o E A, we fix a unitary corepresentation ( 19' of the class a and an orthonormal basis of the underlying Hilbert space, so that (p' is given by a unitary matrix E Â} V a = (VZ)i,j=1,...,d a • By Corollary 10, the set {yi'i I i j =1, 2, • • • , dc„ is a vector space basis of Ao . (Note that in the proof of Corollary 10 the counit was not used.) We define two linear mappings e : A0 --+ C and S : A0 -+ A0 by e(v) = kJ and S(y) = (qi )* for a E A, i,j = 1, 2, • • • , da . Using the relations = vagqi and Ei (v)*va Ei v7,40* = bik one easily checks that e and S satisfy the counit and antipode axioms, respectively. Therefore, Ao is a Hopf algebra. Since the conjugate (p a of vcc is irreducible and unitarizable by Lemma 45, va belongs also to some class of A, so that Ao is a *-algebra. By Definition 11(i), L is a *-homomorphism. Hence Ao is a Hopf *-algebra. Since Ao fulfills condition (ii) in Theorem 27, Ao is a CQG algebra. Now let 13 be an arbitrary CQG algebra such that 13 is a dense *= / A [B. Then the irreducible unitary corepresubalgebra of A and sentations of 8 are also irreducible unitary corepresentations of A. Hence B C Ao by the Peter-Weyl decomposition (2) of B and the definition of Ao We prove that B = Ao . Assume on the contrary that B A o . Then there exists a E A such that C(u0 ) n B = {0}. By the uniqueness of the Haar functional (see Subsect. 11.2.2), the restriction of the Haar state h of A to Ao is the Haar state of Ao . By (25), h(b*a) = 0 for all b E B and a E C(0). Approaching a* by elements of B (because B is dense in A) and using the continuity of the state h on A, we get h(a*a) = 0 for a E C(0). This contradicts Proposition 29. Thus we have proved that B = A o . Because the matrices u" and uv, v E I, are invertible by Definition 11(ii.2), the class R defined in the prod! of Lemma 41 is contained in Ro) so A is a *-subalgebra of Ao . Repeating the above reasoning, it follows that A is also a CQG algebra such that LA = i6A1- A and A is dense in A by Definition 11(ii.3). As shown in the preceding paragraph, this implies that A = Ao . This completes the proof of Theorem 39. ,

...

11.4.4 Another Definition of CQG C*-Algebras An equivalent definition of CQG C*-algebras is the following Definition 11'. A compact quantum group C*-algebra is a unital C* -algebra A which satisfies the conditions (0 of Definition 11 and (ii)': The linear spaces d(A)(A01) and A(A)(10A) are dense in AA.

11.5 Finite-Dimensional Representations of GL q(N)

435

Lemma 41 above says that Definition 11 implies Definition 11'. From Theorem 1.2 in [Wor7] it follows that Definition 11' implies Definition 11, so that both definitions of a CQG C*-algebra are equivalent. In the proof of Theorem 40 only the conditions (i) and OW are essentially used. Therefore, this proof shows also that each CQG C*-algebra in the sense of Definition 11' admits a unique Haar state. Definition 11 (with condition (ii) rather than (fi)') is better adapted to the construction of examples of CQG C*-algebras, because the standard compact quantum matrix groups (see Subsects. 9.2.4 and 9.3.5) are built from a unitary fundamental corepresentation u = (uii ). But in contrast to that, Definition 11' gives an elegant axiomatic approach to CQG C*-algebras. For the next definition, let M(B) denote the multiplier C*-algebra of a C*-algebra B (see [Mu], 2.1.5). Let £(1 - ) and C(7-0 be the C*-algebras of bounded resp. compact operators on a Hilbert space Definition 13. A unitary corepresentation of a CQG C* -algebra A on a Hilbert space 7-t is a unitary element y of the multiplier algebra M(C(7-0A) such that (id 2i)y = y12v13.

In particular, each unitary element y of the C*-algebra £(7-0 gA - satisfying (id 0 20y= y 12 y13 is a unitary corepresentation of A, because £(7-0 gA - is a C*-subalgebra of the multiplier algebra M(C(7-00A). Clearly, if the Hilbert space 11 is finite-dimensional, then M(C(7-0A) = £(7-0 0 A and Definition 13 reduces to the definition of a unitary corepresentation as used above.

11.5 Finite-Dimensional Representations of

aL q (N)

In classical representation theory (see, for instance, [Zhe], Chap. XVI) all finite-dimensional analytic irreducible representations of the group GL(N, C) can be constructed by means of characters of the subgroup of diagonal matrices and the invariance with respect to the Borel subgroups of triangular matrices. In this section we extend this procedure to the quantum group GL q (N) and show that 0(GL q (N)) is a cosemisimple Hopf algebra when q is not a root of unity. Throughout this section we assume that q is not a root of unity.

11.5.1 Some Quantum Subgroups of GL q (N) Let DN be the diagonal subgroup of CL(N,C). Its coordinate Hopf algebra 0(DN) is the commutative algebra C[ti, tr I , • • • , tN, tA of all Laurent polynomials in N indeterminates t1, t2, • • • , tN with comultiplication 1(t i) = 404 and counit e(ti) = 1. Obviously, there exists a surjective Hopf algebra homomorphism irD : 0(GL q (N)) —> 0(DN) determined by rD(uii) = 8iitil j = 1, 2, • • , N. Therefore, by Definition 9.11, DN is a quantum subgroup of GL q (N), called the diagonal subgroup.

436

11. Corepresentation Theory and Compact Quantum Groups

Next we define the quantum Borel subgroups Bq+ and Bq- of GL q (N). Let 1+ (resp. 1 - ) be the two-sided ideal of the algebra 0(GL q (N)) generated by the elements 4 , j > j (r esp. j < j). It is easily checked that /± is a Hopf ideal of 0(GL q (N)). Hence the quotient (9(13) 0(GL q (N))11 + is again a Hopf algebra. Its structure is described a little more explicitly as follows. As algebras, 0(N) and C9(N) have iN(N + 3) generators z ij , j < j, z 1 and zij , i > j, z 1 , respectively. The defining relations are zii z,V = 1 zii = 1 and the relations (9.17) for 0(Mg (N)) when we set t4 = 0, i > j (resp. t4 = 0, j < j) therein. Under the latter replacement the Hopf algebra structure of O(Bi) (resp. of O(N)) is derived from that of A = O(GLq (N)). There are surjective Hopf algebra homomorphisms 7r B ± : 0(GL q (N)) O(B) such that 7r ± (D-1) = 7 B:

Bq

( 14) = Z ii 7

z NN -1 and <

3 = z-2. 2• 7 i > j,

> j,

for

0(B),

7)3,7 (ui ) = 0, i < j,

for

0(N).

7rE: (Uji =

Hence B q+ and B q- are quantum subgroups of the quantum group GL g (N), called the quantum Borel subgroups. From (9.17) it follows in particular that the elements z11, • • • , zNN, zj, • • - , zN -N 1 pairwise commute.

11.5.2 Submodules of Relative Invariant Elements Here we introduce some general notions which will be used for the construction of irreducible representations of GL q (N). Let C.9(G q ) be the coordinate Hopf algebra of a quantum group Gq and let C9(X g ) be the coordinate algebra of a left (resp. right) Gq-quantum space X q . Recall that the latter means that there exists a left coaction 0/, : 0(X q ) -+ C9(Gq )00(X q ) (resp. a right coaction : 0(X q ) 0(X000(G q )) which is an algebra homomorphism.

Definition 14. Let x be an element of C9(Gq ). An element a E C9(X q ) is called left (resp. right) relative Gq-invariant with respect to x if OL(a) = x0 a (resp. OR (a) = a 0 x). The set of all left (resp. right) relative Gq -invariant elements with respect to x is denoted by 0(Gq \X q ;x) (resp. 0(X q lGq ; x)).

In the case x = J. the spaces 0(Gq \X q ;x) and C9(X q lGq ;x) are just the algebras of left- and right-invariant elements of C9(X g ), respectively. Let X q be a left Gq -quantum space with left coaction çL : 0(X, g ) 0(Gq ) 0 0(X g ) and a right Hg-quantum space with right coaction : 0(Xq ) 0 (X q ) 0 O (Hq ). We say that X q is a two-sided (Gq ,I1q )-quantum space if the coactions 0/, and OR commute, that is, if

(95 .L 0 id) OR = (id 0 OR) 0 OL• The following rather simple facts are crucial for what follows.

(41)

11.5 Finite-Dimensional Representations of GL q (N)

437

Proposition 46. Let X g be a two-sided (Gg , Hg )-quantum space and let x E 0(Gg ) and x' E 0(Hq ). Then 0(Gg \X g ;x) is a right 0(Hg )-subcomodule of

0(X g ) and 0(Xg 1Hg ;x') is a left 0(Gg )-subcomodule of 0(X g ). Proof. For a E 0 (G g \X g ; x), we have (q5L id) o Ma) = (id OR ) o L (a) = x 0 R(a) by (41), hence OR(a) E 0(Gg \X g ;x) 0 0(Hg ). The proof for 0(X g IHg ;x') is similar. 0 Now let Kg be a quantum subgroup of Gg with surjective Hopf algebra homomorphism lrKq : 0(Gg ) 0(K g ). Then the formulas Lit-q

(7rK g id) o

and

Ricci := (id ® T-R

-q

) 0 4A

(42)

define a left coaction L Kq and a right coaction RK q of 0(Kg ) on the algebra 0(Gg ) such that Gg becomes a left and right Kg-quantum space. In this way, Gg is a two-sided (Kg , GO-quantum space with coactions OIL = LI{ q = Zi and a two-sided (Gg , Kg )-quantum space with coactions OIL = and OR = • Applying Proposition 46 to these particular cases we obtain Corollary 47. For any element x E 0(Kg ), the subspace 0(Kg \Gg ;x) is a

right 0(Gg )-subcomodule of the right coernodule 0(Gg ) with respect to the comultiplication. Likewise, 0(Gg i Kg ; x) is a left 0(Gg )-subcomodule of 0(Gg ). 11.5.3 Irreducible Representations of GL q (N)

In this subsection we apply the above notions and facts to the quantum group Gg =GLg (N) and its quantum subgroups Kg = DN B. Let P {A = Alf]. + • • • + ANEN I Xi, - • - AN E Z1 be the set of integral weights of the Lie algebra gl(N, C) and let P± = {A = A1€1 + • • • + ANEN E P I A i > - - • > AN} denote the subset of dominant integral weights. For any A = A 1 €1 + • • • + ANEN E P, we define group-like elements

tA =

- • - ekiv E 0q (DN)

and

Clearly, we have t A tm = 0+0 and zA zo = z A+0 for A, 47 we know that the spaces

0(DN\GL q (N);t A )

and

E 0(B q- ).

= 4.'1 • • • z

E P. From Corollary

0(B qT\GL g (N); z A )

are right 0 (G L (n-subcomodules of the comodule 0(G L g (N)). Next we describe some elements of these spaces. For any matrix A -= (aii ) E MN(No) the monomial

UA:= (U1) a 1 I OiD a 12.

(U N1 r 1 n (liD a 21 (11D a22

• • •

(

satisfies the condition L 13, (U A ) = t" (A) UA with a(A) := P, so that uA E O(DN\GLq(N);tcqA)).

Ulkl YINN

Ei (Ej aii ) Ei

E

438

11. Corepresentation Theory and Compact Quantum Groups

Relative invariants for the Borel subgroup Bq- are obtained by means of quantum minor determinants. Let On be the set of n-tuples I = , in} of integers from {1, 2, • • • , N} such that j 1 < i2 C • • • C in . Recall that the quantum n-minor 1.31-.1j for I, J E f2n was defined by formula (9.18). In the case I = {1, 2, • - , n} we write Dj := Dip Since 2A(D j) = > 1 D1 0 DIJ (u: ) = 0 for i < j, we get L B -(DJ) by Proposition 9.7(ii) and E i 7riç (D1 ) 0 DÇ = Dj and so Dj E 0(B,27\GL q (N); zAn), where is an algebra A n = ei + 62 + • • • + En is the fundamental weight. Since homomorphism, it follows that Di, Di, • • • VI E (.9(Bq- /GL q (N); ?),

(43)

for arbitrary mi-tuples Ii E f2m, , j = 1,2, • • • , r, where A = Ami + • • • + A my.. Clearly, an integral weight A E P is of the form A = A mi + Am, + - • - + Arns with N > m 1 > m 2 > • • • > m, > 0 if and only if it can be written as A = AlEi + • - • + AN6N with A 1 > A2 > • • > AN > O. The set (A1, • • • , AN) is obtained from (mi, • • • , m8 ) by transposing the corresponding Young diagram. For such A E P, let T = {Trp I 1 < r < N, 1 < p < Ar } be a tableau of elements from the set {1, 2, - - • , N}. The tableau T is called semistandard of shape A if Tr _ Lp < Trp , < Trp ,

1 < p < s, 1 C r < N,

2 < r < 772p , 2
The set of all semistandard tableaux of shape A is denoted by TabN(A). For each T E TabN (A) we define the standard monomial DT by setting DT

DA • • • DJ. E C9(GL q (N, C)))

where Jp = {T1p, • • • , Trnpp}, 1 < p < s. By (43), DT E C9(N, \GL q (N); ?). Proposition 48. (0 Suppose that A E P+ and s is an integer such that AN> —s. Then the set {D q-- sDT T E TabAA +s(ei + • • • +EN)» is a basis of the vector space 0(N, \GL q (N,C); ?). (ii) If A E P is not in P+ , then 0(B7q7/GL q (N,C); z A ) = {0}. Proof. The proof is given in [NYM], Theorem 2.5.

By Theorem 9.18, there is a dual pairing (•, -) of the Hopf algebras Ug (ON ) and 0(aL q (N)). By Proposition 1.16, such a pairing turns the algebra 0(GL q (N)) into a Uq (g1N )-bimodule with actions defined by (1.65). Proposition 49. Let A E P. An element a E 0(Dn \GL q (N);t A ) belongs to C3(N\GL q (N);?) if and only if a is annihilated by the right action of all generators Ek E Uq (g1N), k = 1, 2, • • • ,N — 1, that is, a.Ek a(o)a(2) = 0 for k == 1,2, • • • ,N — 1. E(kl Proof. The proof follows from Theorem 2.2 in [NYM]. (Note that our dual pairing of Uq (g1N ) and 0(GL q (N)) differs from that in [NYM].)

11.5 Finite-Dimensional Representations of GL q (N)

439

For A E P± , let TA denote the corepresentation of 0(GL q (N)) on the right 0(GL q (N))-comodule 0(B.7- \GL q (N);?). Then, by Proposition 1.15, the equation i"),(f) := (id® noTA, f E Uq (g1N), defines a representation t of the algebra Uq (g1N ) on the finite-dimensional vector space 0(i3 - \G q (N); ?). In the next proposition and in the proof of Theorem 51 below we shall work with highest weight representations of Uq (g1N ) with respect to the sequence of simple roots -ai , - • — aN (see also Remark 7 in Subsect. 6.2.4). Proposition 50. For any A E P+ TA iS an irreducible corepresentation of 0(G L q (N)) and is an irreducible representation of Uq (g1N ) with highest weight A with respect to the simple roots • , -oe N of glN .

Proof. It is sufficient to prove the assertion for A such that Ai > A2 > • • • > AN > 0 (otherwise we multiply all elements of 0(.13,7\GL q (N); zA) by Dq-8 , s E N, and apply Proposition 48(i)). Let A =-- ri Ai + r2A2 + - - • ± rN AN,• r1, r2,•• , rN E No. Let us consider the standard monomial 1-) TN- 1 a :- D TN Dr2 {1, 2} Dr{ '1 } and set V := t(Uq (g1N ))a. The re• " •

striction of i"), to V is a finite-dimensional representation and hence a weight representation of Uq (g1N ). Since (Kr', = 8kiq 6k1 by the formulas from Sect. 9.4, we obtain 1),(K i-1 )a E a (i) a( 2)) = ea. From the relation A(Fi ) = F 0 1 + 0 Fi (see Subsect. 6.1.2) we get =

n+1

(K71 K i+i )e(n- j+1) 0

Fi 0 1 0(j-1)

Z-d j--=1

Since (Fi , = 0 and (Ki l , ulic) = 0, k 1, by the formulas from Sect. 9.4, we derive that (Fi )a = O. Hence a is a highest weight vector for the representation TA ry with weight A with respect to the simple roots -ai, • • • — CtN• By Proposition 48(i), the representation space of TA has a basis labeled by the standard tableaux of shape A. From classical representation theory it is known that the same is true for the irreducible representation of the Lie algebra gl(N, C) with highest weight A. As noted in Subsect. 7.3.3, the dimensions of irreducible representations of gl(N, C) and Uq (g1N ) with the same highest weight coincide. Since contains a subrepresentation with highest weight A as shown above, it follows that tA is irreducible. The irreducibility of TA obviously implies the irreducibility of TA. LI 11.5.4 Peter-Weyl Decomposition of 0(GL q (N)) The main result of this subsection is the following. Theorem 51. Suppose that q is not a root of unity. Then the coordinate Hopf algebra 0(GL q (N)) decomposes into a direct sum of coefficient coalgebras C(T),) of irreducible corepresentations TA, A E RF, of 0(GL q (N)):

0(GL q (N)) =

ED,E ,+ C(T),).

(44)

440

11. Corepresentation Theory and Compact Quantum Groups

Proof Throughout this proof highest weights always refer to the simple roots - al, • • • - fitN. From the defining relations for Uq (g1N ) we see that there is an algebra anti-homomorphism 0 of Uq (g1N ) such that 19(Ei ) = ,N. We O(Fi) = Ei, j = 1, 2, • • • ,N -1, and 9(Ki ) = Ki , j = 1,2, define a representation p of Uq (g1N ) on 0(aL q (N)) by p(f)a = a.0(f), f E Uq (g1N ), a E 0(GL q (N)). Let a E 0(GL q (N)) be a highest weight vector for p. By definition we have p(Fk)a = a.0(Fk) = a.Ek = 0 for k = 1, • • • ,N - 1. From the definition of O(DN\GL q (N);t") it follows that an element b G 0(GL q (N)) belongs to O(DN\GL q (N); t A ) if and only if = qÀib , j = 1,2, • • • , N. Therefore, a E 0(DN\GL q (N);t A ) for some A E P± and hence a E 0(B q- \GL q (N); zA ) by Proposition 49. Clearly, we have 0(47- \GL q (N);?) C C(L), so that a E C(2-7),). On the other hand, let 0(Mq (N)) 3 be the span of monomials in uji of degree less than or equal to s. The algebra 0(GL q (N)) is the sum of finitedimensional left 0(GL q (N))-subcomodules Dq- TO(M q (N)),, r,s E No , and so of finite-dimensional p-invariant subspaces. Therefore, by Theorem 7.8, p is a direct sum of finite-dimensional irreducible representations. Each irreducible component contains a highest weight vector. Since such vectors belong to some C(T),) as shown in the preceding paragraph and C(T),) is p-invariant, it follows that 0(GL q (N)) =-- EAERE C(TA). If A, p, E P+ and A )2, the rep-

resentations 11A and î are inequivalent (because they have different highest weights). So the corepresentations TA and T are also inequivalent. Hence, by Corollary 9, the sum EA C(TA) is actually a direct sum. Theorem 51 has a number of important corollaries. Recall that, as always in this section, q is not a root of unity. Corollary 52. Any irreducible corepresentation of 0(aLq (N)) is equivalent to one of the corepresentations TA , A E P+ . Proof. If T is an irreducible corepresentation not equivalent to some TA ) then we have C(TA) n C(T) = {O} for all A E P± by Corollary 9. Because of (44), this is impossible.

Corollary 53. The Hopf algebra 0(aL q (N)) is a cosemisimple Hopf algebra and the direct sum (44) is its Peter-Weyl decomposition. Proof. Combine Theorem 51 and the implication (i) -(vi) of Theorem 13. 0

Corollary 54. The dual pairing (•,.) of Uq (g1N) and 0(GL q (N)) from Theorem 9.18 is non-degenerate. Proof. The proof is analogous that of Corollary 23.

Another immediate consequence of Theorem 51 is the existence of a unique Haar functional h on 0(aLq (N)). From formula (11.3) we know that O. h is given by h(1) = 1 and h(a) = 0 for a E C(TA), E P+,

11.5 Finite-Dimensional Representations of

GL q (N)

441

11.5.5 Representations of the Quantum Group U(N) In this subsection we assume that q is a real number such that q O ±1. Recall from Subsect. 9.2.4 that the Hopf *-algebra C9(Uq (N)) is just the Hopf algebra 0(GL q (N)) equipped with the involution defined by (ti i )* = ,

N, and (DV)* -= Dq . Further, as noted in Example S(u3i:), j , j = 1, 2, • 11.7, C9(Uq (N)) is a CMQG algebra. Therefore, all results on the quantum group GL q (N) obtained in the previous subsections and the theory of CQG algebras developed in Sect. 11.3 are valid for the quantum group Uq (N). Most of the corepresentation theory of Uq (N) can be derived by combining these two sources of results. Some of them will be sketched in the following. From Proposition 29, the Haar functional h of the cosemisimple Hopf algebra 0(Glig (N)) is a faithful state of the *-algebra A := 0(Uq (N)). Hence the equation (a, b) := h(a* b), a, b E A, defines a scalar product on A. Formula (36) says that the character f l of A is given by h(-) = (K2 0 , -), where 2p = Ek=i (N - 2k + 1)ck. Now let A e P+ be a dominant integral weight. We choose an orthonormal basis {ei } of the space V), := C9(B\GL q (N), z A ) with respect to the scalar product (-, -) on A such that ei is a weight vector with weight A(i) e P for the representation /IA of Ug (g1N ) on VA. Let Ni l be the matrix elements of the corepresentation TA Of A with respect to the basis fed. Then, w = (u`i ) is an irreducible unitary matrix corepresentation of A. The unique intertwiner FA = ((FA)0) E Mor (0, (0)") such that Tr FA = Tr FA-1 (by Lemma 30) is given by (FA ) ii = q 2 (P ,A (i))6ii (45) We shall prove formula (45). Using Proposition 6.6 and (45) we get

(f, S 2

= (S2 (f), = (1(2pf K2-pl 1 U )q 2("(.1)) = (1{2p f, •) = (f, (FA) in uATii )

for f E Uq (g1N ) and so (FA)in unA i = S 2 (u,L)(F),)5 by Corollary 54. Thus, E Mor (0, (0)). Since Tr FA = Tr IV, the matrix FA defined by (45) is indeed the intertwiner from Lemma 30. Next we combine formula (45) with the Schur orthogonality relation (25) from Subsect. 11.3.2. Then we conclude that the set ui),"i IA E P+, i, j = 1, 2, • • • , dim VA} is an orthogonal vector space basis of A and {

IIu 112 = dilq-2(p,A(i)). The number

q2(p,A(0) Tr 1), = Y.‘ d--d mi)EP(À)

q-2(p,A(i)) L--dx(i)ePoo

(46)

442

11. Corepresentation Theory and Compact Quantum Groups

may be considered as a quantum analog of the dimension of VA • It coincides with the quantum dimension (see Subsect. 7.1.6) of the corresponding representation of Uq (g1N ). Here P(A) is the set of all weights of the representation TA of Uq (g1N ) taken with multiplicities. Finally, let us turn to the character x(A) := Ei 14, of the corepresentation TA, A1 > ••• >AN >O. It is an element of the algebra A = O(Uq (N)). Since t)i" • • • P1;1' for MO = Ale]. + • + ANfiv, IrD(X0,) coincides 7rD(q,) = t A(i) with the corresponding expression in the classical case. The latter is known to be the Schur function S A (t) S A (t 1 ,• • ,tN) (see [Mac]). Thus, we have n-1 1 472-3 7 IrD(X(A)) = SA(t) and d), = S),(q

lq -n-F1) .

11.6 Quantum Homogeneous Spaces 11.6.1 Definition of a Quantum Homogeneous Space Recall that a quantum space for a Hopf algebra may be viewed as a generalization to quantum groups of a space on which a group acts. If the action is transitive, such a space is usually called a homogeneous space. The following definition gives a generalization of the latter notion to quantum groups. Let Gq be a quantum group given by its coordinate Hopf algebra 0(Gq ) and let 0(X q ) be the coordinate algebra of a right Gq-space with coaction ço : 0 (X q ) —+ 0(X q ) 0(G q ). Definition 15. We say that X q is a right quantum homogeneous G q -space if there exist a subalgebra X of 0(G q ) such that A(X) C X 0 0(Gq ) and an algebra isomorphism 0 : 0(X q ) X such that Ao 0 = (0 0 id) o p.

Any subalgebra X of 0(Gq ) such that A(X) C X 0 0(Gq ) is obviously a right quantum Gq -space with right coaction A X. Definition 15 says that up to isomorphisms the right quantum Gq-spaces of this form are precisely the right quantum homogeneous G q-spaces. If X q is such a quantum homogeneous Gq-space, then f := eo0 is a character of the algebra 0(X q), that is, f (xy) = f(x)f(y) for x, y E 0(X q ) and f(1) = 1. This means that any quantum homogeneous G q-space contains at least one "classical point". Left quantum homogeneous Gq-spaces are defined similarly (by requiring that A(X) C 0(Gq ) 0 X and A o 0 = (id 0 0) o ço). Example 10. The quantum vector space CqN is a right and left quantum A homogeneous GL q (N)-space by Proposition 9.11.

Example 11. Let q be not a root of unity. By Proposition 4.31, the quantum 2-sphere 'S Si,, are right quantum homogeneous SL q (*spaces.

A

11.6 Quantum 'Homogeneous Spaces

443

11.6.2 Quantum Homogeneous Spaces Associated with Quantum Subgroups In differential geometry it is well-known that a homogeneous manifold for a Lie group G is isomorphic to the coset space H\G (resp. G I H) of G with respect to the stabilizer subgroup H (see, for example, [Wal, 3.58). Then, omitting technical details, functions on H\G (resp. G I H) can be identified with those functions on G which are invariant under the left (resp. right) action of the subgroup H on G. Thus, it is natural to construct quantum homogeneous G q -spaces in a similar manner. Let _Kg be a quantum subgroup of Gq (see Definition 9.11) with coordinate Hopf algebra 0(1(q ) and surjective Hopf algebra homomorphism 71-K : 0(Gq ) ---+ 0(Kq ). For notational simplicity we abbreviate 0c := 0(Gq ) and OK := 0(K g ). Let Licq and RK g be the left and right coactions of OK on OG, respectively, defined by liK g := (7K 0 id) o z and RK, 2, := (id 0 ir o 6. Definition 16. The elements of the sets

0 K\OG := {a C OG

Lic q (a) = 1 0 a},

OG/OK := {a E OG I RK g (a) = a ® 1} 1 OK\OG/OK

:= OK\OG n OGIOK

are said to be left-invariant, right-invariant and biinvariant, respectively, with respect to the quantum subgroup 1(q . The sets 0 K\OG and OG I OK are called coordinate algebras of the right resp. left quantum homogeneous Gq -spaces .K.q \Gq resp• Gq 1Kq . Since LK, 9, and RK g are algebra homomorphisms, OK\OG, OG/OK and OK\OG/OK are subalgebras of OG Proposition 55(ii) says that Kq \Gq and Gq IKq are indeed right and left quantum homogeneous Gq -spaces. •

Proposition 55. (i) S(OK\OG) g OG/OK, S(OOOK) g OK\OG. ,i(OK\OG) C OK\OG OG, 4(0G/ 401C) C OG OGIOK. (iii) Setting C := {a E OG I 7ric (a) = e(a)1}, we have

OK\OG --= {a E Cl46(a) E C OG}, CC /OK = fa E C I .6(a) C OG C}, OK\OGIOK = {a E C

E C C}.

Proof. (i): Let a E OK\OG, that is, LK .7 (a) = 1 0 a. Since 7 rK is a Hopf algebra homomorphism, 7rKoS=So7rK . Thus, we have RK g (S(a))

=

(id 0 7ric. ) 0 ,i(8(a)) =

E(S = (S

E S(a(2)) 710(am))

S)(a(2) 0 7ric(a(1))) = (S S) O o T(1

TO

0 a) = S(a) 1,

that is, S(a) E Cc/OK. The second inclusion is proved similarly.

LK q (a)

11. Corepresentation Theory and Compact Quantum Groups

444

00 follows from Corollary 47 applied with x = 1. (iii): We verify the first equality. Let a E OK\OG. Applying id. E to (ric 0 id) o 46(a) = 1 0 a yields T-K(a) = E(a)1. Hence, OK \OG C C. By (ii), 46(a) E OK\OG 0 OG • Conversely, suppose that a E C and L(a) E CO 0c• Then (IrK 0 id) 0 L(a) = (a(1))1 a(2) = I 0 a. Thus, a E OK\OG. We illustrate the above notions by restating some results from Chap. 4. Example 12 (Gq = SLq (2)). Throughout this example we assume that q is not a root of unity. Recall that the group Kg of diagonal matrices in SL(2, C) is a quantum subgroup of Gg = SLg (2). From Subsect. 4.2.2 we know that

OK\OG =

A[0, n],

A[n, O] OK\OGIOK = C[bd.

0G/0K =

,

That is, the algebra OK\OG is generated by the elements bc, ac, db, the algebra OG/OK is generated by bc, ab, cd and the algebra OK\OG/OK possesses the single generator bc. Hence 0 K \°c coincides with the algebra generated by the elements = —1,0, 1, from (4.80). Therefore, by Proposition 4.31, the right quantum homogeneous G q-space OK\OG is just the quantum 2sphere 4,0 . It is not difficult to show that the other quantum 2-spheres Q(, are not of the form OH\OG for some quantum subgroup Hg of Gg . Let us note that .6(bc) is neither contained in OK\OG/OK 0 Oc nor in OG OK\OGIOK, so the subalgebra OK\OG/OK of OG is neither a left nor a right quantum homogeneous Gq -space.

4),

Proposition 56. Let 0( 1(g ) be a cosemisimple Hopf algebra with Haar functional hK . Then the linear mappings P1 := (h.K 0 7ri- ® id) o 46 and Pr := (id hK o IrK ) o 46 are projections of OG onto OK\OG and OGIOK, respectively. The projections Pi and Pr commute and PIP, is a projection of OG onto OK\OGIOK• Proof. Since IrK : OG —÷ OK is a Hopf algebra homomorphism, we obtain PIP!

=

071- K 0 id) o o (hK 0 -TrK 0 id) 0 21

= (hKorK0kKoirifOid)0(460id)0.6 =

It

(hK hK

id)(6 g id) o (rK 0 id) o =

is obvious that Pi (a) = a if a E OK\OG. For a E (.9G 7 we have

(7ric

id)

0

.6(pi (a)) =

(7rK

id) 046

OK

7rK

(hic 0 IrK 0 id) o .6(a)

id)(rK 0 id 0 id) 0(z 0 id) 046(a)

OK 0 id 0 id)(46 0 id) 0 (rK 0 id) o 46(a) 1 0 OK 0 71-K 0 id) o 46(a) = 1 0 Pi (a),

so Pi (a) E OK\OG. Thus, we have shown that Pi is a projection of OG onto OK\OG. Similar reasoning proves the assertions concerning P. One verifies that PiP, and Pf .Pi are both equal to (hic 0 7rK 0 id 0 hK 07rK)(id 2,) o Hence PIP, = Pr.Pi. The remaining assertions follow easily.

11.6 Quantum Homogeneous Spaces

445

11.6.3 Quantum Gel'fand Pairs Let us first recall the corresponding classical notion. A pair (G K) of a locally compact group G and a compact subgroup K is called a Gel'fand pair if the algebra L'(K\G I K) of integrable K-biinvariant functions with respect to the convolution product is commutative. If G is compact, this is equivalent to the requirement that the dimension of the subspace of K-biinvariant elements in the coefficient space of each irreducible representation of G is at most one. In the following we shall generalize this concept to quantum groups. Throughout this subsection we assume that Kg is a quantum subgroup of a quantum group Gg such that the Hopf algebras 0(Kg ) and 0(Gg ) are cosemisimple. As we have seen in Example 12, the comultiplication .6 of OG does not map OK\OG/OK to OK\OG/OK OK\OG/OK. For this reason we introduce a new comultiplication 46b i on OK\OG/OK by 4613i :=

(id

hic

id)

71‘

0 (.6

id)

0 = (id 0

Pi ) o = (Pr 0 id) 0,6.

Proposition 57. The vector space Cbi := OK\OG/OK is a coalgebra with comultiplication 2ibi and counit e[Cbi. The restriction to Cbi of a left- (resp. right-) invariant linear functional on OG is also left- (resp. right-) invariant with respect to the comultiplication 2ibi. On CIA we have the identity 2ibi 0 S To

(S S) 0 2ibi•

Proof. We sketch the proof of the first assertion. Let a c Chi . By Proposition 56, PrP/ (a) = a . From the relations ,6 o Pr = (id 0 Pr ) 0 .6 and 2, o P1 = (Pi 0 id) 0,6 and the fact that Pr and P1 are commuting projections we get Lb(a) = (Pr 0 id) o o PrP/ (a) = (13,r .Pi 0 Pr) 0 2ibi (a)

= (id 013/) 0 .6 o PrPi (a) = ( PI ®P1 13,-.) o

These identities imply that 46bi(a) E Cbi Cbi. Some computations show that both expressions (.6b i 0 id) 0 i6bi and (id 0 46bi ) 0 i6bi are equal to (id hK o 71-K 0 id 0 hK 0 7rK 0 id) 0,6(4 ), so 2ibi is coassociative. We have (E.

0 id)

=

( E. 0 id)(id (1/K ric

=

hK o 7:-K id) o (.6 id) ,i(a)

id)(E 0 id 0 id) o (.6 0 id) o

( hK 0K id)

0

.6(a) =

id)(1 0 a) = a

for a E OK\OG/OK. Similarly, (id 0 e) o46b i (a) = a. Hence for CIA. This proves that Cbi is a coalgebra. The comultiplication

2ib1

Efebi

is a counit D

is not an algebra homomorphism in general.

Proposition 58. The following conditions are equivalent: (j ) The coalgebra OK\OG I 0 K with comultiplication 2ib1 is cocominutaave. (ii) dim (OK\OG/OK nC((p)) < 1 for all irreducible corepresentations of OG .

446

11. Corepresentation Theory and Compact Quantum Groups

In the proof of Proposition 58 we will use the following Lemma 59. Let be a corepresentation of0(Gq ) on a finite-dimensional vector space V such that 0 K \OG 10K n c () {0}. Then there exists a basis of V such that the corresponding matrix y = (vii) of (ig satisfies the condition = 1,2, • • r} for some r 5_ dim V. OK\OGIOK nC(W) = Lin {v I Proof. Clearly, (pK := (id 0 -FK ) o(p is a corepresentation of OW on V. Put := {v E V I cp K (v) = v 1 } . Since 0(1Q is cosemisimple by assumption, (plc is a direct sum of irreducible corepresentations. Hence there exists a (PK invariant subspace V2 of V such that V = V1 EDV2 and (pK [V2 does not contain the trivial corepresentation. Take a basis {el, • • • , e ri } of V such that e i E 1717 i < r, and ei E V2 i > r, and let (vij ) be the matrix corepresentation of (p with respect to this basis. Since (pK(ei) = E j ei T- K(v ii ) and V1 and V2 are WK-invariant, we get -n-K(vii ) = 0 if j < r, i > r and if j > r, i < r. Moreover, 7K(vii) = 6ii -1 if i, j < r, because the elements of V1 are (pK-invariant. From these facts it follows that vii E OK\OG/OK for all i, j < r. Since 1172 does not contain the trivial corepresentation, P/ Pr (v ii ) = Em h ic (n-K (v im ))y mn hic (T -K (v ni )) = 0 for all i,j > r by (3). Since PLPT is a Projection of OG onto O K \O G /O K by Proposition 56, we conclude that OK\OG/OK fl Lin {v j, j > r} = {0}. {0}. Proof of Proposition 58. (i) —.(ii): Assume that OK\OG/OK fl C ( ) We then choose a matrix (v in Lemma 59. For i,j < r we then have E OK\OG/OK and -n-K (vii ) = Sii •l, so we obtain )

=

Ern,n

hK(71-K(Vrri n))Vim Vnj

=

E n=1

yin

Vnj-

Since the matrix elements vni,n are linearly independent and r o bj = Zbi by (i ), the latter is only possible if r = 1. (ii)—÷(i): Since the Hopf algebra Oc is cosemisimple, OK\OG/OK is the sum of subspaces OK\OG/OK nC((p) for all irreducible corepresentations (p of Oc. By (ii) and Lemma 59, oK\oGioic nc(v) = C - vu and hence = Em,„ virnhic(irK(vmn))0vni-=vilOvii. Hence r 0 Aibi zbi. I=1 Definition 17. The pair (Gq ,Kq ) is called a quantum Gerfand pair if the equivalent conditions in Proposition 58 are satisfied. If in addition the algebra C9(Ifq )\0(Gq )10(1‘q ) is commutative, then the quantum Gelfand pair is called strict.

A condition for (Gq ,Ifq ) being a quantum Gel'fand pair is contained in Proposition 60. Suppose that there exist a bijective coalgebra anti-homomorphism bc : 0(G q ) 0(Gq ) and an injective coalgebra anti-homomorphism 6K 0( 1(q ) —÷ 0(Ifq ) such that 7r-K 0 6G = (5K 0 7rK and 5G(a) = a for all a E 0 K\OG 10K. Then (Gq , Ifq ) is a quantum Gerfand pair. Proof. The proof is given in [F1].

11.6 Quantum Homogeneous Spaces

447

(2)). Let .9 be the algebra automorphism of 0(S L q (2)) determined by 0(a) = a, 0(b) = e, 0(e) = b, 0(d) = d (see Proposition 4.5(i)). Then the mappings 6G := 0 and 5 K := id have the properties required in Proposition 60, where Kg denotes the diagonal subgroup of Gq . If q is not a root of unity, then the Hopf algebra 0(SL q (2)) is cosemisimple by Theorem 4.17. Therefore, since OK\OG/OK = C[bc] (see Example 12) is commutative, (G q , K g ) is a strict quantum Gel'fand pair if q is not a root of unity. Example 13 (G q = S

11.6.4 The Quantum Homogeneous Space Uq (N-1)\/q (N) Let us briefly recall the corresponding classical situation. It is well-known that the unitary group U(N) acts transitively on the unit sphere Scicv-1 of the complex vector space CN and that the stabilizer of the point eN = (0, • • • , 0, 1) c 41-1 in U(N) is just the subgroup U(N-1). Hence the right coset space U(N-1)\U(N) can be identified with the unit sphere SV 1 . In the following we study a quantum analog of the homogeneous space U(N-1)\U(N). In the course of this we use the representation theory of the quantum group GL q (N) developed in Sect. 11.5. Throughout this sub0, ±1. Whenever we speak about section we suppose that q is real and q highest weights for representations of Uq (gln ), we refer to the simple roots • • • , -an (see also Remark 6.7 and Proposition 50). Let us abbreviate G q := Uq (N) and K g := Uq (N-1). We shall denote the generators of 0(.1Cq ) by yi3:, 1 < j j < N -1, and its quantum determinant by Dq . There exists a unique surjective Hopf algebra homomorphism 7-K : 0(G q ) 0(K q ) such that ,

7K( 1 L ii ) = yip i i ; N -1, 7K(4) r -K(u lkv ) --= 6kN1, k = 1,2, • • • , N.

(Indeed, it suffices to check that the corresponding elements satisfy the defining relations of Uq (N).) Thus, _Kg is a quantum subgroup of Gq . From Sect. 11.5 we recall that for any A E P+ there exists a unique irreducible corepresentation TA of 0(G q ). Let t be the matrix elements of TA with respect to some basis of the carrier space. We fix an index i and set VR (A) = Lin {t I j = 1,2, • • • ,d),} and VL(A) = Lin {t I j =1, 2, • • , }, 4 = dim TA. Then VR(A) and Vi(A) are right and left 0(Kq )-comodules with coactions RA g = (id 0 K ) o zl and Vic = (n-K 0 id) 0 4 respectively. Proposition 61. The right O(ICq )-comodule VR(A) and the left 0(K q )comodule Vi(A) decompose into a direct sum of irreducible subcomodules as

VR (A) = sED t, VR(A,

Vi

= 01, Vii(A71 1 ),

(47)

where in both cases the summations are over all dominant integral weights = Pifi + • - • + PN-ieN-1 such that .À1 >

> A2 >U2 > * • > AN-1 >

AN-1

> AN.

(48)

448

11. Corepresentation Theory and Compact Quantum Groups

17R (A, it) and VL (A,,u) are right and left irreducible 0(Kg )-subcomodules of VR (A) and VL(A), respectively, with highest weights p. Outline of Proof. Clearly, 7rK(Dg ) = V the one-dimensional 0(K g )comodule C • Dql is equivalent to C • (Dq ) 1 , 1 E No. Therefore, by Proposition 48 (i), we may assume that AN > O. For each dominant integral weight ,u satisfying condition (48) we introduce the standard monomials ... -/-02 na2 nbi nai vA = D {1,-..,N} AN DbN -1 {1,—,N-1 {1,-..,N-2,N } } Da' '{1,2}'{1,N } '{1}'{N}I

(49)

where ak = Ak — Ak and bk = Ak — Ak+i, k

1,2, - • • , N — 1. One verifies that vi, is a highest weight vector of weight ,u + (E kN-11 ak)eN for the corresponding representation of Ug (gliv_ i ) on VR(A). Thus, there exists a unique irreducible 0(Kg )-subcomodule VR(A, p,) in VR(A) containing vii, as a highest weight vector of weight p,. This proves that Epi, vR (A, p,) c VR(A). Since both sides have the same dimensions, the first equality in (47) follows. The second equality is proved similarly. D We shall say that the right (resp. left) 0(G g )-comodule VR (A) (resp. VL(A)) is of class 1 with respect to the quantum subgroup Kg if it contains a nonzero Kg -fixed vector, that is, 4 , (v) = v 0 1 (resp. Vi(v) = 1 0 v). Corollary 62. The left 0(Gg )-comodule V L (A) is of class 1 with respect to Kg if and only if A = lei —P€N, 1,1' E No. In this case, the standard monomial VO

= (Dq ) -1 ( D { 1,...,N-1}) e (u iv ) 1 -a SOLliv ) 1 (u iv )1 '

/

is up to a complex multiple the unique Kg -fixed vector in VL(A). Proof. The proof follows from Proposition 61 and formula (49).

0

Proposition 63. The algebra 0(Kg \Gg ) of left K g -invariant elements of 0(Gg ) decomposes into a direct sum of irreducible right 0(Gg )-comodules as

0(Kg \Gg ) =

e 0

Ci(Tici-vcN)•

Here Ci(n ,Ei-PEN) is the irreducible right 0(Gg )-subcomodule of the coefficient coalgebraC(TiEl _ I I EN ) which contains S(u ) I' (unt as its highest weight vector of weight lei — PEN. Proof. Since the Hopf algebra 0(Gg ) is cosemisimple by Corollary 53, the right 0(Gg )-comodule 0(Kg \Gg ) decomposes into a direct sum of irreducible components. In order to determine this decomposition explicitly, we first note that 0(Kg \Gg ) C erit=0C(Tlei-veN) by Corollary 62. For A = /e l — l'EN we choose a basis e l , • • - ,ek of the carrier space of TA such that the vector e l g (N,1)-invariant and the space Lin { e2, • • - , ek} is Ug (N-1)-invariant. isU Then one easily verifies that a matrix coefficient Oi of TA is left Uq (N —1)invariant if and only if i = 1. Moreover, each left Uq (N-1)-invariant element in C(T),) is a linear combination of ti'j , j = 1, 2, . • • , k. Let C1 (TA),

11.6 Quantum Homogeneous Spaces

449

A = 1E 1 - l'eN, be the set of left Uq (N-1)-invariant elements in C(L). Then 0(1-Cq \Gq ) = eli'7, =0 C1 (71/„_/, , N ). The comultiplication /1 realizes the irreof 0(Gq ) on Ci (Thi _ ii, N ). By Corollary ducible coreprese'ntation El 62, S(4,) 1'(ur)1 is the unique highest weight vector in Ci(Tui-vEN)All results developed so far are still valid for the quantum groups Gq = GL q (N) and Kq = GLq (N-1) if q is not a root of unity. From now on we shall use the *-structures of 0(Uq (N)) and 0(Uq (N-1)) too. Since 7rK : 0(Gq ) C9(Kq ) is a *-algebra homomorphism, 0(I-Cq \Gq ) is a *-subalgebra of 0(Gq ) and a right *-quantum space for the Hopf *algebra 0(Uq (N)). By construction, 0(Kg \Gq ) is a right quantum homogeneous Uq (N)-space. Definition 18. The *-algebra 0(Kg \Gq ) is called the coordinate algebra of the quantum sphere related to Uq (N). Our next aim is to describe the *-algebra 0(Kq \Gq ) in terms of generators j = 1,2, • • • , N. Since LiK q (zi) and relations. Let us abbreviate ; = u 7K(u) = 1 zi , the elements ; and z: = ( 41 )* = S(u) belong to 0(Ifq \Gq ). The right coaction (pR of C.9(Gq ) on Zi E 0(Ifq \Gq ) is given by the comultiplication, that is, we have ,

/4, j = 1,2, • • • , N.

(PR(z3) =

Proposition 64. (i) 0(K g \Gq ) is the *-subalgebra of 0(Gq.) generated by the elements z i , j = 1,2, • • • , N. (ii) The generators z1, Z2 7 • • ' 1 zN satisfy the relations z i zi = qzi z i , 2

Zjrc Zk ZkZi: + (1 — q )

E jcic

i < j; 3

j

z;z i = qzj z, i j; k = 1, • • • , N;

E k=1 zkzz =

(50) 1. (51)

As a *-algebra, 0(Ifq \Gq ) is generated by N elements z 1 ,z 2 , • • • , zN with defining relations (50) and (51). (iii) The set {4 1 • (znml • • (z k4 ) 7" I s i ,m i E No} and the set {(4) 7n 1 • • • (z1v )nN z1 1 • • • Z N S N Si Mi G No } are bases of the vector space

0(Kg \Gq ). Proof. (0: The elements (z k 4 ) l'zi = S(4) / ' (un i are highest weight vectors for the 0(Gq )-comodules C1(Ti f ,-1 , , N ). Hence these comodules and so 0(1-Cq \Gq ) by Proposition 63 are contained in the *-subalgebra generated by

zi , = 1,2, • • • ,N• (ii): The relations (50) and (51) follow easily from the commutation relations for the elements u in the *-algebra 0(Uq (N)). That the relations (50) and (51) are already the defining relations of the *-algebra 0(Kg \Gq ) can be shown by using appropriate *-representat ions of 0(Uq (N)) (see [VS2]). (iii) can be proved by means of the diamond Lemma 4.8.

11. Corepresentation Theory and Compact Quantum Groups

450

Note that because of (50) the relations (51) can be rewritten as Z je ZZ = 4.4 — ( 1

q2 )

j
j-1) Z * i Z.7

2(N -

zk = 1 .

The restriction of the Haar functional of 0(Gq ) to the subalgebra 0(1‘q \Gq ) determines a right-invariant linear functional h on 0(.1fq \Gq ) such that h(1) = 1. From Proposition 63 it follows immediately that h is the only functional with this property. We want to describe the functional h more explicitly. First we note that the diagonal subgroup DN of GL q (N) becomes a quantum subgroup of Uq (N) if we introduce a *-structure on 0(DN) by (t i )* := We have

0 (If q \G q ) = IED),Ep 0 (.1 q \G q )

(52)

where 0(.1fq \Gq )), =- {a E 0(Ifq \Gq ) I R D , (a) = a 0 tAl are the weight subspaces. Clearly, 0(ifq \Gq )),=0 is just the algebra 0(Ifq \Gq 1DN ) of D invariant elements in 0(Ifq \G q ). The functional h on 0(Ifq \Gq ) vanishes on all subspaces 0(.1fq \Gq ),), from (52) with A 4 0. Thus, it suffices to determine the functional h on the subalgebra 0(.1fq \GSDN). It is easily seen that the subalgebra 0(Ifq \GSDN) is spanned by the monomials (znml • • • (z7v )mNz N mN • • • ziml, mk E No . Proposition 65. (i) The algebra 0(Ifq \Gq 1DN) is commutative and generated by the elements (k = E ik zi4 k =-- 1,2, • - • , N —1. (ii) On 0(If q \Gq IDN ), the functional h is given by h((4)7n1

z ri)

* ) 7nN z mN (z N

( 172; q2) 1911

(q2; q2)

(q2 ; q2) in

where m mi + - • - + mN + N —1. (Hi) For any polynomial p(Cl• • • 7 (N -- 1 ) C 0(Ifq \Gq 1 DN) the value h(p) can be expressed in terms of q-integrals as h(p)

1,-,2.

)N-1

(1_ 47 2)N-1.

f l [_1 N

0

0

(2

10. .-P((11 •

7 C_N - 1 )dq 2 (1 • • •

dq2(N _ 1 .

Proof. The proof is given in [NYM].

El

From Propositions 5800 and 63 one can derive Corollary 66. (Gq ,l(q ) is a quantum Gelfand pair and we have

0(Ifq \Gq 11(q )=

(J'

,

47—cc/v.

The matrix coefficients := tiire fiv are called quantum zonal spherical functions on Ifq \Gq . In the next proposition they are explictly described in terms of the little q-Jacobi polynomials (see Subsect. 2.3.3).

11.6 Quantum Homogeneous Spaces

451

Proposition 67. The quantum zonal spherical functions O le are given by

pit (CN = pi

((

17 1

I 17 2)

N 1; q N-2 1 fir I q 2 ) z *1N'-/

if

1>,

if

>

where (N- 1 -= EiN VZZ = 1 - zN4, and pi are the little q-Jacobi polynomials.

LI

Proof. The proof is given in [N11114].

Proposition 67 is a quantum analog of classical results on zonal spherical functions for the unitary group U(N) (see [VK1], Chap. 11).

11.6.5 Quantum Homogeneous Spaces of Infinitesimally Invariant Elements

The quantum homogeneous spaces treated in the preceding subsections were constructed by means of quantum subgroups. However, this method does not yield quantum deformations of all compact Riemannian symmetric spaces, since quantum groups do not have as many quantum subgroups as the corresponding Lie groups have Lie subgroups (see [114]). In particular, quantum analogs of the classical symmetric spaces SU(N)ISO(N) and SU(N)IUSp(N) cannot be obtained by means of quantum subgroups. Moreover, as we know from the study of quantum 2-spheres (see Example 12 above and Sect. 4.5), even if there exists a corresponding quantum subgroup, we do not get all interesting deformations of the classical space in this manner. In this brief subsection we sketch the "infinitesimal" method for the construction of quantum homogeneous spaces. Let us first explain the underlying idea in the classical situation. Functions on a homogeneous space I K are usually considered as functions on the Lie group G which are right-invariant with respect to shifts by elements from K. But they can also be defined as those functions on G which are annihilated by the right action of elements from UM, where t is the Lie algebra of K. We assume that LI and A are Hopf algebras equipped with a dual pairing (.,.):11xA-+ C. By Proposition 1.16, the algebra A becomes then a 14bimodule with left and right actions of 14 on A given by X.a = (id 0 X)A(a) and a.X = (X ® id)A(a), X E 11, a E A. Recall also that a coideal of U is a a linear subspace I of 14 such that A(I) CI 0 U + U 01- and E(.1)-= {0}. Definition 19. Let I be a coideal of U. The elements of the sets

2-A := fa E A

X.a = 0, X E 1A1

:= {a E A I a.X = 0, X E

:= 1A n A 1

are called infinitesimally left-invariant, right-invariant and biinvariant with respect to I, respectively.

452

11. Corepresentation Theory and Compact Quantum Groups

Proposition 68. Let I be a coideal of U. Then the sets 1A, A1 and 1A1 are

subalge bras of A and we have ZA(1A) C A® IA, z(A1) C AI 0 A.

Moreover, u.1.4 = 1A and .41.0 = Al. Proof. The fact that A1 is a subalgebra follows immediately from the relation ab.X = E(a.X (1) )(b.X (2) ), a,b E A, X E I, combined with the assumption z(I) = / 0 /I + 11 0 /. For a E A1 and X E I we have 0 = Z1(a.X) -= E a( i ).X 0 a( 2). Hence ZA(a) E A1 0 A. The equation a.XY = (a.X).Y implies the equality Ai. u = Al. The assertions for IA and 1A1 are derived 111 similarly. If A is a coordinate Hopf algebra 0(Gq ), then by Proposition 68 the algebras 1A and A1 are left resp. right quantum homogeneous Gq-spaces. Example 14 (Gq = SL q (2)). Let A = 0(SLq (2)) and 11 = tIq (512 ). Let X p , P E C Wool, be the element of Ûq (S12) defined by (4.84) resp. (4.85). Since z1(X) -= K -1 0 Xp + Xp 0 K, the one-dimensional space ip := C.Xp is a coideal of U. If q is not a root of unity, then Proposition 4.31 says that the right A-quantum space AT, is isomorphic to the coordinate algebra O(S). Let us emphasize once more (see Example 12) that only the quantum 2-sphere A Sq2 „3 can be obtained by means of a quantum subgroup of SL q (2).

11.6.6 Quantum Projective Spaces In this subsection we outline how a family of quantum homogeneous Uq (N)spaces can be constructed by using the coideal approach. These spaces are quantum analogs of the complex projective space (U(N -1) x U(1))\U(N). Throughout this subsection q is a real number such that q 0, ±1. Let A be the Hopf *-algebra 0(Uq (N)) and let ti be the Hopf algebra 1,1(GL q (N)) of L-functionals on the coquasitriangular Hopf algebra A (see Subsect. 10.1.3 for details). Since q is real, II(GL q (N)) becomes a Hopf *algebra, denoted /1(Uq (N)), with involution determined by (i +1)* = S (j ±?) (see (10.48)). By construction, II is a Hopf subalgebra of the dual Hopf algebra A° . Hence the evaluation of functionals f e Li at elements a E A gives a dual pairing of Hopf *-algebras 11 = ti(Uq (N)) and A = O(Uq (N)). Suppose that r and s are nonnegative real numbers such that (r, s) (O, 0). Let er,8) denote the subalgebra of II generated by the elements Oil - l -NN ,

ril - / NN )+ 17-si+N1 ± -1:77s riN - (r - s)(/1 - ril ),

( 53)

1 +1; + Arsrik , 2 < k < N - 1,

(54)

1+ : rkk 1 -1-r i +ki ± -V7-5 1 -kN 1, i,

i1

2
Lemma 69. er,8) is a coideal of U such that S(f)*

E t (" ) for f

(55) E

11.6 Quantum Homogeneous Spaces

453

Proof. The assertion is easily obtained from (10.40) and (10.47). Note that the coideal t ( ' , $) depends only on the ratio sir. Let us write co if s = 0 and N rs-Fr = q", a E R, if r > 0 and s > O. We also set o's) is also denoted by F. For arbitrary a, a = —co if r = 0. The coideal —co < a < +co, we define

er

13`q,'

:= {a E 0(Uq (N)) a.X = 0 for all X E tu l.

By property (1.41) of the dual pairing of the Hopf *-algebras 11 and A we have a* .f = (a.S(f)*)* , a E A, f E U. Therefore, since t' is invariant under the mapping * o S by Lemma 59, the algebra Bq is a *-subalgebra of 0(Uq (N)). From Proposition 68 it follows that B q' === Apa is a right quantum homogeneous Uq (N)-space. The harmonic analysis on the quantum homogeneous space BZ has been studied extensively in the paper [DN], where the proofs of the following two propositions may be found. Proposition 70. (i) The elements

s(u)*u; of 0(Uq (N)) belong to B qr's . They generate the algebra Bqr , s and satisfy the relation (x ii )* = xii. (ii) The right 0(Uq (N))-comodule B q' decomposes as a direct sum =

1=0

v( iEi _IEN)7

where V(lE i lEN) is the irreducible right 0(Uq (N))-subcomodule generated by the highest weight vector x im . The irreducible decomposition of the comodule B°q- is analogous to the irreducible decomposition of the space of functions on the complex projective space U(N)I(U(N —1) x U(1)) (see EVKli, Chap. 11). For this reason we call Bq' the coordinate algebra of the quantum projective space CP 4 . We now consider biinvariant elements of 0(Uq (N)) with respect to the coideal t a and define

11Z := fa E 0(Uq (N)) I X.a = a.X = 0 for all X E

1.

Then, 'HZ is a *-subalgebra of O(Uq (N)). An element of is called a quantum zonal spherical function of Bq' if it belongs to the coefficient coalgebra C(L) for some A E P+ . Proposition Tl. (i ) If we set 7-(7(A) := 7-C7 = M

(TA), then we have

00

.•1***. 1=0

The algebra 7-tZ is commutative dimensional.

n

'1-tt"(ifi — ifN). .1

and the spaces 11Z(lf i — lE N ) are one-

11. Corepresentation Theory and Compact Quantum Groups

454

(ii) If cr E 111, then the algebra 7-(7 is generated by the element s (cr) := X1N + where

±- N1 q7+1±-11

(q2cr+1 q-2cr-1)11

= qu(u)*u.; + q - c r(u li )*u.7 + (41 )* + (u)*u7 E O(Uq (N)).

The assertion of Proposition 71(i) can be expressed by saying that the pair (11, Fr) is an infinitesimal quantum GelIand pair. The preceding exposition and the treatment of the quantum 2-spheres in Subsect. 4.5.4 have shown how one-parameter families of quantum symmetric spaces with the same classical counterpart can be obtained by the infinitesimal approach. This method makes it possible to construct quantum analogs for most of the classical compact Riemannian symmetric spaces (see [DO] and the references therein). The L-functionals and the reflection equation are essentially used in these constructions.

11.7 Notes Cosemisimple Hopf algebras occurred in [Sly], Chaps. 11 and 14, see also [La]. The important Theorem 22 on the cosemisimplicity of the coordinate Hopf algebras 0 (G q ) is due to T. Hayashi [H3]. The theory of compact quantum (matrix) groups and their representations was developed by S. L. Woronowicz in his pioneering papers [Won]] and [Wor3]. In Sect. 11.3 we followed the algebraic approach of M. S. Dijkhuizen and T. H. Koornwinder [DK1] thus avoiding the use of C*-algebras. The discrete quantum groups in [ER] and [VD4] are in fact equivalent to the CQG algebras (see [Ks] for a comparison). The universal CQMG algebras A(E) are in [VDW]. The polar decomposition of the antipode is in [BS]. Both approaches to compact quantum groups C*-algebras are due to S. L. Woronowicz [Wor5], [Wor71. Our proof of the existence of the Haar state follows [VD3]. There exists an extensive literature on compact quantum group algebras and C*-algebras, see [An], [Lei, [LS2], [MN], [Nag], [Riel], [Rie2], [Soil], [Wan1], [Wan2], [Y] • Multiplicative unitaries have been studied in [BS] [B] [Wor81. The representation theory of GL q (N) is taken from the paper [NYM]. The first well-studied quantum homogeneous spaces were the quantum spheres of P. Podleg [Pod]] and of L. L. Vaksman and Y. S. Soibelman [VS2]. In our treatment of the quantum spheres in Subsect. 11.6.4 and of the quantum projective spaces in Subsect. 11.6.6 we have followed the papers [NYM] and [DN], respectively. Quantum Gel'fand pairs were introduced in 1K031 and studied in [Vail], [Vai2], [FI]. The comultiplication ,Abi appeared in [Vai2]. As mentioned in the Notes to Chap. 4, the infinitesimal method for the construction of quantum homogeneous spaces was first used by T. H. Koornwinder [Ko4], [DK21 in the case of the quantum 2-spheres. It was considerably generalized in [N] and [DN] (see also [Sag] and [Dij2]).

Part IV

Noncommutative Differential Calculus

12. Covariant Differential Calculus on Quantum Spaces

Nowadays differential forms on manifolds have entered the formulation of a number of physical theories such as Maxwell's theory, mechanics, the theory of relativity and others. There are various physical ideas and considerations (quantum gravity, discrete space-time structures, models of elementary particle physics) that strongly motivate the replacement of the commutative algebra of Coe-functions on a manifold by an appropriate noncommutative algebra and the study of "noncommutative geometry" there. Differential forms also appear to be a proper framework for doing this. The basic concept in this context is that of a "differential calculus" of an algebra. It allows us to introduce differential geometric notions and carries in this sense the geometry of the "noncommutative space" which may be thought to be behind the algebra. The three final chapters of this book deal with covariant differential calculi on coordinate algebras of quantum spaces and quantum groups. In the first two sections of this chapter we develop general notions and facts on covariant differential calculi on quantum spaces, while the final section is concerned with the construction of such calculi. In particular, a, covariant differential calculus on the quantum vector space CqN is treated in detail. In Subsects. 12.1.1, 12.2.1 and 12.2.2, X denotes an arbitrary algebra. In the rest of this chapter we assume that X is a left quantum space (or equivalently, a left A-comodule algebra) for a Hopf algebra A with left coaction ço:X—>A® X. We retain the convention to sum over repeated indices.

12.1 Covariant First Order Differential Calculus 12.1.1 First Order Differential Calculi on Algebras It is well-known that the exterior derivative d of a C"- manifold M satisfies the Leibniz rule and maps the algebra Coe (M) to the 1-forms on M. If M is compact, any 1-form is-a finite sum of 1-forms f.dg-h with f,g,h E C(M). These facts give the motivation for the following general definition.

Definition 1. A first order differential calculus (abbreviated, a FODC) over an algebra X is an X -bimodule F with a linear mapping d : X—>F such that

458

12. Covariant Differential Calculus on Quantum Spaces

(i) d satisfies the Leibniz rule d(xy) = x.dy + dx.y for any x, y E X, (ii) F is the linear span of elements x.dy.z with x,y,z E X.

We shall say that two FODC 1-1 and 112 over X are isomorphic if there exists a bijective linear mapping V) 112 such that 11)(x•d 1 y.z) = x•d2y-z for x, y, z E X. We shall use self-explanatory notation such as X•dX := Lin{x-dy x, y E X}, X•dX•X := Lin{x-dy-z x, y, z E X}, etc. If F is a FODC, then by (i ) we have x•dy•z = x•d(yz) xy•dz for x, y, z E X and hence F = X•dX = dX•X. Sometimes we omit the dot and write xdy for x-dy. Condition (i) means that d is a derivation of the algebra X with values in the bimodule r. Conversely, if d is a derivation of X with values in an X-bimodule rb then F := X -dX-X is obviously a FODC over X. Let us emphasize that in contrast to the case of classical differential forms for a general FODC the 1-forms dx-y and y-dx are not necessarily equal, even when the underlying algebra X is commutative. As noted above, the 1-forms on a compact Ce()-manifold .A4 constitute a FODC over the algebra C'(,A4). Having this classical picture in mind, we consider the bimodule F in Definition 1 as a variant of the space of 1-forms over the algebra X. We refer to elements of F simply as 1-forms and to the mapping d as the differentiation. We illustrate the generality of the notion of a FODC by a simple but instructive example.

Example 1 (X = C[x]). Let X be the algebra C[x]' of all polynomials in one variable x. Fix a polynomial p E X. Let F be the free right X-module with a single basis element denoted dx. That is, the elements of F are expressions dx. f with f E X and the right action of X is given by the multiplication of X. There is a unique X-bimodule structure on F such that xdx = dx•p. We denote this X-bimodule by fp. It is easily seen that fp becomes a FODC over X with differentiation d defined by

d (E anxn) =

EE Ti

Ti

cEnsi-dx.xi

i±j=n-1

Two such FODC fp and rfi are isomorphic only if p = p. Since dx is a right X-module basis of F, for any f E X there is a unique element a(f) E X such that df = dx a( f). Because f (x)dx = dx. f (p(x)) by definition, we obtain

a(xn)=

E

p(x) 1 x3

i+j=rz— 1

Let us consider the special case p(x) = qx, where q E C, q 1. Then the

X-bimociule fp is characterized by the equation f(x)dx = dxf (qx) and a(f) is just the q-derivative Dq (f) = (f (qs) — f (x)) I (qx — x) (see Subsect. 2.2.1). Another interesting situation is obtained when p(x) c, where c E C[x],

12.1 Covariant First Order Differential Calculus

459

C O. Then we have f (x)dx = dx- f (x c) and 0(f) = (f (x f (x)). Note that in both cases 0(f) becomes the ordinary derivative of f when q 1 and c 0, respectively. Take a fixed 1-form 77 0 of and define dn f = f77 - rj f, f E X. Then Xp ,71 : X•dn X•X is a FODC over X with differentiation d n . If i = rds with r, s E X, then we have dn f =-- dx.r(p(x))0(s)(f (p(x)) - f (x)). A Definition 2. A FODC F over a *-algebra X is called a *-calculus if there exists an involution p p* of the vector space r such that (x.dy.z)* = z*.d(y*).x* for x,y,z E X. Proposition 1. A FODC F over a *-algebra X is a *-calculus if and only if Ei si .dyi = 0 with x i , yi E X always implies that Ei d(yn.x: = O. Proof. The only if part is trivial. Conversely, if the latter condition is fulfilled, then (E i si dyi )* := Ei d( y )-x, x i , yi E X, gives a well-defined(!) antilinear mapping of F which has the desired properties. 111 12.1.2 Covariant First Order Calculi on Quantum Spaces

If the algebra X is a quantum space for a Hopf algebra A, it is natural to look for covariant FODC. The precise definition of this notion is given as follows. Definition 3. A first order differential calculus F over a left quantum space X with left coaction : A® X is called left-covariant with respect to A if there exists a left coaction q : A 0 F of A on F such that

(i) Oxpy) = v(x)0(p)v(y) for all x, y c X and p e F, (ii) 0(dx) = (id 0 d)c,o(x) for all x c X. Conditions (i) and (II) mean that the coa.ction 0 of A on F is compatible with the coaction of A on X and with the differentiation d, respectively. Condition (ii) says that the mapping d intertwines the coactions (p, and 0, that is, d E Mor(vI 0), or equivalently that the following diagram is commutative: X

d

ça A 0X

id 0 d >

14)

A®F

For instance, if x i , • • • , x n are elements of X such that v(x i ) = t4 0 xi , then 0(dx i ) = uii 0dxi by (ii). That is, (ii) implies that the differentials dsi transform under 0 just as the elements si do under v. However, the differentials ds i need not be linearly independent if the s i are.

Example 2 (X = C[s] - continued). The algebra C[x] can be interpreted as the enveloping algebra U(g) with g = R. Hence, by Example 1.6, X is

12. Covariant Differential Calculus on Quantum Spaces

460

a Hopf algebra with comultiplication given by A(x) = xe1 ± 10x. If we consider X as a left quantum space for the Hopf algebra X itself with respect to the comultiplication, then the FODC fp from Example 1 is left-covariant if and only if p(s) = s. (In order to prove this, apply to the equation scis = cis • p(s) and use conditions (i) and (ii).) Let A = CZ be the group Hopf algebra of Z (see Example 1.7). We write A as the algebra C[z, of Laurent polynomials in z with comukiplication A(z) = zgz. Clearly, X is a left quantum space of A with coaction determined by (,o(x) = zgx. Then the FODC fp is left-covariant if and only if p(x) = qx for some q E C. Thus we see that in both cases the covariance requirement is an essential restriction of the wealth of possible FODC. Note that given a FODC over X there is at most one linear mapping as in Definition 3. Indeed, if such a q exists, then by (i ) and :f A (ii), we have . xi dyi )

=

.w(xi)(id

(1)(y) .

(1)

The right-hand side of (1) and so q is uniquely determined by ço and d. An intrinsic characterization of the left-covariance of a FODC which avoids the use of the coaction is given by Proposition 2. For any FODC f over X the following assertions are equivalent: (i) I' is left-covariant with respect to A. (ii) There is a linear mapping ç : such that for all s,y G X .4 0 we have 0(xdy) = v(x)(id00,o(y). Ei si dyi = 0 in implies that Ei (p(xi )(id c) w (y ) = O in A0 F. Proof.

is true by definition and is trivial. (i ): By (Hi), equation (1) defines unambiguously(!) a linear mapping qS : F A f which obviously satisfies condition (ii) of Definition 3. By the properties of the coaction w, we have (A

(e

id)0(xdy) = (A id)ko(x)(id 0,o(y)] = [(A id)(p(x)1Rid id d)(A id)w(y)] = [(id (p)(p(x)i Rid id 0 d)(id 0 ;0)(i, (y)] = (id 0)[(,o(x)(id 0,o(y)] = (id 0)0(sdy), id)0(xdy) = (e id)Ro(x)(id d)(y)] = (6 id)(,a(x)•(e d)w(y) -=

that is, 4 is a left coaction of A on F. To prove condition (0 of Definition 3, let p. = Ei xi dyi . Using the Leibniz rule for d and the fact that ço is an algebra tomomorphism (because X is a quantum space), we obtain

12.2 Covariant Higher Order Differential Calculus

461

E.0(xxid(yiy)) E. 0(xxiyi dy)

0(xpy)

E ,o(xxi )(id

d)cp(yiy) -

E.,(xxiyi)(id

d)(p(y)

= ço(x) (E i 49 (x2)(id d)40 (Yi)) ço(y) =

12.2 Covariant Higher Order Differential Calculus 12.2.1 Differential Calculi on Algebras The de Rham complex of a C"-manifold is formed by extending the exterior derivative to the algebra of differential forms. A differential calculus may be viewed as a substitute of the de Rham complex for arbitrary algebras. Definition 4. A differential calculus (abbreviated, a DC) over X is a graded algebra pA = 6)7.7 o rAn (that is, rA is a direct sum of vector spaces rAn and the product A of f" maps rAn x rAm into r^(n+m)) with a linear mapping rA(n+i),) such that: rA of degree one (that is, d : rAn d : r^

(i) d2 = 0, (ii) d(p A()

dp A ( (-1)np A d( for p (iii) rAo — x and l'An = Lin {x 0 A ds i A n E N. =

and ( E _rA , A dxn I xo , • • •,x n E X} for

E FAn •-•

If condition (iii) of Definition 4 is dropped, then we obtain the definition of a differential graded algebra. The notation p A e reminds us of the cap product of forms in differential geometry. For the product by elements x E rAo = X we shall write simply x p and px, p E rA. Condition (ii) is called the graded Leibniz rule. By induction on n one proves that condition (hi) in Definition 4 can be replaced by (iii)' rA0 = X and _T'An = X•drA(n -1 ) for ri E N. By a differential ideal of a DC rA over X we mean a two-sided ideal {0} and J is invariant under J of the algebra rA such that J n PA0 the differentiation d. Suppose that J is a differential ideal of a DC PA over X. Let 7r : rA ij denote the canonical map of rA to the quotient algebra := rA 1J. We define El(r(p)) 7r(dp), p E rA. The condition n rA0 = folj ensures that PA° = rA0 = X. Then Îv is again a DC over X with differentiation d , called the quotient of PA by the differential ideal J. For any DC over the algebra X the following identities hold:

r^

d(xodxi A (xo dx i A

••• A

dxn ) = dxo A

dx1 A • • - A

dxn ,

(2)

• • • A dxn ) A (xn +1dxn +2 A •• • A dXn+k)

= — 1) 11 SoSi dX2A • - - AdXn-Fic

— 1) n—r SodXi A. - •Ad(sr xr +i)A• • • AdXn±k r=1

(3)

462

12. Covariant Differential Calculus on Quantum Spaces

Indeed, (2) and (3) are easily verified by induction using the Leibniz rule and the equation d2 = O.

Definition 2'. A DC l'^ over a *-algebra X is a *-calculus if there exists an involution p p* of the vector space l'A which coincides with the involution of the *-algebra X on rA0 = iv and satisfies the relations (pi, A pk )* = (_onk pz A pn* and d(p*) = (dp)* for any pn c rA n e roc an d p e rA . Pk Note that the involution of a *-calculus FA (if it exists) is always uniquely determined by the involution of the *-algebra X, because FA is generated as an algebra by rAo = X and dX.

12.2.2 The Differential Envelope of an Algebra The aim of this subsection is to define a DC 12(X) over X which is universal in the sense that any other DC over X is isomorphic to some quotient of f2(X). The calculus f2(X) itself is very large and so is not of interest. We shall need it only as a technical tool in order to construct differential calculi as its quotients (just as free algebras are used to construct algebras in terms of generators and relations). Roughly speaking, the main idea behind the definition of the DC f2(X) can be described as follows: f2(X) is the free algebra generated by elements x c X and symbols dx, x E X, subject to the Leibniz rule dx•y =-- d(xy) x•dy. By the latter relation, any element of f2(X) can be written as a sum of monomials s0dx1 A • • • A dx, with xo, x 1 , • • • , x n E X, where A denotes the product of the algebra (2(X). Thus it suffices to define the product and the differentiation of f2(X) for such monomials. Since we want (2(X) to be a DC, the formulas (2) and (3) must hold. They will give us the corresponding definitions for monomials. The above idea is realized by the following precise mathematical construction. Let X := X/C•1 be the quotient vector space of X by C • 1 and let "î denote the equivalence class x + C • 1 of x E X. We set CX)

m en := $ fr(X),

where Q° (X) := X,

(271 (X) X et- cDn, n c N.

n--=0

We write xodsi A • • • A dxn for xo ±n . It can be shown (see, 0•• for instance, [Con], [CQ] or [Ka]) that the formulas (2) and (3) (with the obvious interpretations for n = O and k = 1) define a differentiation d and a product A, respectively, such that fl(X) becomes a differential calculus over X. Clearly, Q(X) is a *-calculus if X is a *-algebra. Define a linear map Let TA be another DC over X with differentiation Q(X) FA by 'Or(x) x and Cxodsi A • • • A dxn) := xoasi A • • A axn, n E N. (The definition of f2n (X) as a tensor product ensures that IP is welldefined!) From the identities (2) and (3) it follows that g := ker //) is a differential ideal of Q(X) and that re^ is isomorphic to the quotient DC

a.

ir2(X)/Ar.

12.2 Covariant Higher Order Differential Calculus

463

The first order part Q' (X) of f2(X) is obviously a FODC over X and an arbitrary FODC over X is isomorphic to a quotient of ,0 1 (X). The DC 0(X) is called the differential envelope or the universal differential calculus of the algebra X. Likewise, the FODC Q' (X) is called the universal first order differential calculus of X. 12.2.3 Covariant Differential Calculi on Quantum Spaces First we extend Definition 3 and Proposition 2 to higher order calculi.

Definition 3'. A differential calculus F A over the left quantum space X is called left-covariant with respect to A if there exists an algebra homomorphism AOF A which is a left coaction of A on FA such that OA (x) = (p(x) for x E 1 A0 =-- X and O A (dp) = (id 0 d)q5A (p) for p E 1 A • Proposition 2'. For any DC FA over X the following statements are equivalent: (0 FA is left-covariant with respect to A. (ii) There exists a linear mapping O A T A -3 A® F A such that the restriction of O A to X is equal to cp and for any xo , x i , • • , xn E X we have (sodxi A • • • A din ) = (p(x0)(id d)(p(xi) - • - (id 0 cl)(p(xn).

(iii) Ei xt,dit

A --• A

dsin = 0 in FA always implies that d)(p(x1)• ..(id

d)(itn ) = 0 in A® I'.

2

Using formulas (2) and (3) with k = 1, Proposition 2' is proved in a similar manner as Proposition 2. For the proof of the main implication (iii) —3 (i ) the coaction OA is defined as in (ii). From the latter we see in particular that the mapping OA in Definition 3' is uniquely determined by the coaction (p of A on X and the differentiation d. Since (id0d)cp(s) = (id0d)(p(1) for x E X, conditions (iii) of Propositions 2 and 2' are satisfied for the DC f2(X) and the FODC 01 (X). Thus, we obtain Corollary 3'. The universal DC f2(X) and the universal FODC f21 (X) over X are left-covariant.

Now suppose that F is a FODC over X with differentiation dl . In general there are many higher order calculi with first order part F. (A trivial one is obtained by setting _Finn = {O} for n > 2.) We want to define the "most free" differential calculus over X whose first order calculus is the given F. For this purpose we set N{>xdy1 E 01 (X) I Exd 1y1 = o in F}. From the Leibniz rule for d1 it follows easily that At is a X-subbimodule of Q 1 (X) and the FODC F is isomorphic to the FODC Q 1 (X)/N. Let Jr := f2(X)AN .A.0(X)+f2(X)AciATAf2(X) be the differential ideal of the DC f2(X) generated by N. Then 1- 11,\ := 12(X)1,71- is a DC over X whose first order pa rt

464

12. Covariant Differential Calculus on Quantum Spaces

is f2'(X)I.Ar and so is isomorphic to F. It is clear from the construction that the following universal property: If TA =erirAn is another the DC 1' DC over X such that FA' is isomorphic to T, then 1-1^ is isomorphic to a quotient of ru^. Because of the latter property, fu^ = f2(X)1,7r is called the universal differential calculus of the first order differential calculus F. Proposition 4. (i) If r is left-covariant, then so is F. (ii) If X is a *-algebra and F is a *-calculus, then Fu^ is also a *-calculus. Proof. (i): Let 0^ denote the coaction of A for the left-covariant DC (1 (X).

The characterization of left-covariance of r given in Proposition 2(iii) means precisely that 0^ (g) C A0 . By Definition 3' applied to the left-covariant DC r2(X) we have 0"4:1 = (id 0 Or and so 0^ (cW) C A 0 dAr . Since 0" is an algebra homomorphism, it follows from the preceding that O A (Jr ) C A 0 Jr . Therefore, 0^ passes to the quotient ru^ = r2(X)/Jr and defines a left coaction of A on /7,1,,' . Hence ru^ is left-covariant. (ii): Using the Leibniz rule and Definition 2 for the *-calculi 12 1 (X) and I', we conclude that IV is *-invariant. This implies that the differential ideal Jr is *-invariant. Therefore, the involution of the *-calculus P(X) passes to and hence F is also a *-calculus. 111

ruA

Remark. All notions and results concerning left-covariant FODC and DC on

a left quantum space established above have their counterparts for rightcovariant FODC and DC on a right quantum space. They are obtained if we replace "left" by "right", "id 0 d" by "d 0 id", "A 0 r" by "I' 0 A" and "A0 rA" by "FA 0 A" in Definitions 3 and 3', Propositions 2, 2', and 4, and Corollary 3'.

12.3 Construction of Covariant Differential Calculi

on Quantum Spaces 12.3.1 General Method

The purpose of this subsection is to develop and discuss a general result (Proposition 5) concerning the existence of left-covariant FODC on quantum spaces. Let Ixd iE/ be a family of linearly independent generators of the algebra X and let {4} i ,j ,ku be a set of elements of X. Proposition 5. Under the assumptions (5) and (7) stated below, there exists a unique left-covariant FODC r over X such that {dxi}j E I is a free right X module basis of F and the X -bimodule structure off is given by the relations Xi.dXj = dXk•X lici , i,j E

I.

(4)

12.3 Construction of Covariant Differential Calculi on Quantum Spaces

465

Assumption (5) expresses the covariance of the defining relations (4) of the FODC F and (7) requires two consistency conditions, one for the X-bimodule structure of F and one for the differentiation d. Note that the structure of the FODC F is completely described by the equations (4) and the property that {dxi}i c i- is a free right X-module basis of F.

Proof. Let X0 = Lin{x i i E I} and (po (preY o . Let T: X0 0 X0 —) X0 0 X denote the linear mapping defined by T(x i 0 x i ) = xk E I. We assume that Xo and T E Mor ( Po (Po, (Po

(P(Xo) C A

(P)•

(5)

To begin the construction of F, let y be the free algebra with generators xi , i E I. Let dX0 denote a vector space with a basis indexed by the set I. The basis elements are denoted by ax i , j E I. The vector space F := yodxo oy is a Y-bimodule with the obvious right and left actions of y on P. For simplicity we write y•Elx i •z for y 0 z. Since y is the free algebra generated by there are a unique algebra homomorphism b 0 : A 0 Y which extends the linear map w 0 : X0 —> A0 X0 and a well-defined linear mapping P such that a(1) = 0, a(x i) ax i and :Y 71

11(xi 1

"Xi n

= E

X»

,,•axini _xim+

,.

•

ain

m=1

for i, i i , • • , i n E I. Then Y is a left quantum space of A with coaction 00 . Define a coaction qo : clX0 —> AOdX0 by 00 (ax) := (id0a)Vo(zi). It is easily seen that a satisfies the Leibniz rule and 5 := /Po 00 P is a > left coaction of A on P such that c Mor (00 , V)). Hence i; is a left-covariant FODC over 3) with differentiation Now we suppose that J is a two-sided ideal of Y such that the algebra X is (isomorphic to) the quotient algebra y j and 00 (j) c A 0 3. Note that such an J always exists, since we assumed that the algebra X is generated by the set lx i l ic i. Let it : Y —› X be the corresponding quotient map and take elements y i/ci C Y such that 71- (y) = xt. Using once again that y is the free algebra with generators fzi lie j, it follows that there exist linear mappings :y y i i E I, such that pii (yz) =--- p(y)p(z) for y, z E y , pii (1) = kJ ]. and (xi) = y. Then N := Lin{z-ax i •y — dXk.p(z)yz, y E y,i E I} is a Y-subbimodule of P. Hence F1 := P/J14 is a Y-bimodule. That is, as a vector space F1 is isomorphic to dX 0 0 y and the Y-bimodule structure of is determined by the relations

a

—

a.

,

xi •axi

j E I.

(6)

Let N0 := dX0•,7 Linfax i .z IzEJ,iE II and let 30 be a set of generators of the two-sided ideal J. We assume that zo•axi E dX0.„7 and azo E dX0 •,7 for z0 E

go and

i E I.

(7)

466

12. Covariant Differential Calculus on Quantum Spaces

Clearly, .M0 is a y-subbimodule of f1 . The first condition of (7) implies that becomes C M. It is obvious that riJ c No. Hence F :--= F be the a bimodule for the quotient algebra X = yij. Let r i : quotient map. The second condition of (7) and the Leibniz rule for imply that d(r(y)) :=- ri(a(y)), y E y, defines unambiguously a linear mapping d :X f. Then F is a FODC over X with differentiation d. We can consider Mo also as a subset of .T;- rather than of fi, since MinAro = and ?,b0 (J) C NI. From assumption (5) and the facts that ir(y) = .4) C A 0 (go + MO. Since the A 0 3. we conclude that i,b(x i -ax i j E I, Y-subbimodule Ari is generated by the elements xi -jlxi and Mo is a Y-subbimodule of fi we obtain 7,b(Mi) C A 0 (Aro + Mi). Since also O(Aro) C A Aro , we have ip(ri + Aro ) c A 0 (N-0 + MO. Hence the coaction //) of A on f passes to the quotient f/(Ari + Aro) r to define a coaction clo there. The map q satisfies the conditions of Definition 3 for f, since 1,b does for P. Thus we have shown that F is a left-covariant FODC over X. From the preceding construction it is clear that fclxilifi is a free right X-module basis of f and the relations (4) hold.

rum

a

axk

4

Let F be the left-covariant FODC from Proposition 5. Then, for any x E X there are uniquely determined elements ai(x) E X such that

dx =

dxi •ai (x).

(8)

The mappings ai : X X are called the partial derivatives of the FODC F. Let us turn to the universal higher order DC f of r. Recall from Subsect. 12.2.3 that FiLA is the quotient of the universal DC (1(X) over X by the two-sided ideal Jr generated by the defining elements x i -dxj — dxk for the bimodule F and their differentials d(xi •dxi dxk•x) = dxi A dxi +dxk A d4i , i, j E I. That is, the algebra f has the generators x i and dxi , i E I, and the defining relations of Fii^ are those of the algebra X, the relations (4) and

dxi A dxi -= —dxk A dx1Fi , i , j E I.

( 9)

Because of the applications given in the next subsection, we specialize the above procedure. Suppose that the quantum space X is an algebra with generators x i , j = 1, - N, and defining relations written in matrix form as

Buxi x 2 = 0,

(10)

where B is a complex N2 x N2 matrix and x denotes the column vector (x i , • • •, x N )t (see Subsect. 9.1.2 for the corresponding notation). Further, we are linear in the generators x i , that is, we have assume that the elements =X = AZxi for some complex matrix A = (A). Then (4) reads as x i •dxj = Atil dxkai or in matrix notation as

4

j .

12.3 Construction of Covariant Differential Calculi on Quantum Spaces

467

X•dX2 = Al2 dX1 . X2

Now we set Jo := Prikx i xi j n, k = 1,2, • • •,N} and reformulate the key assumption (7) in this case. From the equations (6) and (11) we obtain (B12X1X2) J.X3 = B12A23 xrdX2 'X3 7-- B12 A23Al2aX1 'X2X3, a(Bi2X1X2) = B12(aXi•X2

xrax2)= (B12 +./312Al2)axi-x21

respectively. Therefore, by (10), the assumption (7) is certainly fulfilled if there exists a complex N 3 x N3 matrix T such that (12)

B12A23Al2= T123B23) B12 + B12Al2 =

(13)

0.

12.3.2 Covariant Differential Calculi on Quantum Vector Spaces In this subsection we shall apply the method of the preceding subsection to the following situation: A is the Hopf algebra 0(G q ) with G q q (N),SL q (N),0q (N),Spq (N) and X is the left quantum space with definGL ing relations P_x 1 x2 = O. Here P is the spectral projector of k to the eigenvalue -q -1 (see Sects. 9.2 and 9.3). We suppose that q 2 + 1 4 0 for GL q (N) and SL q (N), (q2 + 1)(1 q 2-N ) 0 for 0q (N) and (1 2 + 1) ( 1 _ c 2N) f(k) for some polynomial for Spq (N). These assumptions ensure that f so X= XL(f; R) is indeed a left quantum space of A by Proposition 9.4. In fact, we have X = O(C) if G q = GL q (N),SL q (N) and X = 0(0,7) if Gq =0q (N). By (9.10), the left coaction o pL of A on X is given by (p(x) = i Oxi, where u = (ui ) is the fundamental matrix of A and x = (x1, • • •, xpi) t is the vector of coordinate functions of X. Then we have (p(X0) C A ® X0 and (idOT) ( (Po O(P0) (xi Oxi) = (idOT)(uluil ®Xk OX/ =U2:k U./ Ankim (San 0Xml ( (Po

(P)T(Xi

xj) = (CPO

(10 )(Xk

AX/)= A ikit unk uL X n,

Xm .

Hence the assumption (5) is fulfilled if A = A+ := q A and if A = A_ . For A -= A+ and A -= A_ both equations (12) and (13) are satisfied with T123 := A23Al2. The relation (13) follows from the equation P_A± 1 --= -q71 P_, while (12) is an immediate consequence of the braid relation for R. Therefore, by the previous considerations, we proved the first assertion of Theorem 6. There exist two. left-covariant FODC 1-1± and F_ over X with respect to A such that the set fclxili=1,-..,N is a free right X-module basis of r± and xi .dx i = q±' (A± 1 )dx k .si . (14) Moreover, we have the following formulas of commutation relations for F±:

468

▪

12. Covariant

Differential Calculus

= 6i3 +

on Quantum Spaces (k ±i)iii,x ka

(15)

(p_)143k a, = o,

(16)

dxi A dxj = —q ±l (k ±1 )Icii dxk A dxi.

(17)

Proof. It remains to verify the formulas (15)-(17). Using (8) and (4) we get

d(si y) = dsi -y -dy--= dxj -y + x -cisi -51(y) = dry q ±1 (k±i)ii.k1 dxi .sk at (y ) .

dzi -ai, (xi y) -=

is a free right X-module basis of F± , the coeffiSince the set cients of dsi on both sides must coincide. This gives (15). Next we turn to (16). Let y E X be such that (P_) ipka/(y) = 0 for all i, j=1, N. By induction it suffices to prove that (p_)ak a,(spy ) = 0 for p = 1, 2, • • , N. The braid relation for implies that fRNÉR1(P--)12 = (P) 2321 I Applying first equation (15) twice, then the identity fRP__ -q -1 P__ and finally the preceding equality we obtain .

P _ )si 3lc ak (x p y) = P _ ) 173k ak( y ) q ±2(p - )/i.ig±1)f 0 mn(k±1.)nr ss ar an (y)

= (p_)icak (y)

. 2.an (y) 7±1.(k±lp4f37 1

= (p_) ipk (y)

q ± 1(_el p _ )

T:L an(y

17±2(fR 21. fR 31. ip )

=q ±2(p_vcr o±Tnk(0±1) .73.si xsar an (y)

( )12)7:Xsaran(Y) q±2 (( p_ )23 mi R23 )ip37-sL r xsaran(y)

= 0.

Finally, note that (17) follows by differentiation of (14).

12.3.3 Covariant Differential Calculus on CN and the Quantum Weyl Algebra In this subsection we treat the covariant FODC F± on X --= O(C) in detail. First we insert the values (9.13) for the numbers Wicji into the formulas (14)-(17) in Theorem 6. Then the commutation relations for the FODC and the defining relations of the algebra O(C) take the following explicit form: (18) x.dx = qdxj•xi + (q 2 1)dxj -xj , j < j,

(19)

xi -dxi = q2 dxj •xi, xj .dxi = q cisi -xi,

j,

(20)

dxi A dx = —tr i dxi A dxi , i < j,

(21)

dzi A dxi = 0,

(22)

sixi =

j,

(23)

12.3 Construction of Covariant Differential Calculi on Quantum Spaces

i < j, aixj = q x -

i

469

(24)

j,

(25)

q2x iai =l+ (q2 -1)E xiai.

(26)

j>i The corresponding relations for the second FODC ./.1_ over X = O(C) are obtained if we replace q by q -1 and the inequality i < j by j < i in the formulas (18)-(26). Of course, the equations (21)-(25) remain unchanged in 1 the FODC 11+ and 11_ are this manner. In particular, we see that for q not isomorphic and that for q = 1 both FODC give the "ordinary" differential calculus on the commutative polynomial algebra C[xi, • • xNI• The above equations (18)-(20) completely describe the bimodule structure of /1+ . The algebra (r+ ).^ of the universal DC associated with r+ admits the 2N generators x i and ds i , j = 1, • • • , N with defining relations (18)-(23). In the case q = 1 the formulas (23)-(26) reduce to the defining relations of the Weyl algebra. This motivates the following ,

Definition 5. The quantum Weyl algebra A q (N) is the unital algebra with

2N generators x l , • • -

v ,a,,. • - ,a,„ and defining relations (23)-(26).

We mention a few algebraic properties of this algebra A q (N).

1. The Weyl algebra A q (N) with q = 1 is simple, that is, it has no nontrivial two-sided ideals. However, the algebra A q (N) is no longer simple if q2 1. Indeed, the kernel of the one dimensional representation given by 1, (1 _ q2 ) _1, xi xi -> 0, a, —> 0, i > 2, is a nontrivial ideal. 2. Let D be the unital algebra with N generators ai,a2,- ,aN and defining relations (24) and let X := O(C). Then the linear map of X 0 O to x•y is a vector space isomorphism. (A proof can A q (N) defined by x y be given by using the diamond Lemma 4.8.) Therefore, we can consider X and D as subalgebras of A q (N) and the set of monomials • -

r _ki

An].

alkiv I rbit. 1

kN n - ••, nN E NO

}

is a vector space basis of Aq (N).

3. The element D := E ixi ai of A q (N) is called the Euler derivation. If fn G X and gn G D are homogeneous elements of degree n, then we have the identities Dfn = [[nlio fn ± fn D where we have set [[n]1 q2 : = ( 172

and gn D = 1 ) 1 ( 72n -

pi llogn ±

Dgn,

1).

4. The partial derivatives ai of the FODC r+ act on the algebra X by

ai (fN(xN)• - -fi(xi))

f N(qxN)• • • f i+i(qx j+i)Dq 2 fi(x i)

• • fl(11)

where fi, - • .1fN are polynomials in one variable and De is the q2-derivative (see (2.43)). It suffices to prove this formula for monomials f(x) =

12. Covariant Differential Calculus on Quantum Spaces

470

First we shift x j:i to the right by using (23) and ai in front of 4i by applying 1 back. [Pci flos iFi -1 and shift x (25). Then we compute ai (xlci) It is well-known that the "ordinary" canonical commutation relations are invariant under symplectic linear transformations. The next proposition contains a quantum analog of this fact. Proposition 7. Let É = (ki ) be the R-matrix and C = (Ci) the matrix of O. the metric for the quantum group Spq (2N) (see Sect. 9.3). Let a E C, a Set yi a q i aN+i-i and YN+i xi for j = 1, 27 • • • , N. Then the generators x i , • • • , x N, satisfy the defining relations (23)-(26) of the algebra • • • , A g (N) if and only if

a,

aN

2N k3lYkYl

qyiyi - a q

i

=0

for

j = 1,2,- • • ,2N.

(27)

k,i=i

The quantum Weyl algebra A q (N) is a left quantum space for the Hopf algebra 0(Sp q (2N)) with coaction ço given by (,n(y0 = uii y i , j = 1, 2, • • • , 2N. Proof. For i, 74 j' we have Cii =-- 0 and (27) reduces to the equations (23)(25). Now we show that the equations (27) for i j' are equivalent to the equations (26). We first rewrite the relations (26) recursively as

=-- 1,2, • • • ,N

= aj x -

aNxN - q 2xN aN

1,

= 1.

These equations are easily verified to be equivalent to yi, yi

2 yi yit = _ q -i a ± (q -2

1)E q k_i ykyk ,, j = 1,2, •••, N.

(28)

kci

On the other hand, inserting the values of C!, and kit' from (9.30) into (27), we see that the equations (27) for i = f are also equivalent to the relations (28). For the second assertion we proceed as in the proof of Proposition 9.4. There exists a unique algebra homomorphism (09 : C(yi )-0(Spq (2N))0C(yi ) such that ço(yi ) = uij Oyi . Let J be the two-sided ideal of the free algebra C(y) which is generated by the elements Pi from the left hand side of (27). Using the relations uCut C and rZu 1 u2 = uiu2rZ (see (9.34) and (9.3)) we passes to the quotient algebra obtain that (p(Jii) = 744 grki. Therefore, D A q (N) = C(yi )/,7 to define a left coaction of 0(Sp q (2N)) on A q (N).

There are two important special cases, where the quantum Weyl algebra A q (N) admits a natural involution. q E R. Then there is a unique algebra involution on Aq (N) such that := x i , j = 1,2, • • • , N. Indeed, since q is real, the defining relations (23)(26) are obviously invariant under this involution. Hence Aq (N) becomes a

Case

1:

12.3 Construction of Covariant Differential Calculi on Quantum Spaces

471

*-algebra. The generators of A(N) can be interpreted as q-boson creation operators a7 = x i and q-boson annihilation operators a i = ai in this way. Case 2: WI = 1. In this case the algebra 0(CqN ) becomes a *-algebra O(R) with involution defined by x7 = xi , i = 1,2, • • • , N, and the FODC 1 -1+ a *-calculus for the *-algebra O(R). The involution of F+ induces theis involution for the generators ai given by ar = —072(N+1-)19i, i = 1,2, • • • , N. Using the assumption WI = 1 one verifies that the relations (23)—(26) are preserved under this involution. Therefore, the algebra A q (N) becomes a * algebra, denoted A q (N; R), such that x1 :- --- x i and ai* = — q 2(N+103. The hermitian elements x i and pi := — 1/--i_ q N+1-i ai of this *-algebra can be viewed as q-analogs of the position and momentum operators, respectively. If one adopts this point of view, it is natural to consider the *-algebra A q (N; R) as a q-deformation of the real quantum mechanical phase space. Let us set a = --I,/-1 - in Proposition 7. Then we have yi = PN+1-i and YN-Ei = Xi therein and the last assertion of Propostion 7 means that the quantum group Spq (2N) acts on the q-deformation A q (N; IR.) of the phase space. -

i

We close this subsection by stating a uniqueness result for the two covariant FODC F± on O(C). Recall from Theorem 6 that the set Idx i l i= i ,...,N is a free right module basis of r+ (and also a free left module basis as follows from the formulas (18)—(20)). The next proposition shows that for N > 3 and for q not a root of unity the latter property characterizes F+ and F_ among all possible left-covariant FODC on O(C). Proposition 8. Suppose that N > 3 and q' 1 for all n E N, n > 2. If F is a left-covariant FODC over X = O(C) with respect to the Hopf algebra A = 0(S L q (N)) such that the set {dxili=1,-.,N is a free right X -module basis of F, then F is isomorphic to F+ or to F_.

Proof. The proof can be found in [PuW]. (In the paper {PuW1, covariant FODC with respect to SU q (N) , q E (-1, 1), are considered, but the proof therein works for SL q (N) if q is not a nontrivial root of unity.) D In the case N = 2 the assertion of Proposition 8 is not true. In this case there are two one-parameter families of such FODC (see [SS1] for details). 12.3.4 Covariant Differential Calculi on the Quantum Hyperboloid

For a nonzero complex number 7, let Xq ,..), denote the algebra with generators x l , x2 and defining relation x1x2 —qx2xi == 7 1-

It is not difficult to check that there is an algebra homomorphism coo : eVy 0(SLq (2))0Xvy defined by ço(x i ) = u.ij eaj , i = 1, 2, such that eVq ,..y becomes a left quantum space for the quantum group SLq (2) with coaction (p. The

472

12. Covariant Differential Calculus on Quantum Spaces

algebra Xq, ..y is called the coordinate algebra of the quantum hyperboloid. Obviously, Xq 2 , 1 is just the quantum Weyl algebra Aq (1). The following proposition says that there exist two distinguished covariant FODC on the quantum space Proposition 9. Suppose that 0 and qn I for n E N, n > 2. Then there are precisely two nonisomorphic left-covariant FODC F over X4,7 with respect to the Hopf algebra 0(S4(2)) such that fclx i ,dx 2 1 is a free right Xq ,..y -rnodule basis of F. The bimodule structures for these two calculi are described by the formulas

-p-l q -1 )dxrx2 + p1

=

+ (1 -p-

1

nv „

p -1 dx i .x 2 + (p - qp-1 )dx 2 •x + (1 - p-1 )7 -1 winv x 2 x i , = pdxj ai + (1- p -1 )7 --i winv(x i ‘2 ) for j = 1, 2, where p := ±(q1 / 2 + q- " 2 ) -1 and w inv

dx1•x 2 qdx2.x1.

This proposition is stated in [SS1]. We shall not carry out its proof. The uniqueness assertion of Proposition 9 is no longer true if q3 = 1 and q 1. Indeed, let q be either 1(-1 + -V7-3) or (-1 Then it can be shown that for any nonzero complex number a there exists a left-covariant FODC Fa over 4,7 with respect to 0(Sli q (2)) such that the set {ds i , dx 2 } is a free right module basis of Fa . The corresponding formulas for the bimodule Fa are

z 1 .dy2 = X2 .11X' =

xi .dxi =

(a -

1- )dxi-x2 + cdx2xi + d Winv X1X2 + (a - q a -1 )dx2 ex i + a winv x2x1l a dx f xi + a Winv (xi ) 2 for j = 1,2,

where de- := -(a + a -1 + 1)7 -1 and winv := dx•.T2 - q dx2-xi

12.4 Notes Differential forms on noncommutative algebras and the differential envelope of an algebra appeared in the work of A. Connes, M. Karoubi, D. Kastler and others (see the references in [Con], [CQ] and [Kaj). The covariant differential calculus on the quantum vector space CqN was obtained independently in [WZ] and [PuW]. These papers were very influential for the study of noncommutative differential calculi, see (for instance) ICSW], ISchj, [0g1, [Z), [3K0), ISc11, [GZ]. First steps towards a q-deformed quantum mechanics have been made in [SW] and [1-1SSWVV]. Apart from quantum vector spaces and quantum groups themselves, covariant differential calculi have been investigated on quantum 2 spheres (see [Pod2], EPod31, [AS ]) . A theory of such calculi on general quantum spaces (if possible) is still at the very beginning. -

13. Hopf Birnodules and Exterior Algebras

This chapter is devoted to some concepts (left-covariant bimodules, bicovariant bimodules) and constructions (tensor algebras, exterior algebras) which are basic tools for the covariant differential calculus on quantum groups in the next chapter. They are also of interest in themselves. The structure theory of covariant bimodules is developed coordinate free in abstract Hopf algebra language. This allows us elegant statements and short proofs of the results. At the end of the subsections the results are reformulated in coordinate form which is more convenient for computations. Throughout this chapter A is a Hopf algebra with invertible antipode S.

13.1 Covariant Bimodules Covariant bimodules can be considered as quantum group analogs of vector fiber bundles over Lie groups which are endowed with left or/and right actions of the groups.

13.1.1 Left-Covariant Bimodules Definition 1. A left-covariant bimodule over A is an A-bimodule 1 -1 which A® 1-1, such that is a left comodule of A, with coaction 2q, : T

p E F.

(1)

Ea(i) p( _ i) b(i) 0 a( 2)p( o)b( 2).

(2)

2q,(apb) = 2i(a),AL(p),A(b)

for

a, b E A

and

In Sweedler's notation, the last condition can be written as

E(apb) (_ 1) 0 (apb)(0)

=

An element p of a left-covariant bimodule f is called left-invariant if 2iL(p) = 1 0 p. The vector space of left-invariant elements of f is denoted by inv T. Lemma 1. Let I' be a left-covariant bimodule over A. There exists a unique linear projection PI, : 1-1 iiiv r such that

PL(aP) = e(a)PL(p),

a E A, p E F.

(3)

474

13. Hopf Bimodules and Exterior Algebras

The map PL is given by PL(P) =

E s(p(_1))P(0)•

For p E F and a E A, we

have (4)

= PL(Pa) = E S(a(i))PL(P)a(2)

adR(a)(PL(P)),

(5)

where adR is the right adjoint action of A on the bimodule F (see Remark 2 in Subsect. 1 .3.4). Proof. Using the formulas (2), (1.30), (1.27), and (1.20) we have ZAL(PL(P))

zAL(s(p(_1))p(0))

=

E s(p ( _2 ) )p ( _i ) s(p(_3 ) )p( 0 ) = EE(p( _ 1) )1

s(p(_2))p(o)

= 1 ® PL(p),

so that PL(p) E inv F. Using the formulas (2), (1.28), and (1.27) we get PL(aP) =

E S((ap)(_1))(ap)( 13) = E S(a(i)P(-1))a(2)P(o) E s(m _ios(a ( oa(2) p(0) = E(a)PL(P).

The relations (4) and (5) follow in a similar way. The second equality in (5) is just the definition of adR(a). If p E inv r, then ,AL(p) = 1 p and hence PL(P) = S(1)P = p, that is, PL is a projection of r ontO P . If P is another projection P : F —■ 1 ,F satisfying (3), then by (4) and (3), P(p)

= E E(p(,)P(PL(p(0))) = E PL(E(p(_1))p(0)) =

PL(P),

which proves the uniqueness assertion. We now begin to describe the structure of left covariant bimodules. Let ro be an arbitrary right A-module, that is, there exists a bilinear map ro X A 3 (w, a) —)w.lac Po such that w a (ab) = (ci.) a a) a b and w a 1 = w for a,b E A and w E ro . It is easy to check that the equations -

a(b w)c = E ahem 0 w ZAL(b

cd) =

c(2))

(6)

cd

( 7)

b(i) ® 1) (2)

define A-bimodule and left A-comodule structures on the vector space rid Aoro such that L becomes a left-covariant bimodule over A. As an example, by computing we verify the right A-module property of (a

cd)bc =

Ea(bc) (i) 0 (4) .1 ((bc)(2)) = E ab(i)c(i) 0 (w< b(2))11 c(2)

= 2 (ab(i)

(.4.)
13.1 Covariant Bimodules

475

r'0 .

Obviously, we have inv (fL ) = I 0 Moreover, from the formulas (6) and (3) we obtain POI 0 w)b) = EE(b(0 )(1 w i b(2) ) = 1 ® w i b, that is, the right A-module structure on ro can be recovered from the left-covariant bimodule Conversely, suppose that is an arbitrary left-covariant bimodule over A. We proceed in the opposite direction and show that up to isomorphism r is of the form just described. For a E A and w E i nv r we put w a a := PL(a) = adR (a)w,

( 8)

where the second equality holds by (5). By Proposition 1.14(i) and Remark 2 after it, this defines a right A-module structure on invf. Let FL := AO inv r denote the left-covariant bimodule over A which is given by (6) and (7) with respect to this right A-module action. We assert that the linear mapping := (id013.00 1, : r 17,, is an isomorphism of the left-covariant bimodules F and FL. To prove this, we let a, b E A and w E inv r and compute

E a( 1 )b( 1 ) PL (a(2)urb(2)) E amb( 1 )06(a(2))PL(wb( 2)) = E ab( 1 )0wab(2) = (a0w)b,

V((aw)b) = (id 0 13L,)z(a),A04))(b) =

(9)

where we used (1), (1.20) and (6). Recall that I' = Ainvr by (4). Therefore, applying (9) with b =- 1, we see that 19 is a bijective linear mapping of and FL . It is obvious that O preserves the left A-module structures and that d L o19 = (id00)0d L . By (9), t9 preserves also the right A-module structures of F and FL. That is, 79 provides an isomorphism of the left-covariant bimodules and 11. For notational simplicity we shall identify r and FL via V, that is, we set aw = a 0 w. Summarizing, the preceding considerations have shown that there is a one-to-one correspondence, given by the formulas (6)-(8), between leftcovariant bimodules r and right A-module structures on the vector space inv r. More precisely, we have proved that r is isomorphic to the left-covariant bimodule A0111vr with structures defined by the formulas (6)-(7). Now we use the assumption that the antipode S is invertible. For a E A and [4) E inv r, we then have aw = E(U., 4 S -1 (a(2)))a(1)• Indeed, setting b = 5-1 (a) and using (6), we obtain

=

a( 1 ) (Go a S -1 (a(3))) a(2))

a(ow (5 -1 (a(3))a(2))

= E so(3) ),,,

s (b p) ))

E s(b(,) ). a (e(b(0)1) S(b)co = aw.

(10)

476

13. Hopf Bimodules and Exterior Algebras

In particular, (10) implies that r = inv r A. Similar considerations as above can be done with A0 i nv f replaced by inv f0A. For any right A-module ro with right action (w, a) w a a, one can verify that the vector space rR := 10 ®A becomes a left-covariant bimodule by defining (11) a(LI) Ø b)c := Ew ci S-1 (a( 2)) Ø ambc, 4AL (44.)

b) :=

b(,)

b(2) .

(12)

inv .r0A Conversely, let r be a left-covariant bimodule over A. Let rR be the left-covariant bimodule with structures given by (11) and (12), where w a a := PL (wa), w E inv Then the mapping 0 : r TR defined by .!".

e(p)

E PL(p o o ci

(13)

provides an isomorphism of the left-covariant bimodules T and FR . We : FR r by only prove that 9 is bijective. Define a linear mapping 0-1 (w0a) = wa. Then 0-1 is indeed the inverse of 0, since by (13), (6), (4), and (8) we have

=

9(91(w

E(PL(p(0))
E P(-3)(PL(P(0)) S 1 (P(-1)) ci P(-2)) Ep( _,PL(p(0) ) =-- p, b)) = 0(wb) = E pi, (wb (3) ) ci s-1(b(2) ) b( , ) EP b(3)) S-1 (b( 2)) b( i ) = w 0 b.

We close this subsection by restating some of the above results in coordinate form. Let I be an index set. We call matrices (74)ijEl and (n)0 E / of elements v ji c A and functionals f c A' pointwise finite if for any i c I, resp. j E I and a E A, all but finitely many terms v resp. (a) vanish. This ensures that all sums over the index set T occurring below are in fact finite sums. Recall the notation f.a = amf(a( 2)) for a E A and f E A'.

r

Proposition 2. Let be a left-covariant bimodule over A and let tw i l iE i be a basis of the vector space inv f . Then the set fw i L E I is a free left A-module basis of and there exists a pointurise finite matrix (n)id E i consisting of , functionals fJ E A' such that for any a,b E A and i,j E

r

fl(ab) =-- f 4(a) f

, fl(1) =

,

wia = (fit.a)wk.

The set fwi

iE I

(14) (15)

s also a free right A-module basis of f and we have awi = waft, o S -1 ).a) .

(16)

13.1 Covariant Bimodules

477

Proof. Since wi a a E in ,F, there exist complex numbers n(a), i, i E .1 1 such that all but finitely many f .1(a) vanish and wi a a = fl(a)coj .

(17)

Clearly, (n)i,jE/ is a pointwise finite matrix with entries fl E A'. The right A-module property of inv r implies (14). Equations (6) and (10) yield (15) and (16). Since : F inv F0 A are bijective mappings, A Ø invr and : F {coi } iE / is a free left A-module basis and a free right A-module basis of F. E 13.1.2 Right-Covariant Bimodules Definition 2. A right-covariant bimodule over A is an A-bimodule F equipped with a right coaction : F —> F 0 A of A on F such that

46R(apb) = 46(a)46R (46(b)

for any

a,b E A and

p E F.

For a right-covariant bimodule F, the elements of the vector space rinv

:= {p

E

F I AMP) = p ® 1}

are called right-invariant. For right-covariant bimodules a similar theory as in Subsect. 13.1.1 can be developed. We will not carry out the details and mention only some necessary modifications. The canonical projection PR : F —> Ti ny is defined by PR(P) = E P(o)S(P(i)) and satisfies

PR(Pa) = 6 (a)PR(P), PR(aP) =

E amPR(p)8(a(2)) adL(a)(PR(P))

for a E A and p E F. The left A-module structure on Fi nv defined by the PR(an) = adL(a)( 7)), determines left adjoint action of A on Tiny , a completely the structure of the right-covariant bimodule I', since F = Finv - A and a(0) = (am 1))a(2)b for a, b E A and n E Finv • 13.1.3 Bicovariant Bimodules (Hopf Bimodules) The main notion of this chapter is introduced by the following Definition 3. A bicovariant bimodule (or a Hopf bimodule) over A is an A-bimodule F together with linear mappings AL : F A® F and 2iR : F F A such that: (I) F is a left-covariant bimodule over A with left coaction LL, (ii) F is a right- covariant. bimodule over A with right coaction dR, (iii) (id o dR) o A L = (A L 0 id) o AR . Condition (iii) means that the left coaction AL and the right coaction AR of A on F commute. This condition justifies the use of the Sweedler notation p(- 1 ) 0 p(0) O p( i) for (id AR) o &(p) = id) o A R (p), p E F.

E

478

13. Hopf Bimodules and Exterior Algebras

The subsequent considerations will determine the structure of bicovariant bimodules. Let ro be a vector space carrying the structures of a right Amodule, with right action denoted by a, and of a right comodule, with right To 0 A, such that the compatibility condition coaction R : To

W(o) < a( i ) ® w( i )a(2 ) = E(w .ia (2) ) (0) 0 a( i )(w a( 2 ))( i )

(18)

is fulfilled for w E To and a E A. Equation (18) is called the Yetter Drinfeld condition. A vector space with these properties is usually called a YetterDrinfeld module. As discussed in Subsect. 13.1.1, the right A-module action on To induces the structure of a left-covariant bimodule over A on r := A 0 To given by (6) and (7). We now define a mapping AR : 0 A by setting -

AR(a w) =

E a(i) w(0) 0 a(2)w(i)

for a e A

and

wE

ro .

(19)

Then r becomes a right A-comodule. It is obvious that AR (a p) = A(a) A R (p) and (id 0 A R ) o A L (p) = (A L 0 id) o A R (p) for p E r and a E A. Thus r is a bicovariant bimodule over A provided we have verified the condition AR(pa) = LX(p)L(a). In fact, the latter is equivalent to the Yetter-Drinfeld condition (18). By (6) and (19), we have

2iR((b O w)a)

AR(b 0 w)A(a)

E AR(ba o , ®w < a( 2 )) E b( i )a( i ) (c.4) a(3))(o) b(1)

w(0))a(1)

b(2)a(2) (co < a(3))(i) (20)

b(2)co( 1 )a(2)

E b(i) a (1) O w( 0 ) a(2) O b(2)w(i)a(2)

(21)

for a, b E A and w E invr. By (18), the right hand sides of (20) and (21) coincide. Therefore, r is a bicovariant bimodule over A. Conversely, let r be a bicovariant bimodule over A. Recall from Subsect. 13.1.1 that w a a := PL(wa) defines a right A-module structure on inv r. By condition (Hi) in Definition 3, we have (2q, 0 id) o 2■ R(w) = (id o AR) o AL (w ) = 1 0 AR(W) for w E inv r, so that Zi R ( inv T) C ,r o A. Let 6R denote the restriction of —R A t-0 inv r • As in Subsect. 13.1.1 we identify r and A 0 invr by setting aw = a 0 w for a E A and w E inv r. Since r is a bicovariant bimodule, the left-hand sides of (20) and (21) coincide. Applying e 0 id to the expressions of the right-hand sides of (20) and (21), we obtain the Yetter-Drinfeld condition (18). Thus, we have proved that there is a one-to-one correspondence, Oven by the formulas (6)-(8) and (19), between bicovariant bimodules F over A and pairs of a right A-module and a right A-comodule structures on the vector space invr satisfying the Yetter-Drinfeld condition (18).

13.1 Covariant Bimodules

479

Next we show that the linear mapping 45: f f defined by 4P(P)

=

s(p(_op(o)s(p(o)

(22)

E 8 -1 (p(Op(o)s

l (P(-1)), we compute

is bijective. Indeed, putting W(p) =

E 0(A-1 ) 00(AI:0 00(AI) E 8(P(2))0S(P(-0P(o)S(P(1))0S(P(-2))) -

SO

CO(P)) =

E s- 1(0(p)(1))0(p)(0)s --- 1(0(p)(_1)) (o E 8 -1 (S(p(_ 2)))(S(p(_1)) P ) S (P(1)))S -1 (S (P(2))) E p(-2)869(_1))p (o)8(p(1))p(2) = P.

Similarly, 0(W(p)) = p for p E T. Hence 0 is bijective with inverse W. w (o) s(w (1) ) = PR (w) and For w E inv .1" and i Ti ny , we have 0(w) = so 0 : in ,1" fi n, and W /I nv —r inv . W( 71) = E 7(o)s -1 (71(_1)) E inv That is, 0 is an isomorphism of the vector spaces inv and We now summarize and reformulate the structure theory of bicovariant bimodules developed above in the coordinate version by the following

Theorem 3 (Fundamental theorem for bicovariant bimodules). Suppose that I" is a bicovariant bimodule over A and let fw i l iE i be a basis of the vector space in„F of left-invariant forms from F. Then there exist pointwise finite matrices y (vii ), jE I and f =Cfp i,jE I of elements vij E A and functionals fI G A' such that for a,b E A and i, j E I we have

(i) w i a = (fk: .a)wk, (ii) ZiR(i) =

(iii) f = (n) is an algebra representation of A (that is, fl(ab) = f it(a) f (b) and fl(1) = (iv)y = (vi) is a matrix corepresentation of A (that is, d(vii ) = 14' 0 v./ and E(v) = (y) the following equality holds: /

3

=---

Ek

(f4.a)vik.

(23)

Moreover, Ini := wi s(v-)} iE/ is a basis of the vector space rim, of rightinvariant forms from f. Both sets {w i } iE i and I i u are free left A-module bases and free right A-module bases off. Conversely, if fwibEI is a basis of a vector space fo and if v = and f =(n)i,jEi are pointwise finite matrices with vii E A and fl E A' satisfying (iii) (v), then there exists a unique bicovariant bimodule F, denoted by the symbol (y, f), such that r0 = Fin, and (i) and (II) hold. -

480

13. Hopf Bimodules and Exterior Algebras

Proof. Clearly, there exists a one-to-one correspondence, given by (17), between right A-module structures on inv .r and pointwise finite matrices f = (fl) i ,jE l satisfying (iii). As noted in the proof of Proposition 2, condition (i) follows from (6). Moreover, there is a one-to-one correspondence, given by (ii), between right comodule structures on 1F and pointwise finite matrices y = (0.) .] satisfying (iv). From (22) we obtain Th = 41g(wi). To cornplete the proof it suffices to check that the Yetter-Drinfeld condition (18) is equivalent to the compatibility condition (23). For j E I and a G A, we compute

Epo (o )
a(i)(wz a(2))(1)

=

E E

Ø v.t (a. fi'),

® (ftc .a)vik .

Hence (18) and (23) are equivalent. The next corollary shows that in the case of the quantum matrix groups the compatibility condition (23) reduces to an intertwining property.

Corollary 4. Let A be the coordinate Hopf algebra 0(Gq ) for one of the quantum groups Gq = GLq (N),SL q (N),0 q (N),Sp q (N) and let u = (u ) denote the fundamental corepresentation of A. Suppose that vii G A and E A', j, j E I, are given such that the matrices y == (vi) and f =--- (f1) satisfy the conditions (iii) and (iv) of Theorem 3. Set AV f(u) for E I and rn,n =1, 2, • • •, N. Then the matrices v and f fulfill equation (23) (and hence they define a bicovariant bimodule (7), f) over A by Theorem (Aimni ) belongs to Mor (1) u, u y). 3) if and only if A

Proof. Using (iii) it is easily seen that (23) is true for ab if it is valid for a and b. Since n(1) = bii, condition (23) holds for a = 1. Thus it suffices to check it for the generators 74 of A and in addition for the inverse Dq-1 of the quantum determinant Dg in the case Gq = GLq (N). From the equations vi

= Akrini Vi k L sr7 1 and

(.fic .u.Dy /.1,

um r y /.1,2471k

we see that (23) holds for all ur8 if and only if A E Mor (y u, u 0 y). If Gq = GLq (N), then, as we have just shown, the relation A E Mor (you, u0y) implies in particular that (23) holds for a = Dg . Multiplying equation (23) for a = Dg by f,.(D q-1 )fis(Dq-1 )D,-2 , summing over j and j and using (iii) we obtain condition (23) for a = Dq-1 . 13.1.4 Woronowice Braiding of Bicovariant Bimodules

One of the most interesting features of bicovariant bimodules is that they possess a natural braiding of tensor products. The definition and properties of this braiding are developed in this subsection.

13.1 Covariant Bimodules

481

Let /11, • • •, rn be bicovariant bimodules over A. It is easy to check that the tensor product ri OA • • • OA rn of A-bimodules is also a bicovariant bimodule f over A with bimodule structure, left coaction ,6/, and right coaction L j defined by

a(P 1 OA

•••

4/.(P 1 OA

4iR(p l- OA

= (aP 1 ) OA

OA Pn )b

•••

OA

Pn )

=

OA

Pn ) =

•••

OA (pub),

E Pt- ir • P7(1-1) 0 (Po) ®A • • • 0A P T(lo)) E (Po) OA • • • OA 4)) 0 P(1) . • Pr(1)

for E fi, • • •,Pn E rn and a, b E A. Clearly, invr = invri ' • OA invrnThis bicovariant bimodule F is called the tensor product of the bicovariant birnodules f1 , • • •, inn and is denoted by /11 OA • • • OA fn . In the case f1 = • • - = fn we simply write (with a slight abuse of notation) (fi )® n for f. By a homornorphisrn of bicovariant bimodules fi and 112 over A we mean an A-bimodule homomorphism T : f1 > 112 which intertwines the left and the right coactions (that is, T(apb) = aT(p)b for a, b E A, p E F; (idOT) = LL 0 T and (T 0 id) 0 LR = 4iFt 0 T). If T is also bijective, it is called an isomorphism of / 1 and r2. —

Definition 4. The linear mapping o

a(P1 OA

P2 )

: f 1 OA 112 —›

r2 OA

ri

given by

:= E pt,PR(4 ) ) ® A PL,(pto ) )pi,

= E pt_2) ,90) s(pi) )

(24)

® A s(Pt--1) )Pto) 2) P

for pl E .rfi and p2 E 112 is called the braiding of the bicovariant birnodules Fi and f2. This terminology stems from the facts that the bicovariant bimodules over A form a braided tensor category (see Subsect. 10.3.4 for this notion) and a is the braiding in this category. We shall not carry out the details of the proof, because we will not need this in the following. One verifies that the linear mapping a-1 : 112 A ri OA -r2 given by 0.-1

(P2 OA

Pl )

:=

E ,q2,40) s- 1(pt_1 ) ) ® A s-4(41) )4,pt_2)

is the inverse of a. Thus, in particular, a is bijective. We give another form of a and o--1 which is more convenient for applications. For a E A, w E 1 f 1 and w2 E invr2 ) we have

akaw 1 OA (2) =

f 1 2 aw(2o) „0, .'6'A k w < W( 1 )),

(au,2 ®A w 1) = E a(w '

i,„ s_1(41)))

®A

(25)

4)

where a is the right adjoint action of A on invrkl k = 1 1 2 1 (see (8)). We verify the formula (25) and set p1 = aw1 and p2 = w2 in (24). Then we have

13. Hopf Bimodules and Exterior Algebras

482

pt.)

=

E

a(2)®a(3)w',

Epi( _2)

Pi )

®A P2 )

--= Ea ( 1)40) 51(41 ) )

so that 0-(pi

OA S(a(2))a(3)w 1 42)

Ea ( i) 40) OA S (41))6(a(2))W 1 42) E a40) OA (W 1 < 4)Similar reasoning shows that for a E A, 77 1 E (roi n, and 7/2 E (1'2)1nvl 2

C( 7/ 1 OA 77 a)

11 2

»4-1)

(26)

77 ) OA NCO)

where bi> 772 := PR (bn2 ) = ad/J (0(i 2 ) for b E A (see Subsect. 13.1.2).

Proposition 5. The mapping a defined by (24) is an isomorphism of the bicovariant bimodules 111 0A 112 and 112 OA ri over A. It is the unique linear map o- : 1-1 OA 112 —0 r2 OA F1 such that a(ace OA n)

an OA W for a E A, w E invfl and

(27)

E ( 1'2)inv

The map a satisfies the braid relation on i OA 112 OA 113 that is,

(o- id)(id c)(o- id) = (id 0 c)(cr

(28)

id)(id 0 a) .

Proof From Subsect. 13.1.3 it is clear that 1 OA 112 is linearly spanned by each of the sets -taw 0A 77 21 , { aw l 0A w2} and tri l ® A 772 b1 with a, b G A, wk E in v f k and Tik E (fk)inv for k = 1, 2. Hence a is uniquely determined by (27) and it suffices to verify all required relations on f1 OA r2 for such elements. We conclude from (25) that (id0a)oZ1L (p) = ZA L oa (p) and a (a = aa (p) for a E A and p E OA 1 12 • Formula (26) in turn implies that (0' 0 id) o R(P) = Z1 R a(P) and a(pa) = o(p)a. We have already noted that a is bijective. Thus, o- is an isomorphism of bicovariant bimodules. For w E 4,1-' 1 and 7/ E (r2)inv, we have E o.;(_ 2) ®W( - i) w(0) = 1 01 (.4) and (0 ® 77(1) ® 77(2) = 1 ® 1 0 1. Inserting this into (24) we obtain (27). OA ® A 7/3 with w 1 inv- 11 wSince the elements p 2 - inv F2 and 173 E (f3) 1 nv generate ri OA T2 OA 1 13 as a left A-module, it suffices to prove the equality (28) for such elements p. Using (25) and (27), we get

En )

-

(0"

id)(id

o-)(o-

id)(p) = E(a

id)(id (7) (4) ®A W 141 41) OA n3 )

E(aOid)(4 ) 0ATI30Auil <40)

E (id 0

c)(0-

id)(id

cf)(P) =

T130A4)

E(id

®A (.4)1

41)

a)(0- 0 id) Ge l OA 773 OA W 2 )

= Ls (id 0 y)(î

(.01 OA co 2 )

= En' ®A c4)0) ®A W 1

449

13.1 Covariant Bimodules

483

which proves the braid relation (28). We close this subsection by giving the coordinate versions for some of the above notions and results. Let {6.0 iE / and {co}i E j be bases of the vector spaces 1 r 1 and invr2, respectively. In the notation of Theorem 3, we write r, = (y, f) and .F2 --= (w, g) and set Id = Loks(v) and ird= 45 1(4). Then each of the four sets OA rin is a free left A-module basis {oil OA oi},{W OA 74} , fig OA (4} , and also a free right A-module basis of ri ®A r2. The bicovariant bimodule OA F2 can be written as (y Ow, f 0g), where y w is the corepresentation of A with matrix elements (y w)'. := vkw4 and f 0 g is the algebra Having these representation of A with matrix elements (f g g) ikin := data, the formulas (i ) and (ii) in Theorem 3 provide an explicit description of the bicovariant bimodule Fi OA r2 in terms of the basis { cul OA 4} of the vector space inv (r, OA F2 ) . Since Zi R (c.4) = 64,2 0 w7, it follows at once from (25) and (17) that the braiding map a acts on elements of this basis by 444

0A

2 0rinikwn2

®A

wic with OTik := f lic (1117) .

0A cup = A R o a (wl belongs to the space Mor (7) ®w,w y), that is,

The equality (a 0 id) o nk rs

Wsn yrk o_isir l

(29)

A 64) implies that a

k E / 7 i7n

By (17) we have the identity col OA cd4a (fri .g.i9.a)Wri OA w.92 • Applying a to both sides, we get 0-77:1 gin. II,

=___ fg a rn

i, k EI ,

j 7 n, E J.

(31)

Let 5- : r2 OA OA r2 denote the braiding of F2 and F1. Inserting the expression AL(q) = A/(.44.2 )2(8(w)) = S(W) irb2, into formula (26) applied to 5- (î4 OA 111) and using first (16) and then (29) and (30) we derive 6-

(74 0A

nk

ilk

(g_A n

E /7 j E

13.1.5 Bicovariant Bimodules and Representations of the Quantum Double

This subsection relates the Yetter—Drinfeld modules to representations of the quantum double. Combined with the one-to-one correspondence between Yetter—Drinfeld modules and bicovariant bimodules shown in Subsect. 13.1.3, this characterizes bicovariant bimodules by representations of the quantum double. Let U be another Hopf algebra and let (., .) be a dual pairing of the Hopf algebras U and A. As noted in Subsect. 8.2.1, (-, -) is an invertible skewpairing of U and A°P. Hence the corresponding quantum double D(14, A°P) is a well-defined Hopf algebra (see Proposition 8.8). Recall that as an algebra,

484

13. Hopf Bimodules and Exterior Algebras

D(U,A°P) is generated by the algebras /I and A°P with cross relations (see (8.23)) (f (1),

a(1))f(2)a(2) = Ea ( i ) fm (f(2) ,a (2) ), f EU, a E A°P.

(33)

By Proposition 1.15, for any corepresentation cp of A, the equation ('(f) = (id 0 f) 0 cp, f E ti, defines a representation ci) of U. Representations of Li of this form are called rational. Now let ro be a right A-module with action a and a right A-comodule with coaction 6. Then the equations Aop (a)(4) :=-- ‘4.) a a and 71-14 f)

:= (f)b) =

define representations 7rAop and 7ru of the algebras A°P and Li, respectively. By construction, 7rzi is rational. Proposition 6. (i ) If 110 is a Yetter-Drinfeld module of A, then 71-Aop and ru are restrictions of a representation of the algebra D(11, A°P) on fo (ii) Suppose that U separates the elements of A. If 7r is a representation of D(11, A°P) on a vector space To such that the restriction 7rrti of 7r to II is rational, then there is a unique Yetter-Drinfeld module structure on To such that 7r rAop = rAop and 7rfli =-- 7ru • .

Proof. Let it be a representation of D(,/, A°P) on To such that it RI is rational, say 7r = for a right coaction 6 of A on T0. Let LA) a a := 7r(a)w, a E A. Applying it for the element in (33) to b.) E To we get

E(f(i), a(1))(w
a(i)(f(2) a(2))

(34)

or equivalently

Eodon((w.,,a (2) ) (0) 0a( i) (w< a(2))(1)) = Eodon(w(0)‹ (35) HU separates the points of A, then equation (35) implies the Yetter-Drinfeld condition (18) and 6 is uniquely determined by 7r ru. This proves (ii). Conversely, if fo is a Yetter-Drinfeld module, then (35) and hence (34) hold. Therefore, it follows that 7r(f a) := 71-u (f)7rAop (a) gives a well-defined(!) D representation it of the algebra D(11 , RP) on Fo . For a representation it of the quantum double D(U,A°P) as in (ii), the associated bicovariant bimodule F can be explicitly given as follows. Let {wi } be a basis of the vector space Fo. There exist pointwise finite matrices f =-- ( ffl and (Ff. ) of functionals f E A' and Fil E Li' such that 7r(a)(4.4 f E U. Since rrii is rational, there E A,, and 7r(f)w =-E U. Then, F = (v, f) is are elements v E A such that Fil (f) = the corresponding bicovariant bhnodule by Theorem 3, where y = (714).

13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules

485

13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules From Examples 1.8 and 1.13 we know that the tensor algebra and the exterior algebra of a vector space are a Hopf algebra and an No-graded super Hopf algebra, respectively. In this section we will show that these results and the corresponding formulas obtained in both examples have generalizations to the tensor algebra and the exterior algebra of a bicovariant bimodule. If not stated otherwise, r denotes a bicovariant bimodule over A and a is the braiding of F®2 = r OA F. 13.2.1 The Tensor Algebra of a Bicovariant Bimodule

First let A be an arbitrary algebra and r a bimodule for A. We set r®° A, F®1 := r and lion r OA • • • OA r (n times). Then the direct sum of vector spaces 00

TA (r)

r®n

n=0 becomes an algebra with multiplication defined by zn yk := x n, OA yk for In C F® and yk E rgk . Here we identify zo OA yo and xo yo for so , yo E F®0 = A. The latter means that the multiplication of zeroth components in TA (r) is just the multiplication in A. The algebra TA (r) is called the tensor algebra of F. Just as for the ordinary tensor algebra of a vector space, the tensor algebra TA (r) has the following important universal property: if 0 1 is a linear mapping of A ED r into an algebra B for which 0 1 IA is an algebra homomorphism and 0 1 (apb) =01 (a)01 (p)0 1 (b) for any a,b C A and p E r, B such that then there exists a unique algebra homomorphism i : TA (r) r is a bimodule 0 1 is the restriction of 0 to A(Dr. In particular, if 0 1 : homomorphism of the A-bimodule r into another A-bimodule F , then there is a unique algebra homomorphism i : TA (r) TA(F) such that 0(a) = a and 0(p) = 01 (p) for a E A and p E F. From now on we suppose that r is a bicovariant bimodule over the Hopf algebra A. Then all F®n are bicovariant bimodules over A. Obviously, F®O = A is a bicovariant bimodule over A with 2iR = 211, := L. Therefore, TA (F) is also a bicovariant bimodule over A with bimodule operations and right and left coactions defined componentwise. Proposition 7. The N o -graded algebra TA(f) becomes a Hopf algebra, denoted again by TA (r), and an No-graded super Hopf algebra, denoted by TA(r), with structure maps 2iT en ST and 464,, 4, s.;,, respectively, such that 46T(a) = 24(a) = 44(4

LT(p) = ,6 1.(P) = ,6 R(p)

+ À6L(p),

ET(a) = 4(a) =-- eA(a), CT(p) = 4(p) =0,

(36)

(37)

486

13. Hopf Bimodules and Exterior Algebras

E SA(P(-1))P(o)SA(P(1))

ST(P) = 51.(P) = for a E

(38)

= A and p E

. By the universal property of TA(r), Proof. We prove the assertion for TA(F). the linear mappings Ali,4 SI, of AF into the algebras TA(F)OTA(F),C and TA(FrP, respectively, defined by (36)-(38), extend uniquely to algebra homomorphisms of TA(F). Here it is understood that TA(f) 0 T(f) carries the No-graded product • given by formula (1.42). By Proposition 1.8 (more precisely, by its No-graded version), it suffices to check that the Hopf algebra axioms are satisfied for the generating set A U F. We omit these straightfor111 ward verifications. ,

In order to generalize the formulas (1,37), (1.38), (1.43) and (1.44) to the present situation, some further notation and preliminaries are necessary. First let us recall Artin's braid groups q3„ corresponding to the Lie algebra sl(n,C) (see Subsect. 6.2.1). Fix n E N, n > 2. Then 23 n, is the group with n - 1 generators s l , s2 , • • • , sn_ i and defining relations

sksk-Fisk = sk-Fisksk+i, S •S si si • =

k = 1, • • • n - 2,

S •S' si si ,

,

2.

The group On plays an important role in knot theory (see, for instance, [Ka], Chap. X). It is the group of equivalence classes of braids on n strands with composition of braids as group multiplication. For k =1, 2, • • • , n - 1, we define an automorphism ak of the bicovariant bimodule Fn by crk = id®A• • •®Ao -OA- • -0Aid, where the braiding a from Proposition 5 stands in the place (k,k+1). Since akak Fiak = cfk+1akak+1 by (28), al, 0. 2, • • •, an-1 satisfy the defining relations of the braid group On . SO there exists a unique group homomorphism r(s) of g3n into the group of automorphisms of the bicovariant bimodule F®n such that r(e) = id and r(sk) = 7k, k = 1,2,•• - n - 1. We give a more explicit description of the homomorphism r. Let tk denote the permutation from the symmetric group 7),2 which interchanges k and k +1 and leaves all other numbers fixed. Each permutation p E Pn can be written as et product p = t ic • •44,) , where 1(p) denotes the length of the permutation p. Then we set r(p) := air ,: • •cri i(p) . It can be shown that the latter is independent of the particular representation p =t il • • •tii(p) of p. Moreover, we have -

7r(PiP2)

= 7 030702) if /(PiP2) = 1030+ l(p 2 ) and P17P2 E Pn

(39)

For n E N and k = 0,1, • • •n we define Bk :=

E

(±1) 1(P) r(p)

(40)

p- lEP,Lk where Pn k is the set of (k, n-k)-shuffles, that is, the set of permutations p E Pn, such that p(1) < • • • < p(k) and p(k + 1) < - • • < p(n).

13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules

487

Let fc.di l ier be a fixed basis of the vector space in„F. Let rh and v.:, j j E ,

be as in Theorem 3. Recall that 2iR(w) = wi v.1 and ph := wi s(v)l iE , is a vector space basis of rinv. For i = (j 11. • •, i) E In we abbreviate the In a similar way we use the product wi , • • •win in the algebra TA (F) by multi-index notation ni and i,j E . Recall from Subsect. 13.1.4 that fc.di liE in is a basis of the vector space inv (F®n ) and that the braiding a leaves iriv (T °2 ) invariant. Thus any operator 7r(p), p E Pn , leaves ii,v (F®n ) invariant. Hence for each B E Lin tir(p) I p E Pn l there exists a pointwise finite matrix (131)i,jEin of complex entries such that B(wi ) = Blwi . For i = (j 17 . • •,ik) E / k and j = (ill • 1jin) E i m , we put T := (ik, - • • 3 il) and (i7i) (ill • ") i, • • •):771Z). Now we are able to give generalizations of the formulas in Examples 1.8 and 1.13. Proposition 8.

For n E N and E I , we have

2iT(Wi) = E (B± n k )("") L4), 0 k=0

(-1) n rir,

(41)

54,(coi ) = (-1) n-2' 1 )71r .

(42)

L ,

ST (w)

ri 46

vT,w j „ ,

4-, Pi) = k=0

Note that there are multi-index summations over j', r E 1k and j" E in - k in (41) and (42). For k = 0 and k = n the summands should be interpreted as 1 0 wi and wr respectively. Proof. We carry out the proof of (42). In order to prove the formula for 24,, we proceed by induction on n. For n = 1 this equation is obviously true. Assume . Put = (i1, i n ). that it holds for all i E In . Let i = (j 1,• •.7 i+1) E n By (42) and Theorem 3(i), 2c4,(win+1 ) = 1 0 win+1 + wm, vrn+ , . Therefore, by the induction hypothesis, equation (42) and the definition (1.42) of the No -graded product., we obtain

i4(wi)

=z

(w)

• 2i1, (win.+1) yi

E(Bri --k)P''''')w r

Carn ®Vimn+i

"W.

E E(-1)/(P)7(ki',in)

(u)

rOvLai,„ win+

lr —kuiruirk+i OVir,C.dittv,rk+i

k=0

(43) where the second sum is over all permutations p E Pn such that p-1 E 'Pnn.• The first summands of the interior sum in (43) correspond to summands (-1) 1 ( 3)7(3) of Bn—±iik withi3-1 _ Pn +1,k, where (i) = p(i) for i = 1, • • • ,n and 23(n + 1) = n + 1. In order to treat the second summands of the interior sum in (43), we observe that wv = v;f4(vnc.4.4 n = v,rolpwin by (29) and hence c.drvirk +' = k+ 18 "-1 crin - 'F28 k 4' • • • cr":". s n' c.d.., with - 3k+inik+2 3k4-2 111 k+3 3n2n+1

13. Hopf Bimodules and Exterior Algebras

488

j " = (ik-1-11• •

',in) and

s"

= (sk + i, • • sa ). Putting

:= (j17 • "IjklMk+i)

and = (Ti, • • rk, rk+1), it follows that the second summands are equal gi s it, to (-1) 1(P)+k (cork+i• • vi.t w if Let /3 pip' G Pn+ i, where \- ' )1 5 • 23 1 := tk+i• • •tn and the restriction to {1, 2, • - n} of p' c Pn+ 1 is p. It is easy to check that 13-1(i) = p--1( i ) for i = 1,2,• • • , k, 23 -1 (k +1) = n + 1 and 1 -1 + 1 ) , 23-1 (i) for i = k + 1, • • , n. Hence 23' is a shuffle permutation 5 (i of Pn+1,k+1 and we have 1 3) = n - k + l(p) = 4p1 ) + 101). Therefore, by (42), (-1) 1( P )+n-k ork+1. • •Grn 7r(P) = ( - 1) 1(73) 7(j3) and the latter is a summand of B:. The preceding shows that there is a one-to-one correspondence (

between summands of (43) and summands of (42) for E in+ 1 . By induction, this completes the proof of the first formula of (42). Formula (38) yields S(w) = - Th for i E I. Since the antipode of the Nograded super Hopf algebra TAM satisfies SI,(xky rn ) = ( -1 ) km514-(Ym)Sli(xk) for xft E ./1 ®k and y, E ./1®m, the second formula of (42) follows. 0

13.2.2 The Exterior Algebra of a Bicovariant Bimodule In this subsection we study quantum analogs of the antisymmetrizer and the exterior algebra for bicovariant bimodules. The basic idea is to replace the flip operator in the corresponding classical constructions by the braiding map

cr • For n E N and k = 0,1, •,n, we define

An := E (-1)/(07(p)

and

Ank := E

pEP

(-1)1(P)7(p).

pEP„k

The map A n is called the quantum antisymmetrizer of f®n. Let k E {1,2, • •, n — 1 } . Each permutation p E Pn admits unique representations p = rpip2 and P P3P4s, where r,s E Pnk and Pi, P3 C Pn (resp. p2, P4 E Pn ) leave the numbers k+1, • - • , n (resp. 1, • • -, k) fixed. Then we have l(p) -= l(r) + l(pi) l(p2) --= - l(p3) + 1(p4) + l(s) and hence (r) 7 (P1) 7 (P2) = 7r(P3)7(p4)7r(s) by (39). From these facts we con1T(P) = clude that Ari = A n k(Ak 0

An-k) (Ak

A n - k) -137;:k •

(44)

We shall use (44) also for k = 0 and k = n, where it is interpreted as An = AnoAn = An Bno and An = Ann An = A n BT. respectively. Let 6 := n = 2 en , where en := ker A. The elements of en can be considered as quantum analogs of symmetric tensors of degree n. From the first equality in (44) we see that 8 is a two-sided ideal of the tensor algebra

TA (r). The quotient algebra re^ := T.,(r)/e Tit (T)/e is called the exterior algebra of the bicovariant bimodule r over A. Clearly, r is again a graded algebra, that is, '

13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules

reA =

co

FAn,

FAO := A,

with

489

F, FArz := rgnierz , n > 2.

n=o The quantum antisymmetrizer A n is a homomorphism of the bicovariant bimodule F®n to itself as a is of F° 2 by Proposition 5. Therefore, 6 is a bicovariant subbimodule of TACT). Hence, the quotient F = TA(T)/6 is also a bicovariant bimodule over A. Taking Proposition 10 stated below for granted, 6 is a Hopf ideal of the No-graded super Hopf algebra T.:49t (f) defined in Proposition 7. Therefore, the quotient TeA = 71:4 9 (nie is also an N o -graded super Hopf algebra with structures induced from 71,34(F). Let (8 2 ) denote the two-sided ideal of the algebra TA(F) generated by 62 = ker (a - I). Since (6 2 ) is a bicovariant subbimodule of TA ( T') and a Hopf ideal of T(T) by the proof of Proposition 10 below, the quotient TA(F)/(6 2 ) is also a bicovariant bimodule over A and an No-graded super Hopf algebra. The following result is needed later in Subsect. 14.4.1.

I"

be a bicovariant bimodule over A and let F be a bicovariant subbimodule of ft (that is, F is an A-subbimodule of r and the left and right coactions of A on F are restrictions of the corresponding coactions on r). Then there exists an embedding F,A C Î , is the identity map on rAo = rAo =- A and the inclusion F c P on FA1 = F, such that 1-1A is an N o -graded super Hopf subalgebra and a bicovariant subbimodule of F.

Proposition 9. Let

Proof. By the universal property of the tensor algebra (see Subsect. 13.2.1), TA (1 )- such there exists an injective algebra homomorphism I : TA(F) that ga) = a and I(p) = p for a E A and p E F. We identify /(C) and for E TA(F), so TA(F) becomes a subalgebra of TACIT Since F is a bicovariant 2 and hence An, = A T, ['ion. (The tilde subbimodule of D , we have a = 6-[F° always refers to the corresponding objects for P.) Thus, 6 = 6 n TA (F) and the assertion follows immediately from the constructions of F,A and Proposition 10. 6 is a Hopf ideal of 774 (11), that is, 6 is a two-sided ideal of the algebra T(F) such that E (6) = {0 } , d1(6) g 60TACT)+TA(T)06 and 514,(e) C e.

Proof. Suppose that C c

= ker A. Since

{wi}iEin is a free left A-module basis of F® n, C can be written as a finite sum = aiwi with ai E A and AC = 0 implies that (An )lai = 0 for all j E in. Let ki denote the k-th summand of the sum for 21.44w1 ) in (42). Since a E Mor (y y) by (30), it follows that (Ak),v1; = v:,(Ak)1:. Using these facts and the second equality of (44) we obtain en

(Ak An_k)Zi(a i )Gi =

Zi(ai )(Bn-k ) i(il 'i") (Ak) , Ws 0 VI: (An-k)1::wr" A(ai )(0k An -OBn-k )fri ' r") w. 0 v:,wrn

= Zi (( An,)fri e ) ai ) we O v t' lwr = 0, n

490

13. Hopf Bimodules and Exterior Algebras

ker A n _ k C so Zgat gki E ker (Ak 0 A n _k) = ker A k 0 ./"° (n-k) f(8) k e k TA(F) TA (r) e n-k . Since Al,(() = L(a)L(w) EL 0 A(aigki by (42), this shows that Al-,(6) C 6 0 TA(T) + TA(f) 0 6. Let po c Pn denote the permutation with P0 (i) = n i 1. Since o- erbni ) = o-rins rh. by (32), we obtain that o-k(m) = (o-n _ k )174 for any i E and k = 1,2, • n - 1. Since o-k = 7r(tk) and an -k = 7r(tn-k) = 7r(PotkPo 1 ) by definition, it follows that (p) (T)) = ir(popp')r j for all p c Pn and hence A(i) = (An ) rti- for i E I. From the latter and (42) we get AS(() = An S(arwi ) = A n (S(w i )S(a1 )) = (-1)(" 42-1 ) A n (rtf)S( )

= ( - 1) ("2-1) T4S((An)la i ) =0, SO SI, (C) C ker An .=

en .

Thus, S.4,(6) C S. Since 6 is a two-sided ideal of TA(F), as noted above, we have proved that 6 is a Hopf ideal of TA (f ). D The construction of the exterior algebra and the above results have their 7r(p) and 2in := counterparts for the symmetric algebra over F. Let Sn := ker Sn . By similar reasoning as above, 2t IT177_2 2in is a Hopf ideal of TA(r). Hence the quotient TA(F)/2t is a Hopf algebra. This Hopf algebra TA(T)01 is called the symmetric algebra of the bicovariant bimodule F. In the case A --= C it is just the symmetric algebra of the vector space F. However, in the important case when f comes from a bicovariant differential calculus as in Sect. 14.6 we have 212 = ker (a + I) {0} if q is not a root of unity (see Subsect. 14.6.2), so the algebra TA(F)Mt is not of interest in this particular situation.

Ep

13.3 Notes The main reference for this chapter is [Wor4]. Yet ter -Drinfeld modules were introduced in [Yt] where they are called crossed bimodules. Their connections to bicovariant bimodules were first observed in [S2]. The relation to quantum double representations (in different forms) appeared in [PoW], [Maj4] and [Tk4]. Propositions 8 and 10 were found in [Dtl] and independently by one of the authors (K.S.).

14. Covariant Differential Calculus on Quantum Groups

This chapter contains the main concepts and results of the general theory of covariant differential calculi on quantum groups. The underlying Hopf algebra structure allows us to develop a rich theory of such calculi which is suggested by ideas from classical Lie theory. In the first two sections left-covariant and bicovariant first order calculi are studied. To each left-covariant FODC F we associate a right ideal 'TZT of ker E and a quantum tangent space Tri . If r is bicovariant, the quantum tangent space Tr carries an analog of the classical Lie bracket and is called the quantum Lie algebra of V. Sections 14.3 and 14.4 deal with higher order left-covariant and bicovariant calculi, respectively. One of the main results is that any bicovariant FODC admits a unique extension to a bicovariant DC on the exterior algebra which becomes then a differential Hopf algebra. Section 14.5 develops a general method for the construction of bicovariant DC on coquasitriangular Hopf algebras. In Sect. 14.6 this method is elaborated for the coordinate algebras of the quantum groups GL q (N), S L q (N), O q (N) and

Spq (N). Throughout this chapter A is a Hopf algebra with invertible antipode.

14.1 Left-Covariant First Order Differential Calculi 14.1.1 Left-Covariant First Order Calculi and Their Right Ideals

Definition 12.3 can also be applied to the Hopf algebra A considered as a left quantum space with respect to the comultiplication. Definition 1. A FODC r over the Hopf algebra A is called left-covariant if

is a left-covariant FODC r over the left quantum space X = A with left coaction = A according to Definition 12.3.

By Proposition 12.2, a FODC r over A is left-covariant if and only if there is a linear mapping AL F —) A TT such that AL(adb) = A(a)(id 0 46(b)

for all

a, b E A.

(1)

If such a mapping AL exists, then, by the proof of Proposition 12.2, AL is a left coaction of A on r which satisfies AL(apb) = A(a)AL(p)A(b) for

492

14. Covariant Differential Calculus on Quantum Groups

a, b E A and p E F. Thus, according to Definition 13.1, F is a left-covariant bimodule over A. Let F be a left-covariant FODC over A. First we restate some results obtained in Subsect. 13.1.1 in the present context. We define a linear mapping inv F by setting wr(a) = PL (da). If no confusion can arise, we omit wr A the subscript F and write simply ci.)(a). From Lemma 13.1 we get w(A) = 11F. Asshown in Subsect. 13.1.1, F = Aw(A) = w(A) A and any basis of the vector space w(A) is a left A-module basis and a right A-module basis of F. Since ,6L(da) = a( 1 ) da(2) by (1), it follows immediately from the definition of PL and (13.4) that

w(a) =

E S(a(o)da (2) and da =

amw(a (2) ) for a c A.

(2)

From the formulas (13.8), (2), and (13.3) and the Leibniz rule we obtain

w(a)

b = PL (E S(a(0)da( 2).b) =

E E(S(a( 1)))PL (da(2)•b) = PL (da•b)

= PL(d(ab)) — PL(adb) = w(ab) E(a)w(b) = w(ab). Recall the notation

a :=

a — E(a)1. Hence, by (13.5) and (13.6), we have

adR(b)(w(a)) ----z w(a) i b = LON, bw(a)c = >

( 3)

bc( l)w(ãc(2) ).

(4)

In order to characterize left-covariant FODC's as quotients of the universal FODC, we use another realization of the universal FODC over A which is better adapted to the Hopf algebra structure of A.

Example 1 (The universal first order differential calculus over A revised). Let f-1 1 (A) := A 0 ker E. Let us write aw(b) instead of a 0 6, where a, b E A. In particular, w(b) denotes the element 1 0 b for b E A. Taking into account the formulas (2) and (4), we introduce an A-bimodule structure of fl l (A) and a linear mapping d : A (21 (A) by c(bw(a)) := cbw(a), (bw(a))c bc(i) w(etc (2) ) and da := E a( i)w(a( 2)) for a, b, c E A. From the relations -

a•db + da•b =

E abmw(b( 2)) + E a(ow(a(2))b

E ab(ow(b( 2)) + E amb( 1)ci.)(a( 2)b(2)) E a( i)E(a( 2))b( 1)w(b( 2)) E a( 1 )b( i)w(a( 2)b(2)) = d(ab), w(a) = e(a( 1 ))44)(a( 2)) = E S(a(1))a(2)w(a(3)) = E S(a(i))da(2) (5) we see that d. satisfies the Leibniz rule and that f-1 1- (A) =

(2 1 (A) is a FODC over the Hopf algebra

A.

A dA, that is,

14.1 Left-Covariant First Order Differential Calculi

493

We prove that 0 1 (A) is the universal FODC over A. Let 1" be another We first show that the linear mapping FODC over A with differentiation r given by 7,b(a.db) = a • a is well-defined. For this we suppose : 0 1 (A) that Ei ai •dbi = 0 in 0 1 (A) with ai, bi E A. By the definitions of d and ,r-21 (A), we have E E.2, az- bi(1) w(bi(2) ) = E E.2 az- bi(1) Obi(2) = 0 in A 0 ker e. S id)(id 21) and using the Hopf Applying the mapping (m id)(id =0 algebra axioms, we compute Ei ( ai bi — ab i0 1) --- O. Hence Ei in r and 71; is well-defined. It is straightforward to verify that the FODC f is Isomorphic to the quotient FODC 0 1 (A )/ M, where W is the A-subbimodule kerli) of 0 1 (A). Therefore, f2 1 (A) is indeed the universal FODC over A. Of course, the universal FODC over A is unique up to isomorphism. Hence the universal FODC 0 1 (A) defined above and the universal FODC constructed in Subsect. 12.2.2 are isomorphic. By Corollary 12.3', the FODC Q 1 (A) is left-covariant. Moreover, formula (5) shows that w(a) is nothing but the 1-form PL (da) w o i (A ) ( a), a E A, in accordance with the notation A introduced above.

a.

For a left-covariant FODC

r over A

we define

:= {a E ker e w F (a) = O}.

(6)

7Zri gives a bijection The next proposition shows that the mapping 1" between left-covariant FODC over A and right ideals of ker E . Proposition 1. (1) Let R, be a right ideal of ker E . Then Ar := A 0).(-21(.A)(R) is an A-subbimodule of f2 1 (A) and the quotient f := if21 (A)/Ar is a leftcovariant FODC over A such that 1 ' = 1. (ii) We suppose that r is a left-covariant FODC over A. Then R. r, is a right ideal of ker E and the FODC r is isomorphic to the quotient FODC -

(21

(A) IA wf 21 (A) (R r) •

Proof. (i): Throughout the proof of (0 we write w(-) for w ni (A )(.). Since R. is also a right ideal of A, we have (4.) (a)c = Ec(o w(ac( 2) ) E A w(1) by (4) for a E R (C ker e) and c E A. That is, Al. = A w(R) is an A-subbimodule of (2 1 (A). Hence r = oi(A)I N. is a FODC over A. Let d i, : 01 (A) A 0 Q' (A) denote the coaction for the left-covariant FODC 0 1 (A). Since d (aw(b)) = d(a)(1 ®w(b)), we have 24,() g Ao.w. Therefore, di, passes to a left coaction of A on the quotient r = 'WAIN * which obviously has all the properties required in Definition 12.3. This shows that the FODC 1" is left-covariant. By the definition of d on the quotient FODC it is clear that R , C Te r. Conversely, suppose that wr(a) = 0 for some a E kere. Then w(a) E N = A w(R.). By the definition of f2 1 (A), this means that 10a E A0 R. which in turn implies a E Te. Thus, R. = R.1-'. (ii): Since r is left-covariant, (3) holds. Therefore, if a E R. and b E ker E 7 then cair. (a) = 0 and hence wr(ab) = adR(b) (W r (a)) = 0 by (3). Thus, R. is a right ideal of kere. The second assertion of (II) follows from (i). El

494

14. Covariant Differential Calculus on Quantum Groups

Proposition 2. Suppose that A is a Hopf *-algebra and f is a left-covariant FODC over A. Then f is a *-calculus if and only if S(RT)* C 1Zr •

Proof. Let

r be a *-calculus and let a E 'RE . Using the identity (S(a)*)

E S(a( 2))* S(a( 1 ))*, the formulas

=

(2) and (1.39), properties of the involu-

tion, and finally the Leibniz rule, we compute

wr(S(a) * ) = E S(S(a(2)) * )d(S(a(1)) * )

Ea ( 2 ) *( dS(a(0 ))*

= E((dS(a(1)))a(2))* --= E(d(S(a (1) )a(2) ) = (E(a)d1 wr(a)) * —wr(a)* = o, ( 7) so S(a)* E R. Thus, S(R. F)* CR T, . Conversely, suppose that S(R.r)* C R.r and write w for wo i( A ). Recall from Subsect. 12.2.2 that the universal FODC over A is a *-calculus. Therefore, by the calculation (7) applied to the *-calculus (2 1 (A) and by the assumption, we have (AO)* = —w(S(a)*) E (.0 (RT ) for a E IZT, so that w(R E )* C w(R. r). Since R.F is a right ideal, cd..)(7Z E )A C Ack)(7Z. r) by (4). Hence we get (Aw(TZT))* = w(RT)*A* C w(Te r )A C Aw(R.r), so the involution on (2 1 (A) passes to the quotient f2 1 (A)/Aw(1Z r) which therefore becomes a *-calculus. Since this quotient FODC is isomorphic to f by Proposition 1, I' is also a *-calculus. .

14.1.2 The Quantum Tangent Space

If f is the "ordinary" differential calculus on a Lie group G, then we have = (ker = ff E A1 f(e) = (df)(e) = 0} and the tangent space at the unit element can be considered as the vector space of all linear functionals on C°° (G) annihilating R.r, and the constant functions. This motivates R.F

Definition 2. For a left-covariant FODC f over A, the vector space

Tr := {X E A' I X(1) = 0 and X(a) = O for a 1ZT 1

(8)

is called the quantum tangent space toi. In what follows we will omit the subscripts f in Ti and RT Propositions 3 and 4 below show that T admits properties analogous to those of the classical tangent space at the identity of a Lie group. However, the commutation relations between elements of T are not necessarily quadratic as in the classical case (see Subsect. 14.1.4 for an example). .

Proposition 3. There is a unique bilinear form (.,.) : T x

(X,a.db) = E(a)X(b)

for

r=

a,b E A,

X E T.

C such that (9)

w(A) and T form a nondegenerate dual pairing The vector spaces inv with respect to this bilinear form (that is, (co), X) = 0 for all X E T implies cd.) = 0 and (w, X) = 0 for all cd.) inv implies X = 0), and we have

r

14.1 Left-Covariant First Order Differential Calculi

(w( a), X) = X(a) for a E A, X E T.

495

(10)

Ei

:= X(E i *Obi ). Proof. For ( = ai dbi E r and X E T, we define (X, We have to prove that this value does not depend on the particular representation a idbi of the form (. For this it suffices to check that the right-hand side of the defining equation vanishes if ( = 0. Indeed, by (13.3), = 0 yields 0 --- PL (() = e(ai )PL(dbi ) w(E i e(a)b), so e(a) E R. and hence X(E i e(a)b) = 0 by the definition of T. Thus, the bilinear form (., .) is well-defined. Since r = A dA, the bilinear form (., .) is uniquely determined by (9). Formula (10) follows at once from (9) and (2). The assertions concerning the dual pairing are easily verified.

Ei

Ei

Ei

It is clear from (8) and the fact that inv dual pairing that

r and T

form a nondegenerate

dim inv f = dim T = dim ker OZ.

(11)

r. r

We call this (cardinal) number the dimension of the left-covariant FODC Let us fix the following notation which will always be kept in the following: {Xi}i c i denotes a basis of the vector space T, Iwi l ie/ is the dual basis of 1 (that is, (X i , w) = (5ii for j j E I) and jE/ is the family of functionals from Proposition 13.2. ,

Proposition 4. With the above notation, we have for a and b in

da =

A that (12)

X i (ab) = e(a)Xi (b)+Ei X j (a)fi (b).

(13)

Proof. Since (Xi , w(a)) = Xi (a) by (10), we have w(a) = Ei Xi (a)w, and hence da = a( i )(.4.)(a( 2)) = E i a(i)Xi(a(2))wi = E i (Xi .a)wi which proves (12). Using (12) and the Leibniz rule, we get

E (X i .ab)(.4.4 = d(ab) = a•db + da•b =

.a(Xi .b)w i

E.a Xi.b)w ( i+ 2

213

Equating the coefficients of wi one obtains

X i .ab = a(Xi .b)+Ei (X i .a)(fii .b),

E I.

Applying e to this equation and using the identity e(f.c) = f(c) for f E A' and c E A, we obtain (13). Formula (12) indicates that the linear mapping Lx i: A A defined by Lx i (a) := Xi .a might be regarded as a quantum analog of a left-invariant vector field in classical Lie theory (see Subsect. 14.4.2 for more details).

496

14. Covariant Differential Calculus on Quantum Groups

Suppose now that the dimension of the vector space T is finite. Then we conclude from (13) and (13.14) that the functionals Xi and f; belong to the dual Hopf algebra A' (see Subsect. 1.2.8) and we have

A(Xi ) = 6

Xi + Xk

fik , 46 (fn = fØf:

(14)

in A°. Hence the linear span of functionals E, X i and g, i, r, s c I, is a subcoalgebra of A°. The antipode of Xi in the dual Hopf algebra A° is given by (15) -X k S(fh. Indeed, by (1.26) and (14), we have the relation

mAo(id S)1(X) -=

(60 S(Xi) + Xk S(fi k )) -=- O =

which in turn implies (15). Proposition 5. A finite-dimensional vector space T of linear functionals on a Hopf algebra A is a quantum tangent space of a left-covariant FODC if and only if X(1) = 0 and Zi(X) X E T 0 A° for all X c T

Proof. The necessity of the second condition follows at once from (14). To prove the sufficiency part, let R = fa E ker s X(a) 0 for all X c Tj. From the assumption d(X) --E0XET0 A° for X E T it follows that R. is a right ideal of ker e. By Proposition 1, there exists a left-covariant FODC F over A such that R, = 7Zr. Since T is finite-dimensional, the definition of TZ implies that T = {X E A' I X(1) = 0 and X(a) = 0 for a E 7Z}. That is, T is the quantum tangent space of F.

Proposition 6. Let

r

be a finite-dimensional left-covariant FODC over a Hopf *-algebra A. Then F is a *-calculus if and only if its quantum tangent space T C A' is *-invariant, where A' is endowed with the involution

Proof. By (1.40), the functional f* is defined by f* (.) = f (S(-) * ) for f E A'. If F is a *-calculus and X E T, we have X* (a) = X (S(a)*) = 0 for all a E by Proposition 2, so that X* c T. Conversely, if T is *-invariant and a E R,, then X (S(a)*) = X* (a) ---- 0 for all X E• T. Since T is finite-dimensional, this implies that S(a)* E R., so r is a *-calculus again by Proposition 2. 0 14.1.3 An Example: The 3D Calculus on SLq (2) -

Throughout this and the next subsection, A is the Hopf algebra 0(514(2)) and we assume that q2 1. Let a = n b = t4 c uf, d = u3 denote the entries of the fundamental matrix of A and let E, F, K be the generators of Oq (s12), see Subsect. 3.1.2. Recall from Subsect. 4.4.1 that there is a dual pairing of the Hopf algebras 6,7 (512) and A =-- 0(SL Q (2)) determined on the generators by the equations ,

,

, (E c) = (F, b) = 1 and zero otherwise. (16) (K , a) = q -1/2 , (K, = /q2

14.1 Left-Covariant First Order Differential Calculi

497

We define three linear functionals X 0 , Xi , X2 on A by A x i := (1 _ q-2)-1 ( 6 ____ K4) . .x.0 := q -112FKI x- 2 :____ q 112E7---1 By the definition (3.12) of the comultiplication of

2■Xi = E 0 X 3 ± Xi 0

K 2 for j=0, 2

iTg (s12) we have

and 2■X1 = E ® Xi +X1 0 K4 . (17)

Therefore, by Proposition 5, T := Lin {X0 , X i , X2 } is the quantum tangent space of a (unique up to isomorphism) left-covariant FODC r over A. This is the so-called 3D-calculus of S. L. Woronowicz. Let {w0,(.4.)1,w2 } be a basis of inv r which is dual to the basis {X0 , X1 , X2} of T. Then, by (12), we can write 2

(18) dx = E(Xi.x)wi, x E A. j=0 Since X0(b) = X2(e) = Xi(a) = 1 and X0 (a) = X0(e) = X2(a) = X 2 (b) = X i (b)= X2 (c)= 0 because of (16), we obtain Wci =----

w(b), W2 = w(c), wi = w(a) = -q -2 w(d),

(19)

da = bw2 + aw i , db = aw0 - q2 bw i , dc = av1 -Fdw2 ,dd= -q2 dwi±cwo. (20) Comparing (17) with (14) we see that fl --= K 2 for j = 0,2, na = K4 and f ;7 = 0 otherwise. This leads to the following commutation rules between matrix entries and 1-forms:

(via = q-l awi , wi b z--- qbw i , wi c=q-l avi , wi d=qdwi for

j = 0,2, (21)

(22) cvia =- q -2 aw1, wib = q2 bwi, w i c= q-2 av 1 , wid= q2 dwi. Let R, := B. A be the right ideal of kerE, where B is the vector space generated by the six elements a + q -2 d- (1+ q-2 )1, b2 , c2 , bc,

(a -1)b,

(a -1)c.

(23)

Using (16) one easily verifies that the functionals X0, X2 1 X1 annihilate these elements, hence R, C R, r. Since dim (ker E )/ R, < 3 by the definition of R, and dim (ker E )/ /RE = dim T = 3 by (11), we conclude that R, = R,r, that is, R, is the right ideal of kerE associated with the FODC r. The six generators s from (23) satisfy the condition S(x)* E R, for the Hopf *-algebras 0(SL q (2,R)), iqj =--- 1, 0(SUq (2)) and 0(SUq (1,1)), q E R. Hence, by Proposition 2, r is a *-calculus for all three Hopf *-algebras. From the commutation rules in the algebra Ûq (512) it follows that the basis elements of the quantum tangent space T satisfy the relations

q2 X1 X0 7

C 2X0X 1 = (1 + q2 )X0,

(24)

q2 X2 X1 - q -2 X1X2 = (1 + q 2 )X2 7 (25) qX 2 X0 - q -1 Xo X2 = —q -1 Xi . (26) In the limit q -p 1 all the above formulas give the corresponding formulas for the ordinary first order differential calculus on SL(2).

14. Covariant Differential Calculus on Quantum Groups

498

14.1.4 Another Left-Covariant Differential Calculus on SL q (2) We put

Xo := q -5/ 2 FK 5 , X2 := q5/2 EK5 , Xi := (1 - q -2)- 1( E.

K 4 ).

Note that X1 is the same as in Subsect. 14.1.3. Now we have

LX E

Xi + Xi

K 6 + (q -2 1)X1 X j, j= O,2.

By Proposition 5, there exists a left-covariant FODC F over Sid g (2) with quantum tangent space T := Lin {X0 , Xi , X21. This FODC F can be developed in a similar manner as in Subsect. 14.1.3, so we only mention the necessary modifications. Formulas (18)-(20), (24) and (25) remain valid without change. In (21) we replace q by q3 and q-1 by q-3 , and in (23) the two last generators of R must be replaced by (a q -2 )b and (a - q -2 )c. Instead of (22), we now have wia ==

q-2 aw1 (q-2

1)bo.)2,

wic = q-2 cw i + (q-2 1)dW2 1

= q2dw 1 (q -2 1)cw o . wib = q2 bw 1 + (q-2 - 1)ac.4.)0 , The commutation relation (26) becomes the equation q5 X2X0

q-5 X0X2 = -C 1 X1 2q -2 (q - q -1 )X?

(q - q ) 2 )q.

1 of the FODC F is just the As in Subsect. 14.1.3, the classical limit q ordinary first order calculus on SL(2). Note that in the case q = 1 the commutation relations of X0 , X1 and X2 for both FODC in Subsects. 14.1.3 and 14.1.4 give the relations of the classical Lie algebra sl(2,C).

14.2 Bicovariant First Order Differential Calculi 14.2.1 Right-Covariant First Order Differential Calculi At the end of Subsect. 12.2.3 we explained how Definition 12.3 and Proposition 12.2 have to be changed in the case of a right quantum space X and a right-covariant FODC F on X. We apply' these modified versions in

Definition 3. A FODC F over A is said to be right-covariant if F is a rightcovariant FODC over the right quantum space X = A with right coaction = A or equivalently if there exists a linear mapping AR : F F A such that (27) AR(adb) = A(a)(d id),6(b) for a, b E A. Suppose that F is a right-covariant FODC over A. Then F is in particular a right-covariant bimodule over A. We introduce a linear mapping n PR(da). Clearly, 71(A) = /inv . Since AR(da) = A Fin , by setting n(a) E da( i ) a( 2) by (27), it follows that li(a) =

E da(1).S(a(2)) and da = Eri(a (i) )a(2) for a

E

A.

14.2 Bicovariant First Order Differential Calculi

499

Bicovariant First Order Differential Calculi

For a Hopf algebra it is quite natural to look for FODC which are compatible with the comultiplication from the left and from the right. These are the bicovariant F0 DC's. Definition 4. A first order differential calculus over a Hopf algebra A is called bicovariant if it is both left-covariant and right-covariant. By Subsects. 14.1.1 and 14.2.1, a FODC F over A is bicovariant if and 4 and AR:r- 4 1-'0A only if there exist linear mappings A L : such that the equations (1) and (27) are satisfied. Suppose that r is a bicovariant FODC over A and let A L and AR be the corresponding mappings. From (1) and (27) we obtain (id 0 AR) 0 AL (adb) ==

(A L 0 id) o A R (adb)

E (id 0 AR)(a(i)b(i) 0 a(2)db(2)) E a( l )b( l ) a( 2)db( 2 ) a(3)b( 3), EGLI L id)(amdb (i) Oa(2 )b( 2) )

E a( i )b( i )

a (2 )db(2)

a(3)b( 3)

for a, b G A. Since r = A-clA, the preceding shows that (id 0 AR) 0 AL = (AL 0 id) o 2IR, so condition (iii) in Definition 13.3 is fulfilled. The validity of the two other conditions in Definition 13.3 have been already mentioned in Subsects. 14.1.1 and 14.2.1. Therefore, r is a bicovariant bimodule over A and the whole theory developed in Chap. 13 applies to F. The following formulas describe the right coaction AR on a left-invariant form w(a) and the left coaction AL on a right invariant form n(a) in terms of the right and left adjoint coactions of A on itself, respectively: A R(w (a)) = (w id) (Ad R(a)) , A

L (7) (a))

= (id 077)(AdL (a)),

a E A. (28)

We verify (for instance) the first equality and compute AR(w(a))

E AR(S(a( 1 ))da( 2)) = E A(S(a(1)))(d 0 id)A(a(2)) E S(a(2))da(3) S(a (1) )a (4) = E w(a(2) ) S(a(o)a( 3) id)(AdR(a)).

(c.i)

Next we characterize bicovariant FODC among left-covariant FODC in terms of their right ideals.

Proposition 7. Let r be a left-covariant FODC over A with associated right ideal R.. The FODC F is bicovariant if and only if R. is invariant under the right adjoint coaction (that is, AdR(R) C R.0 A).

Proof. If

r is bicovariant, then (28) holds. Since R. = {a

E kere I w(a) = OE

(28) implies that AdR(R.) C R. 0 A. Conversely, assume that AdR(R) C

500

14. Covariant Differential Calculus on Quantum Groups

R 0 A. By Corollary 12.3', the universal FODC 0 1 (A) is left-covariant. Similarly, 0 1 (A) is right-covariant (see the Remark at the end of Subsect. 12.2.3) and hence bicovariant. From formula (28) applied to the bicovariant FODC 0 1 (A) and from the AdR-invariance of R it follows that ZAR(A wrp(A)(TZ)) Ç Aci.)(21(..4)(7?)0A. Therefore, the right coaction /AR of (A) passes to the quotient 01 (A)/M, where )f :=-- A w pi ( A ) (R.) , the FODC and the quotient FODC 0 1 (AN is right-covariant. Since 1" is isomorphic is also right-covariant. to 0 1 (A)1 AT by Proposition 1(ii),

r

r

Let be a bicovariant FODC over A. We state two useful formulas for which will be needed later. For any a, b E A, we the braiding a of OA have a(w(a) OA w(b)) = (29) w(b ( 2) ) OA w(aS(b(i))b(3)),

r

r

(w OA w)AdR(a) = (id — (7)(a) OA w)Z(a)-

(30)

Formula (29) follows at once from (13.25) combined with (28) and (3). Using the Hopf algebra axioms, (30) is easily derived from (29). 14.2.3 Quantum Lie Algebras of Bicovariant First Order Calculi

r

Unless stated otherwise we suppose in this subsection that is a bicovariant FODC over A with right ideal R and quantum tangent space T such that dim T < a). For X and Y in T, we define a linear functional [X, Y] on A by setting

[X ,171(a) := (X 0 Y)(AdR(a)),

a E A.

(31)

Since the right ideal R is AdR-invariant by Proposition 7 and the elements of T annihilate R because of (8), it follows from (31) that [X ,Y] also annihilates R. Since X(1) --= Y(1) =-- 0, we get [X, r(1) = O. Hence we have [X, Y} E T by (8). Thus, C : X 0 Y --* [X, Y.] defines a linear mapping of T T to T. The assertions of Proposition 8 below show that the mapping C can be viewed as a quantum analog of the classical Lie commutator bracket. First we need some notation. Let (-, -) be the bilinear form on T x from Proposition 3. There exists a unique bilinear form (-, -) 2 on (T T) x such that inv(r

r)

(X

Y, w 1 OA 2)2 = (X, W 1 )(37, w2 ) for w

,

w2

inv r, X, Y E T (32)

Clearly, the bilinear form (-, -) 2 is nondegenerate as (-, -) is. Since the braiding a maps inv (i" OA II) into itself, its transpose at with respect to the nondegenerate bilinear form (-, -) 2 is a well-defined map of T 0 T into itself. Let mA, : A' A' A' denote the multiplication map of the algebra A', that is, mAi(f g)(a) = fg(a) = >f(a( I ))g(a(2)) for f g E A' and a E A. Recall from Subsect. 14.1.2 that T is contained in the dual Hopf algebra A°, because we have assumed that dim T <

501

14.2 Bicovariant First Order Differential Calculi

Proposition 8. For arbitrary elements X,Y,Z E T we have:

[X, Y] = adR(Y)X = XY — rnAlcr t (X 0Y). (ii) [X ,[Y, [[X, Y], Z1—E i [[X, Z4, Y] with Ei Z®Y T 0T, then C() = 0. (iii) If a t () = for some

:=

at (7®

z).

Proof. (i): First let f, g E A' and a E A. By the definition of the Hopf algebra structure of A° (see Subsect. 1.2.8), we have

(adR(g)f)(a) =

E s(g( i) )fg (2) (a) =-- E g(i)(S(am))f(a(2))g(2)(a(3)) = (f

(33)

g)AdR(a).

Comparing with (31), we get [X,Y]= adR(Y)X. Let us write a t (X 0Y) as a finite sum Ei ®X i with X 1 , Yi, E T. Using (10), (29) and (32) we compute

ct (X

Y)(a

= (E =

b) = Ei Yi(a)X2(b) = Ei (Yi , w(a))(Xi , (.4)(0)

, w(a) OA W(b)) 2 = (X

Y, cf(w(a) OA W(b)))2

(X, w (b (2))) (Y , w(d, S (b (1))b (3))) =

E Yo.)Nx(b(2))37(2)(s(b(ob(3))

= E y(i) (a)S(Y(2) )XY(3) (b) for a , b E A. That is, we have o-t (X

mAl at (X 0 Y) =

Y) =

E

® adR(Y(2))X and so

E YwadR(Y(2))X

E(Y( l ) s(Y(2))xY(3) - E(y( os(ypoxy(3) ) = XY —

adR(Y)X,

(34)

which yields the second equality of (i). (iii) follows at once from (0. (ii): Since adR is a right action of A° by Proposition 1.14, we have

[[X 1 311 Z] = (adR(Z) o adR(Y))X = adR(Y Z)X = [X, YZ]. Inserting the expression YZ = [Y, relation {X,Z iYil=[[X,Z i],Yi ] from

(35)

Ei Zir from (34) and using the

(35)

we obtain the desired identity. El

If the braiding a is the flip operator, then Proposition 8 says that T is an "ordinary" complex Lie algebra with bracket [X, Y] = XY YX. In the general case, assertions (ii) and (iii) are quantum versions of the Jacobi identity and the antisymmetry, respectively. For this reason, we call T the quantum Lie algebra of the bicovariant FODC F. Corollary 9. (0 For any g E , we have adR(g)T C T. (II) Suppose that Bo is a subset of A' such that the subalgebra B generated by B o separates the elements of A, that is, g(a) = 0 for all g E 8 implies a = O. If I' is a left-covariant FODC over A such that dim Tr < oc and adR(g)Tr Ç Tr for any g E Bo, then f is a bicovariant FODC.

502

14. Covariant Differential Calculus on Quantum Groups

Proof. (i): Since AdR(R,) C R.0 A by Proposition 7, it follows from (33) that adR(g)X(a) =-- 0 for X E T and a E A, so that adR(g)X E T. (ii): The assumptions and formula (33) imply that AdR(Ter) C 7Zr, 0 A, 1=1 so F is bicovariant by Proposition 7. Example 2. The 3D-calculus from Subsect. 14.1.3 is not bicovariant by Corollary 9(i), since adR(F)Xi = (q -1- + q -3 )FK 5 is not in T.

For the conclusion of Corollary 9(ii) it does not suffice that the assumption adR (g)T C T holds for all functionals g = Xi and g = g i,r,s E I. For instance, the left-covariant FODC over SL q (2) with quantum tangent space T = Lin {e - K 2 } has this property, but it is not bicovariant. Corollary 10. Suppose that the dual Hopf algebra A° separates the elements of A. Then a finite-dimensional linear subspace T of A° is the quantum Lie algebra of a bicovariant FODC over A if and only if adR (g)T C E for g E A° and X(1) = 0 and ,6(X) e 0 X E T® A° for X E T .

Proof. Combine Proposition 5 and Corollary 9. As an interesting application of Corollary 10 we prove Proposition 11. If the Hopf dual A' separates the elements of A, then for

any central element c of A° there exists a bicovariant FODC over A with quantum Lie algebra

E(c) := Lin -{x„ := E c(2) (a)c(i) c(a) e f a E Al.

(36)

Proof. Let us write .6(c) = Ei c2 ® bi such that the set {N} is linearly independent. From its very definition we see that the vector space T(c) is the linear span of elements ci ci (1)e. Hence T(c) is a finite-dimensional subspace of A° and we have E(c) = ff E A° I = 0 for b E 7-e(c) e C.11, where R.(c) := {b E ker Ix a (b) = 0 for a E A } . For a, x E A and b E R,(c) C•1, we have ('6 (Xa)

6 Xa b 0

X)

= E C(3) (a)c(i )(b)c(2)(x) - c(a)e(b)e(x) 6(b) (E c( 2)(a)c( i)(x) - c(a)e(x)) = E c( i ) (b)c(2) (xa)

c(xa)e(b) = x xci (b) = 0,

so that ,6(xa ) E 0 xa G T(c) 0 A° for any a E A. Next we show that T(c) is adR-invariant. Let g E A'. Since c is central in A°, we have i6( 2)(g)(1 c 1) = (1 c 0,6(2) (g). Applying the mapping (m12 0 m34)(5 0 id id 0 S)(id ,6 0 id) to this equality we get

E S (g ( 1) )cmg ( 2) 0 c(2) = E c(i) O g ( i)c(2)S (g ( 2)). Using the latter formula, some properties of the dual pairing of A' and A

and again the centrality of c we compute

14.2 Bicovariant First Order Differential Calculi

503

adR (g),,, = E,(2) (a)8(g(0 ),( ,)g(2) — c(a) g (i)E = E g(i) c(2) 8(g(2) )(a)c(i)

c(a)g(i.)E

= E g (a(i) S(a(3) ))c(2) (a(2) )c(i) — c(a)g(1)E = E g (a(1) 8(a(3) ))„,,,, 2 , + E g(a(i) (a(3) ))c(a(2) ) c(a)g(1)E E g (a(i) S(a(3) ) x a (2) E T(c). Thus T (c) is adR-invariant. Since obviously X(1) = 0 for all X G T(c), we CI have shown that T(c) has the properties required in Corollary 10.

The quantum Lie algebras of the bicovariant FODC F±,, on the Hopf algebras C9(Gq ) constructed in Subsect. 14.6.1 will be of the form (36). Let us make the structure of the quantum Lie algebra more transparent. Proposition 12. The basis elements Xi, i E I, of T satisfy the commutation

relations = X i iX— c4i77.. nX„,X m = qXk

[Xi ,

(37)

7

where fm i (vy)

and Clci = Xi (vt)

for

i, j,n,rn,k E I.

(38)

Proof. The first equalities in (37) and (38) follow immediately from Proposition 8 (i ) and (13.29). We prove the formula for Clci and compute

E a (1) X,(a (2) )wi

12

( 3

)=

Edam 0 a( 2 ) = (d Ø id)(a) = 4R(da)

= E ,R(a ( 1 ) xm.(a (2) ) w,n) = Ea (i) x,n (a(3) ),,

a( 2)v;,_,

for a E A. Applying PI, 0 id to both sides of this equality, using formula (13.3) and equating the coefficients of wi, we obtain

E Xi(a(1))a(2)

a( 1 )14Xm (a( 2)),

a E A.

Applying Xi to (39) and then using (13) and the formula for o

XiXi(a)

(39) ,

get

xi(a ( 1 ) )Xi(a ( 2) ) = E xi(a(ov:)zn(a(2)) E E(a ( i ) )xi (472)x,n(a (2) ) + E xn(a ( i ) )fr(v)x,n(a ( 2) ) X nX in (a)

x;(14)Xk(a) + 0-

which gives the above expression for C

and completes the proof.

Moreover, applying the functional nc to (39) and taking (13.14) and (13.29) into account, we obtain

=

(40)

504

14. Covariant Differential Calculus on Quantum Groups

We specialize (33) to the functionals Xi and f's . From the formulas (40), (37), (14) and (15) we easily derive

adR(Lni)X,n, =am Xi and

adR(Xi)Xi = Ciki Xk.

(41)

By (41), the matrix coefficients crrir of the braiding a and the structure constants CY,ci of the quantum Lie algebra T are computed from the right adjoint action of the dual Hopf algebra A' on T.

14.2.4 The 4D+ - and the 4a..-Calculus on S.L q (2) As in Subsect. 14.1.3 we use the Hopf algebra Ûq(s12) and the dual pairing of Ùq (512) and A := 0(SL q (2)) given by (16). In order to treat both calculi F+ and F_ at once, we introduce the character E._ of the algebra A defined by = -6ij, i j = 1, 2. It is easy to check that 2i(E_) = E_ g E_ I E2 = E and that E_ belongs to the center of the dual algebra A'. Moreover, we set ,

E+ := E. Let T± be the vector subspace of A' spanned by the four linear functionals

:= E±K -2 _ E7 X2 := 2E± FK - 1 1

x-3 :=

X4 := E±K2 (q-q- ycie.±FE Note that q -1 + qX 4 + (q + q-1)1 = E± ( 17 4r1)2Cq where

Cq := F E + (q _

(qic 2 + .7 -1 K -2 )

is the quantum Casimir element of (4(512 ). In particular, T+ contains the element Cq e(Cq )1. From Proposition 5 and the formulas

LX5

2,x4 =

E ® ± ® E±K-2 , E ® Xi ± Xi E± ± Xi Xi = 2 7 3, E x4 + 7 t 1)2(r 1 X1 ® E± FE ± X4 ® E±K 2

(4

+( ti

r

q - 1)2q - 1/2 (X2 E±EK ± X3® E±KF)

we conclude that T± is the quantum tangent space of a left-covariant FODC F± over A. The FODC T± is called the 4D±-calcu1us on SL q (2). Let R,± 13 •A be the right ideal of ker E generated by the vector space 8± spanned by the following nine elements:

b2 , c, b(a - d), ca - d),

a2 + q2 d2 - (1 ± q 2 ) (ad + q - 1 be),

x±b, s±c, s±(a - d), x±(a + q -2 d - (1 + q -2 )1). Here we abbreviated x± := q2a±dT(q -1 -Eq3 )1. Using (16) one can check that the functionals X1, X2 ) X3 7 X4 annihilate these nine elements, so R.± C Ter± Since dim (ker E )/ R,± < 4 and dim T± = 4, it follows that R.± is the right ideal R./-1 of the FODC F±.

14.2 Bicovariant First Order Differential Calculi

505

The vector space 13± is AdR-invariant. (Indeed, it is not difficult to verify that the generators listed above span an AdR-invariant subspace.) From the formula Ad(j) = x(2)y(2) S((xy) (0 )(xy)( 3) we see that the right ideal 7Z.±= BA is AdR-invariant as 13+ is. Therefore, by Proposition 7, f± is a bicovariant F0 DC. From Proposition 2 we derive that 1-1± is a *-calculus for the three real forms SL q (2, R), 1q1 = 1, SU(2) and SUq (1,1), q E R.

E

Remark. Let U(T) be the subalgebra of Ùq (512) generated the elements of the quantum Lie algebra T+ and the unit element. Contrary to what is occasionally asserted in the literature, U(T) is different from Ùq (512 ). In fact, the set of monomials Fi KiEi such that i + j + / is odd is linearly independent modulo U(T+ ). But U(T) still separates the points of 0(SL q (2)) when q is not a root of unity. 14.2.5 Examples of Bicovariant First Order Calculi on Simple Lie Groups In this subsection we shall construct two examples of bicovariant FODC on simple Lie groups which are different from the ordinary differential calculus. For SL(N,C) it can be shown that the N bicovariant FODC obtained in this manner are just the classical limits of the 2N bicovariant FODC for the quantum group SL (N) developed in Sect. 14.6. Suppose that G is one of the Lie groups SL(N,C), O(N,C) or Sp(N,C). Let A = 0(G) be the coordinate Hopf algebra of G (see Example 1.3), g the complex Lie algebra of G, and {X1 , - • • , X n } a basis of g. We shall use the dual pairing of the Hopf algebras U(g) and 0(G) described in Example 1.6.

Example 3. Let C = E =1 aii X i X i be a nonzero fixed quadratic element of the enveloping algebra' U(g). Then we have ,i(Xi ) = e 0 Xi ± Xi 0 e and

zi(c)=E0c+coe+E, ,i x,

Xhau ctii

)•

(42)

Hence, by Proposition 5, the (n + 1)-dimensional subspace T := g JED C•C of A° is the quantum tangent space of a left-covariant FODC F over A. Because of Proposition 4, the FODC F can be easily determined explicitly. For notational simplicity, let us set X0 := C. Then we can write da =

(Xi .a)wi,

a E A.

(43)

For the commutation rules of forms and functions we need the functionals from Proposition 13.2. COmparing the formulas (14) and (42), we obtain = Enk=i (a ik + aki)Xkl fi? = 60iE and ft = 6ijE for i = 1, 2, • - • ,n. Therefore, by (13.15), woct = atoo, wia = awi +

.a)wo for i = 1, 2, . . . , n.

(44)

506

14. Covariant Differential Calculus on Quantum Groups

Suppose now that C is a Casimir element of U(g). Since C lies in the center of U(9), it is clear that adR(y)C = e(y)C and adR(y)x E g for x E g and y E U(9). This implies that T is adR-invariant. Therefore, by Corollary 9(ii), f is bicovariant. Let us emphasize that in contrast to the ordinary differential calculus on G the differentiation (43) involves the second order differential operator X0 = C, and forms and functions do not commute as A seen from (44).

Example 4. For G = SL(N,C), let ( E C be such that ( 1 and ( N 1. For G = 0(N, C), Sp(N,C), we set := —1. Let e( denote the multiplicative Set X6 := linear functional on A := 0(G) defined by e(u) = e, := e(X i , i =1, 2, . , n, and T= Lin {Xi = 0, 1, • - , n}. We have = E 0 X: ± X 7ç

E( .)4 X,;,

(45)

for j = 1,2, - - • , n. Since the functional e( belongs to the center of A°, the linear subspace T of A° is adR-invariant. Thus, T is the quantum Lie algebra of a bicovariant FODC I" over A by Corollary 9 (u i). explicitly. From (14) and (45) we see that Let us describe this FODC = 6( for i := 1,2, n. Let Ak be the linear 6ojE( and fi° span of elements // ill • • • ujit . For a E Ak we have e<-.a = ( k a and so we obtain

r

woo, = ( k awo +

i=1

(X.a)wi and wi a = ( k awi for j = 1, 2, • • • n. (46)

Combining the first equality of (46) with (12), we conclude that

da = woa awo,

a E A.

The quantum Lie derivatives Li (a) := Xl.a satisfy the twisted Leibniz rule

L i (ab) = ( k aLt (b)+ (mLi(a)b,

a E Ak, b E

14.3 Higher Order Left-Covariant Differential Calculi In this section f is a left-covariant FODC over the Hopf algebra A with associated right ideal R.

14.3.1 The Maurer—Cartan Formula The Maurer—Cartan equations for left-invariant differential forms on Lie groups have the following quantum analog.

Proposition 13. If FA = Ep nco rAn is an arbitrary DC over A such that rAi = f , then we have

clw(a) = —

w(a (l) ) A w(a(2)),

a E A.

(47)

14.3 Higher Order Left-Covariant Differential Calculi

Proof. Differentiation of the identity tion

E S(a( 1 ))a( 2) = E(a)1 yields

E dS(a(0)•a(2) + E S(a(i))da(2) = 0 dS(a) = -

507

the equa-

and so

E S(a(1))da(2)•S(a(3)).

Hence, by the graded Leibniz rule and the relation d 2 ---- 0 in 11 ^, we get

E d(s(a ( 1 ) )da(2) ) = E dS(a(1)) A da(2) = - E S(a( 1 ))da( 2) A S(a(3))da(4) = - E w(a(1)) A (4)(a(2)).

dw(a) =

0

Equation (47) is called the Maurer-Cartan formula of the DC /1^ . Let us give two coordinate versions of this equation. From (10) it follows that w(a) = EiXi (a)wi and hence E w(a( 1 )) A w(a( 2 )) = E ii Xi X3 (a)w, A wi . Therefore, the Maurer-Cartan formula (47) can be expressed as N-- Xi (a)dwi = -37, XiXi(a)wi A wi,

ac

A.

(48)

For the second variant of formula (47), let us take elements a i E A such that Xi(ai) = 6ii and set dici = X i Xi (ak), i, j, k E I. Inserting a -- ak into (48) and applying (37) to ak, we then obtain dwk = --elci wi A wi , k E

cr i i = ck. I, and clici _knm 7.3 nm

)

k, i, j E I.

(49)

Moreover, since w(a) = 0 for a c R, formula (47) implies also that

E w(a( 1 )) A w(a( 2)) ------ 0,

a c R.

(50)

Note that the right ideal R depends only on the FODC F, while the cap product A in (50) is taken in F^. We shall see in Subsect. 14.3.3 that (50) is the only obstruction for the construction of a higher order DC which contains F as its FODC.

14.3.2 The Differential Envelope of a Hopf Algebra In Subsect. 14.1.1 we gave a second construction of the universal first order calculus for a Hopf algebra. In this subsection we extend this idea to the universal higher order calculus. Let f2° (A) := A and Q(A) := A 0 (kerE)®n for n E N. We shall write aw(a l ) A • - • A w(an) for the tensor a 0 a l 0 • • • 0 an E fln (A). As in Example 1, we set w(a) := 1 0 d for a E A. On the vector space 12(A) := EEC 0 Q(A) we define a product A and a linear mapping d: 11(A) (A) -) 11(A) (A) by (aw(a l )A ... Aw(an)) A (bw(b 1 )A...Aw(b m ))

:= E caw w(a l b( 2))A• • •Aw(anb( 7,1_ 1 )) A w(b 1 )A• • •Aw(bm ),

(51)

14. Covariant Differential Calculus on Quantum Groups

508

d(ar.i.;(a l )A

Aw(an )) :=

Eam w(a(2) )

A w(a 1 )A.

Aw(a n )

+E(-1) k at.ii(a 1 )A...Aw(a k-1 )Aw(al(VAw(al(c2))Aw(a k+1 )A... Aw(an ) (52) k.i

ker e and a, b E A. (In order to motivate these formulas, let us recall that for any left-covariant DC over A we have w(b)c = c(o w(bc( 2 ) ), da = a(i)w(a(2)), dw(a) = — Ew(a(1)) A w(a(2)) by the formulas (4), (2), and (47), respectively, for a, c E A and b E ker e.) for a l , - - - , an, b l , • • • ,

bm

E

Proposition 14. With the above definitions, Q(A) is a bicovariant differential calculus over A and the differential envelope of A.

Proof. It is easily seen that 12(A) is an algebra. The associativity of the product follows from the fact that L is an algebra homomorphism. The unit element 1 of A = (A) is the unit of Q(A). It suffices to verify the Leibniz rule for elementary tensors x n = aw(a l ) A • • • A b.)(an) and y, = bw(b 1 ) A • • • A w(b m ). In order to do this we fix x n . From (51) and (52) we see that d(xn b) = (dxn )b+ (-1)nx n A dbn . By induction on m we prove that d(x n A ym ) = dxn A ym (-1)nx n A dym . Using the Leibniz rule, another simple induction argument shows that d 2 (x) = 0 for all xn , so d2 = O. Thus, Q(A) is a differential calculus over A. Next we prove that Q(A) is the universal DC over A. Let F^ be an arbitrary DC over A with differentiation Each p E Qn (A) is a finite sum p= aidal • • • da. let us show that p = 0 in Qn(A) implies that

a.

Ei

Ei Waal A • • • A C- lain = O. Indeed, arguing E

az(1 0 — ®1)12(1®a

as in Example 1, it follows that

— a2i 1)23

(1

i 0 an i — an

1) n ,n +1 = O.

Ei

The element on the left-hand side is a sum of a i 0 ail 0 • 0 ail., and an element from the set A010A®•••®A+•--+A®•••0A0 1. This in turn implies that a i aat A • • • A da = O. By the fact proved in the preceding paragraph, there is a well-defined linear map V) : Q(A) F^ such that 7,b(a) = a and 7P(adai A • • A da n ) = aaal A • • • A C - lan . Using the formulas (12.2) and (12.3) applied to both differential calculi Q(A) and TA we derive that Q(A)Iker 7/) is a DC which is isomorphic to F^. Hence Q(A) is the universal DC over A. As noted in Subsect. 12.2.3, the universal DC of a left (resp. right) quantum space of A is left- (resp. right-) covariant. Therefore, the FODC Q(A) is bicovariant. D

Ei

In what follows, the symbol Q will always refer to the universal DC of A.

14.3.3 The Universal DC of a Left-Covariant FODC In Subsect. 12.2.3 we defined the universal DC Q/Jr- of a FODC F over an arbitrary algebra. in this subsection we realize this DC for a Hopf algebra A as a quotient of the tensor algebra TA(F).

14.3 Higher Order Left-Covariant Differential Calculi

509

Recall from Subsect. 12.2.3 that Jr is the differential ideal of the universal DC f2 generated by the A-subbimodule Ar of f2 1 such that F is isomorphic to Q'/N. Since N = At pi (R.) by Proposition 1, we have Jr := f2(.4,; (2 1(R.)Q+ Qdws.2 1(1?.),Q. Let Ir := flw r2i(R.),r2 be the two-sided ideal of f2 generated by wn i (R.). It is not difficult to verify that the mapping aiwo i(a) A

••

A w,r2i (ain ) ir —0

E aiwf (al) • • • .„-• (ani )

defines an isomorphism of the quotient algebra 12//r and the tensor algebra TA ( T') of the A-bimodule F. By (47), under this isomorphism the quotient Jr /Tr goes into the two-sided ideal SR, of the algebra TA(f) generated by the elements S(x) := Ewr(x ( 1 ) ),(x ( 2 ) ), X E . (53) Since 2 I jr is isomorphic to (f2/.2-r)/(Jr/if ), we obtain a canonical isomorphism r of f2IJr onto the quotient algebra ./7,\ := TA(F)/SR . Clearly, Fu" r=o F^ n , where F^1 F FAn := F ®n icSjz' and SR :=--- SR, n F®n . We define dp := dr -1 (p) for p E F.^ . Then FILA is a DC over A which is isomorphic to 12/Jr by construction. Thus, FuA is the universal differential calculus of the first order calculus F. By Proposition 12.4, rut\ is left-covariant as F is. In terms of bases, the mapping S : inv(F ®2 ) defined by (53) can be written as s(x) = E (54) E ,

Lemma 15. For z E R. and a E A we have S(x)a =

Ea mS(xa(2) ),

S(xa) =

E S(a(1))45(x)a(2)•

(55)

Proof. Using formula (4) combined with the fact that w w r vanishes on the right ideal R. C ker E we obtain

w(x( ow(x (2 ) )a E a(ow(x( i)a( 2))1.4.)(x( 2)a( 3))

S(x)a =

E a( 1 )[w(x( i )a( 2))w(x( 2)a(3 )) — w(E(X(2))x(i)a(2))10(a(3))

w(a(2))w(e(x(i) )S(2) a(3))

+ E(x(1))E(x(2))w(a(2))w(a(3))]

E a(i) [S(xa( 2)) — — (.4)(xa(2))44)(a(3))

+ E(x)w(a(1))w(a(2))]

E a(i) s(xa (2) ) and so S(xa) =

ES(a (1) )a(2) 8(xa(3) ) = E S(a(1))S(x)a(2)•

1=1

Recall that 8,2, = Asmdet by definition. By (55), 8 = AS(R) is an A-subbimodule of T° 2 and hence a left-covariant bimodule of A. Clearly,

510

14. Covariant Differential Calculus on Quantum Groups

S(R.) is just the left-invariant part in,(4) of Slz . Therefore, the results of Subsect. 13.1.1 and formula (55) imply the following

Proposition 16. (0 Each basis of the vector space S ( R) is a free left A-

module basis of 4. (ii) Let B be a linear subspace of A such that R. = B.A. Then we have = AS(B)A. If for any a E A and b E B there exist finitely many ai E A and ei E 13 such that S(b)a = Ei aiS(ei ), then S(13) = S (1Z ).

The previous construction becomes explicit in the following example. Example 5 ( The 3D-calculus on SL g (2)). Let A = 0(SL / (2)) and let F be the FODC over A from Subsect. 1413. Assume that q2 -1. We compute the elements S(x) (see (54)) for the six generators of the right ideal R. listed in (23). Let x = be. Then we have S(x) = X i (uri- u82 )Xi (u72- unwi wj . By the dual pairing (16) of Ûq (512) and A, the only nonvanishing terms Xi (î4u2.5 ) are Xo (ulu3) q and X2 (î4i4) = 1. There are only two corresponding nonzero terms Xj(uu1), namely X2(Au) = 1 and Xo (ulul) = q -1-. Thus we get S(x) = qw0(.4)2 +17 -1 w2wo. Similar computations for the other five elements in (23) show that 8(5) is spanned by the elements 2 2 2 2 4 4 W, (56) q W0W21 W1W0 q WOW15 W2W1 q W1W2 44) 2 W2WO of the tensor algebra TA(F) over F. From the relations (21) and (22) we see that for each element S(x) from (56) and any a E A we have S(x)a = aaS(x) with some a E C. Therefore, it follows from Proposition 16 that the six elements in (56) form a basis of the vector space 8( 1 .) and a free left Amodule basis of 4. Hence FA2 = F° 2 /4 has the free left A-module basis two A cv2, wo A CV1 ) Wi A (4)21. From (56) we get the equations wi A w = 0 for j = 0,1,2, (57) w2 A wo -- q2 w0 A w27 wi A wo -q 4 w0 A wi W2 Aw1 = -q4wi A w2 (58) in FA2 . The 3-form (.4)0 A w1 A w2 is a free left-A-module basis of FA3 • Clearly, FAn - {0} for n > 4. From the equations (19), (47), (57), and (58) we derive the Maurer-Cartan formulas dwo =__ (q2 q4 )wo A w1, dwi —wo A w2, dw2 = ( 172 q4) wi A w2.

Summarizing the preceding, we have obtained a complete description of the universal higher order differential calculus ./.1,1,` of the FODC F. In particular, we see that its classical limit q -* 1 is just the ordinary DC on SL(2). A Example 6 (The FODC on Sli g (2) from Subsect. 14.1.4). Let F and R. be as in Subsect. 14.1.4 and suppose that q 12 1. The elements x1 := a + q'd a2 ( 1 ± q 2)a q -2 1 and 13 := (a q-2 )bc belong to (1 q -2 )1, x2 the right ideal 7"?.. One computes S(exi- q6 x2) = (q6 + q4 1 )WOW2 w2w0 and 8 (x 3 ) = (q-2 -1) (q 3w0w2 + q -7 w2w0). Therefore, we have w0w2 E S(7"?.) and hence wo A.c4i2 = 0 in 1A2 • One also verifies that wi A wj = 0, j = 0,1,2. Hence all 3 forms of 1-1,/,` vanish, so the higher order calculus F ' does not give the ordinary higher order calculus on SO) when q -> 1. -

-

14.4 Higher Order Bicovariant Differential Calculi

511

14.4 Higher Order Bicovariant Differential Calculi 14.4.1 Bicovariant Differential Calculi and Differential Hopf Algebras Let F be a FODC over A. In general, there are many differential calculi over A which have F as their first order parts. For instance, if TA = aoic 0 rAn is a DC with TA1 = F, then so is any quotient FA /J of FA by a differential ideal J such that J n (A e F) = {O} (for example, J = EBnce_ k rAn for some k > 2). A natural higher order calculus with first order part F is the universal DC /-1^ considered in Subsect. 14.3.3. If F is bicovariant, then there is another distinguished DC FeA with FODC F built on the exterior algebra (see Subsect. 13.2.2) over the bicovariant bimodule F. The aim of this subsection is to construct this differential calculusr and to prove that both bicovariant calculi P and /IA give rise to differential Hopf algebras. Definition 5. A differential Hopf algebra is a differential graded algebra FA = enco 0 rAn with differentiation d which is an N o -graded super Hopf algebra with comultiplication .6 satisfying the condition d® otl=tlo d, where d o FA o rn FA orn is defined by d® (pi op2) = dPi OP2 ( -1 ) n PlOdP2 for E r An and p2 E rA .

Note that cl° otl = tlod is the "dual" relation to the graded Leibniz rule m o d® = d o m, where m denotes the multiplication map for the algebra FA. By differentiating the identity E p(i)S(P(2)) = 5 (P(1))P(2) = e(p)1 for p E FAn it follows easily that dS = Sd in any differential Hopf algebra. The bicovariant DC's FEA and 1T2 defined below and related constructions (see the last paragraph in Subsect. 14.6.2) appear to be the most important DC's with given bicovariant FODC F. A formal reason is that the definition of the exterior algebra / 1‘\ over F is completely similar to the classical case if the permutation group is replaced by the braid group. More deeper, these DC's are based on the braiding a which is a fundamental concept in quantum group theory. Theorem 17. Let 11 be a bicovariant FODC over A and let .1""` EB ncc 0 rAn be the exterior algebra over the bicovariant bimodule F with the N o -graded super Hopf algebra structure defined in Subsect. 13.2.2. Then there exists a unique linear mapping d: ft‘' --1-1‘` whose restriction to A is the differentiation d i. of I' such that l'e^ is a bicovariant differential calculus over A with differentiation d and a differential Hopf algebra. Proof. A convenient way to define the differentiation d on the exterior algebra is Woronowicz' method of extending the bicovariant bimodule F. Let /70 be the free left A-module with one generator, denoted e, and let i; be the left A-module /70 e F. Any element E f can be written as = be + p with p E F and b E A uniquely determined by p. For such fi and a G A we define pa = bae (bdia + Pa),

(59)

512

14. Covariant Differential Calculus on Quantum Groups

1-L(f3)

= 2i(b)(1 ® e) + 2iL(p),

= 2(b)(e ® 1) + 2iR(p)•

(60)

Then one easily verifies that f becomes a bicovariant bimodule over A with right A-module action of A on 1' given by (59) and left and right coactions Zr, and ZR defined by (60), respectively. From (59) and (60) applied with = p and b = 0 we see that F is a bicovariant subbimodule of P. Hence C by Proposition 13.9. 17- \ by dfi := 0 A r) (-1)5 A e We define a linear mapping d : Pe" for r) C r An . Obviously, d satisfies the graded Leibniz rule. Since e is leftand right-invariant by (60), we have 5-(e OA 0) = e OA e in P° 2 by (13.27) and hence e® A e c ker A2 =6 2 ) so that 0 A e = O in t' Since d(d) = enenfi-finene for any C f, it follows that d2 = 0. Thus, feA is a differential calculus over A with differentiation d. Using the fact that 9 is left- and right-invariant, it is easy to check that this DC is bicovariant. From formula (59) applied with p = 0 and b = 1, we have di a = Oa— ae for a C A. Therefore, d rA =_- d1. The proof of the fact that /'‘' is a bicovariant DC is complete as soon as we have shown that dp E /1^ for any p E feA . It suffices to prove this for p C .r‘\ of the form p= adiai A • • - A d i a riwith a, a l , • • • , a, C A, because Fe^ is spanned by such elements. Using the facts that d [A =_- d1, d2 = 0 and that d satisfies the graded Leibniz rule, we derive that dp = da A dai A • • • A dan = di a A d i a l A • • • A di an E Te^ . The latter formula proves also the uniqueness of d. Finally, it remains to show that d® o A = o d. Since F is a bicovariant FODC with differentiation d i = drA, we have 2i(d(a)) = 2q,(da)+L1R(da) = da(i) 0 a(2)= d°(A(a)) for a E A. Since ii is an Noa(i) 0 da(2) + graded homomorphism and d satisfies the graded Leibniz rule, the equality d® o L = ad on F^ n follows by induction on n. The relation d® o A=Aod can also be derived from the definition dp = e A p—(-1)np A e, p E //An , combined with the fact that ,A(e) = 1 0 + 1.

re"

.

As noted in Subsect. 13.2.2, the two-sided ideal (6 2 ) generated by 62 = ker (a — I) is a Hopf ideal of TA (f ). Therefore, the quotient .ri\2 := TA(F)/(e 2 ) is also an No-graded super Hopf algebra. The preceding Proof applies to feA2 as well and shows that Tr2 is also a bicovariant DC over A and a differential Hopf algebra, It is obirious that the DC fe^ is a quotient of 1-1\2 and that /IA has F as its first order part. The bicovariant DC /11\2 is often more convenient to work with rather than fe^ . The reason is that it might be difficult to describe en = ker An for general n, while it is usually easier to determine 6 2 ker (a — I). Now we turn to the universal DC fu^ of the bicovariant FODC F defined in Subsect. 14.3.3. Let R. denote the right ideal Te r .

Theorem 18. The universal differential calculus F t," of a bicovari ant first order calculus F over A is bicovari ant. The ideal SR. is a Hopf ideal of the No -graded super Hopf algebra TA (f ), so the quotient fuA = TA( /1 /SR, is also an No -graded super Hopf algebra. With these structures, Fu^ is a differential Hopf algebra. )

14.4 Higher Order Bicovariant Differential Calculi

513

Proof. By Proposition 12.4(i) and the Remark at the end of Subsect. 12.2.3,

17,'

is left- and right-covariant and hence bicovariant. Let x E R. Recall that Now we prove that SR, is a Hopf ideal of T.s4

(r).

= 1 ® w(a) +

E w(a (2 )) o

S(a (1 ))a (3) ,

a E A,

(61)

by (13.36) and (28). Since AdR(R,) C R. 0 A, we have (id 0 2)AdR(s) = E x( 3) 0 S(x (2 ))s (4) 0 S(x (1 ))x (5 ) E R0 A0 A. Hence, since w(.) vanishes on R-, we obtain

E ,(x (3) ) ® s(i (2) ), (4) ,,(s(i (1) )x (5) ) = 0,

(62)

E,,(, (3) ),,(s(s (2) )s oos(s(x ( ox (5) ) = 0.

(63)

Using definition (1.42) of the No-graded product • and equations (61), (4) and finally (62) we compute =

E 24(w(s(i))) •

_ 1 ® c.4.,(i (i) )w(x (2) ) _ E w(x (3) ) ® w(s(os(s(2))x(4) = Y: + E w(x 2 ) ® s(s ( ox (3) w(x (4) ) + E w(x (2) )w(x (5) ) ® s(x(i))x(3)s(x(4))s(6) =

1 0 S(x) —

,A(x(4) ) ® s(x(3))s(5)w(s(,)s(x(2))x(6))

+ E w(x) ® + E w(x (2) ),4)(x (3) ) ® s(x ( 1 ) )x (4) 1 ® S(x) + E,,(i(3) ) ® s(x (2) )x (4) „,(s(x(2) )x (5) ) +(S 0 id)AdR(x) 1 0 8(x) + (8 0 id)AdR(x) E A 0 8 (R) + 8 (R) 0 A. Thus, it follows that 24(87z ) C T(F) 0 SR, + SR ® From (61) one easily derives that

S'4-(w(a)) =

—

TAT).

E w(a(2))45(a(3))S 2 (a(1)),

Using the Hopf algebra axioms we compute

a E A.

(64)

514

14. Covariant Differential Calculus on Quantum Groups

sEs(x)) = E sEw(s(1))w(s(2))) = E si94w(s(2)))*(w(s(1))) w (x , (1)) , (5), ),5(x _ (6),S2 ,S(4),(h),X(2),S,S(3), }_LS' 2 (x

_

u.,(s (6,),.,(s (2) s(s (5) ), 7) )s(. (8) ),52(s (4) )s(x (3) )s2(x ( 1 ) )

= - Ew(s (4 ) )w(x (2) s(s (3) ). ( 5 ) )s(s (6 ) )s2(x (0 ) +

w(x (3) ),,,(s(s (2) ),40s(s(s ( ox (5) )

_ Ew(x (2) ),,,(s (3) )s(x (4) )s2(x ( , ) ) = 0- Es(s (2) )s(s(s,,os (3) ) = _m(s s)AdR(s)

E S(R)A.

For the third equality we used (64) and for the fourth we applied (13.10) combined with (3) twice. The seventh equality follows from (63) and (53). The preceding formula implies that S(4.5R.) C S. Therefore, SR, is a Hopf ideal of 774 (T) and hence Tu^ = T(F)/SR. is an No-graded super Hopf algebra. Arguing as in the proof of Theorem 17, it follows that d® 0 = 0 d, so that r\ is indeed a differential Hopf algebra. Finally, let us note that the DC and Tv" do not coincide in general as shown by the following artificial but simple example. Example 7. Let r be the universal FODC over A. Then R. = {0} and hence {0} in general (for example, if s E A is Tv" = TA ( T"). Since ker (a — I) group-like, then we have (a — I)(.4.)(s)0Aw(s)) = 0 by (29)), eA is different A from F'.

14.4.2 Quantum Lie Derivatives and Contraction Operators In this subsection we investigate in detail two basic operations on differential forms (Lie derivatives and interior products) for the bicovariant differential calculus /-1 `= T.,.49 (r)/e defined in the preceding subsection. First we introduce a quantum analog of the Lie derivative of differential forms along a vector field. Since is a right comodule for A, we know from Proposition 1.15 that

L1(p) f.p := (id 0 f),R(p),

p E F, f E A',

(65)

defines a left action of the algebra A' on Te^. Note that for a E A = FAO c reA, we have f .a = (id 0 f)zA(a) which is in accordance with the notation from (1.65). Thus, re^ becomes a left A'-module and we have L e = I and L1L 9 = L f9 for f, g E A'. Combined with (37) the latter yields

Lx i Lx i

=

= CtLx,

for i, j

E

/,

Lx preserves the commutator of the quantum Lie that is, the map X L1 (Pi) A L f (2) (p2 ) for f in algebra T. Moreover, we have L f (p i A p2 ) = the dual Hopf algebra A°. By (14), this gives

E

14.4 Higher Order Bicovariant Differential Calculi

515

(66)

Pi, P2 E

Lx,(pi A P2) = Pi A Lx i (P2) + Lxk (Pi) A lifi (p2),

(Strictly speaking, (14) applies only when dim T is finite, since then Xi E A°. But (66) is also valid in the general case and follows then from (14) Definition 6. The map Lx : Lie derivative along X E T.

-) F defined by (65) is called the quantum

Equation (66) can be viewed as a twisted Leibniz rule for Lx i . It is easy to check that for any f E A' the map L f commutes with the right action that is, we have 1:11 the left action LL and the differentiation d of ,

(id

L1) o 46R = 46R

o

Lf,

(id

L f ) 0 461, =

O

Lf

L f d = dL f

,

For instance, we verify the latter by computing L f dp = (id® A6R(dp) (id® f)(dOid) E dp(0) f (P( 1) ) = E d(P(o)f (P(l))) = dLf(P)Moreover, since ,6R(wk) = ‘4.), ® 4, (38) yields L x, (wk) = Ctio.)i

and

(67)

L f; (Wk) = ari twi•

Next we generalize the notion of the interior product of a differential form by a vector field from differential geometry to the calculus f. Proposition 19. For any r E I there exists a unique linear mapping i(X r ) : Fr\ FA(n-i) for n E No , where Fen of degree -1 (that is, (X,-) FAn r A(_1) {0 }) such that i(X,-.)(w) (X,,,) for all c4.; c inv F and i(XT)(Pi A p2) = ( -1 ) 72 P1 A i(Xr)(P2)

i(Xs)(Pi) A L.N2 (P2)

(68)

for p i C FAn and p2 E F. On _1" \ , the mappings d, Lx, and (X,-) satisfy the identity Lx, = i(X r )d + di(Xr ). (69)

Proof. In order to get well-defined maps i(X r ) on _/", we first define these mappings for the tensor algebra TA (F) over F and then pass to the quotient Fe^ = TA(F)le. We freely use the multi-index notation established in Sect. 13.2. Let us set f := f • ft: for i = (ii, • • , j) and t = (k1, .,k) in ra• Further, recall that TA (F) is a left A'-module by Proposition 1.15 with action given by f.p := (id 0 A6R(p), f E A', p E TA(F). Let B 1 I and Bn := (-1) n+1 B; n _ 1 for n c N, n 2, where .13,7 n _ i is given by (13.40). We define a linear Mapping i r : TAU") -) TAU") of Clegree -1 by 4(0 ai(Bn ) r) we for = aiwi E F®n 7 ai E A, r E I. (Note that there are summations over j c In and t E P -1 .) Since An -iBn -=- ( -1 ) n+ l An by the second equality of (13.44), we have An—lir(0 =

(70)

ai (Bn)r' r) (An-i)twt =

Therefore, if (E en, then (A n ) 1179 ai = 0 and hence 4(0 E en-1 by (70). Hence i r ((S) ç 6 and ir passes to the quotient F = TA(F)/e '

14. Covariant Differential Calculus on Quantum Groups

516

reA. Since obviously Pn,n-1 = to define a linear mapping i(X,) : 1 yields le, 42 _1 1 tn-2tn-1) • • • tl •• tn-lb the definition (13.40) of B

Bn = (_1)n+11-

Cr n- lan-2

• • 0- 2 + an - an-2 ' • ' al

and hence .731,...,n+ k = ( -1 ) n Bn-f4,--,n+k + an+k-i. • • • anBi,...,n for lc, n E N, where the lower indices of B denote the variables on which B acts. Formula (13.31) implies that (Bri )fii = (Bn )M for E,1 C P. Let r, s C I. Since = wig(v) = aco, E I, by (13.29) we obtain the relation g.we =-

(e,8) ' (crk • • • ai ) (,, e) (.44/ for it C /k . Let anwn and btt.4.4 be finite sums with a", b C A for n C I, Ik and n, k C N. Using the preceding facts and the definition of i r we compute (-1)nanw n ir (bt we ) + i8(an con)g.(bt wt) = (-1)nanw n ir (btwe) + an (Bn ),ç,‘"5) wi(g.bt )(ff.we) = (-1)nanw n b t i r (we) + an (Br)' s) (fip( 1) .bt )wp(ak• • .0-1) (xt' n (g.bt )wnt (Bk) (p' r) wt + an (a k• • • a 1) ( xti

we ,

(B)'

(-1)na

= )(P,e,r) ,,,

+( crn+k- i.• • •an 13 1,...,ni(nl,e) w(P,r)i

=

= an (f:i•b t )ir(wwwe) = ir(a n wnb t we)•

Passing to the quotient in the preceding equation, we obtain (68) for arbitrary elements pl E l' A n and p2 E rAk• With obvious modifications this proof works for k = 0 and n = 0 as well. By linearity, (68) holds for all p2 C re^. The uniqueness assertion is clear. Using (68), (66) and the graded Leibniz rule for d, it is straightforward to check that (69) holds for the product pl A p2 as soon as it holds for pl and for p2 • So it suffices to prove the identity (69) applied to a c A and wk, II separately. Since i(X r )(c4.4) = 5i r by definition, (69) is obviously truekE for a E A. From the definition of i(Xr ) and the formulas (49), (67), and (68) we obtain

di(X T )(wk) i(X T )(dwk)

= d(Sk r 1) i(XT )(—clici wi A wi) =

—

=

Ctrwi

= Lxr (wk)•

D

We define i(X) for any X in the quantum Lie algebra T of r by requiring i(X) is linear. Then (69) is still valid for X C T. that the map X Definition 7. The form i(X)(p) C 11 \ is called the contraction of the form p E

re"

by X ET .

Equation (68) can be regarded as a twisted Leibniz rule for i(Xr ). This formula is useful for computations. For instance, it implies that i(Xr )(ap) = brjWi l etc. ai(X,)(p) for a E A and p E ./1\ 7 i(Xr )(Wi A wj) = 4:‘,01

14.5 Bicovariant Differential Calculi on Coquasitriangular Hopf Algebras

517

14.5 Bicovariant Differential Calculi on Coquasitriangular Hopf Algebras In this section A denotes a coquasitriangular Hopf algebra equipped with two universal r-forms r and s. Further, let u=(uii ) i ,i=1 ,...,d be a matrix corepresentation of A, that is, u is a matrix of elements u i E A such that ii(uii )=uik Ou ll and (t4)= 5 . Our aim is to construct bicovariant differential calculi over A by using the L-functionals associated with r resp. s and u. First let us repeat some notation and facts from Sect. 10.1. For this reason, let r be an arbitrary universal r-form on A. The L-functionals associated with r and u are the linear functionals / ±ii on A defined by 1+1(a) = r(a,14)

and 1 -1(a) = tquii , a),

a e A.

We introduce matrices R = (R), ii ------- () and D = (Dii ) by

R ni3:, 7 i : =--- r (ui,„ , u..);.,,, ) , .h.ii nm ..= r(u,S(

)), Dii :=--

V iVki c , . Z—iek

3

(71)

That is, R and D are the matrices appearing in (10.37) and (10.19), respec7-7.From the latter tively. Hence, by (10.20), we have S2 (uji ) = (D- 1) inunm Li equality and the properties of r and f we obtain the relations

= -kit'fi 1

( ulic ) =

(72)

— feci / = (k-1 )V 7 Sr3 i )(Uk/ )— il

(73)

IV RT:Tiri = binbkm 7

(74)

RP3F rtZ.rii =

1

1v 13 i737:M = (D-1 )ik(-k —i ) i nk Dn

(75)

where as usual fi = T 0 R. By (74), il is just the matrix i?' defined by (10.14). Recall from Subsect. 10.1.3 that the matrix L± =7.-- Y IP gives a representation of the algebra A. Let 1_ + '' be the contragredient representation of L± and let xi' be the contragredient corepresentation of u. That is, LI ,' and uc have the matrix elements (L± ,c» = S(l ii) and (uc) ji = S(u3i:), respectively. The starting point for the construction of bicovariant DC is the crucial observation that the pairs

Fr := (u,L - '')

and

.1-l; := (tic , L±)

(see Theorem 13.3 for this notation) are bicovariant bimodules over A. In order to prove that they are indeed bicovariant bimodules, by Theorem 13.3 it remains to verify the compatibility condition (13.23). Setting f ai-' = S (1 ki ) and using equation (10.5) and Proposition 10.2(v), we compute

518

14. Covariant Differential Calculus on Quantum Groups

L—dk 2

E s(lji)(a(o )ulica(2) = E t(ulc , S(a( 1)))11 1:a( 2) E r(ulc , a(o)u ilca(2) = E amujk r(4,a( 2))

3

amujk S(/ -i k )(a (2) ) = E k (flic .a)u3;,.

This proves condition (13.23) for Fr . The proof for 1'; is similar. In the following we will need the matrices LI = R = (R A i ) and = ) for the matrix corepresentation u and for both universal r-forms r and s. We shall express the dependence on the universal r-forms by adding r resp. s as a lower index. Since /7 and F. are bicovariant bimodules as just shown, 1 70Ars (li c 0

r12

Ill 14- 0 1-)

is also a bicovariant bimodule over A. By Theorem 13.3, the structure of the bicovariant bimodule F12 is described as follows. The vector space i1v-T 1 12 of left-invariant elements of F12 has a basis {O j , j = 1,2, ... d} such that the right and left actions of A on the A-bimodule F12 satisfy the relations

0,j a = (fn .a)0„,n ,

where

(76)

and the right coaction of A on this basis is given by where eir = (tic u) ini = S(utr )uT.

ZAR(Oii) = Onm vr,

(77)

We set 0 := T40 ii , where the matrix D = (Di) is defined by (71) for one of the forms r or s. Using again condition (10.5) one easily shows that ZA R (0) = 001. Since 0 G i 1v F12 is left-invariant, 0 is a biinvariant (that is, left- and rightinvariant) element of F12. We now define

da = Oa — a0,

a E A.

(78)

Further, let T be the span of linear functionals Xii

Din'

-

i,j = 1, 2,

Proposition 20. F := A.dA.A is a bicovariant FODC over the Hopf algebra A with differentiation d and quantum Lie algebra T.

Proof. It is clear that F is an A-subbimodule of F12 which is invariant under the left and right coactions of A and that d satisfies the Leibniz rule. Hence F is a FODC over A. Since 0 is biinvariant, we have 46R(da) =

E 0a( i ) a( 2) E a( i)0 0 a(2) = (d 0 id)46(a)

and similarly 4 L (da) = (id 0 46(a) by (78). Thus, f is bicovariant. Let WO be a basis of the vector space T. Let Xii = atili Y„. Setting 0„ := drii 190) we derive from (78) that da = = En (Yn .a)On and so w(a) =

14.5 Bicovariant Differential Calculi on Coquasitriangular Hopf Algebras

519

Y72 (a)0,2 for a E A. By (8) and (10), this implies that {O n } is a basis of i„,,F and that {17 } is the corresponding dual basis of Tr . In particular, T = Lin {Yn } is the quantum Lie algebra of F. 0 Of course, the dimension of the quantum Lie algebra T may be less than d2 . For instance, if A is commutative and r and s are the trivial universal r-form e e on A, then all Xii are zero. Let us assume in the remainder of this section that the linear functionals X , i , j = 1, 2, • • • , d, on A are linearly independent. Then the quantum Lie algebra T and the FODC F have dimension d2 . Since w(a) = 8(a( 1 )) da( 2) = Xii (a)00 for a E A, it follows then that Oii E AAA = F. Hence we have r = r12 and the set {00 } is a free left A-module basis of F. Moreover, it is the dual basis to {Xii }. Let a denote the braiding of roA r (see Subsect. 13.1.4). We denote by o- nik r the matrix coefficients of a with respect to the basis {000.400 of inv (roAr) (that is, 19- (000A6ki) = cr7cir8 6nm0A0„) and by C,!fir„.., the structure constants of the quantum Lie algebra T with respect to the basis {Xij } (see Subsect. 14.2.3). These quantities are determined by Proposition 21. , nm,rs ti, k1

fp, ;1 \pi \sm k s yys fz_ytyk xn1 -"r Jyr )wi k s i3P

D') (fir )

C nm kl ,rs

( 17' sg,s1( 17' s) .7pk — D7.7.6 knbiin.

Proof. From the second formula of (71) and Proposition 10.2(v) we obtain

4 S 2 (4)) = s(u ij ,

S(Vi )(S(7410) = Vi(S2(742))

,

(14- ) = )

Using this fact and the formulas (13.29), (77), (13.14), (76), (72), and (73) we compute nm,rs

k1

= gs(v r) = f l (s(unk ))f fsw (ur) = i;.,:( s( uns(itziw)( s(upkn i;V(ulin )s(10 (un (it zTYPn(k- s);Ppk(frnrrn(its)Ywsi.

=

Since Cnk ni — D rs e(vr) by (38), the second l rs = Xrs (Vrc irn ) = formula follows at once from the previous computation. 0 Under the assumption the matrix Rr is a scalar multiple of R., we now determine the spectral decomposition of the braiding map a. Let us write R instead of R r . For A = (Ar),B = E .C(Cd x Cd ) we define

(Br)

du

B)

737.:-kL ,ir s :=

fqirm

3pp

w11 'j

Clearly, if R. = aRT , then fl,3 = ccli-4 by (74). Hence the formula for ar."'" remains unchanged if we set R. — Rr = R therein. Therefore, we tj,k1 have a = , fi). Using formula (74) it follows that

14. Covariant Differential Calculus on Quantum Groups

520

CA1,B1)0(A2,B2)

= CA2A1,B1B2) and

4(/,/) = I.

(79)

the spectral decomposition of it (that is, Ei P = I, Pi2 = Pi and PP, =-- 0 for j 74 j). By (79), P := «Pi, Pi ) are also projections with E i Pii = I and Pii P,,, = 0 if (i, j) 74 (n, m). Hence the spectral decomposition of a is given by

Let

î = Ei A i Pi be

( 80)

a- =(k', fl) =

We briefly turn to another variant of the preceding construction. In order to define a bicovariant FODC F', we consider the bicovariant bimodule

F21 _ Fs OA

= (u uc L; 011).

Let us denote all structure quantities for F21 and F' by the same symbols as for F12 and F but equipped with a prime. It is obvious that 0' E oji is a biinvariant element of 11. Setting d'a := O'a - ak for a E A, we obtain a bicovariant FODC F' :=-A.A over A. For the FODC F' we have

flijm = S(is- 7)/;.1„

and

= S(/; 27,)/itin - 60 6.

(81)

Assume that the functionals X, i j = 1, 2, • - • , d, are linearly independent. Then the matrix coefficients of the braiding map a' with respect to the basis OA 0;ci l and the structure constants of the quantum Lie algebra with respect to the basis {Xi'i } are given by the formulas ,

(-1

C/ n kim,rs

zi,k1

( \np( bp \ix ( k -"s)zx k -"T )kyl -"r

-1 ly

I sw

(f \rw pm ,

rw

= ( 14 07 (RrYky(Rr ).9 „ (Rs) mp

e

Ors Onk 0

An advantage of the FODC F over F' is the fact that the 1-forms 00 transform under the right coaction LR just as the left-invariant 1-forms S(4)du i ldo. Indeed, by (28), we have for anyw(t4) PL(du) bicovariant FODC over A that ZiR (wii ) = S(uni )uyi. The bicovariant bimodules F12 and E21 and hence the bicovariant FODC F and F' over A are always isomorphic. It can be shown that there is an isomorphism of F21 to r12 and so of F' to F which maps (fIr )On' m to O. Now let A be a Hopf *-algebra. If both universal r-forms r and s are real ui)* 3 = S(un, i , j = 1, 2, ..., d, then the FODC F and F' are *-calculi. In order to prove this, first note that the reality of r resp. s implies that Di = D. From this fact and the formulas (10.47) we obtain (Xii )* = X» and (X)* = X , so F and r are indeed *-calculi by Proposition 6. Finally, let us turn to the higher order calculi. If we know the braiding map a, we can form the exterior algebra reA = TA(F)/6 over the bicovariant bimodule F (see Subsect. 13.2.2). In the present case the differentiation d of the bicovariant DC ./" from Theorem 17 can be expressed by the No-graded commutator and

(

14.6 Bicovariant Differential Calculi

dp = 0

A

on Quantized Simple Lie Groups

p— (-1)". p A 0 for p E ri An ,

nE

No.

521

(82)

Indeed, since the latter defines a bicovariant DC on Fe^ with first order part r, it must be the DC from Theorem 17. The differentiation of the bicovariant DC riej 2 = TA(F)/(e2) is also given by formula (82). All assertions of this paragraph remain valid when F is replaced by r'. Summarizing the preceding, we have developed a general method for the construction of bicovariant differential calculi over a coquasitriangular Hopf algebra A. These DC depend on a matrix corepresentation u =(u .;) and on two universal r-forms r and s of A. For many quantum groups there is a natural fundamental corepresentation u of A. But even if u is fixed, different choices of the universal r-forms r and s may still give nonisomorphic calculi in general.

14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups In what follows,

A denotes the Hopf algebra O(G g ) for one of the quantum

groups Gq = GLq (N), S L q (N),0 q (N), Sp q (N) and u (uji ) i ,j _ i ,...,N is the fundamental corepresentation. In this section we elaborate the method from the previous section in this situation. Complete proofs of most of the unproven assertions in the following treatment can be found in the papers [SS2], [SS3]. 14.6.1 A Family of Bicovariant First Order Differential Calculi Let us say that a complex number z is admissible if z L 0 for Gg = GL q (N), for G g = S L q (N) and z2 = 1 for Gg = 0q (N), Sp q (N). By ThezN = orem 10.9, the Hopf algebra A = 0(Gq ) is coquasitriangular with universal r-form r z and

r z (uji , unm ) = zR'A,

j, n, m = 1 7 2 7

7 N7

where z E C is admissible and the matrix R = (R A 2 ) is given by the formulas (9.13) resp. (9.30). Let s and y be admissible parameters and set z xy. We denote the FODC F defined in Subsect. 14.5 in the case r = rx , s = ry by F± and in the case r = (r x )21, s = (i)21 by F-- , z. From the description given below it will be clear that the FODC F = F±,, depends only on the product xy, so this notation is justified. It can be shown that the corresponding generators Xii , j , j = 1, 2, • • • , N, of the quantum Lie algebra T are linearly independent. Hence 1--1± ,z is an N2-dimensional bicovariant FODC over A. First we describe the bicovariant bimodule F±,z more explicitly. Recall from Subsect. 10.1.3 that the functional / ±ii goes into 17ii if the universal rform r is replaced by f21 • Thus, we have the following equality of bicovariant bimodules

522

14. Covariant Differential Calculus on Quantum Groups

= (tic

u,L,.±.

L).

Let us repeat the main structure data of the bicovariant bimodule f±,, according to Theorem 13.3. The matrix entries of LI r and LT ry, c are the functionals determined by the formulas (72) resp. (73) if I is replaced by ilk resp. y ik therein. Since xy = z, these formulas yield the commutation relations ra

(6±1 PM 9 Ir. jAi rs = ,,,±1(1k±1\ik ,41 )js UkUnm Jpn 1 11

(83)

between the basis elements Oii , j j = 1, 2, • • , N, of the vector space of leftinvariant elements of F±,, and the generators 'es . The right coaction of A on this basis is given by ,

ZAR(Oii) = Onm Set43ur. A convenient form of the corresponding functionals f4.?;,, is obtained as follows. Let Ec be the character of the algebra A such that Ec (uii ) = (6ii for j = 1, 2, • • N, where ( 0 for GL q (N), (N = 1 for SL q (N) and ( 2 = 1 for 0q (N) and Sp q (N). Obviously, Ec belongs to the center of the algebra denote the L-functionals with respect to rzo and u for . Further, let some fixed admissible parameter zo . Then the bimodule structure of F± is determined by the equations ,

eij a, = (fni3:m .a)Onm ,

where

fnrn = Eei +i n S(i Tm )

:= ZZ 0-2 .

(84)

Next we turn to the differentiation d of the FODC / 7±, z . From Example 10.3 we know that the matrix D = (Dii ) from (10.19) is a scalar multiple of the diagonal matrix (D i6ij ), where D i = q — 21' for Gq = GLq (N), SL q (N) and D i = q 2 /31 for Gq = °q (N), Spq (N). Therefore, after scaling the basis D iOii. Thus the elements Oii, the biinvariant element 0 of F+,z is 0 := differentiation d of the FODC /7±, z is defined by

Ei

da =

Oa — a0 =Y.

(Xii.a)0ii ,

where {Xii Iî j = 1, 2, • • • ,N} is a basis of the quantum Lie algebra of From (84) we immediately derive that ,

Xii =

En Dn Ec ins(iT

Before we continue the study of the bicovariant FODC 17 , we briefly comment on isomorphic calculi among them. First let us note that the FODC r of Sect. 14.5 is isomorphic to f = F±,,. (The map (0 1 ) inm Onn, Oiii ex.) For the quantum groups 0q (N) and tends to an isomorphism of f and Spq (N) the bicovariant bimodules 4,, and r_,, and so the corresponding such that FODC are isomorphic. (There is an isomorphism of 1-14_,, and (c—i ) nr krr Oii .) Similarly, the FODC Eh, and /"_,z for GL(2) and SL q (2) are isomorphic. Let us write f+ := 17+,1 and 17_ := 1"+,_. 1 for

14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups

523

the quantum groups 0q (N), Sp q (N) and SLq (2). (It can be shown that the FODC I-1+ , 1 and on SL q (2) defined above are indeed isomorphic to the FODC /-1± and respectively, defined in Subsect. 14.2.4, so the notation .r± is justified.) Summarizing, we have constructed so far the following bicovariant FODC:

r+,_ 1

- one-parameter families r+,, and T_, z , z E CVO}, for GL g (N),N>3; - a one-parameter family r±,z , z E CVO}, for GL q (2); - 2N calculi r±,z, = q-2 1 for SL q (N), N > 3; - two calculi 11F and L.. for 0q (N), Sp q (N) and SL q (2). It can be shown that different FODC of this list are not isomorphic at least if q is not a root of unity. Note that all these FODC have dimension N 2 . We close this subsection with some remarks about these FODC and Ft (see Sect. As noted in the paragraph before last, the FODC 14.5) are isomorphic. Hence their quantum Lie algebras coincide and the functionals Xiiri from (81) give another basis of the quantum Lie algebra of = T±, z . In the above notation, these functionals are written as

r

i +n — 21C• 23 =EtS(1 7n )1

4

i3

7

r = r±,z•

i7 — - 1 27 • • • 7 N.

(85)

Since S(l ni)/±in is equal to the functional Note that Ec = E if z = of defined by (10.39) for the universal r-form r = r zo , the basis elements the quantum Lie algebra of the bicovariant FODC are just the entries of the matrix E(L - El. This shows the importance of the matrix L = S(L - )L+ introduced in Subsect. 10.1.3 for the study of bicovariant differential calculi. Further, by Proposition 10.16(ii), .

r±,z

c:= Ec Ti LD -1 =

(86)

is a central element of A'. From formula (85) we see that c - c(1)E belongs to the quantum Lie algebra of .r+ , z and

tl(c) = Ej ak 0 Yk71, 1 where Ykn, := E(S(l k2 )/ ±in (D -1 ) i

(87)

If q is not a root of unity, it can be shown that the functionals Yk n,, k, n = 1,- - -,N, are linearly independent. Therefore, by (87) and the first paragraph of the proof of Proposition 11, the vector space T(c) defined by (36) coincides with the linear span of functionals X = Ec /ii . That is, the quantum Lie algebra of the bicovariant FODC F+ ,, is of the form TN, where c is the central element of A' given by (86). An analogous result holds also for the FODC and the universal r-form (r, ) 2 1 , because the passage from r to F21 interchanges the functionals 1+1 and Finally, we turn to the question of *-calculi. Suppose that the parameters q, s and y are real and A is one of the Hopf *-algebras in Proposition 10.10(i). Then the universal r-forms r and s are real and hence, as discussed in Sect. 14.5, the corresponding FODC r±,z , z = sy, are *-calculi.

r_,,

524

14. Covariant Differential Calculus on Quantum Groups

Next we assume that 141 = izi = Izol = 1, z 2 and A denotes one of the Hopf *-algebras from Proposition 10.10(ii). Then the universal r-forms r zo and (,.)21 on A are inverse real. Since z2 = zg, we have C = and hence C c* = ec. Therefore, it follows from the formulas (10.48) and (85) that all functionals (X 'i )* = (e l)* - 6iie = knm (D -1 )7X k/ r,, again belong to the quantum Lie algebra of Therefore, by Proposition 6, F± ,, is a *calculus. In particular, r±,z4 is a *-calculus by applying the preceding with z = 4. Setting z = +1 and zo = +1 we conclude that both FODC Ti+ and are *-calculi for 0q (n, 71), O q (71 1 n + 1) and Sp q (N; 14.6.2 Braiding and Structure Constants of the FODC

r±,,

The matrix coefficients (0-±)T of the braiding o-± and the structure constants (C±),nkim ,„ of the quantum Lie algebra of the FODC F± with respect to the bases Oii OA OM} and {X.0} are given by {

Proposition 22. ( , ±) ritir,nrs

(R fi„ rwzn(r? ±1 )172 (r? ±i )i9 (k f1 4;1 Dk_1 1)191

(c±) ri klm,r5 = (rRT2) ,yukn (i• ±1) xwrm (ik±1)gD y

LIruknulmurs

Proof. As noted in Example 10.3, D = cD i 6ii for some constant c 0. Since S2 (um n ) = an-l um n Dm by Propositions 9.10 and 9.13, we get zo ( r i) pwikp,-D i Di w ipp = fzo (uti l upk)D w—ip i p k-1D p

=

z o (S2 (un 7

S2 (ukP )) = f zo (u7 ,

=

ZoOk —i\wk )11:7 •

Moreover, for F+, z we have and i?' s y rz. Inserting these facts and equation (75) into the formulas in Proposition 21 we obtain the above formulas in the case of 17+ , z. The proof for F- i z is similar. 111

Let us turn to the spectral decomposition of the braiding map a±. Since RT. = (x/y)± 1 R., the assumption made in Sect. 14.5 is fulfilled and formula (80) gives the spectral decomposition of ± when the spectral decomposition of R known. Recall that a± then has precisely the eigenvalues E I, if fk± 1 possesses the eigenvalues A i , i E I. In the case Gq = GL q (N), S Lq (N) the eigenvalues of k are q and so a := o-± has the eigenvalues 1 , -q2 , --q -2 and satisfies the equation (a - I)(cr + q 2 1)(a + q 2 1)

0.

For the quantum groups Gq = O(N), Sp(N) the matrix Ik has the eigenvalues q,- q -1 ,Eq 6- N , hence a := 0-± has the eigenvalues 1, - q2, _ q - 2 7 N 9 q - N -q€N -2 , -4 - €N+ 2 and we have the equation (0._ 1-)( 0. +112.0(0. ±q -2 1-)(0._e_0( 0._ q -Nn(0. +q 6N-2 .0(0. +q -EN+2 .0 = 0.

14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups

525

In both cases the spectra of g+ and cr_ coincide and for q not a root of unity the operator g±+I is always invertible in contrast to the classical situation. The last formula shows another peculiarity that appears for the quantum groups 0q (N) and Spq (N). Recall that the space of 2-forms for the DC TeA from Theorem 17 is (T OA r)/ ker(o- - ./). Let T denote one of the FODC /-1± for Op (N) or Spq (N). In the classical limit q ---4 1 the eigenspaces of g± for the eigenvalues q N and q -N belong also to the symmetric part of T OA T. Thus, it seems to be natural in this case to take the quotient T A2 = (F 0A 1 )132 as the space of 2-forms, where 32 := ker(a - I) e ker (g - q N I) eker (a - q -N I). This definition has the advantage that the dimension of the space of leftinvariant 2-forms in ,r 2 is ( N22 ) just as in the case of an N2-dimensional rAn is then no longer an No classical first order calculus. But T A super Hopf algebra, because the comultiplication of TA(T) does not-grade map 32 to 32 0 A + A 0 32 when qN 1. Another disadvantage of this definition is that the dimensions of the spaces of higher order left-invariant forms inv rAn 7 n > 3, are smaller than in the classical case.

14.6.3 A Canonical Basis for the Left-Invariant 1-Forms In the preceding subsections we worked with the artificially chosen basis { aii } of the vector space i n, (f+ z ). A more natural and intrinsic subset of this space is formed by the projections

w(u) ==_ PL(duii-) __=_ S(u ik )du;i of the differentials duii of the generators ttii . The purpose of this subsection is to decide when the set fwiil is a basis of the vector space inv (F± z ) and ' to determine the basis transformation {9i.i } -0 fwiil in this case. Since {0 } is a basis of in, (T ±,z ), there is a complex N2 x N2 matrix = ((A±, z )Tir) such that

= (A±,,Yir 0,-,,,. It is clear that the set -PO is a basis of inv W I z ) if and only if the matrix ' A+, z is invertible. We shall use the notation A := q - C I ,

A + := q + q-1 and

B = (Bni) := (Drib„,,bii ),

where as earlier D i = q -22 for Gq =aL q (N), SL q (N) and Di =q2Pi for G q =0q (N), Sp q (N). Case 1 (Gq .--- aLq (N), 51 Lq (N)). Let us abbreviate z := E iAl i q -22 , -- The transformations A± ,, are s + 1 - q - 2 and s_ : =-- s - (2 2 N 4_ q2N2. -

A+ ,z = 4-1 zi ± (z - 1)B, A,z . _ Aq -2N- l z -l i + (.2.-1 — 1)B. The set ILwii i i, j = 1, 2, - • • , N} is a basis of the vector space inv(r± z ) if and only if z± lz± s. If this is fulfilled, the inverses of A±, z are given by

526

14. Covariant Differential Calculus on Quantum Groups

ATiz = qz -'1A - 1 / 7 • ki + (1 '''.- Z)a-f- , ZB),

where a±,,

Ai

_ q2N+i zA -i (i+ k1--.3 , -1

)ce- , z-B),

(z±ls± - s) -1 .

Case 2 (Gq = 0q (N), Spq (N)). Setting A± :-- A±,1 and kt := Eq N—E _ Eti e—N , we have

A+ = AEq N- €1. - A(C 1 )IC,

A_ = -Acq N-( / - 2B + A(C)IC.

The set fwii ( ti , j = 1, 2, • • • , NI is a vector space basis of in,r+, resp. inv f _, if and only if q 2N— 2 E 1, resp. 2+0+20 -1 O. In these cases the inverses AV are

A)I + (c -i )i k cti _ Aa+B ) ,

A -+1

7((EqN-E

A-1 =

--y((fe - A)I + (C -1 ) ti k ti + (A -

where A --.1 6qc-N 7 a , (2 + AA + 20 -1 ) --1 , 7 := Eqf - N A -1 (ti _ A)-'.

We close this subsection by considering the two FODC over 0(GL q (N)) which are obtained from the family r±,z in the particular case z = 1.

Example 8 (The calculi T +,1 and I' i for Glig (N)). Let Gq = GLq (N) and z =--- 1. In this case (83) holds with Oii replaced by wii , because the above transformation A+, 1 is a complex multiple of I. Therefore, we have dui..ur 3 8

= ui w . ur = ui ur fk±1\nmfrZ±Vci n

nj $

nmi

/ kx k

=,__ (1- ±1.vir un u rnik±10F/ Inm k x k

)38S(ux)d y tti

1j$Wsi Y _ (f?,±1)ir „nd,„m(k±1\ lc/ nm uk ill — Jis

(88)

or in matrix notation dill - U2

=--- q21 U1 - C11-12121 •

Thus for both calculi .r±, 1 we have linear commutation relations between the matrix entries u; and their differentials duii . For any higher order DC of f±,1 it follows from (88) by applying d that dui A du2 =---- —q21 dui A du2M1

.

Recall that by Proposition 9.11 the quantum space O(C) can be considered as a subalgebra of 0(aL q (N)) by identifying the generators x i and 74. Since (k±i)ili = eiski sii, under this identification (88) yields

dxi-xi = q±1- (k ±lAim x„dx. That is, on the subalgebra 0(q) the FODC left-covariant FODC FT from Theorem 12.6.

F±,i

of Glig (N) induces the A

14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups

527

14.6.4 Classification of Bicavariant First Order Differential Calculi It might be natural to find all bicovariant FODC over A for which the differentials du3i- of the matrix entries uji generate the left module of 1-forms. The "ordinary" differential calculus on the coresponding Lie groups obviously possesses this property. We first give some equivalent formulations of this condition and then we state the classification theorem. Lemma 23. For any left-covariant FODC F over ditions are equivalent: (i) F = Lin fa duii a E A, i, (ii) inv r -----.-- Lin {wii

A the following three con-

1,2, • • • ,N},

j = 1, 2, • • ,NI,

(iii) ker E = Ter ± Lin {14 - 6ii 1 I

j = 1, 2, • • • , N}.

Proof. (i)-)(ii) is clear, since PL(adu i ) = e(a)PL(dui ) e(a)w by (13.3). 00-00: Let a G ker e. By (ii), wr (a) = ci..i jj with a ii e C. Thus we 1)) = 0 and so a - aij (14 - 1 ) have co r (a - aij (ujj which proves (iii). (iii) -(i): Let a E A. By OW, Et is of the form a = aí(4 - 6kil) with a kjE C and x E Te r. Thus, wr(a) = w(ã) = akiwki = akiS(undu jz. and hence I" = A ci.) r (A) = Lin la du i }. EJ Theorem 24. Suppose that q is not a root of unity. Then there exist up to isomorphism precisely the following bicovariant FODC r over A such that dim F > 2 and F = Lin { a duji a E A, i,j = 1,2, • • ,N}:

- A = 0(aLi g (N)),N>3 : F±,z with z± 1-5±74 z,z c CVO}, - A = 0(G.4(2)) : 1-1± with z - 1( 1+ q2) 1+ 0, z E CVO; - A = 0(8L q (N)), N > 3: F± , z with z± 15± 74 s, zN q -2 ; ,,

- A = 0(84(2)) and A = 0(Spq (2)) : f+ and F_; - A = 0(0q (N)) and A = 0(8pq (N)), N > 4 : f+ and F_ with 2+ vA + 20 -1 740.

Two FODC over A of this list with different parameters are not isomorphic. For the quantum group 0q (3) under the assumptions of Theorem 24 there are precisely four such bicovariant FODC. These are the two 9-dimensional FODC f± from Sect. 14.6.1 and two additional 4-dimensional FODC developed in [S S3]. Theorem 24 and the preceding assertion show in particular that under the above assumptions for the quantum groups Sid q (N), °q (N) and Sp q (N) there is no bicovariant FODC whose dimension coincides with the dimension of the corresponding classical Lie group. Further, all calculi occurring in Theorem 24 have dimension N2 and belong to the family of FODC F±, z which was constructed by the general method of Sect. 14.5. The conditions z± 1 5± 5 for GL q (N) and Sli g (N),N>3, and 2 + /2A + 212A-1 74 0 for 0q (N) and Spq (N), N > 4, are only needed to ensure that

528

14. Covariant Differential Calculus on Quantum Groups

the corresponding FODC r±,z satisfy the assumption r =Lin {a du} or equivalently inv f = Lin {} (see Subsect. 14.6.3). If N is fixed, then except for the quantum group aL q (N) there exist only finitely many q for which these conditions apply. For all other values of q the list in Theorem 24 contains exactly 2N FODC for SL q (N), N > 3, and 2 FODC for 0q (N) and Spq (N), N > 4.

14.7 Notes The poineering work on covariant differential calculi on quantum groups is due to S. L. Woronowicz {Wor2J, [Worzli. Many results (but not all) in Sects. 14.1-3 appeared in the fundamental paper [Wor4]. The 3D-calculus in [Wr2] was the first example of a noncommutative differential calculus on quantum groups. The short approach given in Subsect. 14.1.3 follows the paper [SS4]. The results in Subsects. 14.3.2 and 14.3.3 on higher order left-covariant calculi are also from [SS4]. Proposition 11 is taken from [BM1]. The higher order bicovariant calculus 1'e^ was obtained in [Wor4j The fact that it yields a differential Hopf algebra was first observed in [Brz], but the proof therein works only for in. Complete proofs have been provided since then by [Dt1], one of the authors (KS.) and probably others. Note that our proof in the text was based on Proposition 13.10. Theorem 18 seems to be new. Quantum Lie derivatives and contraction operators appeared first in [SWZ11 and [AC], but a complete proof of Proposition 19 seems to be still missing in the literature. The method for the construction of bicovariant differential calculi developed in Sect. 14.5 is new at least in the general setting of coquasitriangular Hopf algebras. In the special situation of Subsect. 14.6.1 it gives the procedure used in [Jur] and [CSWW], where bicovariant calculi for the standard quantized simple Lie groups have been constructed in this manner. The two bicovariant calculi in Example 8 of Subsect. 14.6.2 have been found in [Mani.) and also in [Mal]. Theorem 24 was proved in {SS2J and [SS3]. There is now an extensive literature on differential calculi on quantum groups. In addition to the papers mentioned above, an (incomplete) list includes [Bd], [BMDS], [Ct], [CW], [DSWZ], [HS], [IV], [MH11], [Sud2], [Sud3], ISWZ2], [Ts].

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Index

Subject Index Action - adjoint 34, 181 - left 27,34 - right 27, 34 Algebra 7 - associative 6 Birman-Wenzl-Murakami 293 crossed product 349, 383 Drinfeld-Jimbo 163 free 8 h-adic 26 Hecke 293 homomorphism of 7 Hopf 4,13 - involution of 20 left comodule 33 left module 33 - multiplication of 7 - opposite 8 - right comodule 32 - right module 33 - unit of 7 - universal enveloping 18, 160 *-Algebra 20 - Hopf 20 - left comodule 33 - right comodule 33 Antipode 13 - inverse of 20, 247, 334 - order of 15 - polar decomposition of 426 - square of 15, 164, 247,334 341 Bargmann-Fock realization 151, 153 Basis at q = 0 226 Baxterization 296 Bialgebra 11 - braided 373

- comatched pair of 362 - coquasitriangular 331 cotriangular 331 double crossed coproduct 363 double crossed product 360 dual 22 FRT 304 - homomorphism of 12 - matched pair of 359 - quasitriangular 243 - triangular 243 *-Bialgebra 20 Bicharacter 24,336 Biedenharn-Elliott identity 88 12 Biideal Bimodule 28 - bicovariant 477 - - braiding of 481 homomorphism of 481 — tensor product of 481 - left-covariant 473 - right-covariant 477 Birman-Wenzl-Murakami algebra 293 Bosonizat ion 383 Braid - relation 247, 482 - group 173, 486 Braided - Hopf algebra 373, 376 - line 374 - matrix algebra 368 - product 373, 376 — algebra 378 - tensor category 378 - vector algebra 382 Braiding 373, 378,481 Brauer-Schur-Weyl duality 291

546

Index

Cartan - matrix 159 159 - - symmetrized - sub algebra 157 Category 376 - algebra in 378 - tensor, monoidal 377 - - braided 378 Character 422 Clebsch-Gordan coefficient 74,211 - reduced 219 Coaction - adjoint 34 - left 29,34 - right 29,34 Coalgebra 9 - cocommutative 9 - comultiplication of 9 - coopposite 9 - cosimple 398 - dual 22 - homomorphism of 9 - twisting of 357 *-Coalgebra 20 Coassociativity 9 Coboundary 355 2-Cocycle 354,357 - cohomologous 355 - counital 357 - invertible 357 - left 354 - right 354 Coefficient - Clebsch-Gordan 74, 211 - coalgebra 400 - q-binomial 39 - Racah 83 Coherent state 152 Coideal 9 Comodule 29 - algebra 32 - coalgebra 352 - left 30 - morphism of 30 - right 30 Compact quantum group - algebra 415 - C*-algebra 429,434 - matrix algebra 415 Cornultiplication 9 Contraction of a differential form 516 Convolution product 10

Coordinate algebra 98, 100, 304, 308, 310, 315, 320, 324, 388, 449 Copra duct 9 - smash 350 Corepresentation 29, 395, 429 - character of 406 - conjugate 398 - contragredient 398 - equivalent 30 - fundamental 305 - intertwiner 28, 397 - irreducible 30, 398 - isomorphism of 28 - matrix 30,397 - tensor product of 32, 397 - trivial 29 - unitarizable 402 - unitary 30, 114, 402 Counit 9 Crossed coproduct coalgebra 353, 382 Crossed product algebra 349, 383 Crystal basis 227, 231 - global 230,231

3D-Calculus 497,510 4D±-Ca1culs50 Diamond lemma 103 Differential - calculus 461 - - first order 457 - - left-covariant 463 - quotient 461 - - universal 463, 508 - - universal of a first order 463, 509 - *-calculus 462 - envelope 463, 508 - graded algebra 461 - Hopf algebra 511 - ideal 461 Double crossed - copro duct bialgebra 363 - product bialgebra 360 Drinfeld-Jimbo algebra 163 - center of 192, 194 - h-adic 166 - rational 165 Dual pairing 16, 21, 120,327 - nondegenerate 17, 120, 410, 440 Element - biinvariant

105, 443, 451

Index - canonical 256 - group-like 12 - invariant 30,324, 443, 451 - locally finite 181 - primitive 12 - twisted primitive 129 Exterior algebra - of a bicovariant bimodule 488 - of the quantum Euclidean space

Hecke algebra

293

Heisenberg - double 351 - representation 350 Hexagon relation 90 4, 13 Hopf algebra - braided 373, 376

320

- of the quantum plane 102 - of the quantum symplectic space 324

- of the quantum vector space Euler derivation 469

310

First order differential calculus

457

-

bicovariant 499 - classification of 527 - construction of 517, 521 - quantum Lie algebra of 501 - left-covariant 459, 491 dimension of 495 quantum tangent space of 494 quotient 493 right ideal associated with 493

right-covariant 464,498 - universal 463,492 Flip operator 6 Fock representation 149 Fourier transform on SUg (2)

- coquasitriangular 331 - cosemisimple 402 - deformation of 26 - differential 511 - dual 22 - factorizable 372 - h-adic 26,60 16 - homomorphism of - No-graded super 23 - quasi-cocommutative 248 - quasitriangular 61, 243 - super 24 - triangular 243 Hopf - *-algebra 20 - bimodule 477 16 - ideal

Intertwining operator

28, 30

Jordan-Schwinger realization

138

424 KMS condition Kostant partition funtion 176

117

FRT bialgebra 304

Leibniz rule 458 - graded 461 - twisted 506, 515, 516

Function - basic hypergeometric 41 - coordinate 5 - g-exponential 47

L-functional 276, 284, 342, 345

Linear form

- g-gamma 48 - g-trigonometrical 47 - spherical 129 Fundamental matrix 302

- AdR-invariant 335 - dominant 179 - integral 179

Linear functional - central 422 111, 187 - invariant - left-invariant 403 - right-invariant 128, 403

Gel'fand-Tsetlin basis 214 GNS representation 421 Group algebra 19

Haar functional 112,404 - faithfulness 419 Haar state 114, 420, 431

L-operator 275

Matrix coefficient, element

h-Adic

30, 74,

276, 396 Maurer-Cartan formula 507 Modular automorphism group Module 27

- algebra

26 - Drinfeld-Jimbo algebra 166 - Hopf algebra 26 - topology 25 Harish-Chandra homomorphism

547

- algebra 33,349 193

- coalgebra 33

425

548

Index

- isomorphism of 28 - left 27 - morphism of 28 - right 27 - simple 28 - Verma 178 - Yetter-Drinfeld 478

Multiplicative unitary Multiplicity of weight

427 177, 203

No-Graded super Hopf algebra 489, 511,512

23,

Partial derivative 466 176, 295 Partition function Pentagon equation 427 Peter-Weyl decomposition 110, 404, 410,439 Poincaré-Birkhoff-Witt theorem 160, 176

Polynomials - big q-Jacobi 52 - little q-Jacobi 51 - q-Hermite 50

Product - braided

373, 376 - covariantized 368 - smash 350 q-Binomial coefficients 39 q-Coherent state 152 q-Differentiation 44 q-Factorial 37 q-Integral 46 q-Number 37 q-Oscillator algebra 133 - symmetric 134

Quantum - algebra -

-

53 antisymmetric subspace 273 antisymmetrizer 488 Borel subgroup 436 Casimir element 54, 190, 248 codouble 359,363

- determinant 99,312 - dimension 207, 442 - double 252, 356, 361, 483 - Gerfand pair 446,450 - - infinitesimal 454 Quantum group

- compact 317,326 - GL q (N) 315 - GL q (N;R) 316

- inhomogeneous 387,392 320 - 0q (N) - CMN; R) 326 0q (N; v) 326 - Oq (n, n) 325 - 0q (n, n 1) 325 315 - SL q (N) 316 - SL q (N;R) - S L q (2) 100 102 - SLi g (2,R) 323 - S0q (N) 326 - S0q (N;R) - Sp q (N) 320 - Sp q (N; R) 325 - Sp q (N; v) 326 317 - Stlq (N) SUq (N; v) 317 102 - SUq (1, 1) 102 - SUq (2) C*-algebra 118 326 - USp q (N) - Uq (N) 316

Quantum - Euclidean space 320 - homogeneous space 442,451 - hyperboloid 472 - Killing form 189 - Lie algebra 501 - Lie derivative 515 - matrix space 98, 304, 310 - orthogonal sphere 322 - plane 101 - projective space 453 - space 32,436 125 -- isomorphism of - left 32,438 — right 32,436 - sphere 449 - 2-sphere 126 129 - spherical function

- subgroup 105,323 273 - symmetric subspace 324 - symplectic space - tangent space 494 - trace 206 - unitary group 316 - vector space 308, 310 - Weyl algebra 469 Yang-Baxter equation 61, 245, 247, 297, 333, 337

- zonal spherical function 450,453 space 33 *-Quantm 358 Quasi-Hopf algebra

Index

Racah coefficients 83 Real form 21, 58, 102, 172, 316, 325 - compact 172 Realificat ion 363 Reflection equation 346, 368 Representation 27 adjoint 35 character of 203 28, 200 completely reducible conjugate 35 cyclic 69, 145, 234 Fock 149 intertwiner of 28 - irreducible 28 N-finite 140 of class 1 217 of type 1 62, 178, 213 rational 31, 484 self-dual 271 semicyclic 70, 145 tensor product of 28, 69 weight 64, 177 with highest weight 61, 70, 141, 177, 199 with lowest weight 70, 141 *-Representation 28, 123, 420 R-Matrix 61, 247, 264 - of a vector representation 270 - universal 61, 243, 254, 259, 261 Root 157 - negative 158 positive 158 simple 158 subspace 157 - vector 175 Rosso form 188, 257 Schur orthogonality relations 419 Serre relations 160 Shuffle 19 Skew-copairing 358 Skew-pairing 250 - invertible 250 Subcoalgebra 9 Sweedler notation 10,31 Sweedler's Hopf algebra 19, 244, 337 Symmetric algebra 19 - of a bicovariant bimodule 490 Tensor - algebra

- - of a bicovariant bimodule 485

549

8, 19, 24 - - of a vector space - product 8 algebra coalgebra 9 of bicovariant bimodules 481 of corepresentations 32, 397 28, 69 of representations - operator 93, 219 Theorem 160, 176 - Poincaré-Birkhoff-Witt - Wigner-Eckart 93, 219 295 Transfer matrix 375 Transmutation 83 Triangular condition Universal - enveloping algebra 18, 160 - r-form 331 - - factorizable 372 - inverse real 336, 341 -- real 336,340 -- regular 366 61, 243, 254, 259, 261 - R-matrix Vector - representation 210, 267 64 - weight - - highest 64, 177 *-Vector space 20 Verma module 179 Weight 64, 177 - highest 64, 177 177, 203 - multiplicity of - representation 64, 177 64, 177 - space - vector 64 Weyl group 158 93, 219 Wigner-Eckart theorem Yang-Baxter equation 61, 245 - quantum - rational 297 297 - trigonometric Yetter-Drinfeld condition 478 Yetter-Drinfeld module 478, 484 289 Young diagram

Index

550

Index

of Symbols

An , A n k

AdL, AdR

A' A"P

Jr' 13,"13

Ag A[[h]] A[m 7-1.] A(R) A(R) A g (N) A g (N 2t, 2.1n [a], [a]q [ a] , [ a] q (a; q), (a; q),, a v adL, adR a. f

BWMr (p, q) 93„ Zg Cq

488 34 16 22 9 8 14 134 133 25 105 303 310 469 471 490 37 37 38 27 34, 181 36 486 293 453 486 174 54

c"i j,i121+i

82

Cg (t, j)

74 74 47 5 400 445 271, 317 310 25 25 8 11, 303 47 45

i

C, (11 , 12 , 1; i, m) Cosg z C(T) p) C( v) Cbi

c

(

,

311 99, 312 99 438 254 252

D (A, 8; a) D co (A, B; 0 .)

d id°

ai Ec,

Ear Er n)

252 359 457 511 466 157 175

Er

174 47 169

e(z)

106 47

E (z)

Élq - (Z)

Ern) For

174 175 170 422 106 36 313 5 157 21 157 50 293 420 453 51 8 515

Fr

fz f .a G L (N) gii

lin(x q) Hr (q) 7- ( A) 7--C; h(x q)

i(X r ) id

3,7

164

KA I K2p

K() KG

s)

r

Li,-

Lf

Lin

{

u ji }

L±,

r+ L r(A, 13) £(10 1+ , 1 - , 1; 1 i+i

14

i

,xn]

176 274 6 19 13 452 275 514 11 346 276, 346 285 10 27 342 346 276

Index [t, j)

77 285 412 310 377 397 13 7 38 208

0(S p :1 1 , v)) 0(SUq (2)) 0(SUq (N)) 0(Stlq (N; v)) 0(SUq (1, 1)) 0(Uq (N)) P, P± P(Gq )

ni AA' x [ Tri n] q, [7,1 1

208

p.„(,a , P)(x ;

n + , n_ 0(.13)

159 436

((*

101

O(C)

310 435 313 316 392 13 97 310 318 320 325 326 326 325 325 316 125 97 99 102 314 316 325 318 325 324 326

tzi M (n, r) Mq (N)

Mor (X, Y) Mor (u, y) M N (I C) 771

[nr

]

4

!

O (D N)

0(G L q (N)) 0(GL q (N R)) 0(IGq ) Q(M N (1K)) 0(Mq (2)) 0(Mq (N))

0(0q (N)) 0(0 :1 1 ) 0(0q (N; R)) 0(0q (N; u)) 0(0 :1 1 (6, 14) 0(0q (n,n + 1)) 0(04n, n))

(9(11 :1 1. ) O(S) 0(S L(2)) 0(S L q (2)) C9(SL q (2,1' ) 0(S L q (N)) C9(S L q (N; 0(80q (N; R)) 0(Sp q (N)) 0(Sp q (N; • )) )

0(Sp) 0(Sp q (N; v))

d q)

P + , P_ 39 P(g), Pr (0) Pnk q (0) Pn(x; a, bl q)

(21 Q+ Qq Q(q) qi

R

(7) Rep (G) Rep(_4(0) Rvw vw R q (1112131 112123 7 1) R,

rvw rz ryx

ST sisSill q Z 50q (N)

Spec (so), Spec (3,) SR, 8;1 8(.1) Sing Z TalA

TA

Tn rrV

551 326 102 317 317 102 316 179 410

52 273, 309 274 289 20, 486 210 51 164 319 165 161 89 5 232 246, 266 246 83 269, 309, 318 61, 243, 261 493 244 264 331 331 332 339 372 3, 13 426 370 322 485 47 323 195 509 509 47 68 178 199 179

Index

552

199 61

Twi Li

TA (11 ), Ts84 (11 ) T_ Tr T(c) ti( ti) U(g) Uh(g) Uh (5 12)

(g1 N)

q (S12)

Ûq (s12) • (512(R)) item Uq (sliv) Uq (s1(N,R) C7q (soN)) Uq (su2)) Uq (Sill ,1)) •

(SU

N

q• (511(a

• • •

I

al))

Uq (b +), Uq (b_) Ug,(2)

291 485 504 494 502 106 18, 160 166 60 163 53 57 59 164 173 288 59 59 172 172 168 161

• (13) Ùq (g)

189 164

• (g) • (13)

281

Uq(n+) Uq(1—) Uf ( n+)

UV3 (11.--)

U(R) V[[11]] vii va (v, f) Wa

XL(f, R), XR(f; R) Xq,-y z ?'

hq

A

r+1 r-

286 168 168 169 285 5, 11, 303 100, 303 25, 202 30 27 479 158 307 471 437 192

473 477 517

invr Fjflv rC

rr,

rq (n)

48

AO*

3,9 158 442 10 158 473 477 485 3,9 485 7 498 100 102

A(C)

310

A(0 :1 1 )

320

A(0 :1 ( 6, u))

326

A(8p)

324

A(Sp iliv (6, u))

326 24 462 492 525 488 481 483 250 6 43

bi i

6(n)

i6+ L • R

•

46 T

ET ET

77(.)

A(V) Q(X)

On (X)

w(*) Lou

en

o. a2,7 71 .

201(a, 0; -y; q, z)

er

2cpi (a, b; c; q, z)

ou j

E a( i )

408 40 36, 396 395 463 518

a(2)

10

E v(0 ) v ( 1 )

31

f(x)d q s

46

461 457 467, 504

FA 2 rAn

461

( . 7 ')

11‘`

488

1*,

Rh\

463

HI

26 16 500