Quantum Groups and Their Representations (Theoretical and Mathematical Physics)

A.Klimyk, K.Schmudgen QUANTUM GROUPS AND THEIR REPRESENTATIONS Springer-Verlag, Berlin Heidelberg 1997

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. Table of Contents Part I. An Introduction to Quantum Groups 1. Hopf Algebras 3 1.1 Prolog: Examples of Hopf Algebras of Functions on Groups 3 1.2 Coalgebras, Bialgebras and Hopf Algebras 6 1.2.1 Algebras 6 1.2.2 Coalgebras 8 1.2.3 Bialgebras 11 1.2.4 Hopf Algebras 13 1.2.5* Dual Pairings of Hopf Algebras 16 1.2.6 Examples of Hopf Algebras 18 1.2.7 *-Structures 20 22 1.2.8* The Dual Hopf Algebra A ° 1.2.9* Super Hopf Algebras 23 1.2.10* h-Adic Hopf Algebras 25 1.3 Modules and Comodules of Hopf Algebras 27 1.3.1 Modules and Representations 27 1.3.2 Comodules and Corepresentations 29 1.3.3 Comodule Algebras and Related Concepts 32 1.3.4* Adjoint Actions and Coactions of Hopf Algebras 34 1.3.5* Corepresentations and Representations of Dually Paired 35 Coalgebras and Algebras 1.4 Notes 36 2. q-Calculus 37 2.1 Main Notions on q-Calculus 37 2.1.1 q-Numbers and q-Factorials 37 2.1.2 q-Binomial Coefficients 39 2.1.3 Basic Hypergeometric Functions 40 41 2.1.4 The Function 1ϕ0(a;q,z)

2.1.5 The Basic Hypergeometric Function 2ϕ1 2.1.6 Transformation Formulas for 3ϕ2 and 4ϕ3 2.1.7 q-Analog of the Binomial Theorem 2.2 q-Differentiation and q-Integration 2.2.1 q-Differentiation 2.2.2 q-Integral 2.2.3 q- Analog of the Exponential Function 2.2.4 q-Analog of the Gamma Function 2.3 q-Orthogonal Polynomials 2.3.1 Jacobi Matrices and Orthogonal Polynomials 2.3.2 q-Hermite Polynomials 2.3.3 Little q-Jacobi Polynomials 2.3.4 Big q-Jacobi Polynomials 2.4 Notes 3. The Quantum Algebra Uq(sl2) and Its Representations 3.1 The Quantum Algebras Uq(sl2) and Uh(sl2) 3.1.1 The Algebra Uq(sl2) 3.1.2 The Hopf Algebra Uq(sl2) 3.1.3 The Classical Limit of the Hopf Algebra Uq(sl2) 3.1.4 Real Forms of the Quantum Algebra Uq(sl2) 3.1.5 The h-Adic Hopf Algebra Uh(sl2) 3.2 Finite-Dimensional Representations of Uq(sl2) for q not a Root of Unity 3.2.1 The Representations Tωl 3.2.2 Weight Representations and Complete Reducibility ( 3.2.3 Finite-Dimensional Representations of U q (sl 2 ) and Uh(sl2)

42 43 44 44 44 46 47 48 49 49 50 51 52 52 53 53 53 55 57 58 60 61 61 63 65

3.3 Representations of Uq(sl2) for q a Root of Unity 3.3.1 The Center of Uq(sl2) 3.3.2 Representations of Uq(sl2) 3.3.3 Representations of U qres (sl 2 )

66 66 67 71

3.4 Tensor Products of Representations. Clebsch-Gordan Coefficients 3.4.1 Tensor Products of Representations Tl 3.4.2 Clebsch-Gordan Coefficients 3.4.3 Other Expressions for Clebsch-Gordan Coefficients 3.4.4 Symmetries of Clebsch-Gordan Coefficients 3.5 Racah Coefficients and 6j Symbols of Uq(sl2) 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

72 72 74 78 81 82 82 84 84 85 88 90 90

3.6 Tensor Operators and the Wigner-Eckart Theorem 3.6.1 Tensor Operators for Compact Lie Groups ( 3.6.2 Tensor Operators and the Wigner-Eckart Theorem for U q (sl 2 )

92 92 93

3.7 Applications 3.7.1 The Uq(sl2) Rotator Model of Deformed Nuclei 3.7.2 Electromagnetic Transitions in the Uq(sl2) Model 3.8 Notes 4. The Quantum Group SLq(2) and Its Representations 4.1 The Hopf Algebra O(SLq(2)) 4.1.1 The Bialgebra O(Mq(2)) 4.1.2 The Hopf Algebra O(SLq(2)) 4.1.3 A Geometric Approach to SLq(2) 4.1.4 Real Forms of O(SLq(2)) 4.1.5 The Diamond Lemma 4.2 Representations of the Quantum Group SLq(2) 4.2.1 Finite-Dimensional Corepresentations of O(SLq(2)): Main Results 4.2.2 A Decomposition of O(SLq(2)) 4.2.3 Finite-Dimensional Subcomodules of O(SLq(2)) 4.2.4 Calculation of the Matrix Coefficients 4.2.5 The Peter-Weyl Decomposition of O(SLq(2)) 4.2.6 The Haar Functional of O(SLq(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 O(SUq(2)) 4.3.3 The Fourier Transform on SUq(2) 4.3.4 *-Representations and the C*-Algebra of O(SUq(2)) 4.4 Duality of the Hopf Algebras Uq(sl2) and O(SLq(2)) 4.4.1 Dual Pairing of the Hopf Algebras Uq(sl2) and O(SLq(2)) 4.4.2 Corepresentations of O(SLq(2)) and Representations of Uq(sl2) 4.5 Quantum 2-Spheres 4.5.1 A Family of Quantum Spaces for SLq(2) 4.5.2 Decomposition of the Algebra O( S q2ρ )

94 94 95 96 97 97 97 99 101 102 103 104 104 105 106 108 110 111 113 113 114 117 117 119 119 123 124 124 126

4.5.3 Spherical Functions on S q2ρ

129

4.5.4 An Infinitesimal Characterization of O( S q2ρ )

129

4.6 Notes 5. The q-Oscillator Algebras and Their Representations 5.1 The q-Oscillator Algebras Aqc and Aq 5.1.1 Definitions and Algebraic Properties 5.1.2 Other Forms of the q-Oscillator Algebra ( 5.1.3 The q-Oscillator Algebra and the Quantum Algebra U q (sl 2 )

132 133 133 133 136 137

5.1.4 The q-Oscillator Algebras and the Quantum Space M q 2 ( 2)

140

5.2 Representations of q-Oscillator Algebras 5.2.1 N-Finite Representations 5.2.2 Irreducible Representations with Highest (Lowest) Weights 5.2.3 Representations Without Highest and Lowest Weights 5.2.4 Irreducible Representations of Aqc for q a Root of Unity

140 140 141 143 145

5.2.5 Irreducible *-Representations of Aqc and Aq 5.2.6 Irreducible *-Representations of Another q-Oscillator Algebra 5.3 The Fock Representation of the g-Oscillator Algebra 5.3.1 The Fock Representation 5.3.2 The Bargmann-Fock Realization 5.3.3 Coherent States 5.3.4 Bargmann-Fock Space Realization of Irreducible Representations of ( U q (sl 2 ) 5.4 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(g) 6.1.4 Some Algebra Automorphisms 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 Poincare-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 Poincare-Birkhoff-Witt Theorem 6.2.4 Representations with Highest Weights. 6.2.5 Verma Modules 6.2.6 Irreducible Representations with Highest Weights 6.2.7 The Left Adjoint Action of Uq(g) 6.3 The Quantum Killing Form and the Center of Uq(g) 6.3.1 A Dual Pairing of the Hopf Algebras U q (b + ) and U q (b − ) op

147

157 157 157 161 165 167 168 171 172 173 173 174 175 177 179 180 181 184 184

6.3.2 The Quantum Killing Form on Uq(g) 6.3.3 A Quantum Casimir Element 6.3.4 The Center of Uq(g) and the Harish-Chandra Homomorphism 6.3.5 The Center of Uq(g) for q a Root of Unity 6.4 Notes 7. Finite-Dimensional Representations of Drinfeld— Jimbo Algebras

187 189 192 194 196 197

148 149 149 150 152 153 154

197 197 200 202 203 204 205 207 208 211 212

7.1 General Properties of Finite-Dimensional Representations of Uq(g) 7.1.1 Weight Structure and Classification 7.1.2 Properties of Representations 7.1.3 Representations of h-Adic Drinfeld-Jimbo Algebras 7.1.4 Characters of Representations and Multiplicities of Weights 7.1.5 Separation of Elements of Uq(g) 7.1.6 The Quantum Trace of Finite-Dimensional Representations 7.2 Tensor Products of Representations 7.2.1 Multiplicities in Tensor Products of Representations 7.2.2 Clebsch-Gordan Coefficients ( 7.3 Representations of U q (gl n ) for q not a Root of Unity ( 7.3.1 The Hopf Algebra U q (gl n ) ( 7.3.2 Finite-Dimensional Representations of U q (gl n ) 7.3.3 Gel'fand-Tsetlin Bases and Explicit Formulas for Representations 7.3.4 Representations of Class 1 7.3.5 Tensor Products of Representations 7.3.6 Tensor Operators and the Wigner-Eckart Theorem 7.3.7 Clebsch-Gordan Coefficients for the Tensor Product Tm ⊗ T1 7.3.8 Clebsch-Gordan Coefficients for the Tensor Product Tm ⊗ Tp

214 217 218 219 220 221

7.3.9 The Tensor Product Tm ⊗ T1 for q ±1 → 0 7.4 Crystal Bases 7.4.1 Crystal Bases of Finite-Dimensional Modules 7.4.2 Existence and Uniqueness of Crystal Bases 7.4.3 Crystal Bases of Tensor Product Modules 7.4.4 Globalization of Crystal Bases 7.4.5 Crystal Bases of U 'q (n− )

224 225 226 227 228 229 230

7.5 Representations of U q (g) for q a Root of Unity

232

7.5.1 General Results 7.5.2 Cyclic Representations 7.5.3 Cyclic Representations of the Algebra U ε (sll +1 ) 7.5.4 Representations of Minimal Dimensions 7.5.5 Representations of U ε (sll +1 ) in Gel'fand-Tsetlin Bases 7.6 Applications 7.7 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

232 234 235 237 238 240 242 243 243 243 246 247 250

212 213

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(sl2) 8.3.2 The Universal R-Matrix for Uh(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-Matrices 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 qext ( g)

250 254 257 258 259 259 261 264 267 267 269 272 275 275 277 281

8.5.4 L-Functionals for Vector Representations of U q (g)

283

8.5.5 The Hopf Algebras U(R) and U qL ( g)

285

8.6 An Analog of the Brauer-Schur-Weyl Duality ~ 8.6.1 The Algebras U q (so N ) Uq(soN)

288 288

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 8.7.1 Baxterization 8.7.2 Elliptic Solutions of the Quantum Yang-Baxter Equation 8.7.3 R-Matrices and Integrable Systems 8.8 Notes 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 χL(f; R) and χR(f; R) 9.2 The Quantum Groups GLq(N) and SLq(N) 9.2.1 The Quantum Matrix Space Mq(N) and the Quantum Vector Space C qN 9.2.2 Quantum Determinants 9.2.3 The Quantum Groups GLq(N) and SLq(N) 9.2.4 Real Forms of GLq(N) and SLq(N) and *-Quantum Spaces 9.3 The Quantum Groups Oq(N) and Spq(N)

289 291 293 294 295 297 298 300 303 303 303 307 309 310

311 313 316 317

9.3.1 The Hopf Algebras O(Oq(N)) and O(Spq(N)) 9.3.2 The Quantum Vector Space for the Quantum Group Oq(N) 9.3.3 The Quantum Group SOq(N) 9.3.4 The Quantum Vector Space for the Quantum Group Spq(N) 9.3.5 Real Forms of Oq(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(Gq) 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 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 11. Corepresentation Theory and Compact Quantum Groups 11.1 Corepresentations of Hopf Algebras 11.1.1 Corepresentations 11.1.2 Intertwiners 11.1.3 Constructions of New Corepresentations 11.1.4 Irreducible Corepresentations 11.1.5 Unitary Corepresentations

318 320 323 324 325 327 330 331 331 331 337 342 349 349 352 354 357 359 362 363 365 365 370 376 377 380 382 386 388 390 394 395 395 395 397 397 398 401

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 11.4.4 Another Definition of CQG C*- Algebras 11.5 Finite- Dimensional Representations of GLq(N) 11.5.1 Some Quantum Subgroups of GLq(N) 11.5.2 Submodules of Relative Invariant Elements 11.5.3 Irreducible Representations of GLq(N) 11.5.4 Peter-Weyl Decomposition of O(GLq(N)) 11.5.5 Representations of the Quantum Group Uq(N) 11.6 Quantum Homogeneous Spaces 11.6.1 Definition of a Quantum Homogeneous Space 11.6.2 Quantum Homogeneous Spaces Associated with Quantum Subgroups 11.6.3 Quantum Gel'fand Pairs 11.6.4 The Quantum Homogeneous Space Uq(N-1)\Uq(N) 11.6.5 Quantum Homogeneous Spaces of Infinitesimally Invariant Elements 11.6.6 Quantum Projective Spaces 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

402 402 404 408 415 415 419 420 422 426 427 429 429 431 433 434 435 435 436 437 439 441 442 442 443 445 447 451 452 454 457 457 457 459 461 461 462 463 464 464 467

12.3.3 Covariant Differential Calculus on C qN 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 Left-Covariant Bimodules 13.1.2 Right-Covariant Bimodules 13.1.3 Bicovariant Bimodules (Hopf Bimodules) 13.1.4 Woronowicz' Braiding of Bicovariant Bimodules 13.1.5 Bicovariant Bimodules and Representations 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 14. Covariant Differential Calculus on Quantum Groups 14.1 Left-Covariant First Order Differential Calculi 14.1.1 Left-Covariant First Order Calculi and Their Right Ideals 14.1.2 The Quantum Tangent Space 14.1.3 An Example: The 3D-Calculus on SLq(2) 14.1.4 Another Left-Covariant Differential Calculus on SLq(2) 14.2 Bicovariant First Order Differential Calculi 14.2.1 Right-Covariant First Order Differential Calculi 14.2.2 Bicovariant First Order Differential Calculi 14.2.3 Quantum Lie Algebras of Bicovariant First Order Calculi 14.2.4 The 4D+- and the 4D--Calculus on SLq(2) 14.2.5 Examples of Bicovariant First Order Calculi on Simple Lie Groups 14.3 Higher Order Left-Covariant Differential Calculi 14.3.1 The Maurer-Cartan Formula 14.3.2 The Differential Envelope of a Hopf Algebra 14.3.3 The Universal DC of a Left-Covariant FODC 14.4 Higher Order Bicovariant Differential Calculi 14.4.1 Bicovariant Differential Calculi and Differential Hopf Algebras 14.4.2 Quantum Lie Derivatives and Contraction Operators 14.5 Bicovariant Differential Calculi on Coquasitriangular Hopf Algebras 14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups 14.6.1 A Family of Bicovariant First Order Differential Calculi 14.6.2 Braiding and Structure Constants of the FODC Γ± , z 14.6.3 A Canonical Basis for the Left-Invariant 1-Forms 14.6.4 Classification of Bicovariant First Order Differential Calculi 14.7 Notes Bibliography

468 471 472 473 473 473 477 477 480 483 485 485 488 490 491 491 491 494 496 498 498 498 499 500 504 505 506 506 507 508 511 511 514 517 521 521 524 525 527 528 529

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

545

Subject Index - dual 22 - FRT 304 - homomorphism of 12 - matched pair of 359 - quasitriangular 243 - triangular 243 *-Bialgebra 20 Bicharacter 24, 336 Biedenharn-Elliott identity 88 Biideal 12 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 Bosonization 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 Cartan - matrix 159 -- symmetrized 159 - subalgebra 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 Comultiplication 9 Contraction of a differential form 516 Convolution product 10 Coordinate algebra 98, 100, 304, 308, 310, 315, 320, 324, 388, 449

Coproduct 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±-Calculus 504 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 - coproduct 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 - 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 320 - of the quantum plane 102 - of the quantum symplectic space 324 - of the quantum vector space 310 Euler derivation 469 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 SUq(2) 117 FRT bialgebra 304 Function - basic hypergeometric 41 - coordinate 5 - q-exponential 47 - q-gamma 48 - q-trigonometrical 47 - spherical 129 Fundamental matrix 302 Gel'fand-Tsetlin basis 214 GNS representation 421 Group algebra 19 Haar functional 112, 404

- faithfulness 419 Haar state 114, 420, 431 h-Adic - algebra 26 - Drinfeld-Jimbo algebra 166 - Hopf algebra 26 - topology 25 Harish-Chandra homomorphism 193 Hecke algebra 293 Heisenberg - double 351 - representation 350 Hexagon relation 90 Hopf algebra 4, 13 - braided 373, 376 - coquasitriangular 331 - cosemisimple 402 - deformation of 26 - differential 511 - dual 22 - factorizable 372 - h-adic 26, 60 - homomorphism of 16 - N0 -graded super 23 - quasi-cocommutative 248 - quasitriangular 61, 243 - super 24 - triangular 243 Hopf - *-algebra 20 - bimodule 477 - ideal 16 Intertwining operator 28, 30 Jordan-Schwinger realization 138 KMS condition 424 Kostant partition function 176 Leibniz rule 458 - graded 461 - twisted 506, 515, 516 L-functional 276, 284, 342, 345 Linear form - AdR-invariant 335 - dominant 179

- integral 179 Linear functional - central 422 - invariant 111, 187 - left-invariant 403 - right-invariant 128, 403 L-operator 275 Matrix coefficient, element 30, 74, 276, 396 Maurer-Cartan formula 507 Modular automorphism group 425 Module 27 - algebra 33, 349 - coalgebra 33 - isomorphism of 28 - left 27 - morphism of 28 - right 27 - simple 28 - Verma 178 - Yetter-Drinfeld 478 Multiplicative unitary 427 Multiplicity of weight 177, 203 N0 -Graded super Hopf algebra 23, 489, 511, 512 Partial derivative 466 Partition function 176, 295 Pentagon equation 427 Peter-Weyl decomposition 110, 404, 410, 439 Poincare-Birkhoff-Witt theorem 160, 176 Polynomials - big q-Jacobi 52 - little g-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 - Gel'fand pair 446, 450 -- infinitesimal 454 Quantum group - compact 317, 326 - GLq(N) 315 - GLq(N;R) 316 - inhomogeneous 387, 392 - Oq(N) 320 - Oq(N;R) 326 - Oq(N; ν) 326 - Oq(n,n) 325 - Oq(n,n+1) 325 - SLq(N) 315 - SLq(N;R) 316 - SLq(2) 100 - SLq (2,R) 102 - SOq(N) 323 - SOq(N;R) 326 - Spq(N) 320 - Spq(N;R) 325 - Spq(N; ν) 326 - SUq(N) 317 - SUq(N;ν) 317 - SUq(1,1) 102 - SUq(2) 102 - C*-algebra 118 - USpq(N) 326 - 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 -- isomorphism of 125 - left 32, 438 -- right 32, 436 - sphere 449 - 2-sphere 126 - spherical function 129 - subgroup 105, 323 - symmetric subspace 273 - symplectic space 324 - 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 *-Quantum space 33 Quasi-Hopf algebra 358 Racah coefficients 83 Real form 21, 58, 102, 172, 316, 325 - compact 172 Realification 363 Reflection equation 346, 368 Representation 27 - adjoint 35 - character of 203 - completely reducible 28, 200 - 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 -- of a vector space 8, 19, 24 - product -- algebra 8 -- coalgebra 9 -- of bicovariant bimodules 481 - - of corepresentations 32, 397 -- of representations 28, 69 - operator 93, 219

Theorem - Poincare-Birkhoff-Witt 160, 176 - Wigner-Eckart 93, 219 Transfer matrix 295 Transmutation 375 Triangular condition 83 Universal - enveloping algebra 18, 160 - r-form 331 -- factorizable 372 -- inverse real 336, 341 -- real 336, 340 -- regular 366 - R-matrix 61, 243, 254, 259, 261 Vector - representation 210, 267 - weight 64 -- highest 64, 177

*-Vector space 20 Verma module 179 Weight 64, 177 - highest 64, 177 - multiplicity of 177, 203 - representation 64, 177 - space 64, 177 - vector 64 Weyl group 158 Wigner-Eckart theorem 93, 219 Yang-Baxter equation - quantum 61, 245 - rational 297 - trigonometric 297 Yetter-Drinfeld condition 478 Yetter-Drinfeld module 478, 484 Young diagram 289