HOPF ALGEBRAS An Introduction
Sorin DGsc5lescu Constantin Ngst5sescu +rban Raianu
HOPF ALGEBRAS
PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS Zuhair Nashed Universityof Delaware Newark, Delaware
Earl J. Taft Rutgers University New Brunswick, New Jersey
EDITORIAL BOARD M. S. Baouendi University of California, Sun Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology
Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University
S. Kobayashi Universityof California, Berkeley
David L. Russell Virginia Polytechnic Institute and State University
Marvin Marcus University of California, Santa Barbara
Walter Schempp Universitat Siegen
W. S. Massey Yale University
Mark Teply University of Wisconsin, Milwaukee
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano, Integral Formulas in Riemannian Geometty (1970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood, trans.) (1970) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation ed.; K. Makowski. trans.) (1971) 5. L. Nariciet a/., Functional Analysis and Valuation Theory (1971) 6. S. S. Passman, Infinite Group Rings (1971) 7. L. Domhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular RepresentationTheory (1971,1972) 8. W. Boothby and G. L. Weiss, eds., Symmetric Spaces (1972) 9. Y. Matsushima. Differentiable Manifolds (E. T. Kobavashi, trans.). (1972) . L. E. Ward, ~ r . , . ~ o ~(1972) olo~~ A. Babakhanian, Cohomological Methods in Group Theory (1972) R. Gilmer, Multiplicative ldeal Theory (1972) J. Yeh, Stochastic Processes and the Wiener Integral (1973) J. Barns-Neto, lntroduction to the Theory of Distributions (1973) R. Larsen, Functional Analysis (1973) K. Yano and S. Ishihara, Tangent and Cotangent Bundles (1973) C.Procesi, Rings with Polynomial Identities (1973) R. Hermann, Geometry, Physics, and Systems (1973) N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) J. DieudonnB. lntroduction to the Theow of Formal GrouDs . (1. 973), 21. 1. Vaisman, dohomology and ~ifferentialForms (1973) 22. 6.-Y. Chen. Geometw of Submanifolds (1973) Finite ~imensional~ultilinear~ l ~ e b(in r atwo parts) (1973, 1975) 23. M. 24. R. Larsen, Banach Algebras (1973) 25. R. 0.Kuiala and A. L. Vitter. eds.. Value Distribution Theow: Part A; Part 8: Deficit t by ~ i i h e l mtoll (1973) and ~ e z b uEstimates 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) 28. B. R. McDonald, Finite Rings with Identity (1974) 29. J. Satake. Linear Algebra (S. Koh et al.. trans.) (1975) 30. J. S. Golan, ~ocalizationof on commutative dings (1975) 31. G. Klambauer, MathematicalAnalysis (1975) 32. M. K. Agoston, Algebraic ~opology(1976) . 33. K R. Goodearf, Ring Theory (1976) 34. L. E. Mansfield, Linear Algebra with Geometric Applications (1976) 35. N. J. Pullman. Matrix Theorv and Its Ao~lications11976) 36. B. R. ~ c ~ o n a l ~de, o m e t r i c ~ l ~ edbirear Local dings (1976) 37. C. W. Groetsch. Generalized Inverses of Linear Operators (1977)' J. E. ~uczkowskiand J. L. Gersting, Abstract ~lgebra(1977) C. 0. Christenson and W. L. Voxman, Aspects of Topology (1977) M. Nagata, Field Theory (1977) R. L. Long, Algebraic Number Theory (1977) W. F. Pfeffer, Integrals and Measures (1977) R. L. Wheeden and A. Zygmund, Measure and Integral (1977) J. H. Curtiss, lntroduction to Functions of a Complex Variable (1978) K. Hrbacek and T. Jech, lntroduction to Set Theory (1978) W. S. Massey, Homology and Cohomology Theory (1978) M. Marcus, lntroduction to Modem Algebra (1978) E. C. Young, Vector and Tensor Analysis (1978) S. 8. Nadler, Jr., Hyperspacesof Sets (1978) S. K. Segal, Topics in Group Kings (1978) A. C. M. van Rooii. Non-Archimedean Functional Analvsis 11978)' 52. L. Corwin and R.-~zczarba,Calculus in Vector spaces (19?9) 53. C. Sadoskv, Interpolation of Operators and Singular Integrals (1979) 54. J. Cronin, ~ifferential~quations(1980) 55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980)
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I. Vaisman, Foundations of Three-DimensionalEuclidean Geometry (1980) H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1980) S. B. Chae, Lebesgue Integration (1980) C. S. Rees et a/., Theory and Applications of Fourier Analysis (1981) L. Nachbin, Introductionto Functional Analysis (R. M. Aron, trans.) (1981) G. Onech and M. Omch, Plane Algebraic Curves (1981) R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysls 11981\ ..--'I
94. 95. 96. 97. 98. 99. 100. 101. 102. 103.
W. L. Voxman and R. H. Goetschel, Advanced Calculus (1981) L. J. Corwin and R. H. Szczarba, MultivariableCalculus (1982) V. I.Istrijtescu, lntroduction to Linear Operator Theory (1981) R. D. Jawinen, Finite and Infinite Dimensional Linear Spaces (1981) J. K. Beem and P. E. Ehdich, Global Lorentzian Geometry (1981) D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981) J. W. Brewer and M. K Smith, eds., Emmy Noether: A Tribute (1981) K H. Kim, Boolean Matrix Theory and Applications (1982) T. W. Wieting, The MathematicalTheory of Chromatic Plane Ornaments (1982) D. B.Gauld, Differential Topology (1982) R. L. Faber, Foundationsof Euclidean and Non-Euclidean Geometry (1983) M. Canneli, Statistical Theory and Random Matrices (1983) J. H. Camrth et a/., The Theory of Topological Semigroups (1983) R. L. Faber, Differential Geometry and Relativity Theory (1983) S. Bameff, Polynomials and Linear Control Systems (1983) G. Karpilovsky, Commutative Group Algebras (1983) F. Van Oystaeyen and A. Verschoren, Relative lnvariants of Rings (1983) I.Vaisman, A First Course in Differential Geometry (1984) G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) T. Albu and C. Ni3stdsescu, Relative Finiteness in Module Theory (1984) K. Hrbacek and T. Jech, lntroduction to Set Theory: Second Edition (1984) F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1984) B. R. McDonald, Linear Algebra Over Commutative Rings (1984) M. Namba, Geometry of Projective Algebraic Curves (1984) G.F. Webb, Theory of NonlinearAge-Dependent Population Dynamics (1985) M. R. Bremner et a/., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) A. E. Fekete, Real Linear Algebra (1985) S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) A. J. Jeni, Introduction to lntearal Equations with ADDlications (1985) G. ~arpilovsky,Projective ~epreseniationsof ~ i n i t Groups e (1985) L. Nariciand E, Beckenstein, Topological Vector Spaces (1985) . . J. Weeks, The Shape of Space (1985) P. R. Gribik and K 0. Kortanek, Extremal Methods of Operations Research (1985) J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) G. D. Crown et a/., Abstract Algebra (1986) J. H. Carmth et a/., The Theory of Topological Semigroups, Volume 2 (1986) R. S. Doran and V. A. Belfi, Characterizationsof C'-Algebras (1986) M. W. Jeter, Mathematical Programming(1986) M. Altman. A Unified Theorv of Nonlinear O~eratorand Evolution Eauations with ~p~lications (1986) A. Verschoren. Relative Invariants of Sheaves (1987) . , R. A. Usmani, .Applied Linear Algebra (1987) P. Blass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) J. A. Reneke et aL, Structured Hereditary Systems (1987) H.Busemann and B. B. Phadke, Spaces with Distinguished Geodesics (1987) R. Harte, lnvertibility and Singularity for Bounded Linear Operators (1988) G. S. Ladde et a/., Oscillation Theory of Differential Equations with Deviating Arguments (1987) L. Dudkin et a/., Iterative Aggregation Theory (1987) T. Okubo, Differential Geometry (1987)
113. D. L. Stancland M. L. Stancl, Real Analysis with Point-Set Topology (1987) 114. T. C. Gard, lntroduction to Stochastic Differential Equations (1988) 115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988) 1 1 6. H. Strade and R. Famsteiner. Modular Lie Alaebras and Their Representations (1988) 117. J.A. Huckaba, ~ornmutative'~in~s with ~ e r i ~ i v i s o(1988) rs 118. W. D. Wallis. Combinatorial Desians (1988) 1 1 9. W. Wi@aw,.Topological Fields (1988) . 120. G. Karpilovsky, Field Theory (1988) 121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rinas 11989) 122. W. ko.howski, Modular Function Spaces (1988) 123. E. Lowen-Colebunders. Function Classes of Cauchy Continuous Maps (1989) 124. M. Pavel. Fundamentals of Pattem Recoonition 11989) 125. V. ~aksh'mikanthamet a/., Stability ~ n a l ~ofi son linear Systems (1989) 126. R. Sivaramakrishnan. The Classical Theorv of Arithmetic Functions (1989) . . 127. N. A. Watson, parabolic Equations on an lilfinite Strip (1989) 128. K. J. Hastinas. introduction to the Mathematics of Operations Research (1989) 129. B. Fine, Algebraic Theory of the Bianchi Groups (19i39) 130. D. N. Dikranjan et a/., Topological Groups (1989) 131. J. C. Moraan 11. Point Set Theorv 11990) 132. P. ~ i l eand r A. '~itkowski,problems in ath he ma tical Analysis (1990) 133. H. J. Sussmann. Nonlinear Controllabilitvand O~timalControl (1 . 990) , J.-P. Florens et a/., Elements of ~ a ~ e s i ~tatisbcs an (1990) N. Shell, Topological Fields and Near Valuations (1990) B. F. Doolin and C. F. Martin, lntroduction to Differential Geometry for Engineers
(1990) S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1990)
J. Okninski, Semigroup Algebras (1990) K. Zhu, Operator Theory in Function Spaces (1990) G. B. Price, An lntroduction to Multicomplex Spaces and Functions (1991) R. B. Darst, lntroduction to Linear Programming (1991) P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) T. Husain, Orthogonal Schauder Bases (1991) J. Foran, Fundamentals of Real Analysis (1991) W. C. Brown, Matrices and Vector Spaces (1991) M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structures of Rings (1991) C. Small, Arithmetic of Finite Fields (1991) K. Yang, Complex Algebraic Geometry (1991) D. G. Hoffman et a/., Coding Theory (1991) M. 0. Gonzdlez. Classical Comolex Analvsis (1992) . , 152. M. 0. ~ o n z d l ecomplex i ~ n a l ~ s(1 i 99i) s 153. L. W. Bagoett. FunctionalAnalysis (1992) 154. M. ~nied&ich, Dynamic programming (1'992) 155. R. P. Aganval, Difference Equations and Inequalities(1992) 156. C. Brezinski. Biorthogonalityand Its Applications to NumericalAnalysis (1992) 157. C. Swarfz, An lntroduction to FunctionalAnalysis (1992) 158. S. B. Nadler, Jr., Continuum Theory (1992) 159. M. A.Al-Gwaiz, Theory of Distributions (1992) 160. E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) 161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (I992) 162. A. J. Jeni, Integral and Discrete Transforms with Applications and Error Analysis 11 897) \'---I 163. A. Charlier et a/., Tensors and the Clifford Algebra (1992) 164. P. Biler and T. Nadzieja, Problems and Examples in Differential Equations (1992) 165. E. Hansen, Global Optimization Using Interval Analysis (1992) 166. S. Guene-Delabriere. Classical Sequences in Banach Spaces (1992) 167. Y. C. Wong, IntroductoryTheory of Topological Vector Spaces (1992) 168. S. H. Kulkami and B. V. Limaye, Real Function Algebras (1992) 169. W. C. Brown, Matrices Over Commutative Rings (1993) 170. J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) ' 171. W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (I993)
172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195.
E. C. Young, Vector and Tensor Analysis: Second Edition (1993) T. A. Bick. Elementary Boundary Value Problems (1993) M. Pavel, Fundamentalsof Pattern Recognition: Second Edition (1993) S. A. Albeverio et a/., NoncommutativeDistributions (1993) W. Fulks, Complex Variables (1993) M. M. Rao. Conditional Measures and Applications (1993) A. Janicki and A. Wemn, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1994) P. Neittaanmeki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) J. Cmnin, Differential Equations: lntroduction and Qualitative Theory, Second Edition (1994) S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) X. Mao. Ex~onentialStabilitv of Stochastic Differential Eauations 11994) . . 6.S. Thomson, Symmetric properties of Real Functions11994) J. E. Rubio. Ootimization and Nonstandard Analvsis (19941 J. L. ~ u e s et'al., o Compatibility, Stability, and ti eaves (1995) A. N. Micheland K. Wang, Qualitative Theory of Dynamical Systems (1995) M. R. Damel. Theory of Lattice-OrderedGroups (1995) 2. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) L. J. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) L. H. Erbe et ab, Oscillation Theory for Functional Differential Equations (1995) S. Agaian etal., Binary Polynomial Transfomls and Nonlinear Digital Filters (1995) M. I.Gil', Norm Estimations for Operation-ValuedFunctions and Applications (1995) P. A. Grillet, Semigroups: An Introduction to the Structure Theory (1995) S. Kichenassamy, Nonlinear Wave Equations (1996) V. F. Kmtov, Global Methods in Optimal Control Theory (1996)
196. K. I. Beidaret a/., Rings with Generalized Identities (1996)
197. V. I. Amautov et a/., lntroduction to the Theory of Topological Rings and Modules (1996) 198. G. Sierksma, Linear and lnteger Programming (1996) 199. R. Lasser. lntroduction to Fourier Series (1996) 200. V. Sima. Algorithms for Linear-QuadraticOptimization (1996) 201. 0. Redmond, Number Theory (1996) 202. J. K. Beem et a/., Global Lorentzian Geometry: Second Edition (1996) 203. M. Fontana et al., Priifer Domains (1997) 204. H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997) 205. C. Q. Zhang, lnteger Flows and Cycle Covers of Graphs (1997) 206. E. Spiegeland C. J. O'Donnell, IncidenceAlgebras (1997) 207. B. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Control (1998) 208. T. W. Haynes etal.. Fundamentalsof Domination in Graphs (1998) 209. T. W. Haynes et a/., Domination in Graphs: Advanced Topics (1998) 210. L. A. D'Aloffo et a/., A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998) 21 1. F. Halter-Koch, Ideal Systems (1998) 212. N. K. Govilet a/., Approximation Theory (1998) 213. R. Cross, Multivalued Linear Operators (1998) 214. A. A. Maftynyuk, Stability by Liapunov's Matrix Function Method with Applications (1998) 215. A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces (1999) 216. A. lllanes and S. Nadler, Jr.. Hyperspaces: Fundamentals and Recent Advances (1999) 217. G. Kato and 0. Stmppa, Fundamentalsof Algebraic Microlocal Analysis (1999) 218. G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis (1999) 219. D. Motreanu and N. H. Pavel. Tangency. Flow Invariance for Differential Equations, and Optimization Problems (1999) 220. K. Hrbacek and T. Jech, lntroduction to Set Theory, Third Edition (1999) 221. G. E. Kolosov, Optimal Design of Control Systems (1999) 222. N. L. Johnson, Subplane Covered Nets (2000) 223. B. Fine and G. Rosenberger. Algebraic Generalizations of Discrete Groups (1999) 224. M. Vath, Volterra and Integral Equations of Vector Functions (2000) 225. S. S. Miller and P. T. Mocanu, Differential Subordinations(2000)
226. R. Li et a/.. Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods (2000) 227. H. Li and F. Van Oystaeyen, A Primer of Algebraic Geometry (2000) 228. R. P. Aganval, Difference Equations and Inequalities: Theory, Methods, and Applications, Second Edition (2000) 229. A. 6. Kharazishvili, Strange Functions in Real Analysis (2000) 230. J. M. Appell et at.. Partial Integral Operators and lntegro-DifferentialEquations (2000) 231. A. I. Prilepko et a/.. Methods for Solving Inverse Problems in Mathematical Physics 12000) 232. k van Oystaeyen, Algebraic Geometry for Associative Algebras (2000) 233. D. L. Jaoerman. Difference Eauations with Aoolications to Queues 12000) 234. D. R. ~ h k e r s o net at., coding Theory and 'Cryptography: The ~sseniials,Second Edition. Revised and Expanded (2000) 235. S. D&scalescuet a/., Hopf Algebras: An Introduction(2001) 236. R. Hagen et at., C-Algebras and NumericalAnalysis (2001) 237. Y. Talpaert, Differential Geometry: With Applications to Mechanics and Physics (2001) Additional Volumes in Preparation
HOPF ALGEBRAS An Introduction
Sorin Di!isciilescu Constantin N3sti9sescu Serban Raianu University of Bucharest Bucharest, Romania
M A R C E L
MARCEL DEKKER, INC. D EK K E R
Library of Congress Cataloging-in-PublicationData Dascalescu, Sorin. Hopf algebras : a n introduction I Sorin Dascalescu, Constantin Nastasescu, Serban Raianu. p. cm. - (Monographs and textbooks i n pure and applied mathematics ; 235) Includes index. ISBN 0-8247-0481-9 (alk.paper) 1. Hopf algebras. I. Nastasescu, C. (Constantin). 11.Raianu, Serban. 111. Title. IV.Series
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Preface A bialgebra is, roughly speaking, an algebra on which there exists a dual structure, called a coalgebra structure, such that the two structures satisfy a compatibility relation. A Hopf algebra is a bialgebra with an endomorphism satisfying a condition which can be expressed using the algebra and coalgebra structures. The first example of such a structure was observed in algebraic topology by H. Hopf in 1941. This was the homology of a connected Lie group, which is even a graded Hopf algebra. Starting with the late 1960s, Hopf algebras became a subject of study from a strictly algebraic point of view, and by the end of the 1980s, research in this field was given a strong boost by the connections with quantum mechanics (the so-called quantum groups are in fact examples of noncommutative noncocommutative Hopf algebras). Perhaps one of the most striking aspect of Hopf algebras is their extraordinary ubiquity in virtually all fields of mathematics: from number theory (formal groups), to algebraic geometry (affine group schemes), Lie theory (the universal enveloping algebra of a Lie algebra is a Hopf algebra), Galois theory and separable field extensions, graded ring theory, operator theory, locally compact group theory, distribution theory, combinatorics, representation theory and quantum mechanics, and the list may go on. This text is mainly addressed to beginners in the field, graduate or even undergraduate students. The prerequisites are the notions usually contained in the first two year courses in algebra: some elements of linear algebra, tensor products, injective and projective modules. Some elementary notions of category theory are also required, such as equivalences of categories, adjoint functors, Morita equivalence, abelian and Grothendieck categories. The style of the exposition is mainly categorical. The main subjects are the notions of a coalgebra and comodule over a coalgebra, together with the corresponding categories, the notion of a bialgebra and Hopf algebra, categories of Hopf modules, integrals, actions and coactions of Hopf algebras, some Hopf-Galois theory and some classification results for finite dimensional Hopf algebras. Special emphasis is
PREFACE put upon special classes of coalgebras, such as semiperfect, co-Frobenius, quasi-co-Frobenius, and cosemisimple or pointed coalgebras. Some torsion theory for coalgebras is also discussed. These classes of coalgebras are then investigated in the particular case of Hopf algebras, and the results are used, for example, in the chapters concerning integrals, actions and Galois extensions. The 'notions of a coalgebra and comodule are dualizations of the usual notions of an algebra and module. Beyond the formal aspect of dualization, it is worth keeping in mind that the introduction of these structures is motivated by natural constructions in classical fields of algebra, for example from representation theory. Thus, the notion of comuItiplication in a coalgebra may be already seen in the definition of the tensor product of representations of groups or Lie algebras, and a comodule is, in the given context, just a linear representation of an affine group scheme. As often happens, dual notions can behave quite differently in given dual situations. Coalgebras (and comodules) differ from their dual notions by a certain finiteness property they have. This can first be seen in the fact that the dual of a coalgebra is always an algebra in a functorial way, but not conversely. Then the same aspect becomes evident in the fundamental theorems for coalgebras and comodules. The practical result is that coalgebras and comodules are suitable for the study of cases involving infinite dimensions. This will be seen mainly in the chapter on actions and coactions. The notion of an action of a Hopf algebra on an algebra unifies situations such as: actions of groups as automorphisms, rings graded by a group, and Lie algebras acting as derivations. The chapter on actions and coactions has as main application the characterization of Hopf-Galois extensions in the case of co-Frobenius Hopf algebras. We do not treat here the dual situation, namely actions and coactions on coalgebras. Among other subjects which are not treated are: generalizations of Hopf modules, such as Doi-Koppinen modules or entwining modules, quasitriangular Hopf algebras and solutions of the quantum Yang-Baxter equation, and braided categories. The last chapter contains some fundamental theorems on finite dimensional Hopf algebras, such as the Nichols-Zoeller theorem, the Taft-Wilson theorem, and the Kac-Zhu theorem. We tried to keep the text as self-contained as possible. In the exposition we have indulged our taste for the language of category theory, and we use this language quite freely. A sort of "phrase-book" for this language is included in an appendix. Exercises are scattered throughout the text, with complete solutions a t the end of each chapter. Some of them are very easy, and some of them not, but the reader is encouraged to try as hard as possible to solve them without looking at the solution. Some of the easier
PREFACE
v
results can also be treated as exercises, and proved independently after a quick glimpse at the solution. We also tried to explain why the names for some notions sound so familar (e.g. convolution, integral, Galois extension, trace). This book is not meant to supplant the existing monographs on the subject, such as the books of M. Sweedler [218], E. Abe [I], or S. Montgomery [149] (which were actually our main source of inspiration), but rather as a first contact with the field. Since references in the text are few, we include a bibliographical note a t the end of each chapter. It is usually difficult to thank people who helped without unwittingly leaving some out, but we shall try. So we thank our friends Nicol6.s Andruskiewitsch, Margaret Beattie, Stefaan Caenepeel, Bill Chin, Miriam Cohen (who sort of founded the Hopf algebra group in Bucharest with her talk in 1989), Yukio Doi, Josk Gomez Torrecillas, Luzius Griinenfelder, Andrei Kelarev, Akira Masuoka, Claudia Menini, Susan Montgomery, Declan Quinn, David Radford, Angel del Rio, Manolo Saorin, Peter Schauenburg, Hans-Jiirgen Schneider, Blas Torrecillas, Fred Van Oystaeyen, Leon Van Wyk, Sara Westreich, Robert Wisbauer, Yinhuo Zhang, our students and colleagues from the University of Bucharest, for the many things that we have learned from them. Florin Nichita and Alexandru StZ.nculescu took course notes for part of the text, and corrected many errors. Special thanks go to the editor of this series, Earl J . Taft, for encouraging us (and making us write this material). Finally, we thank our families, especially our wives, Crina, Petrufa and Andreea, for loving and understanding care during the preparation of the book.
Sorin Dkciilescu, Constantin N k t k e s c u , Serban Raianu
Contents Preface
1 Algebras and coalgebras 1.1 Basic concepts 1.2 The finite topology 1.3 The dual (co)algebra 1.4 Constructions in the category of coalgebras 1.5 The finite dual of an algebra 1.6 The cofree coalgebra 1.7 Solutions to exercises 2
Comodules 1.1 The category of comodules over a coalgebra 2.2 Rational modules 2.3 Bicomodules and the cotensor product 2.4 Simple comodules and injective comodules 2.5 Some topics on torsion theories on M~ 2.6 Solutions to exercises
3
Special classes of coalgebras 3.1 Cosemisimple coalgebras 3.2 Semiperfect coalgebras 3.3 (Quasi)co-Frobeniusand co-Frobenius coalgebras 3.4 Solutions to exercises
4
Bialgebras and Hopf algebras 4.1 Bialgebras 4.2 Hopf algebras 4.3 Examples of Hopf algebras
vii
...
CONTENTS
vlll
4.4 4.5
Hopf modules Solutions to exercises
5 Integrals 5.1 5.2 5.3 5.4 5.5 5.6 5.7
6
7
The definition of integrals for a bialgebra The connection between integrals and the ideal H*"' Finiteness conditions for Hopf algebras with nonzero integrals The uniqueness of integrals and the bijectivity of the antipode Ideals in Hopf algebras with nonzero integrals Hopf algebras constructed by Ore extensions Solutions to exercises
Actions and coactions of Hopf algebras 6.1 Actions of Hopf algebras on algebras 6.2 Coactions of Hopf algebras on algebras 6.3 The Morita context 6.4 Hopf-Galois extensions 6.5 Application to the duality theorems for co-Frobenius Hopf algebras 6.6 Solutions to exercises Finite dimensional Hopf algebras 7.1 The order of the antipode 7.2 The Nichols-Zoeller Theorem 7.3 Matrix subcoalgebras of Hopf algebras 7.4 Cosemisimplicity, semisimplicity, and the square of the antipode 7.5 Character theory for semisimple Hopf algebras 7.6 The Class Equation and applications 7.7 The Taft-Wilson Theorem 7.8 Pointed Hopf algebras of dimension pnwith large coradical 7.9 Pointed Hopf algebras of dimension p3 7.10 Solutions to exercises
181 181 184 189 192 194 200 22 1
233 233 243 25 1 255 267 276
CONTENTS A The category theory language A.l Categories, special objects and special morphisms A.2 Functors and functorial morphisms A.3 Abelian categories A.4 Adjoint functors
B C-groups and C-cogroups B. 1 Definitions B.2 General properties of C-groups B.3 Formal groups and affine groups Bibliography Index
ix
Chapter 1
Algebras and coalgebras 1 .
Basic concepts
Let k be a field. All unadorned tensor products are over k. The following alternative definition for the classical notion of a k-algebra sheds a new light on this concept, the ingredients of the new definition being objects (vector spaces), morphisms (linear maps), tensor products and commutative diagrams.
-
-
Definition 1.1.1 A k-algebra is a triple (A, M, u), where A is a k-vector A and u : k A are mo~phismsof k-vector spaces space, M : A @ A such that the following diagrams are commutative:
C H A P T E R 1. A L G E B R A S A N D C O A L G E B R A S
W e have denoted by I the identity map of A , and the unnamed arrows from the second diagram are the canonical isomorphisms. ( I n general we will denote by I (unadorned, i f there is no danger of confusion, the identity I map of a set, but sometimes also by Id.) Remark 1.1.2 The definition is equivalent to the classical one, requiring A to be a unitary ring, and the existence of a unitary ring morphism 4 : k A, with I m q5 Z ( A ) . Indeed, the multiplication a . b = M ( a @ b) defines o n A a structure of unitary ring, with identity element u(1); the role of 4 is played by u itself. For the converse, we put M ( a 8 b) = a . b and u = 4. Due to the above, the map M is called the multiplication of the algebra A, and u is called its unit. The commutativity of the first diagram in the definition is just the associativity of the multiplication of the algebra. I
-
The importance of the above definition resides in the fact that, due to its categorical nature, it can be dualized. We obtain in this way the notion of a coalgebra.
-
-
Definition 1.1.3 A k-coalgebra is a triple (C, A, E ) , where C is a k-vector C @ C and E : C k are morphisms of k-vector spaces space, A : C such that the following diagrams are commutative:
1.1. 1.1.BASIC BASIC CONCEPTSCONCEPTS
33
C®CC®C The maps The Amaps A and and ~ ~are are called called the the comultiplication, comultiplication, and and the the counit, counit, respec-respectively, tively, of of the the coalgebra C. coalgebra C. The commutativity The commutativity of of the the first first diagramisdiagram iscalledcalled coassociativity. coassociativity. || Example 1.1.4 Example 1.1.4 1) 1) Let Let S Sbe be a anonempty set; nonempty set; kS kS is isthe the k-vector k-vector space space withwith basis basis S. S. Then kS Then kS is isa acoalgebra with coalgebra with comultiplication comultiplication A and A counit and counit ~ ~defineddefined by by A(s) A(s) = s= s® s, ®s,~(s) ~(s) fofo r rany any s E sS.S. EThTh is isshowsshows that athat ny veny a ctorctor vespacespace can can be be endowedwith endowed with a ak-coalgebra k-coalgebra structure.structure. 2) 2)Let Let HH be be a ak-vector k-vector space space with with basis basis {cm {cm I Im m ~ ~N}. N}. Then Then H is H isaa coalgebra with coalgebra with comultiplication comultiplication AA and counit and counit ~ ~defined defined byby
for for any any mm ~ ~N N (5iy (5iy will will denote denote throughout throughout the the Kronecker Kronecker symbol). symbol). ThisThis coalgebra is coalgebra iscalled called the the divided divided power coalgebra, power coalgebra, and we andwill we will come back cometo back to itit later.later. 3) 3) Let Let (S, (S, ~) ~) be be a apartially partially ordered locally orderedlocally finite finite set set (i.e. (i.e. for for any any x,y x,y with with x x~ ~y, y,the the set set {z {z ~~ S S~ ~ x x< z
= =(z, (z,z) z)(z,(z, x~z~yx~z~y
~(z, ~(z, y) y) = 5x~5x~ = for for any any (x, (x, y) y) ee ~) ~) The field The field k kis isa ak-coalgebra k-coalgebra with with comultiplication comultiplication AA : :k k~ ~ k@kthe k@k the canon-canonical ical isomorphism, isomorphism, A(a) A(a) = =a a@fofo @r anry an ~ y~~~k,k,andand the the councoun it ~ : it ~k :k~ ~kk the the identity identity map. We map. remark We that remark that this this coalgebra is coalgebra isa~particular a~particular case case of of thethe example in example in 1), 1), for for S Sa aset set with with only only one one element.element. 5) 5) Let Let n n~ ~1 1be be a apositive positive integer, integer, and and MC(n,k) aMC(n,k) ak-vector k-vector space space of of ~. ~.We denote mension mension nn We denote by by (e~y)~,j~n (e~y)~,j~n a abasis basis of ofMC(n, k). MC(n, k).We define We define MC(n, k) MC(n, k)a acomultiplication comultiplication AA
== l~p~nl~p~n
CHAPTER I . ALGEBRAS AND COALGEBRAS
4
for any i,j , and a counit
E
by
In this way, Mc(n, k) becomes a coalgebra, which is called the matrix coalgebra. 6) Let C be a vector space with basis { gi,di I i E N' ). We define A : C - + c @ C a n d ~ : C - +by k
I
Then (C, A, E ) is a coalgebra.
The following exercise deals with a coalgebra called the trigonometric coalgebra. The reason for the name, as well as what really makes it a coalgebra, will be discussed later, after the introduction of the representative coalgebra of a semigroup (the same applies to the divided power coalgebra in the second example above).
Exercise 1.1.5 Let C be a k-space with basis {s, c). We define A : C C @ C a n d & : C - - t k by a ( ~ a(c) E(S) E(C)
)=
-+
S € ~ C + C @ S
c@c-s@s = 0 =
=
1.
Show that (C,A, E) is a coalgebra.
Exercise 1.1.6 Show that on any vector space V one can introduce an algebra structure. Let (C, A, E) be a coalgebra. We recurrently define the sequence of maps (A,),?,, as follows: A, = A, An : C
-
C @ . . . @ C ( C appearing n
An = (A @ I,-')
o An-,,
for any n
+ 1 times )
2 2.
(Throughout the text, the composition of the maps f and g will be denoted by f o g, or simply f g when there is no danger of confusion.) As we know, in an algebra we have a property called generalized associativity. The dual property in the case of coalgebras is called generalized coassociativity, and is given in the following proposition.
1.1. BASIC CONCEPTS
5
Proposition 1.1.7 Let (C,A,€) be a coalgebra. Then for any n any p E (0,. . . , n - 1 ) the following equality holds
2 2 and
Proof: We show by induction on n that the equality holds for any p E (0,. . . , n - 1 ) . For n = 2 we have to show that (A 8 I ) o A = ( I 8 A) o A, which is the coassociativity of the comultiplication. We assume that the relation holds for n . Let then p E (1,. . . , n). We have
. . . .
(IP 8 A I*-p) 0 A, = = (IP 8 A 8 I ~ ~ - (IP-~ ~ ) CXIA 8 I,-,) = =
(1'-I
0
8 ((I@ A) o A) 8In-') 0 An-l
(I+ g ((A. I ) 0 A) IrLPP)0 An-l (I+ @ AI @ In+l-P) 0 ( 1 P - ' .A In-p) = (IP-I a 8 r - p f l ) A, =
0
A*+]
Since for p = 0 we have by definition that A,+l = (Ip 8 A 8 In-p) o A,, by the above relation it follows, by induction on p, that the equality holds I for any p~ (0, . . . , n). Unlike the case of algebras, where the multiplication "diminishes" the number of elements, from two elements obtaining only one after applying the multiplication, in a coalgebra, the comultiplication produces an opposite effect, from one element obtaining by comultiplication a finite family of pairs of elements. Due to this fact, computations in a coalgebra are harder than the ones in an algebra. The following notation for the comultiplication, usually called the "Sweedler notation", but also known as the "HeynemanSweedler notation", proved itself to be very effective. 1.1.8 The Sigma Notation. Let (C, A, E ) be a coalgebra. For an element c E C we denote A(d = c1 8 cz?.
C
With the usual summation conventions we should have written
The sigma notation supresses the index "in. It is a way to emphasize the form of A(c), and it is very useful for writing long compositions involving the comultiplication i n a compressed way. I n a similar way, for any n 2 1 we write
6
CHAPTER 1. ALGEBRAS AND COALGEBRAS
The above notation provides an easy way for writing commutative diagrams, the first examples being the diagrams in the definition of a coalgebra, i.e. the coassociativity and the counit property: 1.1.9 Formulas. Let (C, A, E) be a coalgebra and c E C . Then:
Note that
x
c l , 8 c1, 8 c2 =
x
C1
8~
28 , c2,,
V c E C is just the equality (A 8 I ) o A = ( I 8 A) 0 A, the commutativity of the first diagram in Definition 1.1.3. The commutativity of the second diagram in the same definition may be written as
where 4, : C@k -LC and $1 : k8C 1C are the canonical isomorphisms. Using the sigma notation, the same equalities are written as
The advantage of using the sigma notation will become clear later, when we will deal with very complicated commutative diagrams or very long compositions. I We are going to explain now how to operate with the sigma notation. For this we will need some more formulas.
Lemma 1.1.10 Let (C, A, E) be a coalgebra. Then: 1 ) For any i > 2 we have Ai = (AiFl 8 I ) o A. 2) For any n 2 2, i E (1,. . . ,n - 1 ) and m E (0,. . . , n - i ) we have
Proof: 1) By induction on i. For i = 2 this is the definition of A2. We assume that the assertion is true for i , and then
&+I
(A 8 I i ) 0 A, = (A @ ri)0 (ni-l = (((A 8 lip') 0 = (Ai@I)oA. =
@
I)
A 8 I) o A
1.1. BASIC CONCEPTS
7
2) We fix n 2 2, and we prove by induction on i E ( 1 , . . . , n - 1 ) that for any m E (0, . . . ,n - i ) the required relation holds. For i = 1 this is the generalized coassociativity proved in Proposition 1.1.7. We assume the assertion is true for i - 1 (i 2) and we prove it for i . We pick m E (0, . . . ,n - i ) c (0,. . . , n - i 1 ) and we have
+
A,
>
(Irn@ Ai-l 18I ~ - ~ )-0 ~An-i+l + ~ (by the induction hypothesis) = (17" @ Ai-l @ p - i - m f l ) 0 ( I r n@ A @ In+rn) (by generalized coassociativity) = (I" @ ((Ai-1€4 I ) 0 A) @ In-i-m) 0 A,-i = ( I m@ Ai @ I . - ~ - ~ )0 (using 1)) =
0
Anpi
These formulas allow us to give the following computation rule, which is essential for computations in colagebras, and which will be used throughout in the sequel. 1.1.11 Computation rule. Let ( C ,A, E ) be a coalgebra, i
( i n the preceding tensor product C appearing i
2 1,
+ 1 times) and
-
f:C-c
linear maps such that f o Ai = 7. Then, if n 2 i , V is a k-vector space, and g : C @ ....
+
c-v
(here C appearing n 1 tinzes i n the tensor product) is a k-linear map, for any 1 5 j 5 n 1 and c E C we have
+
This happens because
CHAPTER 1. ALGEBRAS AND COALGEBRAS @Cj+i+l €9.. . @ cn+i+l) = ( ~ j - 18 f 8 ~ n - j + l ) o An+i(~) - g 0 (1j-18 f 8 p - j + l ) 0 ( I H @ A, @ 1n-j+1 ) o An (c) = g o ( 1 j - l 8 ( f 0 A,) @ P - j + l ) 0 An(c) ( ~ j - 1@ 7 ~ n - j + l o An ( c ) -
@
=
~ ~ ( C I @ . . . @ ~ ~ - ~ @ ~ ( C ~ ) B C ~ + ~ @ . . . @ C ~ + ~ )
This rule may be formulated as follows: if we have a formula (*) in which an expression in c l , . . . ,c,+l (from A i ( c ) ) has as result an element in C ( f o Ai = T), then i n a n expression depending o n c l , . . . , cn+i+l (from An+i(c)) in which the expression i n the formula (*) appears for c j , . . . ,cj+i (i+l conpositions), we can replace the expression depending o n c j , . . . ,cj+i secutive by f ( c j ) , leaving unchanged c l , . . . ,cj-1 and transforming cj+i+l,. . . ,cn+i+l I i n cj+l,. . . , cn+l.
Example 1.1.12 If (C,A, E ) is a coalgebra, then for any c E C we have
This is because having in mind the formula ~ E ( c ~=) Cc, ~we can replace i n the left hand side &(c2)c3by c2, leaving cl unchanged. Therefore, C E ( C ~ ) E ( C Z ) C=~ C E ( C ~ ) C Z , and this is exactly c. I We end this section by giving some definitions allowing the introduction of some categories.
Definition 1.1.13 A n algebra ( A ,M , u ) is said to be commutative if the diagram
is commutative, where T : A @ A ---t A 8 A is the twist map, defined by T ( a 8 b) = b 8 a. ii) A coalgebra (C,A, E ) is called cocommutative if the diagram
1.1. BASIC CONCEPTS
is commutative, which may be written as c E C.
C cl 8 c2
=
C cz 8 cl
for any
I
-
Definition 1.1.14 Let ( A , MA,u A ) ,( B ,M B ,u g ) be two k-algebras. The k-linear map f : A B is a morphism of algebras if the following diagrams are commutative
-
zi) Let ( C ,A c , ~ c )(,D ,AD,&,) be two k-coalgebras. The k-linear map D is a morphism of coalgebras if the following diagrams are g : C commutative
The commutativity of the Jirst diagram may be written i n the sigma notation as: a D ( g ( c ) )=
C g(c)l @ g(c)2 = C g ( c d €9 g(c2).
10
CHAPTER 1. ALGEBRAS AND COALGEBRAS
In this way we can define the categories k - Alg and k - Cog, in which the objects are the k-algebras, respectively the k-coalgebras, and the morphisms are the ones previously defined.
Exercise 1.1.15 Show that in the category k - Cog, isomorphisms (i.e. morphisms of coalgebras having an inverse which is also a coalgebra morphism) are precisely the bijective morphisms.
1.2
The finite topology
Let X and Y be non-empty sets and YX the set of all mappings from X to Y. It is clear that we can regard yXas the product of the sets Y, = Y, where x ranges over the index set X. The finite topology of yXis obtained by taking the product space in the category of topological spaces, where each Y, is regarded as a discrete space. A basis for the open sets in this topology is given by the sets of the form ( 9 E yx
I g(xi) = f (xi), 1 5 i 5 n),
where {xi 1 1 5 i 5 n ) is a finite set of elements of X , and f is a fixed element of YX, so that every open set is a union of open sets of this form. Assume now that k is a field, and X and Y are two k-vector spaces. The set Homk(X,Y) of all Ic-homomorphisms from X to Y, which is also a k-vector space, is a subset of y X . Thus we can consider on Homk(X,Y) the topology induced by the finite topology on YX. This topology on Homk(X, Y) is also called the finite topology. If f E Homk(X,Y), the the sets
<
form a basis for the filter of neighbourhoods of f , where {xi ) 1 5 i n ) ranges over the finite subsets of X. Note that L3(f,xl,. . . ,x,) = n
n Q(f,xi)r and O ( f , x l , . . . , x n ) = f + O(0, x i , . . .,x,). i= 1
Proposition 1.2.1 With the above notation we have the following results. a) Homk(X,Y) is a closed subspace of yX (in the finite topology). b) Homk(X, Y), with the finite topology, is a topological k-vector space (the topology of Ic is the discrete topology). c) If dimk(X) < co, then the finite topology on Homk(X,Y) is discrete. Proof: a) Pick f in the closure of Homk(X,Y), and let xl,x2 E X , and X,p E k. The open set U = {g E YX I g(x1) = f (xl),g(xa) = f (x2),g(Xxl +px2) = f (Axl +pxz)) is a neighbourhood of f , and therefore
1.2. THE FINITE TOPOLOGY
11
U r ) H o m k ( X , Y ) # 8. If h E U n H o m k ( X , Y ) ,then h ( x l ) = f ( X I ) , h ( x z ) = f ( x 2 ) ,and h(Xxl px2) = f (Axl p x z ) . Since h(Xxl px2) = Xh(xl)+ph(xz) = X f ( x l ) + pf (x2),we obtain that f (Xxl+px2) = X f ( X I ) + ~f ( x 2 ) ,SO f E H o m k ( X , Y ) . b) If A, p E k, we have to show that the map cr : ( f , g) +-+ X f pg is a continuous mapping from the product space H o m k ( X ,Y ) x H o m k ( X ,Y ) into H o m k ( X , Y ) . Indeed, we can consider the open neighbourhoods of X f + / ~ gof the form N = X f + p g + O(O,xl,..., x,). If we put N1 = f O ( O , x l , .. . , xn) and N2 = g C?(O,xl,.. . , x,), then N1 (resp. N 2 ) is a neighbourhood of f (resp. g). Since O ( 0 ,X I , . . . ,x,) is a k-subspace, it is clear that a ( N 1 x N 2 ) C N , so cr is continuous. c) is obvious.
+
+
+
+
+
+
Exercise 1.2.2 An open subspace in a topological vector space is also closed.
If k is a field, we can consider the particular case when X = V is a k-vector space, and Y = k. In this case, the vector space H0mk(V, k) is the dual V * . We introduce now some notation. If S is a subset of V * ,then we denote
Similarly, if S is a subset of V , then we set
S'
= {U E V * I u ( S ) = 0 ) = { u E
V* I S 5 K e r ( u ) ) .
If S = { u } (where u E V or u E V * ) ,we denote S L = uL. Exercise 1.2.3 When S is a subset of V * (or V ) , S L is a subspace of V (or V * ) . In fact S L = ( S ) I , where ( S ) is the subspace spanned by S .
Moreover, we have S L = ( ( S L ) ' - ) I for , any subset S of V * (or V ) .
+ W L , where W ranges over the finite dimensional subspaces of V , form a basis for the filter of neighbourhoods of f E V * in the finite topology.
Exercise 1.2.4 The set of all f
The following result is the key fact in the study of the finite topology in
v*.
Proposition 1.2.5 i) If S is a subspace of V * and { e l , . . . , e n ) is a finite
subset of V , then n
n
CHAPTER 1. ALGEBRAS AND COALGEBRAS
12
ii) (the dual version of i)) If S is a k-linear subspace of V and { u l ,. . . , u n ) is a finite subset of V * , then
Proof: i) The inclusion
is clear. We show the converse inclusion by induction on n. Let n = 1 and denote in this case el by e. Let x E ( S n e l ) " . If S n e l = S , then x E S' and so x E SL+ke. Hence we can assume that Sne" c S , Sne" # S . Since V * / e l E (Ice)*,and (Ice)* has dimension one, it follows that S / ( S f~ e") is also 1-dimensional. Hence S = ( S n e l ) @ ku, for some u V * ,with u E S and u # Sne". So u # e l , and therefore u(e) # 0. We put X = u(x)u(e)-l. If y = x - Xe, then for any v E S we have v ( y ) = v ( x ) - Xv(e). But v = w + au, where w E S n e l . S o v ( x ) = w ( x ) + au(x) = au(x). On the other hand, Xv(e) = Xw(e) Xau(e) = Xau(e) = au(x)u(e)-'u(e)= au(x). Hence v ( y ) = au(x) - au(x) = 0, and since v E S is arbitrary, we obtain that Y E S". ~ h u s x =Y + X ~ E S1+ke.
+
.n
Assume that the assertion is true for n- 1 ( n > 1). We have S ' n- 1
C kei+ken
n-1
n ef)"+ke, i=l
n
n efne;)' i=l
+ 2 kei = i= 1 n
( S n0 e $ ) l . i=l i=l ii) Since the proof is similar to the one of assertion i), we only sketch the ku 2 ( S n u L ) l . For case n = 1. We put u = u1. We clearly have S' the reverse inclusion, we can assume S n u L # S. Clearly in this case we have u # 0. Since u l = K e r ( u ) , we have that V/u' = V / K e r ( u ) has dimension one. So S / ( S n u l ) E ( S + u " ) / u L I V / u L also has dimension one. There exists e E S , e # S n u L , such that S = ( S n u L ) @ ke. So u ( e ) # 0. If now f E ( S n u L ) l , we put g = f - f(e)u(e)-'u. If x E S , then x = y + Xe, with y E S n u L . But f ( x ) = f ( y ) + X f ( e ) = X f ( e ) , and ( f (e)u(e)-'u)(x) = f (e)u(e)-lu(y)+ X f (e)u(e)-'u(e) = X f ( e ) . So g(x) = X f ( e )- A f ( e ) = 0 , and therefore g E S'. Since f = g+ f (e)u(e)-lu, we obtain f E S L + ku. SL+
=(Sn
=(Sn
=
+
Theorem 1.2.6 i) If S is subspace of V , then (SL)" = S . iz) If S is a subspace of V * , then (SL)" = 3, where 3 is the closure of S in the finite topology. Proof: i) We have clearly that S C ( S L ) " . Assume now that there exists x E ( S L ) I , x @ S . Since S is a subspace, then kx n S = 0. Thus there
1.2. THE FINITE TOPOLOGY
13
exists f E V* such that f ( x ) = 1 and f ( S ) = 0. But f E SL,and since z E (SL)L, we have f ( x ) = 0, a contradiction. Hence (S'-)I = S . ii) S'- is a subspace of V. Hence (5'')'- = n W L , where W ranges over the finite dimensional subspaces of S L . Since WL is an open subspace of V*, it follows that WL is also closed (see Exercise 1.2.2). Hence nW'- is closed, so (SL)' is closed in the finite topology (see also Exercise 1.2.7 below). Since S 5 (SL)'-,it follows that S's (SL)I.Let f E (SL)' and W c V a k-subspace of finite dimension. We show that (f W'-) n S # 0. Clearly i f f E WL then f WL = W'- (because WL is a subspace), and therefore W L ) n S = WL n S # 0 (because it contains the zero morphism). (f Also if W E SL,then (S'-)I C W L , and therefore f E WL. Hence we S L . Thus we can can assume that f $! wL and so it follows that W write W = ( W n S L ) @ W', where W' # 0 and dimk(W1) < oo. Also since f (SL) = 0 and f (W) # 0 it follows that f (W') # 0. Let {el,. . . ,en) (n 2 1) be a basis for W'. We denote by a, = f (e,) (1 5 i 5 n ) , hence not all the a,'s are zero. By Proposition 1.2.5 i), we have
+
+
+
<
Since ei @ SLCB
Sn
n
C kej, then ei $2 ( S n
n
ej'-)+ Hence there exists gi E j#i ef such that gi(ei) = 1. So we have gi E S, and gi(ek) = 6ik. We
j#i
denote by g =
j#i
n
C aigi.
i=l
Hence g E S and g(ek) = ak (1
< k 5 n).
<
Let now h = g - f . Clearly h(ei) = 0 (1 i 5 n), and hence h(W') 7 0. Since h E (SL)'-, then h(SL) = 0, and thus h(W n S L ) = 0. So h(W) = 0, and hence h E WL. In conclusion, g E S n (f W L ) ,and therefore f E 3.
+
Exercise 1.2.7 If S is a subspace of V*, then prove that ( S L ) l is closed in the finite topology by showing that its complement is open. We give now some consequences of Theorem 1.2.6.
-
Corollary 1.2.8 There exists a bijective correspondence between the subSL. I spaces of V and the closed subspaces of V*, given by S Corollary 1.2.9 If S S V* is a subspace, then S is dense in V* zf and only zf S1 = ( 0 ) .
-
Proof: If S is dense in V*, i.e. S = V*, then since 3 = (SL)'-, it is I necessary that SL = (0). The converse is obvious, since { o ) ~= V*.
CHAPTER 1. ALGEBRAS AND COALGEBRAS
14
Exercise 1.2.10 If V is a k-vector space, we have the canonical k-linear map (flv: V (V*)*, $ v ( x ) ( f )= f ( x ) , VX E V, f E V * .
-
Then the following assertions hold: a) The map qbv is injective. b) Irn((flv) is dense i n (V*)*. Exercise 1.2.11 Let V = Vl $ Vz be a vector space, and X = X I $ X2 a subspace of V * ( X i V,',i = 1,2). If X is dense i n V * , then X i is dense in V,',i = 1,2. Corollary 1.2.12 Let X , Y be two subspaces of V* such that X is closed and d i m k ( Y ) < m. Then X Y is closed. I n particular, every finite dimensional subspace of V* is closed.
+
Proof: Since X is closed, we have X = ( X I ) ' . Then by Proposition 1.2.5 ii), we have X + Y = ( X L ) I + Y = ( X I n Y ' ) l , for some subspace Y' of V , and therefore X + Y is also closed, by Exercise 1.2.3. I Corollary 1.2.13 i ) There is a bijective correspondence between the finite dimensional subspaces of V* and the subspaces of V of finite codimension, given by X ++ X I . Moreover, for any finite dimensional subspace X of V * we have d i m k ( X ) = codimk ( X I ) . ii) There is a bijective correspondence between the closed subspaces of V * of finite codimension and the finite dimensional subspaces of V , given by X L . Moreover, for any closed subspace X of V * of finite codimenX sion, we have codimk ( X ) = dzmk ( X I ) .
-
Proof: i) Let X
g V* be a finite dimensional subspace and let { u l , . . . ,~ n
be a basis of X. Then X L = monomorphism 0
n uf
- V/X'
n )
n
=
n K e r ( u i ) . But there exists a
n
$ V / K e r ( u i ) . Since d i m k ( V / K e r ( u i ) )= i=l
1, we have that d i m k ( V / X L ) 5 n = d i m k ( X ) , so X' has finite codimension. , so Conversely, if W V has finite codimension, then W L 2. ( V / W ) *and d i m k ( W L ) = c o d i m k ( w ) < oo. We can now apply Corollary 1.2.8. ii) Let X C V * be a subspace of finite codimension. There exist fl . . . ,fn E
z
n
V * such that V* = X
$
C k f i . Then 0 = V * I = X I
z=1
n
n
n K e r ( f i ) , so i=l
1.2. THE FINITE TOPOLOGY But
-
n
v / ( nK e r ( f i ) )
-
n
-
@ ~ / ~ e r ( f i )k n , i=l i= 1 so 0 --+ X I kn , and therefore dimk ( X I )5 n = c0dimk ( X ).. Conversely, if W V is a finite dimensional subspace, we have v*/w' W * ,so d 2 m k ( V * / w L )= d i m k ( W * ) = d i m k ( W ) . 0
-
I
Exercise 1.2.14 Let X C V* be a subspace of finite dimension n . Prove that X is closed i n the finite topology of V* by showing that d i m k ( ( X L ) L )I n. Exercise 1.2.15 If X is a finite codimensional k-linear subspace of V * , then X is closed i n the finite topology if and only if X I = xi, where xi is the orthogonal of X i n I/**.
-
We denote by k M the category of k-vector spaces. If u : V V' is a map in this category (i.e. u is k-linear) then we have the dual map U* : V'* V * ,whereu*(f) = f o u , f E V'*. Let W V be a subspace. Then
-
c
-
Moreover, if W is finite dimensional, then u ( W ) has finite dimension as a subspace of V ' , and so it follows that the map u* : V'* V * is continuous in the finite topology. We have thus the following result:
-
Corollary 1.2.16 The mapping V V * defines a contravariant functor from the category kM to the category of topological k-vector spaces (k is I considered with the discrete topology). Exercise 1.2.17 If V is a k-vector space such that V = @ V,, where {V, I iEI
i E I ) is a family of subspaces of V , then @ V,* is dense i n V * i n the finite topology.
-
iEI
-
Exercise 1.2.18 Let u : V V' be a k-linear map, and u* : V'* V* the dual morphism o f u . The following assertions hold: i ) If T is a subspace of V ' , then u * ( T L )= u-I ( T ) ' - . iz) If X is a subspace of V1*,then u*(X)' = u - l ( X L ) . iii) The image of a closed subspace through u* is a closed subspace. iv) If u is injective, and Y V" is a dense subspace, it follows that u * ( Y ) is dense i n V * .
CHAPTER 1. ALGEBRAS AND COALGEBRAS
16
1.3 The dual (co)algebra We will often use the following simple fact: if X and Y are k-vector spaces,
C xi 8 yi for i= 1 some positive integer n, some linearly independent (xi)i=l,nin X , and some (yi)i=l,n c Y. Similarly, t can be written as a sum of tensor monomials with the elements appearing on the second tensor position being linearly independent. and t is an element of X 8 Y, then t can be represented as t =
Exercise 1.3.1 Let t be a non-zero element of X 8 Y . Show that there exist a positive integer n, some linearly independent (xi)i=l,n c X , and
some linearly independent ( Y ~ ) ~c =Y ~such , ~ that t =
n
C zi 8 yi
i=l
The following lemma is well known from linear algebra.
Lemma 1.3.2 Let k be a field, M , N , V three k-vector spaces, and the linear maps 4 : M* 8 V -+ H o m ( M , V ) ,4' : H o m ( M , N * ) ( M @ N ) * ,p : M* 8 N* -+ ( M 8 N ) * defined by -+
4 ( f @ v ) ( m )= f ( m ) v for f $ ' ( g ) ( m8 n ) = g ( m ) ( n )for g
E
E
M*,v E V , mE M,
H o m ( M , N * ) ,m E M , n E N ,
p ( f @ g ) ( m @ n= ) f ( m ) g ( n ) for f E M * , g E N * , m E M , n E N .
Then: 2) 4 is injective. If moreover V is finite dimensional, then 4 is an isomorphism. iz) 4' is an isomorphism. iii) p is injective. If moreover N is finite dimensional, then p is an isomorphism.
xi
Proof: i) Let x E M* 8 V with 4 ( x ) = 0. Let x = fi @ vi (finite sum), with fi E M i , v i E V and (vi)i are linearly independent. Then 0 = + ( x ) ( m )= fi(m)vi for any m E M , whence fi(m) = 0 for any i and m. It follows that fi = 0 for any i , and then x = 0. Thus 4 is injective. Assume now that V is finite dimensional. For V = Ic it is clear that 4 is an isomorphism. Since the functors M* 8 (-) and H o m ( M , -) commute with finite direct sums, there exist isomorphisms $1 : M*@ V -t ( M *~ k and : ( H o m ( M ,k ) ) n -+ H o m ( M , V ) , where n = d i m ( V ) . We also have ) ~ , direct sum of n an isomorphism 43 : ( M * 8 k)n 4 ( H ~ m ( M , l c ) the isomorphisms obtained for V = k . Moreover, 4 = (b2&41, thus 4 is an isomorphism too (also see Exercise 1.3.3 below).
xi
)
~
1.3. THE DUAL (COIALGEBRA
17
ii) I f ( X i ) i c I and Y are k-vector spaces, then there exists a canonical iso) ( X i , Y ) . In particular, i f I is a morphism H O ~ ( @ ~ ~ ~ X ~ ,HYo m basis o f M , .then M k(') and we obtain the canonical isomorphisms
--
-- niG1
ul : H o m ( M , N * ) -+ ( ~ o m ( kN*))' , u2 : ( ( k @ N)*)'
+
( M @ N)*.
Since clearly for M = lc the associated.map q5' is an isomorphism, we obtain a canonical isomorphism ug : ( H o m ( k ,N*))'
-+
((k@ N ) * ) ' ,
and moreover 4' = u2u3uli SO $' is an isomorphism too. iii) W e note that p = q5'&, where q50 is the morphism obtained as q5 for V = N * . Then everything follows from the preceding assertions. I Exercise 1.3.3 Show that if M is a finite dimensional vector space, then the linear map $ : M* @ V -+ H o m ( M , V ) ,
defined by $ ( f @ v ) ( m ) = f ( m ) v for f E M * , v E V , m E M , is an isomorphism. Exercise 1.3.4 Let M and N be k-vector spaces. Let the k-linear map:
Q f E M * , g E N * , x E M , y E N . Then the following assertions hold:
a) I m ( p ) is dense in ( M @ N ) * . b) If M or N is finite dimensional, then p is bijective. Corollary 1.3.5 For any k-vector spaces M I , . . . , Mn the map 0 : MT @ . . . @ M; ( M I 8 . .. @ M,)* defined by O ( f l @ . . @ fn)(ml8 . .. @ m n )= f l ( m l ) . . . f,(m,) is injective. Moreover, zf all the spaces Mi are finite dimensional, then 0 is an isomorphism. -+
Proof: The assertion follows immediately by induction from asertion iii) o f the lemma. I
I f X , Y are k-vector spaces and v : X -+ Y is a k-linear map, we will denote by v* : Y * X * the map defined by v * ( f )= fv for any f E Y * . W e made all the necessary preparations for constructing the dual algebra o f a coalgebra. Let then ( C ,A , E )be a coalgebra. W e define the maps M : C* @ C* -, C * ,M = A*p, where p is defined as in Lemma 1.3.2, and u :k C * ,u = ~ * q 5 where , q5 : k --t k* is the canonical isomorphism. -+
-+
,
18
CHAPTER 1. ALGEBRAS AND COALGEBRAS
Proposition 1.3.6 ( C * ,M , u ) is an algebra. Proof: Denoting M ( f @ g ) by f
* g, from the definition we obtain that
for f , g E C * and c E C. From this it follows that for f , g, h E C* and c E C we have
hence the associativity is checked.
We remark now that for a E k and c 6 C we have u(cu)(c) = aa(c). The second condition from the definition of an algebra is equivalent to the fact that u ( 1 ) is an identity element for the multiplication defined by M , that is u ( 1 ) * f = f * u ( 1 ) = f for any f E C*, and this follows directly from I C E ( c ~ )=c C ~ c1E(cz)= C.
Remark 1.3.7 The algebra C* defined above is called the dual algebra of the coalgebra C . The multiplication of C* is called convolution. Most of the times (zf there is no danger of confusion), we will simply write fg instead o f f * g for the convolution product of f and g. I Example 1.3.8 1 ) Let S be a nonempty set, and k S the coalgebra defined i n 1.1.4 1). Then the dual algebra is ( k S ) * = H o m ( k S , k ) with multiplication defined by ( f * g)(s)= f (s)g(s)
for f , g E ( k S ) * ,s E S . Denoting by M a p ( S , k ) the algebra of functions from S to k , the map 0 : ( k S ) * -+ M a p ( S , k ) associating to a morphism f E (kS)*its restriction to S is an algebra isomorphism. 2) Let H be the coalgebra defined in 1.1.4 2). Then the algebra H* has multiplication defined by
for f , g E H " , n E N , and unit u : k n E N.
-+
H*,u(a)(c,) =
for
CY
E
k and
1.3. THE DUAL (C0)ALGEBRA
19
H* is isomorphic to the algebra of formal power series k [ [ X ] ]a, ca.nonica1 isomorphism being given by
The dual problem is the following: having an algebra ( A ,M, u ) can one introduce a canonical structure of a coalgebra on A*? We remark that is is not possible to perform a construction similar to the one of the dual algebra, due to the inexistence of a canonical morphism (A@A)*-+ A*@A*. However, if A is finite dimensional, the canonical morphism p : A* 8 A* -+ ( A8 A)* is bijective and we can use p-' . Thus, if the algebra ( A ,M,u) is finite dimensional, we define the maps A : A* -+ A * @ A *and E : A* --+ k by A = p - l M * a n d & = +u*, where 4 : k* -+ k is the canonical isomorphism, $(f ) = f (1) for f E k*. We remark that if A ( f ) = C,g , 18h,, where g,, h, E A*, then f (ab) = C,g,(a)h,(b)for any a, b E A. Also if (g:, hi), is a finite family of elements in A* such that f (ab)= C, g;(a)hi(b)for any a, b E A, then C ,g, 8 h, = C, gi 8 hi, following from the injectivity of p. In conclusion, we can define A ( f ) = Cg, 8 h, for any (g,, h,) E A* with the property that f (ab)= C,g,(a)h,(b)for any a, b E A.
Proposition 1.3.9 If ( A ,M, u ) is a finite dimensional algebra, then we have that ( A * ,A , E ) is a coalgebra.
xi 8 hi. We let A(gi) = C jg:,j 8 9:;
Proof: Let f E A* and A ( f )= and A(h;)= C jhij 63 hi:,.. Then
('8
')'(f)
=
gi
Cg1@ h:,J hZJ' @
2 ,.I
We consider the map B : A* @ A* @ A* -+ ( A8 A 8 A)* defined by B(u @ v @ w)(a@ b 63 c) = u(a)v(b)w(c)for u,v,w E A*,a, b, c E A. This map is injective by Corollary 1.3.5. But
B(Cg:,, e gi:j 8 hi)(a8 b 8
C)
=
C
g:,j
(a)g::j(b)hi( c )
C H A P T E R 1. A L G E B R A S A N D C O A L G E B R A S
20 and
i
=
f (abc)
and then due to the injectivity of 0 we obtain that
i.e. A is coassociative. We 'also have
hence
xi&(gi)hi
=f,
and similarly
xi~(hi)gi f , =
SO
the counit property
I
is also checked.
Remark 1.3.10 It is possible to express the comultiplication of the dual
coalgebra of an algebra A using a basis of A and its dual basis in A*. Let then (ei)i be a basis of the finite dimensional algebra A and e: E A* defined by ef ( e j ) = Si,j (Kronecker symbol). Then (ef )i is a basis of A*, called the dual basis, and (ej*8 e;)j,l is a basis of A* B A*. It follows that for a n element f E A* there exist scalars (aj,l)j,l such that A ( f ) = Cjtl aj,le58e;. Taking into account the definition of the comultzplzcation, it follows that for fixed s , t we have
f (eset) =
aj,,e;(es)e;(et)= a,,,. i,l
W e have obtained that
Exercise 1.3.11 Let A = Mn(lc) be the algebra of n x n matrices. T h e n
the dual coalgebra of A is isomorphic to the matrix coalgebra M c ( n ,k ) . The construction of the dual (co)algebra described above behaves well with respect to morphisms.
1.3. THE DUAL (C0)ALGEBRA
21
Proposition 1.3.12 i) I f f : C -+ D is a coalgebra morphism, then f * : D* --+ C* is an algebra morphism. zi) If f : A -t B is a morphism of finite dimensional algebras, then f * : B * -+ A* is a morphism of coalgebras. Proof: i) Let d*, e* E D* and c E C. Then
=
x
d*(f (.I)).*( f ( ~ 2 ) ) (f is a coalgebra morphism) = C(f*(d*))(c1)(f*(e*))(cz)
*
= (f*(d*) f*(e*))(c)
and hence f *(d*e*)= f *(d*)f*(e*). Moreover, f * ( ~ g = ) E~ f = EC, SO f * is an algebra morphism. ii) We have to show that the following diagram is commutative.
xi
xi
Let b* E B* , (AA*f *) (b*) = AA*(b* f ) = gi@hi ~i AB*(b*) = pJ Qqj . Denoting by p : A* @ A* 4 (A @ A)* the canonical injection, for any a€A,b€Bwehave
and
CHAPTER 1. ALGEBRAS AND COALGEBRAS
22
which proves the commutativity of the diagram. Also (EA*f *)(b*)= EA* (b*f ) = (brf)(l) = b*(f (1)) = b*(l) = EB* (b*),
I
so f * is a morphism of coalgebras.
Corollary 1.3.13 The correspondences C H C* and f contravariant functor (-)* : k - Cog 4 k - Alg.
H
f * define a
I
Denoting by k - f .d.Cog and k - f .d.Alg the full subcategories of the categories k - Cog and k - Alg consisting of all finite dimensional objects in these categories, the preceding results define the contravariant functors (-)* : k - f.d.Cog -+ k - f.d.Alg and (-)* : k - f.d.Alg -r k - f.d.Cog which associate the dual (co)algebra (there is no danger of confusion if we denote both functors by (-)*). We will show that these functors define a duality of categories. We recall first that if V is a finite dimensional vector space, then the map Ov : V -t V**,Ov(v)(v*) = v*(v) for any v E V, v* E V* is an isomorphism of vector spaces.
Proposition 1.3.14 Let A be a finite dimensional algebra and C a finite dimensional coalgebra. Then: i) OA : A 4 A** is an isomorphism of algebras. ii) Bc : C -t C** is an isomorphism of coalgebras. Proof: i) We have to prove only that BA is an algebra morphism. Let a , b E A and a* E A*. Denote by A the comultiplication of A* and let A(a*) = fi @I gi E A* 63 A*. Then
xi
so BA is multiplicative. We also have that BA(l)(a*)= a*(l) = EA* (a*), so B A ( ~=) & A * , i.e. BA preserves the unit. ii) We denote by A and the comultiplications of C and C**.We have to show that the following diagram is commutative.
a
1.4. CONSTRUCTIONS FOR COALGEBRAS
If p : C** @ C** -+ (C* @ C*)* is the canonical isomorphism and c E C, c*, d* E C* then putting X(Oc(c)) = C, f , 8 g; we have P(@~C)(C))(C*
88) =
Cp(fi8 gz)(c*@ d*) 2
=
Oc(c* * 8 ) '
= (c* * d*)(c)
and
=
c*(cl)d*(cz)= (c* * d*)(c)
proving the commutativity of the diagram. We also have that
showing that ~ c k . 0=~EC and the proof is complete.
Example 1.3.15 By Exercise 1.3.11 that M,(lc)* ceding proposition shows that M c ( n ,k)* 2 M,(k).
1.4
2
I M c ( n ,k ) . The pre-
I
Constructions in the category of coalgebras
Definition 1.4.1 Let ( C I A l & be ) a coalgebra. A k-subspace D of C is called a subcoalgebra if A ( D ) C D @ D. I It is clear that if D is a subcoalgebra, then D together with the map A D : D -+ D @ D induced by A and with the restriction E D of E to D is a coalgebra.
24
CHAPTER 1. ALGEBRAS AND COALGEBRAS
Proposition 1.4.2 If (Ci)iEr a family of subcoalgebras of C , then CiEI Ci is a subcoalgebra. Proof:
I
A(CiEI ci)= CiEIA(ci)G CiEIci8c.c- (CiEI C i ) @ ( CZ .E I C . ). 2
In the category k-Cog the notion of subcoalgebra coincides with the notion of subobject. We describe now the factor objects in this category.
Definition 1.4.3 Let ( C , A , E )be a coalgebra and I a k-subspace of C . Then I is called: i ) a left (right) coideal if A ( I ) C C 8 I (respectively A ( I ) G I 8 C). ii) a coideal if A ( I ) G I 8 C C 8 I and € ( I ) = 0. I
+
Exercise 1.4.4 Show that if I is a coideal it does not follow that I is a left or right coideal. Lemma 1.4.5 Let V and W two k-vector spaces, and X vector subspaces. Then (V 8 Y )n ( X 8 W ) = X 8 Y .
G
V, Y
CW
Proof: Let ( x j )j E J be a basis in X which we complete with ( x j )j e J t up to a basis of V . Also consider ( Y , ) , , ~ a basis of Y, which we complete with ( Y , ) ~ , to ~ get a basis of W. Consider an element
in (V 8 Y )n ( X @ W ) ,where aj,, bjp,c j p ,d j p are scalars. Fix jo E J , po E P and choose f E V * ,g E W * such that f ( x j o ) = 1, f ( x j ) = 0 for any j E J U J ' , j # jo, and g ( y p o )= l , g ( y p )= 0 for any p E P U P 1 , p # po. Since q E V @ Y , it follows that (f 8 g ) ( q ) = 0. But then denoting by q5 : k 8 k -+ k the canonical isomorphism, we have q5(f @ g ) ( q ) = bjop0, hence bjop, = 0. Similarly, we obtain that all of the bjp,cj,, d j p are zero, and thus q = 0. It follows that (V 8 Y )n ( X 8 W ) 5 X 8 Y. The reverse inclusion is clear. I
Remark 1.4.6 If I is a left and right coideal, then, by the preceding lemma A ( I ) G ( C 8 I ) n ( I 8 C ) = I 8 I , hence I is a subcoalgebra. I The following simple, but important result is a first illustration of a certain finiteness property which is intrinsic for coalgebras.
1.4. CONSTRUCTIONS FOR COALGEBRAS
25
Theorem 1.4.7 (The Fundamental Theorem of Coalgebras) Every element of a coalgebra C is contained i n a finite dimensional subcoalgebra. Proof: Let c E C . Write A z ( c ) = C c i @ x i j @ d j with , linearly independent i ,.i
ci's and dj's. Denote by X the subspace spanned by the xij's, which is finite dimensional. Since c = ( E @ I @ € ) ( A 2 ( c ) )it, follows that c E X. Now
and since the d,'s are linearly independent, it follows that
Since the ti's are linearly independent, it follows that A ( x i j ) E C @ X . Similarly, A ( x i j )E X @ C , and by the preceding remark X is a subcoalgebra.
I The following lemma from linear algebra will also be useful. Lemma 1.4.8 Let f : Vl -+ V. and g : Wl -+ W2 morphisms of lc-vector spaces. Then K e r ( f @ g ) = K e r ( f ) 8 Wl Vl @ K e r ( g ) .
+
Proof: Let ( v , ) , ~ A , be a basis of K e r ( f ) ,which we complete with ( v , ) , ~ ~ ~ to form a basis of V;. Then ( f ( v , ) ) , ~is~a ~linearly independent subset of V2. Analogously, let ( w p ) p E B 1be a basis of K e r ( g ) which we complete with ( w p ) p ~ to ~ ,a basis of Wl. Again ( g ( w p ) ) p E ~is, a linearly independent family in W2. Let
c,pv, @ wp
q=
E
K e r ( f @ g).
aEAiUA2
OEBIUBZ
Then aEAlUA2 OEBiUB2
By the linearly independence of the family ( f ( v , ) @ g ( ~ ~ ) ) , it~ ~ ~ , p follows that cap = 0 for any cu E A2,P E B 2 . Then q E K e r ( f ) @ WI VI 8 K e r ( g ) and we obtain that
+
K e r ( f 8 g ) C K e r ( f ) 8 WI The reverse inclusion is clear.
+ VI 8 K e r ( g ) I
Proposition 1.4.9 Let f : C --+ D be a coalgebra morphism. Then I r n ( f ) is a subcoalgebra of D and K e r ( f ) is a coideal i n C .
26
CHAPTER 1. ALGEBRAS AND COALGEBRAS
Proof: Since f is a coalgebra map, the following diagram is commutative.
Then A o ( I m ( f )) = A o ( f( C ) ) = ( f €9 f ) A c ( C ) c ( f @ f ) ( C @ C ) = f ( C ) @ f ( C ) = I m ( f ) @ I m ( f ), so I m ( f ) is a subcoalgebra in D. Also ADf ( K e r ( f ) ) = 0, so ( f @ f ) A c ( K e r ( f ) )= 0 and then
I
by Lemma 1.4.8, hence K e r ( f ) is a coideal.
The construction of factor objects, as well as the universal property they have, are given in the next theorem.
Theorem 1.4.10 Let C be a coalgebra, I a coideal and p : C -+ C / I the canonical projection of lc-vector spaces. Then: i ) There exists a unique coalgebra structure o n C / I (called the factor coalgebra) such that p i s a morphism of coalgebras. ii) I f f : C 4 D i s a morphism of coalgebras with I K e r ( f ) , then there exists a unique morphism of coalgebras 7 : C / I -+ D for which Tp = f . Proof: i) Since ( p @ p ) A ( I )C ( p @ p ) ( I @ C + C @I ) = 0, by the universal property of the factor vector space it follows that there exists a unique linear map & : C / I --+ C / I @ C / I for which the following diagram is commutative.
This map is defined by h ( 2 ) = CFi @ G, where E = p(c) is the coset of c modulo I . It is clear that
--
( K @ I ) ~ (=E()I @ A)A(z)=
xZi@z@~
1.4. CONSTRUCTIONS FOR COALGEBRAS is coassociative. Moreover, since & ( I )= 0, by the universal prophence erty of the factor vector space it follows that there exists a unique linear map 2 : C / I -+ k such that the following diagram is commutative.
We have Z(Z) = E ( C ) for any c E C , and then
x,
It follows that ( C / I , T ) is a coalgebra, and the commutativity of the two diagrams above shows that p is a coalgebra map. The uniqueness of the coalgebra structure on C / I for which p is a coalgebra morphism follows from the uniqueness of and 5. ii) From the universal property of the factor vector space it follows that -there exists a unique morphism of Ic-vector spaces f : C / I D such that f p = f , defined by f ( ~ = ) f ( c ) for any c E C . Since -+
and E D ~ ( z= )
ED(^ (c)) = E C ( C ) = S ( C )
it follows that f is a morphism of coalgebras.
I
Corollary 1.4.11 (The fundamental isomorphism theorem for coalgebras) Let f : C -+ D be a morphism of coalgebras. Then there exists a canonical isomorphism of coalgebras between C / K e r ( f ) and I m ( f ) . I Exercise 1.4.12 Show that the category k - Cog has coequalizers, i.e. if f , g : C --+ D are two morphisms of coalgebras, there exists a coalgebra E and a morphism of coalgebras h : D -+ E such that h o f = h o g . We describe now a special class of elements of a coalgebra C.
28
CHAPTER 1. ALGEBRAS AND COALGEBRAS
Definition 1.4.13 A n element g of the coalgebra C i s called a grouplike element i f g # 0 and A(g) = g @ g . T h e set of grouplike elements of the coalgebra C i s denoted by G(C). I
The counit property shows that ~ ( g=) 1 for any g E G(C). Moreover, we show that they are linearly independent. Proposition 1.4.14 Let C be a coalgebra. T h e n the elements of G(C) are linearly independent. Proof: We assume that G(C) is not a linearly independent family, and look for a contradiction. Let then n be the smallest natural number for which there exist g , g l , . . . ,gn E G(C), distinct elements such that g = xi=,,, aigi for some scalars ai. If n = 1, then g = a l g l and applying E we obtain al = 1 and hence gl = g, a contradiction. Thus n 2 2. Then all ai are non-zero (otherwise we would have such a linear combination for a smaller n). We apply A to the relation g = x i = , , , aigi and we obtain
Replacing g , it follows that
Since the elements g l , . . . ,g, are linearly independent (otherwise again we would obtain one of them as a linear combination of less then n grouplike elements), it follows that for i f. j we have aiaj = 0, a contradiction. I
If A is a finite dimensional algebra, then the grouplike elements of the dual coalgebra have a special meaning. Proposition 1.4.15 Let A be a finite dimensional algebra and A* the dual coalgebra of A. T h e n G(A*) = A l g ( A , k), the algebra maps from A t o k. Proof: Let f E A*. Then f is a grouplike element if A ( f ) = f 8 f , and taking into account the definition of the dua.1 coalgebra, this implies that f (ab) = f (a)f (b) for any a , b E A. Moreover, f (1) = ~ ( f = ) 1, SO f is a I morphism of algebras. Exercise 1.4.16 Let S be a set, and k S the grouplike coalgebra (see Example 1.1.4, 1 ) . Show that G ( k S ) = S .
1.4. CONSTRUCTIONS FOR COALGEBRAS
29
We can now give an example of a coalgebra which has no grouplike elements.
Example 1.4.17 Let n > 1 and C = M c ( n , k ) the matrix coalgebra from Example 1.1.4.5). Then C is the dual of the matrix algebra K ( k ) (from Example 1.3.1 I ) , hence G ( C ) = A l g ( M n ( k ) ,k ) . O n the other hand, there are no algebra maps f : M,(k) -+ k , since for such a morphism K e r ( f ) would be an ideal (we will use sometimes this terminology for a two-sided ideal) of M1,(k), so it would be either 0 or M n ( k ) . But K e r ( f ) = 0 would imply f injective, which is impossible because of dimensions, and K e r ( f ) = M n ( k ) is again impossible because f (1) = 1. Therfore G ( C ) = 0. I Exercise 1.4.18 Check directly that there are no grouplike elements i n M C ( n ,k ) if n > 1. We study now products and coproducts in categories of coalgebras.
Proposition 1.4.19 The category k - Cog has coproducts. Proof: Let (Ci)iEI be a family of k-coalgebras, 6BiEICi the direct sum of this family in kM and qj : C j -t BiEICi the canonical injections. Then there exists a unique morphism A in kM such that the diagram
is commutative. Also there exists a unique morphism of vector spaces E for which the diagram
30
CHAPTER 1. ALGEBRAS AND COALGEBRAS
is commutative. It can be checked immediately, looking at each component, that (eiEICi,A,&)is a coalgebra, and that this is the coproduct of the family (Ci)iEIin the category k - Cog. I Before discussing the products in the category k - Cog we need the concept of tensor product of coalgebras. Let then (C,A c , EC) and (D, AD, ED) twocoalgebrasandA:C@D-+C@D@C@D,~:C@D+kthemaps defined by A = ( I @ T @ I ) ( A ~ @ A DE )= , ~%(Ec@ED), where T(c@d) = d@c and q5 : k @ k 4 k is the canonical isomorphism. Using the sigma notation we have A(c@d)= C c l @ d l @ c z @ d z
for any c E C, d E D. We also define the maps rc : C @ D -+ C, r o : C @ D -+ D by rc(c@d) = C E D ( ~ ) , T ~ ( C =@ cC(c)d ~ ) for any c E C, d E D.
Proposition 1.4.20 (C@D , A, E) is a coalgebra and the maps rc and r~ are morphisms of coalgebras. Proof: From the definition of A it follows that
and
A)(C c1 @ dl B c2 B d2)
( I @ A)A(c @ d) = (ICBD @
= E c i 8 dl @ c2 @ dz @ c3 @ d3,
showing that A is coassociative. We also have
and analogously C E(CI@ dl)(cz @ dz) = c @ d, showing that C 63 D is a coalgebra. The fact that rc is a morphism of coalgebras follows from the
1.4. CONSTRUCTIONS FOR COAL,GEBRAS relations (Tc 8i r c ) A ( c8d ) =
c I & D ( d i )'8 ~ 2 & D ( d 2 )
and E C T C ( C '8 d ) = E C ( C ) E D (= ~ E) ( C
@ d).
I Similarly, T D is a morphism of coalgebras. We can now prove that products exist, not in the category k - Cog, but in the full subcategory of all cocommutative coalgebras k - C C o g . This result is dual to the one saying that in the category of commutative k-algebras, the tensor product of two such algebras is their coproduct. Proposition 1.4.21 Let C and D be two cocommutative coalgebras. T h e n C 8 D , together with the maps T C and T D is the product of the objects C and D i n the category Ic - C C o g . Proof: Let E be a coalgebra, and f : E -+ C,g : E -+ D two morphisms of coalgebras. We prove that there exists a unique morphism of coalgebras 4 : E -+ C @ D such that the following diagram is commutative.
We define Then
4 :E
~ C $ ( X= )
-+
C 8 D by d ( x ) =
f ( x 1 ) 64 g ( x 2 ) for any
f
f
(xI)&D(g(.2)) =
hence rc4 = f , and analogously coalgebra map. We have
.rrD4 =
3:
E E.
(xl)&~(x= 2 )f ( x )
g. We show now that
4 is a
CHAPTER I . ALGEBRAS AND COALGEBRAS
32 and
But E is cocommutative, hence
= 1 x 1 8 (x2)2 '8 (x2)18 5 3 =
and from here it follows that Moreover,
XI
8 53 8 X2 8 X4
A+(x)= (4g 4)AE(x).
thus €4 = E E . It remains to prove that 4 is unique. For this, note that if 4' : E -+ C @D is a morphism of coalgebras with .rrC4'= f and .rrDcP1 = g , then
since clearly (rc8 x D ) Ais the identity. Therefore
4'
= ( f 8 ~ ) A= E 4.
I
Another frequently used construction in the theory of coalgebras is the one of co-opposite coalgebra. Let (C,A, E ) be a k-coalgebra and the map A c O p : C -t C 8 C , ACOp = T A , where T : C 8 C + C 8 C is the map defined by T ( a 8 b) = b 8 a.
Proposition 1.4.22 ( C , A C O p ,E ) is a coalgebra. Proof: Immediate.
1.5. THEFINITE DUAL
33
Remark 1.4.23 The coalgebra defined in the previous proposition is called the co-opposite coalgebra of C and it is denoted by CCOp.This concept is dual to the one of opposite algebra of an algebra. We recall that if ( A ,M , u ) is a k-algebra, then the multiplication MT : A@A -, A and the unit u define an algebra structure on the space A, called the opposite algebra of A. This I is denoted b y AOp. Proposition 1.4.24 Let C be a coalgebra. Then the algebras (CCoP)* and (C*)Opare equal. Proof: Denote by MI and M2 the multiplications in (CCOp)* and (C*)"P. Then for any c*,d* E C* and c E C we have
MI(c*@ # ) ( c ) = (c* @ G ) ( T A ( c ) = ) ~c*(cz)d*(ci)
which ends the proof.
I
We close this section by giving the dual version for coalgebras of the extension of scalars for algebras. Let (C,A , E ) be a k-coalgebra and 4 : k -, K a morphism of fields. We define A' : K@kC -t ( K @ ~ C ) @ K ( K @and ~C) E' : K@kC 4 K by At(a@c)= E ( a @ c l ) @ ( l @ cand a) ~ ' ( a @= c )a4(e(c)) for any a E K , c E C . The following result is again easily checked
Proposition 1.4.25 ( K @h: C , A', E') is a K-coalgebra.
I
1.5 The finite dual of an algebra We saw in Proposition 1.3.9 that for any finite dimensional algebra A, one can introduce a canonical coalgebra structure on the dual space A*. In this section we show that to any algebra A we can associate in a natural way a coalgebra, which is not defined on the entire dual space A*, but on a certain subspace of it. Let then A be an algebra with multiplication M : A @ A -, A. We consider the following set
A" = { f
E
A* I K e r ( f ) contains an ideal of finite codimension)
We recall that a subspace X of the vector space V has finite codimension if d i m ( V / X ) is finite. It is clear that if X and Y are subspaces of finite codimension in V , then X n Y also has finite codimension, since there exists an injective morphism V / ( X n Y) -+ V / X x V / Y . Then if f , g E A0
34
CHAPTER 1. ALGEBRAS AND COALGEBRAS
+
+
it follows that K e r ( f ) n Ker(g) Ker(f g) and so f g E A". Also for f E A O , aE k we have K e r ( f ) C K e r ( a f ) , so af E A" too. Thus A" is a k-subspace in A*. It is on this subspace that we will introduce a coalgebra structure associated to the algebra A.
Lemma 1.5.1 Let f : A -, B be a morphfism of algebras and I an ideal of finite codimension in B . Then the ideal f - I ( I ) has finite codimension in A. Proof: Let p : B -+ B / I be the canonical projection. Then the map p f : A -+ B / I is a morphism of algebras and Ker(pf ) = f - l ( I ) . Then A/ f - I ( I ) E I m ( p f ) 5 BII which has finite dimension. I
Lemma 1.5.2 Let A, B be algebras and f : A -+ B a morphism of algebras. Then: i) f *(BO) A", where f * is the dual map o f f . iz) If we denote by 4 : A* @ B * -+ (A@B)* the canonical injection, we have 4(A0 @ B O )= ( A @ B)". iii) M*(AO)C_ $(A0 @ A"), where M is the multiplication of A and $ : A* @ A* -+ (A @ A)* is the canonical injection. Proof: i) Let b* E BOand I be an ideal of finite codimension in B which is contained in Ker(b*). Then f -'(I) is an ideal of finite codimension in A by Lemma 1.5.1 and fP1(I) & Ker(b*f ) = Ker(f*(b*)). It follows that f*(b*) E A". ii) Let a* E A", b* E B0 and I , J ideals of finite codimension in A, respectively B , with I C Ker(a*),J g Ker(b*). Then A @ J I @ B & Ker(4(a* @ b * ) ) , and since A @ J + I @ B is an ideal in A @ B and A / I @ B/J, which is finite dimensional, it (A @ B)/(A @ J I @B ) follows that 4(a* @ b*) E (A €3 B)", so 4(A0 @ BO)C (A @ B)". Let now h E (A @ B)" and K an ideal of finite codimension of A @ B with K Ker(h). We define I = {a E Ala @ 1 E K}, which is an ideal of A, and J = {b E BI1 @ b E K), which is an ideal of B. Since I is the inverse image of K via the canonical algebra map A 4 A @ B , sending a to a @ 1, from Lemma 1.5.1 we deduce that I has finite codimension in A. Analogously, J has finite codimension in B , and moreover A @ J I@ B is an ideal of finite codimensioninA@B. Clearly A @ J + I @ B C K , s o h ( A @ J + I @ B ) = O . Then since (A @ B)/(A @ J I @ B ) r? A I I @ B I J, there exits an making the following diagram commutative
+
-
+
+
+
1.5. THE FINITE DUAL
where p1 gi p~ are the canonical projections. Since A / I and B / J have both finite codimension, there exists a canonical isomorphism 8 : ( A l l ) *63 ( B /J)* 4 (AII @ B / J)*. Then there exist (r,), ( A / I ) *(S,), , G ( B / J ) *with , h = B(C,y, ~36,). Then
c
=
C~ ( y i4 @
) ( P I (a)€4 p j
(b))
@(xi
and SO h = yipi@bip~). But yip1 E A* 8i I C Ker(yipr),hence yip1 E A", and analogously yip^ E BO. We have obtained that h E #(A081B O ) . Consequently, we also have that ( A@ B)" C @(A0 @ B O ) and , the equality is proved. iii) Let a* E A0 and I a finite codimensional ideal of A with I C Ker(a*). Then A@I+I@Ais a finite codimensional ideal of A@Aand A@I+I@AC Ker(a*M),hence a*M = M(a*) E ( A 8 A)" = $(A0 @ A") by assertion ii) . I We are now in a position to define the coalgebra structure on A". With ) ( A@ the notation of the preceding proposition we know that M * ( A O C A)" = +(A0@ A 0 )where , : A* @A*-t (A@A)*is the canonical injection. By Lemma 1.5.2 the map can be regarded as an isomorphism between A" 8A" and ( A@ A)". Let then A : A" -+A" 8 A", A = M*. We also define the map E : A" -, k by €(a*)= a*(l).
+ +
Proposition 1.5.3 ( A 0 A, , E ) is a coalgebra.
Proof: Consider the following diagram
CHAPTER 1. ALGEBRAS AND COALGEBRAS
We have denoted again by $J the restriction to AO@Aoof the map defined above, by the canonical injection, and by j the inclusion. We prove step by step the commutativity of some subdiagrams. First, (+A)(a*)= (++-'M*)(a*) = M*(a*)= M * j ( a * )for any a* E A*, hence +A = M * j . In order to show that $ J I ( A 8 I ) = ( M @I I)*.JI we note first that if a* E A0 and A ( a * ) = a; 8 bf , then +-lM*(a*) = a; 8 bf , so a*M = M*(a*) = +(a; 8 bf) and then for any a, b E A we have a*(ab) = at (a)bf (b). Then if a*,b* E A" ei a, b, c E A we have
xi
xi
xi =
xi
+l(~a:@b:@b*)(a@b@c) i
=
a: ( a )b: (b)b*(c) i
=
=
a* (ab)b*( c ) ( ( M @ I)*$(a*8 b*))(a@ b @ c )
hence +l(A@ I) = ( M @ I)*+. Similarly +1(I 8 A) = ( I @ MI*+. Also
Then
1.5. THE FINITE DUAL
37
is injective, we obtain ( A @ I ) A= ( I @ A ) A ,the coassociativity and since of A". Let now a* E A0 and A ( a * ) = C i a ; @ bf. Then
for any a E A , and therefore and the proof is complete.
xi&(a:)bf = a*. Similarly, C i~ ( b f ) a f
=
a*
I
B be ,a morphism of algebras. Then f * ( B O )C A" and the induced map f0 : B0 -4 A0 is a morphism of coalgebras.
Proposition 1.5.4 Let f : A
--+
Proof: We already saw in Lemma 1.5.2 that f * ( B O )c A'. In order t o show that f O is a morphism of coalgebras, we have to prove that the following two diagrams are commutative.
The commutativity of the second diagram is immediate, since
for any b* E BO. As for the first diagram, let b* E B O ,A B o (b*)= bf @ c:. Since 11, : A0 @ A" -t ( A @ A)* is injective, in order to show that ( f O@ f " ) A B o = AAO fO it is enough to show that +( f O @ f " ) A B o = f O. But for x, y E A we have
xi
38
CHAPTER 1. ALGEBRAS AND COALGEBRAS
the last equality following directly from the definition of ABO(A = +-lM*, hence +A = M*, then apply it t o b* and then to f (x)@ f ( 3 ) ) . Further on, we have
and the proof is complete. The following result is a consequence of the last two propositions. Corollary 1.5.5 The mappings A H A0 and f ant functor (-)" : k - Alg -+ k - Cog.
I-+
fO
define a contravari-
I
We are going to give now a characterization of the elements of A" which will be useful for computations. To this end, we first remark that A* = Hom(A, k), and since A is an A-left, A-right bimodule, it follows that A* is an A-left, A-right bimodule, with actions given as follows. If a E A and a* E A*, then: - the left action of A on A* by (a a*)(b) = a*(ba) for any b E A. - the right action of A on A* by (a* a)(b) = a*(ab) for any b E A.
--
Proposition 1.5.6 Let A be a k-algebra and f E A*. Then the following assertions are equivalent: 1) f E AO. 2) M * ( f ) f )(A0 8 A"). 3) M * ( f ) E $(A* 8 A*). 4) A f is finite dimensional. 5) f A is finite dimensional. 6) A f A is finite dimensional.
-
- -
Proof: 1) + 2) was proved in Lmma 1.5.2. 2) =+ 3) is clear. 4) Let M*( f ) = af @ bf) with af ,bf E A*. Then for a , b E A we 3) f ) ( a ) = ( C i bf(b)af)(a), that is have f (ab) = Ci ab(a)bf(b), hence (b b f = bf(b)aa. This shows that A f is contained in the subspace of A* generated by and this is finite dimensional.
)(xi
-
--
Ci
-
-
'(a a * ) should be read as " a hits a*", and (a* a ) as "a* hit by a". These conventions were proposed by W. Nichols, and are known as the "Nichols dictionary". = twhit by". They also include " -- = twhits" and
".-
1.5. T H E FINITE DUAL
39
-
-
4 ) + 1) We assume that A f is finite dimensional. Since A f is a left A-submodule of A*, we have a morphism of k-algebras induced by m for any this structure 7r : A A E n d ( A f ) defined by n ( a ) ( m )= a a E A, m E A f . Since E n d ( A f ) has also finite dimension, it follows that I = K e r ( n ) is an ideal of finite codimension in A. But for a E I we have f ( a ) = ( a - f ) ( l ) = 0 , so I C K e r ( f ) and f E A". 3) + 5 ) and 5) + 1) are proved in the same way as 3) + 4) and 4 ) + l ) , working with the right hand structure. 1) + 6) If f E A", let I be an ideal of finite codimension in A with I C K e r ( f ) . Then for a, b E A we have (a f b ) ( I ) = f(bIa) C f ( I ) = 0 , hence A f A C {g E A*lg(I) = 0 ) . But I has finite codimension, so C A completing a basis of I to a basis of A. Denoting by there exist a: E A* the map for which a:(I) = 0, a:(a,) = it follows immediately , , so it has that the subspace {g E A* J g ( I )= 0 ) is generated by ( u : ) ~ = ~and finite dimension. 6) 4) is clear. I
-
--
--
-
- -
*
Remarks 1.5.7 1) The preceding proposition shows that A" zs the biggest coalgebra contained in A* and induced by M . Indeed, we have M* : A* -t ( A @A)* and if X C A* is a coalgebra induced by M , then M* ( X ) C +(X@ X ) . But then M * ( X ) L +(A*@ A * ) ,hence X C M* -'(+(A* @ A * ) )= A". 2) It may happen that A" = 0. For example, let A be a simple Ic-algebra of infinite dimension over Ic (e.g. an infinite field extension of Ic). Then A does not contain any proper ideals of finite codimension, and hence A" = 0 .
I We recall now another result from linear algebra. Lemma 1.5.8 If f l , . . . ,f , be linearly independent elements of V * , then there exist v l , . . . , v, E V with f i ( v j ) = S i , j for any i , j = 1,. . . , n. Moreover, the vi 's are also linearly independent. Proof: We proceed by induction on n. For n = 1 the result is clear. Let now f l , . . . ,f,+l be linearly independent in V*, and appplying the induction hypothesis we find v l , . . . ,v, E V such that f i ( v j ) = 6i,j for any i ,j = 1,. . . , n. Since f i , . . . , f,+l are linearly independent, we have fn+l fn+l(ui)fi # 0, hence there exists a v E V with f.r,+l(v) # Ci=l,,f i t + l ( ~ i ) f i ( v )Then . fn+l(v - z i = l , , v i f i ( v ) ) # 0 and
IVIultiplying then v - Ci=,,,, vi f i ( v ) by a scalar, we obtain a E V with f j ( w n + l ) = Sj,,+l for j = 1 , . . . ,n 1. Let wj = vj - f,+l(vj)wn+l
+
40
C H A P T E R 1. A L G E B R A S A N D C O A L G E B R A S
for j = l , . . . , n . We have f i ( w j ) = f i ( v j ) = 6i,j for any i, j = I , . .. ,n and fn+l ( w j ) = 0 for j = 1, . . . ,n, hence w l , . . . ,wn+l satisfy the required conditions. The last assertion follows by applying the fi's to a linear combination of the vi7s which is equal to zero, in order to deduce that all the coefficients I are zero.
Remark 1.5.9 W e have that
From this we see that A" is a subbimodule of A* with respect to
-
and
-. If we use Exercise 1.3.1 and assume the fi's and gi's are linearly independent, then by Lemma 1.5.8 we obtain that fi E A -- f c A" and
-
gi E f A c A". Hence we remark that if we use (1.1) as the definition for A", the fact that it is a coalgebra follows exactly as in the finite dimensional case treated i n Proposition 1.3.9. The above show that for all f E A", A f A is a subcoalgebra of A", and it is finite dimensional. This means that the Fundamental Theorem of Coalgebras (Theorem 1.4.7) holds i n A". Another consequence is that subcoalgebras of A" are subbimodules of A", and the intersection of a family of subcoalgebras of A" is a subcoalgebra (it contains the subbimodule generated by each of its elements). Thus the smallest subcoalgebra containing f E A" i s A - f -A. Finally, we remark that we have the description of the grouplikes in A" exactly as i n the finite dimensional case (Proposition 1.4.15):
- -
G ( A O )= { f : A
-, k
I
f is an algebra map}.
An important particular case of a finite dual is obtained for the case the algebra A is a semigroup algebra k G , for some monoid G ( k G has basis G as a k-vector space and multiplication given by ( a x ) ( b y ) = ( a b ) ( x y ) for a, b E k , x , y E G ) . As it is well known, there exists an isomorphism of vector spaces
4 : kG -+ ( k c ) * = H n ( k G , k ) ,
$(f)(C aixi) = C i
.if
(xi).
i
Consequently, kG becomes a kG-bimodule by transport of structures via
4:
1.5. THE FINITE DUAL
Definition 1.5.10 If G is a monoid, we call
the representative coalgebra of the monoid G.
I
Note that the coalgebra structure on Rk(G) is also transported via 4. Rk(G) is a kG-subbimodule of kc, and consists of the functions (which are called representative) generating a finite dimensional kG-subbimodule (or, equivalently, a left or right kG-submodule). We have
and the coalgebra structure on Rk(G) is given as follows: if f E Rk(G), and fi, g; E kG are such that f (xy) = C fi(x)gi(y), then A( f ) = C fi @gi. Note also that for any k-algebra A we have
where A, denotes the multiplicative monoid of A. The following exercise explains the name of representative functions.
Exercise 1.5.11 Let G be a group, and p : G + GL, (k) a representation ~ ~ V(p) , be the k-subspace of kG of G. If we denote p(x) = ( f , j ( ~ ) ) let spanned by the {fij)i,j. Then the following assertions hold: i) V(p) is a finite dimensional subbimodule of kG. iz) Rk(G) = C V(p), where p ranges over all finite dimensional represenP
tations of G. We now go back to Exerc,ise 1.1.5 to give the promised explanation of the name "trigonometric coalgebra". The functions sin and cos : R + R satisfy the equalities
and cos(x
+ y) = COS(X)cos(y) - sin(x) sin(y).
These equalities show that sin and cos are representative functions on the group ( R , +). The subspace generated by them in the space of the real functions is then a subcoalgebra of RR((R, +)), isomorphic to the trigonometric coalgebra. Other examples of representative functions include: 1) The exponential function exp : R --+ R, because ex+Y = exeY, hence A (exp) = exp 8 exp.
CHAPTER 1. ALGEBRAS AND COALGEBRAS
42
In general, if G is a group, then f E Rk(G) is grouplike if and only if f is a group morphism from G to (k*, .). 2) The logarithmic function lg : (0, co)-+ R, because lg(xy) = lg(x) +lg(y), hence A(1g) = lg 8 1 1Qi lg, where 1 denotes the constant function taking the value 1. Such a function is called primitive. It is also easy to see that in general, if G is a group, then f E Rk(G) is primitive if and only if f is a group morphism from G to (k, +). 3) Let d, : R -+ R be defined by d,(x) = Since d,(z y) = Cdi(x)dn-i(y) (by the binomial formula), it follows that the dn's are rep-
+
5.
+
i
resentative functions on the group ( R , +), and the subspace they span is a subcoalgebra of RR((R, +)), isomorphic to the divided power coalgebra from Example 1.1.4 2. This explains the name of this coalgebra. We saw in Proposition 1.3.14 that a finite dimensional (co)algebra is isomorphic to the dual of the dual. We study now the connection between a (co)algebra and dual of the dual in the case of arbitrary dimensions. Proposition 1.5.12 Let C be a coalgebra and 4 : C -+ C** the canonical injection. Then Im(q5) C C* " and the corestriction q5c : C -+ C* " of 4 is a morphism of coalgebras. Proof: Let c E C and c*, d* E C*. Then
-
-
+(c) = ~ c * ( c z ) $ ( c l ) It . follows that C* $(c) is finite and hence c* dimensional, being contained in the subspace generated by all 4(cl). This shows that 4(c) E C* O. We prove now that the following diagrams are commutative.
For the second diagram this is clear, since ( E O ~ C ) (= C )~ c ( c ) ( ~ c = ' ) ~ c ( c ) ( E= ) 4 ~ ) .
1.5. T H E FINITE DUAL
43
For the first diagram we will show taht $Ao& = $($c8 $c)A, where $ : C** @ C** -+ ( C * 8 C*)* is the canonical injection. If c E C and c*, d* E C* we have
As $ is injective, it follows that
= ($c@ &)A.
I
Definition 1.5.13 A coalgebra C is called coreflexive if dc is an isomora phism. Exercise 1.5.14 The coalgebra C is coreflexive if and only if every ideal of finite codimension i n C* is closed i n the finite topology. Exercise 1.5.15 Give another proof for Proposition 1.5.3 using the representative coalgebra. Deduce that any coalgebra is a subcoalgebra of a representative coalgebra. Remark 1.5.16 The above exercise, combined with Remark 1.5.9, shows that the intersection of a family of subcoalgebras of a coalgebra is a subcoalgebra (see Corollary 1.5.29 below). The same argument provides a new proof for the Fundamental Theorem of I Coalgebras (Theorem 1.4.7). Exercise 1.5.17 If C is a coalgebra, show that C is cocommutative i f and only if C* is commutative. Proposition 1.5.18 Let A be an algebra. Then the map i A : A --t AO*, defined by i A( a )( a * ) = a* ( a ) for any a E A, a* E A', is a morphism of algebras. Proof: We have first that i A ( l ) ( a * )= a * ( l ) = &Ao(a*),SO i ~ ( 1 =) E A O , which is the identity of A"*. Then for any a, b E A,a* E A" we have
C H A P T E R 1. A L G E B R A S A N D C O A L G E B R A S
44
i ~ ( a b ) ( a *=) a* (ab), and (ia(a)in(b))(a*) =
C iA(a)(a;)iA(b)(b;) C a;(a)b;(b) P
=
P
( + A ( a * ) ) ( a@ b) M* ( a * )( a @ b) = (a*M)(a@b) = a*(ab) = =
where we have denoted A ( a * ) = C pa; 8 b;, and from here it follows that iA(ab) = i A ( a ) i A ( b ) . I
Exercise 1.5.19 If A is an algebra, then the algebra map i A : A -+ A"* defined by i A ( a ) ( a * )= a*(a) for any a E A,a* E A", i s not injective i n general. Definition 1.5.20 A n algebra A is called proper (or residually finite dimensional) i f i A is injective, it is called weakly reflexive if i A is injective, and it is called reflexive i f i~ is bijective. I Exercise 1.5.21 If A is a k-algebra, the following assertions are equivalent: a) A is proper. b) A" is dense in A" in the finite topology. c) The intersection of all ideals of finite codimension i n A is zero. We now have the following contravariant functors (-)O
: k - Alg
-+
k - Cog
In order to work with covariant functors, we will regard these two functors as covariant functors (-)" : k - Alg -+ ( k - Cog)' and (-)* : ( k - cog)' --, k - Alg, where by C0 we have denoted the dual of the category C.
Theorem 1.5.22
(-)O
is a left adjoint for (-)*.
Proof: Let C be a coalgebra and A an algebra. We define the maps
1.5. T H E FINITE DUAL
45
v : H o v L ~ - ~ ~C*) ~ (A' ,H ~ r n k - ~ ~ A') ~(C, by u ( f ) = f * i A for f : C + A0 a morphism of coalgebras and v(g) = go& for g : A -+ C* a morphism of algebras. We prove that the maps u and v are inverse one to each other. For f E H O ~ ~ - ~ , , (A"), C , c E C and a E A we have
which shows that vu = I d . For g E Hornk-~l,(A, C*),a E A and c E C we have
hence also uv = I d . Since the maps u and v are natural, it follows that the claimed adjunction holds. I We remark that if we restrict and corestrict these two functors to the subcategories of (co)-algebras of finite dimension, we obtain the duality of categories described in Section 1.3. In the general case, the above adjunction suggests that many phenomena occur in a dual manner in the categories k - Alg and k - Cog. We show now that there exists a correspondence between the subcoalgebras of a coalgebra and the ideals of the dual algebra. This correspondence is in the spirit of the duality discussed above, since subcoalgebras are subobjects, ideals are subspaces allowing the construction of factor objects, and the notions of subobject and factor object are dual notions.
CHAPTER 1. ALGEBRAS AND COALGEBRAS
46
Proposition 1.5.23 Let C be a coalgebra and C* the dual algebra. Then: i ) If I is an ideal in C * , it follows that I' is a subcoalgebra in C . ii) If D is a k-subspace of C , then D is a subcoalgebra in C if and only if DL is an ideal in C*. In this case the algebras C * / D L and D* are isomorphic. Proof: i) Let ( e j ) j E Jbe a basis in C and let c E I'. Then there exist (cj)jG J C C with A(c) = C j , cj @ e j We show that cj E 'I for any j E J. Choose jo E J and h E C* with h ( e j )= S j ,j, for any j E J. If f E I then fh E I , so ( f h ) ( c )= 0. But ( f h ) ( c )= C j EfJ( c j ) h ( e j )= f (cj,). It follows that f (cj,) = 0 for any f E I, hence cj, E I'. Thus we have that A(c) E 'I @ C . Similarly, one can prove that A ( c ) E C @ I L . Then A(c) E ( I L8 C )fl (C @ I I ) = I L @ 'I by Lemma 1.4.5, and so I L is a subcoalgebra of C . ii) If DL is an ideal in C * , then it follows from i) that D" is a subcoalgebra in C . But D l L = D from Theorem 1.2.6. Conversely, if D is a subcoalgebra, let i : D 4 C be the inclusion, which is an injective coalgebra map. Then i* : C* + D* is a surjective algebra map, and it is clear that Ker(i*) = D'. It follows that DL is an ideal, and the required isomorphism follows from the fundamental isomorphism theorem for algebras.
I A similar duality holds when we consider the ideals of an algebra and the subcoalgebras of the finite dual.
Proposition 1.5.24 Let A be an algebra, and A" its finite dual. Then: i) If I is an ideal of A, it follows that I' n A" is a subcoalgebra in A". ii) If D is a subcoalgebra in A", it follows that D' is an ideal in A.
xi
Proof: i) Let f E I L n A Oand A ( f ) = ui@vi with ui,vi E A" such that ( v ~are ) ~linearly independent. From the preceding lemma it follows that there exist (ai)i A with v i ( a j )= Si for any i,j. Then for any j and any a E I we have aaj E I , hence 0 = f (aaj) = ui(a)vi(aj)= u j ( a ) ,so uj E I L . It follows that A( f ) E ( I I ~ A "@) A 0 .Similarly, A ( f ) E A" @ ( I 1 n A 0 ) and then from Lemma 1.4.5 it follows that A ( f ) E (I' n , A O @ ) ( I L n A"). ii) Let a E DL and b E A. If f E D and A ( f ) = Ciui @ vi with ui, vi E D, then f (ab) = ui(a)vi(b)= 0, hence also ab E D'. Analogously, ba E I D L , so DL is an ideal in A.
xi
xi
Dual results also hold if we look at the connection between the coideals of a coalgebra and the subalgebras of the dual algebra, respectively the subalgebras of an algebra and the coideals of the finite dual.
Proposition 1.5.25 Let C be a coalgebra, and C* the dual algebra. Then: i ) If S is a subalgebra in C*, then S' is a coideal in C . ii) If I is a coideal in C , then I L is a subalgebra in C*.
1.5. THE FINITE DUAL
47
Proof: i) Let i : S -t C * be the inclusion, which is a morphism of algebras. Then i0 : C*" -t So is a morphism of coalgebras, and hence if qbc : C + C*" is the canonical coalgebra map, we have a morphism of coalgebras iOqbc: C -t So. If c E C , then c E Ker(ioqbc) if and only if q5C(c)i = 0, and this is equivalent to f(c) = 0 for any f E S . Thus K e r ( ~ " q 5 ~=) S L . Since the kernel of a coalgebra map is a coideal, assertion i) is proved. ii) Let .rr : C + C / I be the canonical projection, which is a coalgebra map. Then n* : (C/I)* -+ C* is a morphism of algebras. If f E C*, then f E Im(n*) if and only if there exists g E (C/I)* with f = gn. But this is equivalent to the fact that f ( I ) = 0, or f E 1'. It follows that I I' = Im(.rr*), which is a subalgebra in C*. Proposition 1.5.26 Let A be an algebra and A" its finite dual. Then: i) If S is a subalgebra in A, then SLn A" is a coideal in A". ii) If I is a coideal in A", then I' is a subalgebra in A. Proof: i) Let i : S -+ A be the inclusion, which is a morphism of algebras, and i0 : A" -, Sothe induced coalgebra map. Then
so A" n SL is a coideal in A". ii) Let a , b E I' and let f E I . Let A ( f ) = Then f (ab) = x,u,(a)v,(b) = 0, so ab E I'. f (1) = 0 for any f E I, and hence 1 E I'.
x,
U,
@ v, E
Now
A" @ I + I @ A".
E A O ( ~ )=
0 shows that
I
We give now connections between the left (right) coideals of a coalgebra and the left (right) ideals of the dual algebra, respectively between the left (right) ideals of an algebra and the left (right) coideals of the finite dual.
Proposition 1.5.27 Let C be a coalgebra, and C * the dual algebra. Then: 2) If I is a left (right) ideal in C*, then I' is a left (right) coideal in C . ii) If J is left (right) coideal in C , then JL is a left (right) ideal in C*.
xi
Proof: i) Assume that I is a left ideal. Let c E I' and A(c) = ci @ di with (ci)i linearly independent. Fix a j and choose c* 'E C* with c*(ci) = Si,j for any i. If f E I then c*f E I, hence (c*f ) (c) = 0. But (c*f )(c) = ' c*(ci)f (di) = f (d,), so dj E I'. Consequently, A(c) E C @ I I . ii) Assume that J is a left coideal, so A ( J ) C C @J J . Let f E ' J', and c* E C*. Then (c*f ) ( J ) C c*(C)f ( J ) = 0, hence c*f E J'. It follows that JLis a left ideal. The right hand versions are proved similarly. I
xi
48
CHAPTER 1. ALGEBRAS AND COALGEBRAS
Proposition 1.5.28 Let A be an algebra, and A" its finite dual. Then: i ) If I is a left (right) ideal i n A, then I' n A" is a left (right) coideal in A". ii) If J is a left (right) coideal i n A", then J' is a left (right) ideal i n A. Proof: i) We assume that I is a left ideal. If f E I L n A", let A ( f ) = ui @vi with ui, vi E A" and (ui)ilinearly independent. By Lemma 1.5.8 it follows that there exist (ai)i A with u i ( a j ) = Si for any i ,j . If a E I , then aia E A, hence 0 = f ( a i a ) = v i ( a ) , and so vi E I' for any i. We obtained A ( f ) E A" 8 (I' n A"). ii) Assume that J is a left coideal, and let a E J', b E A. Since A ( J ) G A" 8 J , we obtain that f (ba) = 0 for any f E J, hence ba E J I . I
xi
Corollary 1.5.29 Let C be a coalgebra, and ( X i ) i a family of subcoalgebras (left coideals, right coideals). Then nixiis a subcoalgebra (left coideal, right coideal).
XI
= (Cix:)'. But are ideals (left Proof: We have nixi = X t is also an ideal (left ideal, right ideals, right ideals) in C*, thus ideal). Then (CiX I ) l is a subcoalgebra (left coideal, right coideal) in C .
xi
I Remark 1.5.30 The above corollary allows the definition of the subcoalgebra (left coideal, right coideal) generated by a subset of a coalgebra as the smallest subcoalgebra (left coideal, right coideal) containing that set. I Example 1.5.31 The finite dimensional subcoalgebra containing an element of a coalgebra constructed i n Theorem 1.4.7 is actually the subcoalgebra generated by that element. For if Az(c) = Ci,jc i @ x i j 8 d j are as in the proof of the fundamental theorem, and if D is any subcoalgebra containing c, then A 2 ( c ) E D 18D 8 D, and applying maps of the form fi 8 I @ gj to this ( f i is 1 o n ci and zero o n any other cil, and gj is 1 o n d j , and zero o n I any other djt), we obtain that the span of the xij's is contained in D. We give now an application of the fundamental theorem for coalgebras. The following definition is due to P. Gabriel [85].
Definition 1.5.32 Let A be a k-algebra. A is called a pseudocompact algebra if A is a topological k-algebra, Hausdorff separated, complete, and satisfies the A P C axiom: APC: The ring A has a basis of neighbourhoods of zero formed by the ideals I of finite codimension. I Theorem 1.5.33 If C is a k-coalgebra, then the dual algebra C* is a pseudocompact topological algebra.
1.6. THE COFREE COALGEBRA
-
49
Proof: We first prove that the multiplication M : C* x C* C* is continuous. If f , g E C*, let f g wL be an open neighbourhood of f g (W is a finite dimensional subspace of C). It is easy to see that there exist two finite dimensional subspaces WI, W2 of C , such that A(W) L Wl 69 W2. So f W: (resp. g + W;) is an open neighbourhood of f (resp. g). We prove that M(f w;,g w,I) s f g w L . (1.2),
+
+
+
+
+
Indeed, if u E W?, v E W$, we have
Now if c E W, then A(c) = C cl 8 ca E Wl 69 W2. Therefore, (fv)(c) = C f (c1)v(c2) = 0 (since v E W;). Similarly, we have ug(c) = 0 and (uv)(c) = 0. Thus f v ug uv E WL, and (1.2) is proved, and M is continuous. We show now that the set of all ideals I of C* has the following properties: i) all I's have finite codimension (i.e. dim(C*/I) < a). ii) they are open and closed in the finite topology, and they form a basis for the filter of neighbourhoods of zero in C*. Indeed, the set of W L , where W ranges over the finite dimensional subspaces of C is a basis for the filter of neighbourhoods of zero in C*. By Theorem 1.4.7, there exists a finite dimensional subcoalgebra D of C such that W C_ D. Then D L C WL, and I = DIis an ideal by Proposition 1.5.23, ii). Since C * / I D*, it follows that I is finite codimensional. Also since I = D L and dim(D) < CQ, then I is an open neighbourhood of zero in C*. Since I is an ideal (in particular a subgroup) then I is also closed in the finite topology. Now if f , g E C*, f # g, there exists x E C such that f ( x ) # g(x). It is easy to see that (f + xL) n (g xL) = 0, so C* is Hausdorff separated. Finally, since C is the sum of its finite dimensional subcoalgebras D (by Theorem 1.4.7), we have
+ +
--
+
-
= lim Hom(D, k) = lim D* = lim (c*/D~), t t
and therefore C* is complete.
1.6
I
The cofree coalgebra
It is well known that the forgetful functor U : Ic-Alg -t kM (associating to a k-algebra the underlying k-vector space) has a left adjoint T. The functor
CHAPTER 1. ALGEBRAS AND COALGEBRAS
50
T associates to the k-vector space V the free algebra, which is exactly the tensor algebra T(V). We shall return t o this object in Chapter 3, where we show that it has even a Hopf algebra structure. For the moment we will only need the existence of a natural bijection
for any k-algebra A and any k-vector space V. This bijection follows from the adjunction property of T. Due to the duality between algebras and coalgebras, it is natural to expect the forgetful functor U : k - Cog -+ k M to have a right adjoint. Definition 1.6.1 Let V be a lc-vector space. A cofree coalgebra over V is a pair (C,p), where C is a k-coalgebra, and p : C -+ V is a k-linear map such that for any k-coalgebra D, and any k-linear map f : D -+ V there 1 exists a unique morphism of coalgebras 7 : D C , with f = -+
pT.
Exercise 1.6.2 Show that if ( C , p ) is a cofree coalgebra over the k-vector space V, then p is surjective.
Standard arguments show that any two cofree coalgebras over V are isomorphic. The main problem is to show that such a cofree coalgebra always exists. Lemma 1.6.3 Let X and Y be two k-vector spaces. Then there exists a natural bijection between Hom(X, Y*) and Hom(Y, X * ). Proof: Define
4 : Hom(X, Y *) -+Hom(Y, X*) by $(u) (9)(x) = u(x) (y) for any u E Hom(X, Y *), x E X , y E Y and ?I, : Hom(Y, X*) -+ Hom(X, Y*)
by $(v)(x)(y) = v(y)(x) for any v E Hom(Y, X*), x E X , y E Y. Then
hence $4 = I d , and
hence also @$J
= Id.
1.6. THE COFREE COALGEBRA
51
Lemma 1.6.4 Let V be a k-vector space. Then there ehists a cofree coalgebra over V * * . Proof: Let D be a coalgebra. From the preceding lemma there exists a bijection 4 : H o m ( D , V * * ) -+ H o m ( V * ,D * ) defined by 4 ( u ) ( v * ) ( d )= u ( d ) ( v * )for any u E H o m ( D , V * * )d, E D , v* E V * . From the universal property of the tensor algebra it follows that there exists a bijection $1 : H o m ( V * ,D * ) H ~ m k - ~ ~ , ( T ( vD*). * ) , We denote by $ l ( f ) = 7 for any f E H o m ( V S ,D*). From the adjunction described in Theorem 1.5.22 we have a bijection -+
~ any f E H o m k A l g ( T ( V *D ) ,* ) , where $D : defined by 4 2 ( f )= f " 4 for D -+ D*" is the canonical morphism. Composing the above bijections we obtain a bijection
-
q52414(f)= (q5(f))o$D.Let i
I
: V * -+ T ( V * ) be the inclusion, and y : T ( V * ) " -+ V * * ,p = i * j , where j : T ( V * ) " -+ T ( V * ) *is the inclusion. We , is a cofree coalgebra over V * * . Let f : D -+V * * be a show that ( T ( V * ) "p) morphism of k-vector spaces. We show that there exists a unique morphism of coalgebras f : D --iT(V*)" for which f = p f . Let f = 42414(f)= (q5(f))04D. Then for d E D gi v* E V * we have
hence f = p f , and so such f exists. As for the uniqueness, if h E H O ~ ~ - ~ , T~( V( *D) ", )satisfies ph = f , let h = q5nq51q5(f') with f' E H o m ( D , V * * ) .Then from the above computations it follows that ~ 4 ~ 4 ~ 4=( f f' , ' henceph ) = f ' . We obtain that f' = f and I so h = $ 2 $ 1 4 ( f ) = f .
CHAPTER 1. ALGEBRAS AND COALGEBRAS
52
Lemma 1.6.5 Let (C,p) be a cofree coalgebra over the k-vector space V, and let W be a subspace of V. Then there exists a cofree coalgebra over W . Proof: Let D
=
C{EIE c C subcoalgebra with p(E)
C
W ) , and let
.rr : D -+ W denote the restriction and corestriction of p. We show that (D, .rr) is a cofree coalgebra over W. Let F be a coalgebra, and f : F -+ W k-linear map. Then if : F -+ V is k-linear (i is the inclusion), hence there exists a unique h : F -+ C, morphism of coalgebras with if = ph. But
ph(F) = if ( F ) C i(W) = W , so p(h(F)) 2 W, and from the definition of D it follows that h ( F ) C D. Denoting by h' : F -+ D the corestriction of h, we have .rrhl(x) = ph(x) = if (x) = f (x) for any x E F, so ~ h =' f . If g : F -+ D is another morphism of coalgebras with ng = f , then denoting by gl : F --+ C the morphism given by gl(x) = g(x) for any x E F, we clearly have pgl = i f , so h = gl. This shows that g = h' and the uniqueness 1 is also proved.
Theorem 1.6.6 Let V be a k-vector space. Then there exists a cofree coalgebra over V. Proof: From Lemma 1.6.4 we know that a cofree coalgebra exists over V". Since V is isomorphic to a subspace of V**, from Lemma 1.6.5 we I obtain that a cofree coalgebra also exists over V. Corollary 1.6.7 The forgetful functor U adjoint.
:
k - Cog
-+
k M
has a right
Proof: For V E kM we denote by F C ( V ) the cofree coalgebra over V constructed in Theorem 1.6.6. If V, W E kM and f E Hom(V, W), then there exists a unique morphism of coalgebras F C ( f ) : F C ( V ) --+ F C ( W ) for which f p = n F C ( f ) , where p : F C ( V ) V and .rr : F C ( W ) W are the morphisms defining the two cofree coalgebras. From the universal property, it follows that for any coalgebra D and any k-vector space V there exists a natural bijection between Hom(D, V) and f f ~ m k - ~ ~ , (FDC,( V ) ) , hence the functor F C : kM -+ k - Cog defined above is a right adjoint for U. I -+
-+
It is easy to see that for V = 0, the cofree coalgebra over V is k with the trivial coalgebra structure (in which the comultiplication is the canonical isomorphism, and the counit is the identical map of k).
Proposition 1.6.8 Let p : k 0 be the zero morphism. Then (k,p) is a cofree coalgebra over the null space. -+
Proof: Let D be a coalgebra, and f : D -+ 0 the zero morphism (the unique morphism from D t o the null space). Then there exists a unique
1.6. THE COFREE COALGEBRA
53
morphism of coalgebras g : D -+ k for which pg = f , namely g = E D , which is actually the only morphism of coalgebras between D and k. I The cofree coalgebra over a vector space is a universal object in the category of coalgebras. We show now that such a universal object also exists in the category of cocommutative coalgebras. Definition 1.6.9 Let V be a vector space. A cocommutative cofree coalgebra over V is a pair (E,p), where E is a cocommutative k-coalgebra, and p : E -+ V is a k-linear map such that for any cocommutative k-coalgebra D and any k-linear map f : D --t V there exists a unique morphism of coalgebras : D + E with f = I
7
pT.
Exercise 1.6.10 Show that if (C,p) is a cocommutative cofree coalgebra over the k-vector space V, then p is surjective. Theorem 1.6.11 Let V be a k-vector space. Then there exists a cocommutative cofree coalgebra over V. Proof: Let (C,p) be a cofree coalgebra over V, whose existence is granted by Theorem 1.6.6. Denoting by E the sum of all cocommutative subcoalgebras of C (such subcoalgebras exist, e.g. the null subcoalgebra) and let i : E -+ C be the canonical injection. It is clear that E is a cocommutative coalgebra, as sum of cocommutative subcoalgebras. Then (E,pi) is a cocommutative cofree coalgebra over V. Indeed, if D is a cocommmutative coalgebra, and f : D V is a morphism of vector spaces, then since (C,p) is a cofree coalgebra over V, it follows that there exists a unique morphism of coalgebras g : D -+ C such that pg = f . Since D is cocommutative, it follows that Im(g) is a cocommutative subcoalgebra of C , and hence Im(g) 5 E. We denote by h : D --+ E the corestriction of g to E, which clearly satisfies ih = g. Then h is a morphism of coalgebras, and pih = p g = f . Moreover, the morphism of coalgebras h is unique such that pzh = f , for if h' : D -+ E would be another morphism of coalgebras with pih' = f , we would have p(ihl) = f , and ih' : D -+ C is a morphism of coalgebras. From the uniqueness of g it follows that ih' = g = ih, and since i is injective it follows that h' = h, finishing the proof. I -+
For a k-vector space V, we will denote by C F C ( V ) the cocommutative cofree coalgebra over V, constructed in the preceding theorem, In fact, we can construct a functor C F C : kM -+ k - CCog, where k - CCog is the full subcategory of k - Cog having as objects all cocommutative coalgebras. To a vector space V we associate through this functor the cocommutative cofree coalgebra C F C ( V ) . If f : V W is a linear map, and (CFC(V),p) -+
54
CHAPTER 1. ALGEBRAS AND COALGEBRAS
and (CFC(W), T) are the cocommutative cofree coalgebras over V and W, then we denote by CFC(f) : CFC(V) -+ C F C ( W ) the unique morphism of coalgebras for which n C F C ( f ) = f p (the existence and uniqueness of C F C ( f ) follow from the universal property of C F C ( W ) ) . These associations on objects and morphisms define the functor C F C , which is also a right adjoint functor. Corollary 1.6.12 The functor C F C is a right adjoint for the forgetful functor U : k - CCog + k M . Proof: This follows directly from the universal property of the cocommutative cofree coalgebra. I Proposition 1.6.13 The cocommutative cofree coalgebra over the null space is k, with the trivial coalgebra structure, together with the zero morphism. Proof: The cofree coalgebra over the null space is k by Proposition 1.6.8. Since this coalgebra is cocommutative, the construction of the cocommutative cofree coalgebra (described in the proof of Theorem 1.6.11) shows that this is also the cocommutative cofree coalgebra over 0. I We describe now the cocommutative cofree coalgebra over the direct sum of two vector spaces. Proposition 1.6.14 Let Vl, V2 be vector spaces, (Cl, TI), and (C2, ~ 2 the ) cocommutative cofree coalgebras over them. Let
where E ~ , Eare Z the counits of C1 and C2. Then (C1 8 C2, T) is a cocommutative cofree coalgebra over Vl $ V2. Proof: Denote by pl : C1@C2 C1,p2 : C1 @ C2 C2 the morphisms of coalgebras defined by pl ( c 8 e) = ce2(e) and pz (c@e) = eel (c). Also denote by ql : Vl G?V2 Vl and 92 : Vl @ h -4 & the canonical projections. Note l q l and ~ n2p2 = 9 2 ~ . that ~ l p = Let now D be a cocommutative coalgebra, and f : D -+ Vl @ V2 a linear e coalgebra over Vl, it follows map. Since (C1, n1) is a c ~ c ~ m m u t a t i vcofree that there exists a unique morphism of coalgebras gl : D -+ C1 for which ql f = ~ l g l .Similarly, there exists a unique morphism of coalgebras g2 : D -+ C2 for which 92 f = n2g2. We know that C1 @ C2, together with the maps pl,p2 is a product of the coalgebras C1 and C2 in the category of cocommutative coalgebras -+
-+
-)
1.7. SOLUTIONS T O EXERCISES
55
(by Proposition 1.4.21). It follows that there exists a unique morphism of coalgebras g : D -t C1 @ C2 for which p l g = gl and p2g = g2. Then we have q1ng = n l p l g = T l 9 l = q l f , and similarly q27rg = q2f . It follows that n g = f , and hence we constructed a morphism of coalgebras g : D -+ Cl @ Cz such that n g = f. Let us show that g is the unique morphism of coalgebras with this property, and then it will follow that (C1@ C 2 , n ) is a cocommutative cofree coalgebra over Vl $ V 2 . Let g' : D --t C1 @ C2 be another morphism of coalgebras with ng' = f . Then ql f = qlng' = n l p l g l , and from the uniqueness of gl with the property that ql f = n l g l , it follows that plg' = g l . Similarly, we obtain that p2g' = g2. Finally, from the uniqueness of g with p l g = gl and p2g = g2, we obtain that g' = g , which ends the proof. I
1.7
Solutions to exercises
Exercise 1.1.5 Let C be a k-space with basis { s ,c) . W e define A : C C @ C a n d ~ : C - - t k by
--+
S h o w that (C,A, E ) is a coalgebra. Solution: We have
and
,
The counit property is obvious.
Exercise 1.1.6 S h o w that o n a n y vector space V o n e can introduce a n algebra structure. Solution: It is clear if V = 0. Assume V # 0, choose e E V , e # 0, and complete { e ) to a basis of V with the set S = {x,),~I. Define now an algebra structure on V by exi = 5ie
= xi,
xixj = 0, V i ,j E I ,
CHAPTER 1. ALGEBRAS AND COALGEBRAS
56
We remark that this is actually isomorphic to k[Xi I I] (XiXj 1 i , j € I ) ' and it is also the algebra obtained by adjoining a unit to the ring k S with zero multiplication.
Exercise 1.1.15 Show that i n the category k - Cog, isomorphisms (i. e. morphisms of coalgebras having an inverse which is also a coalgebra morphism) are precisely the bijective morphisms. Solution: Let g : C D be a bijective coalgebra map. We have to show that g-' is also a coalgebra morphism. For c E C we have
-
so g preserves the comultiplication. The preservation of the counit is obvious.
Exercise 1.2.2 A n open subspace i n a topological vector space is also closed. Solution: If U is the open subspace and x @ U, then x + U is a neighbourhood of x which does not meet U, hence the complement of U is open. Exercise 1.2.3 When S is a subset of V * (or V ) , SL is a subspace of V (or V * ) . In fact S'- = (S)', where (S) is the subspace spanned by S . Moreover, we have SL= ((SL)')L, for any subset S of V* (or V ) . Solution: Let S be a subset of V *: and x, y E SL, A, p E k. Then if u E S, u(Xx p y ) = Au(x) pu(y) = 0, and so Ax py E. ' -S Now, since S C (S), it is clear that SL 2 (S)'. On the other hand, if x E S'-, any linear combination of elements in S will vanish in x, so equality holds. ((SL)I)'-. Conversely, if x E 5''-, Finally, since S G ( S L ) I , we have 5''we have that u(x) = 0 for all u E (SL)' by definition, so SL= ((SL)')L. The other assertions are proved in a similar way.
+
+
+
>
+
Exercise 1.2.4 The set of all f W'-, where W ranges over the finite dimensional subspaces of V , form a basis for the filter of neighbourhoods of f E V * i n the finite topology. Solution: We know that a basis for the filter of neighbourhoods of f E V *
1.7. SOLUTIONS TO EXERCISES in the finite topology is f + O ( 0 ,X I , .. . , x,). Note that O ( 0 ,X I , . . . , x,) = W L , where W is the subspace of V spanned by X I , . . . , x,.
Exercise 1.2.7 If S is a subspace of V * , then prove that (SL)' is closed i n the finite topology by showing that its complement is open. Solution: I f f @ ( S L ) I ,then there is an x E S1 such that f ( x ) # 0. Then (f + X I )n = 0.
(sL)l
Exercise 1.2.10 If V is a k-vector space, we have the canonical k-linear map d v :v (v*)*,d v ( x ) ( f )= f ( x ) , V x E v, f E V * .
-
Then the followzng assertzons hold: a) The map d v zs znjective. b) I m ( 4 " ) zs dense in ( I / * ) * . Solution: a ) Let z E K e r ( 4 v ) . If x # 0, there exists f E V* such that f ( x ) # 0. Since d v ( x ) = 0, we obtain that f ( x ) = 0, tlf E V * , a contradiction. b) We prove that ( ~ m ( + ~=) )( 0l) . Let f E ( I m ( d v ) ) l C V * . Hence d v ( x ) ( f )= 0 for every x E V . Thus f ( x ) = 0 V x E V , and therefore f = 0.
Exercise 1.2.11 Let V = Vl $ V2 be a vector space, and X = X I $ X 2 a subspace of V * ( X , C V,*, i = 1,2). If X is dense in V * , then X , is dense in V,*, i = 1,2. Solution: Let x E If f E X , then f = f l f 2 , with f l E X I and f i E X2. Since V * = V; $ V,*, we have that f2(V1) = 0 , SO f ( x ) = f l ( x ) f 2 ( x ) = 0. Hence x E X L = ( 0 ) .
+
+
Exercise 1.2.14 Let X C V * be a subspace of finite dimension n. Prove that X is closed i n the finite topology of v* by showing that d i m k ( ( X L ) L5 n.
- -
Solution: Let { f l , . . . , f,)
f)Ker( f,), 2=1 kn, so d i m k ( V / X L ) 5 n, and hence
be a basis of X . Then X L =
and therefore 0 V/XL d i m k ( V / X L ) *5 n. On the other hand, from the exact sequence
we have
0
-
-- -
(v/xL)* v*
(xL)* 0 , Hence d i m k ( ~ ' ) <~ n . Since X C ( X I ) ' ,
so ( v / X L ) * II ( X I ) ' . obtain that X = ( X I ) ' - .
we
CHAPTER 1. ALGEBRAS AND COALGEBRAS
58
Exercise 1.2.15 If X is a finite codimensional k-linear subspace of V * , then X is closed in the finite topology if and only if XXI = xi, where xi is the orthogonal of X in V**. Solution: Assume that X is closed and that d i m k ( V * / X )= n < oo. Let
denote the canonical map. The equality X I = xi actually means $ ( X I )= xi We have 4" ( X I ) = 4 v ( V ) n xi, so C $ ~ ( X ' -C) xi. Let f E V * * , f E xi. Thus f : V * k , with f ( X ) = 0. Since X has finite codimension in V * , it follows from Corollary 1.2.13 ii) that X I has finite dimension as a subspace of V. If we put W = X I , then we have f E ( V * / X ) *= (v*/(xXI)I)* ( ( X I ) * )= * W** W . So there exists a x E W such that f = $v ( x ) ,and thus f E $v ( V )n xi = 4, ( X I ) .Hence $v ( X I ) = xi. Conversely, assume that 4 v ( X X I )= xi. We prove that X = ( X I ) ' - . We have X = K e r ( f ) . If f E xi, there exists x E XXI such that f €xi f = Hence X = K e r ( 4 v ( x ) ) = ( X L ) I . Thus X is closed.
-
--
--
n
XEX"
Exercise 1.2.17 If V is a k-vector space such that V = $ V,, where
{V, I i
iEI
E
I ) is a family of subspaces of V , then $ V,* is dense in V * i n the
finite topology. solution: @ V,' is the subspace X of V * , where i€I
X = { f E V * I f (V,) = 0 for almost all j E I). If x E V , x E
XI,
such that x
# 0, then x
=
C x i , where xi E
V , are
iEI
almost all zero. Since x # 0, there exists io E I such that xi, # 0. Thus there exists f E V * such that f (V,) = 0 for i # i o , and f (xi,) # 0. Clearly f E X , and therefore 0 = f ( x ) = f (xi,), a contradiction.
-
-
Exercise 1.2.18 Let u : V V ' be a k-linear map, and u* : I/'* V* the dual morphism o f u . The following assertions hold: i ) If T is a subspace of V ' , then u* ( T I )= u V 1( T ) I . ii) If X is a subspace of V " , then u* ( X ) I = u-I ( X I ) . iiz) The image of a closed subspace through u* is a closed subspace. iv) If u is injective, and Y C V'* is a dense subspace, it follows that u * ( Y ) is dense i n V * . Solution: i) Let f E u* ( T I ) . Then f = u*( g ) = g o u for some g E
1.7. SOLUTIONS T O EXERCISES
59
T I . If x E uP1(T), then f ( x ) = g(u(x)) = 0, hence u*(TL) C u - l ( ~ ) I . Conversely, let g E uM1(T)'. Define f E V* by f(u(x)) = g(x), and f = 0 on the complement of Im(u). The definition is correct, because if u(x) = u(y), then x - y E Ker(u) C u-l(T). Moreover, f o u = g, since for w E T n Im(u) we have w = u(x), x E u-'(T), and so f(w) = g(x) = 0. It is clear that f E T I , so we have proved that u*(x)' = u-l( X I ) . ii) We have x E u*(x)' f(u(x)) = O V f E X ++ U(X)E X' @ x E u-l(xL). iii) Let X be a closed subspace of V1*.Then
*
u*(X)
=
= = =
=
u*(X) u*(xLL) u-'(x')' (by i)) U*(X)''- (by ii)) u*(X).
iv) We use ii): u * ( Y ) I = u:l(YL) = u-l(O) = 0. Exercise 1.3.1 Lett be a non-zero element of X 4 Y . Show that there exist ~ ,X, , and some a positive integer n, some linearly independent ( x ~ ) ~ = c n
linearly independent ( ~ ~ ) ~ =C1 ,Y, such that t =
C xi 4 yi
i=l Solution: Let n be the least positive integer for which there exists a n
representation t =
C xi 8 yi with (x,),=1,,
linearly independent. We show
i= 1
that (yi)z=l,nare also linearly independent. If not, one of the yi's, say y,, is a linear combination of the others. Thus n-1
yT1=
C aiyi, and then
i= 1
n- 1
n-l
But {xl +alxn, . . . , xn-1 +an-lxn) are obviously linearly independent, and we obtain a representation of t with n - 1 terms, and linearly independent elements on the first tensor positions, a contradiction. Exercise 1.3.3 Show that if M is a finite dimensional lc-vector space, then the linear map q5 : Ad* 4 V -+ Hom(M, V),
CHAPTER 1. ALGEBRAS AND COALGEBRAS
60
defined by q5(f 8 v ) ( m ) = f ( m ) v for f E M * , v E V , m E M , is an isomorphism. Solution: W e know from Lemma 1.3.2 that q5 is injective. Let vi E M , vf
E
M * , i = 1,. . . ,n be dual bases, i.e.
n
C vb(v)vi = v , Qv E M .
i=l
for any f E H o m ( M , V ) we have that f = surjective.
Then
n
$(Cvt 8 f ( v i ) ) ,so 6 is also i=l
Exercise 1.3.4 Let M and N be Ic-vector spaces. Let the k-linear map:
V f E M * , g E N * , x E M , y E N . Then the following assertions hold:
a) I m ( p ) is dense i n ( M @ N ) * . b) If M or N is finite dimensional, then p is bijective.
Solution: a ) W e prove that ( I m ( p ) ) l= ( 0 ) . Let z = be such that z
E
-
C xi @ yi
E M 8N i=l ( ~ m ( ~ )Hence ) l . for any f E M * , g E N* we have
W e can assume that {yi I 1 5 i < n ) are linearly independent. There exists a g E N * such that g(yi) # 0 and g ( y j )= 0 for j # i. R o m (1.3) it follows that f ( x i ) = 0 for every f E M * , thus xi = 0. Since i E (1,. . . , n ) is arbitrary, we have that z = 0. b ) Assume that N is finite dimensional. By induction we can assume that k , and so M * @ N* 2 N k (i.e. d i m k ( N ) = 1). In this case N * M* 63 k N M * . Also (M8 N ) * ( M 63 k)* 2 M*.
-
-
--
Exercise 1.3.11 Let A = M n ( k ) be the algebra of n x n matrices. Then the dual coalgebra of A is isomorphic to the matrix coalgebra M C ( n ,k ) . Solution: I f (eij)lli,jLn is the usual basis o f Mn(Ic) (eij is the matrix having 1 on position ( i ,j ) and 0 elsewhere), let (e&)lli,jln be the dual basis o f A*. W e recall that eije,, = Gjpeiq for any i ,j,p, q. Then Remark 1.3.10 shows that
1.7. SOLUTIONS T O EXERCISES and clearly ( e )= (
1=e
sq)= 6ij
We have obtained that A* is isomorphic to MC(n,k ) .
Exercise 1.4.4 Show that if I is a coideal it does not follow that I is a left or right coideal. Solution: Let k [ X ]be the polynomial ring, which is a coalgebra by A ( X n ) = ( X 63 1
+ 1 8 x)",& ( X n )= O
for n 2 I
Take I = k X , the subspace spanned by X.
Exercise 1.4.12 Show that the category k - Cog has coequalizers, i.e. if f , g : C 4 D are two morphisms of coalgebras, there exists a coalgebra E and a rnorphism of coalgebras h : D -t E such that h o f = h o g. Solution: We show that I = I m ( f - g ) is a coideal of D , then take E = D l 1 and h the canonical projection. If d E I, then we have d = ( f - g ) ( c ) for some c E C, and
Exercise 1.4.16 Let S be a set, and k S the grouplike coalgebra (see Example 1.1.4, 1). Show that G ( k S ) = S . Solution: Let g = C aisi E G ( k S ) . Then A ( g ) = g 8 g becomes
If Ic and 1 are such that Ic # 1 and ak # 0, a1 # 0, let gk and gl be the functions from k S defined by g,(s) = 6,,,, for r E { k , 1). Applying gk @ gl
62
CHAPTER 1. ALGEBRAS AND COALGEBRAS
to both sides of (1.4) we obtain 0 = akal, a contradiction. Thus g = a s ) 1 we obtain g E S. The converse for some a E k and s E S. From ~ ( g = inclusion is clear. Exercise 1.4.18 Check directly that there are no grouplike elements in MC(n,k) if n > 1. Solution: Let x E MC(n,k) be a grouplike, x = C aijeij. Then we have i,j
A(x) = x 8 x, i.e.
Let io # j o , and f E MC(n,k)* such that f (eij) = GiioSjjo. Applying f 8 f to the equality above, we get 0 = a:o jo, hence aiojo = 0. If g E MC(n,k)* is such that g(eij) = SijoSjio, and we apply f 8 g to the same equality, we get aioi, = aiojoajoio= 0. Since io and jo were arbitrarily chosen, it-follows that x = 0, a contradiction. Exercise 1.5.11 Let G be a groupl and p : G
-+ GLn(k) a representation of G. If we denote p(x) = (fij(x))i,jl let V(p) be the k-subspace of kG spanned by the {fijIi,j. Then the following assertions hold: z) V(p) is a finite dimensional subbimodule of kG. ii) Rk(G) = C V(p), where p ranges over all finite dimensional represenP
tations of G. Solution: i) Let f E V(p), and y E G. If f = C a i j fij, then
because
Thus y f = C aijgkjfik E V(p). The proof of f y E V(p) is similar. ii) ( 2 )follows from i) . In order to prove the reverse inclusion, let f E Rk(G) . Then the left kG-submodule generated by f is finite dimensional, say with basis { f ~. . ,. , f,). Write
Then P :G
GLn(k), P(X)= (gij(x))i,j is a representation of G, V(p) is spanned by the gij's, and we obtain that fi = C fj(lc)gij E V(p), thus f E V(P) too. -+
1.7. SOL U'rIONS TO EXERCISES
63
Exercise 1.5.14 The coalgebra C is coreflexive if and only if every ideal of finite codimension i n C* is closed i n the finite topology. Solution: By Exercise 1.2.15, we have to prove that q5c is surjective if and only if for all finite codimensional ideals I of C* we have 'I = 1'. (+) Let I be a finite codimensional ideal of C * . If p E li, then the restriction of p to I is zero, and hence p E C*". Since 4~ is surjective, it follows that p E 1'. Thus I' = I'. (+) Let p E C*". By definition, there exists an ideal I of C*, of finite codimension, such that p I r = 0. It follows that p E 1'- = @c(I'-), hence 4c is surjective. Exercise 1.5.15 Give another proof for Proposition 1.5.3 using the representative coalgebra. Deduce that any coalgebra is a subcoalgebra of a representative coalgebra. Solution: If we take c E C, A ( c ) = C cl 8 c2, and regard the elements of C as functions on C * via q$c,then for f , g E C* we have
This shows that the elements of C are representative functions on C * , and 4c is a coalgebra map.
Exercise 1.5.17 If C is a coalgebra, show that C is cocommutative if and only if C * is commutative. Solution: If C is cocommutative, then for any f ,g E C* and c E C we have ( f * g ) ( c )= C f ( c ~ ) g ( c z=) C f (cz)g(cl)= (g*f ) ( c ) , SO C* is commutative. Conversely, if C* is commutative, then C*" is cocommutative, and the assertion follows from the fact that C is isomorphic to a subcoalgebra of C*". Exercise 1.5.19 If A is an algebra, then the algebra map i A : A -+ A"' defined by i A ( a ) ( a * )= a*(a) for any a E A , a* E A", is not injective i n general. Solution: If A is the algebra from Remark 1.5.7.2), then A"* = 0. Exercise 1.5.21 If A is a k-algebra, the following assertions are equivalent: a) A is proper. b) A" zs dense in A* i n the finite topology. c)The zntersection of all zdeals of finite codimenszon i n A is zero. Solution: B y Corollary 1.2.9, we know that A" is dense in A* if and only if (A0)' = (0). But
(A")'
= {a E A
I f ( a ) = 0,Vf
E A")
=Ker(i~).
64
CHAPTER 1. ALGEBRAS AND COALGEBRAS
Thus a)wb). Denote by 3 the set of finite codimensional ideals of A. Then, by the definition we have that A0 = U 1'. But then IEF
hence we also have b)@c).
Exercise 1.6.2 Show that if (C,p) is a cofree coalgebra over the k-vector space V , then p is surjective. Solution: We know that V can be endowed with a coalgebra structure (see Example 1.1.4, 1). Apply then the definition to the identity map from V to V in order to obtain a right inverse for p. Exercise 1.6.10 Show that if (C,p) is a cocommutative cofree coalgebra over the Ic-vector space V , then p is surjective. Solution: The coalgebra in Example 1.1.4, 1 is cocommutative, so the proof is the same as for Exercise 1.6.2. Bibliographical notes Our main sources of inspiration for this chapter were the books of M. Sweedler [218], E. Abe [I],and S. Montgomery [149], P. Gabriel's paper 1851, B. Pareigis' notes [178], and F.W. Anderson and K. Fuller [3] for module theoretical aspects. Further references are R. Wisbauer [244], I. Kaplansky [104], Heyneman and Radford [95]. I t should be remarked that we are dealing only with coalgebras over fields, but in some cases coalgebras over rings may also be considered. This is the case in [244], [I451 or [88].
Chapter 2
Comodules 2.1
The category of comodules over a coalge-
bra In the same way as in the first chapter, where we gave an alternative definition for an algebra, using only morphisms and diagrams, we begin this chapter by defining in a similar way modules over an algebra. Let then ( A ,M, u ) be a k-algebra.
-
Definition 2.1.1 A left A-module is a pair ( X , p ) , where X is a k-vector space, and p : A €3X X is a morphism of k-vector spaces such that the following diagrams are commutative:
( W e denote everywhere by I the identity maps, and the not named arrow from the second diagram is the canonical isomorphism.) I Remark 2.1.2 Similarly, one can define right modules over the algebra A, the only difference being that the structure map of the right module X is of I the f o m p : X @ A - t X .
CHAPTER 2. COMODULES
66
By dualization we obtain the notion of a comodule over a coalgebra. Let then ( C ,A, E ) be a k-coalgebra.
-
Definition 2.1.3 W e call a right C-comodule a pair ( M ,p), where M is M @ C is a morphism of k-vector spaces such a k-vector space, p : M that the following diagrams are commutative:
Remark 2.1.4 Similarly, one can define left comodules over a coalgebra C , the difference being that the structure map of a left C-comodule M is of the form p : M -+ C 8 M . The conditions that p must satisfy are (A @ I ) p = ( I @ p)p and ( E @ I ) p is the canonical isomorphism. I
2.1.5 The sigma notation for comodules. Let M be a right C comodule, with structure map p : M -t M @ C . Then for any element m E M we denote
d m ) = C m(0)@ m(1)
the elements on the first tensor position (the m ( o )'s) being in M , and the elements on the second tensor position (the m ( l )'s) being in C . If M is a left C-comodule with structure map p : M -4 C @ M , we denote
The definition of a right comodule may be written now using the sigma notation a s the following equalities:
The first formula allows us to extend the sigma notation, as in the case of coalgebras, by writing
2.1. THE CATEGORY OF COMODULES OVER A COALGEBRA =
C m(0)8
h ( l ) ) l@
67
(m(l))z.
The rules for computing with the sigma notation, which were presented in detail in the first chapter, may be easily transferred to comodules. Thus, we can write for any n 2
>
and for any 1 5 i 5 n - 1 we have
For left C-comodules, the sigma notation is the following: if p : M -+ C 4 M is the map defining the comodule structure, then we denote p(m) = 8 r n ( ~for ) any m E M , where this time m ( - l ) are representing elements of C , and m ( ~elements ) in M . Later we will omit the brackets, because even if the sigma notation for comodules will be used in the same time as the one for coalgebras, confusion can always be avoided. The definition of the left comodule may be written using the sigma notation as the following equalities:
C ~ ( m ( - 1 ) ) 7 n (=om ) Also
for any 1
< i < n.
Example 2.1.6 1 ) A coalgebra C is a left and right comodule over itself, the map giving the comodule structure being in both cases the comultiplicationA:C-+C@C. 2) If C is a coalgebra and X is a k-vector space, then X @ C becomes a right C-comodule with structure map p : X @ C -+ X 8 C 8 C induced by A, hence p = I @ A. Thus p(x 8 c ) = C x 8 cl @ cz.
CHAPTER 2. COMODULES
68
3) Let S be a non-empty set, and C = k S , the k-vector space with basis S , endowed with a coalgebra structure as i n Example 1.1.4,l). Let ( M s ) s E s be a family of k-vector spaces, and M = e s E s M s . Then M is a right C-comodule, where the structure map p : M -, M @ C is defined by p(m,) = m, @ s for any s E S and rn, E Ms (extended by linearity). 1 The following result is the analog for comodules of Theorem 1.4.7.
Theorem 2.1.7 (The Fundamental Theorem of Comodules) Let V be a right C-comodule. Any element v E V belongs to a finite dimensional subcomodule. Proof: Let {ci)iEl be a basis for C , denote by p : V structure map, choose v E V , and write
4
V @ Cthe comodule
where almost all of the vi's are zero. Then the subspace W generated by the vi's is finite dimensional. We have
and Cosequently, p ( v k ) = C vi@aijkcj, so W is a finite dimensional subcomodI ule, and v = (I@ E ) P ( V ) E W .
Exercise 2.1.8 Use Theorem 2.1.7 to prove Theorem 1.4.7. We define now the morphisms of comodules. In order to keep to dualizing the corresponding definitions from the case of modules, we also give the definition of a module morphism using commutative diagrams.
-
Definition 2.1.9 i ) Let A be a k-algebra, and ( X ,v),(Y,p ) two left Amodules. The k-linear map f : X Y is called a morphism of A-modules if the following diagram is commutative:
2.1. THE CATEGORY O F COMODULES OVER A COALGEBRA
69
-
ii) Let C be a k-coalgebra, and (M, p), (N, 4) two right C-comodules. The N is called a morphism of C-comodules if the Ic-linear map g : M following diagram is commutative
The commutativity of the second diagram may be written in the sigma notation as: 4(9(c)) =
C s(c(0))
@
C(1).
We can now define the category of right comodules over the coalgebra C . The objects are all right C-comodules, and the morphisms between two objects are the morphisms of comodules. We denote this category by M C . We will also denote the morphisms in MC from M to N by Comc(M, N). Similarly, the category of left C-comodules will be denoted by C ~ For. an algebra A, the category of left (resp. right) A-modules will be denoted by AM (resp. M A ) . Proposition 2.1.10 Let C be a coalgebra. Then the categories
C~
and
Mccopare isomorphic. Proof: Let M E CM with the comodule structure given by the map p : M -+ C @ M , p(m) = C m(-l) @m(o).Then M becomes a right comodule over the co-opposite coalgebra CCop via the map p1 : M -+ M @ CCOp, pl(m) = C m(01 @ m ( - ~ ) .Moreover, it is immediate that if M and N are N is a morphism of left C-comodules, two left C-comodules, and f : M then f is also a morphism of right CcoP-comodules when M and N are regarded with these structures. This defines a functor F : C ~M C C o p . Similarly, we can define a functor G : MCcoP -+ C ~ by , associating to a right Cco"-cornodule M , with structure map p : M -4 M @ CCop,p ( m ) = @ m ( l ) , a left C-comodule structure on M defined by p' : M -+ C @ M, pl(m) = @ m(o). It is clear that the functors F and G I define an isomorphism of categories. -+
-+
C H A P T E R 2. COMODULES
70
R e m a r k 2.1.11 The preceding proposition shows that any result that we
obtain for right comodules has an analogue for left comodules. This is why we are going to work generally with right comodules, without explicitely mentioning the similar results for left comodules. I We define now the notions of subobject and factor object in the category
M ~ . Definition 2.1.12 Let ( M ,p ) be a right C-comodule. A k-vector subspace N of M is called a right C-subcomodule if p ( N ) c N €3 C . I R e m a r k 2.1.13 If N is a subcomodule of M , then ( N ,p N ) is a right Ccomodule, where p~ : N -+ N €3 C is the restriction and corestriction of p to N and N €3 C . Moreover, the inclusion map i : N -+ M , i(n)= n for
any n E N is a morphism of comodules.
I
Let now ( M ,p) be a right C-comodule, and N a C-subcomodule of M . Let M I N be the factor vector space, and p : M -+ M / N the canonical projection, p ( m ) = a, where by Fi we denote the coset of m E M in the factor space. P r o p o s i t i o n 2.1.14 There exists a unique structure of a right C-comodule on M / N for which p : M -+ M / N is a morphism of C-comodules. Proof: Since ( p €3 I ) p ( N ) C ( p €3 I ) ( N €3 C ) p ( N ) €3 C = 0, by the universal property of the factor vector space it follows that there exists a unique morphism of vector spaces 7 : M / N -+ M I N €3 C for which the diagram
is commutative. This map is defined by p(Fi) = Cm(o)€3 m ( ~for ) any m E M. Then ( M / N , p )is a right C-comodule, since
The fact that is a morphism of comodules follows immediately from the commutativity of the diagram.
2.1. THE CATEGORY OF COMODULES OVER A COALGEBRA
71
If we would have a right C-comodule structure on MIN given by w : MIN --+ M / N @ C such that p is a morphism of comodules, then the diagram obtained by replacing 7 by w in the above diagram should be also commutative. But then it would follow that w = from the universal I property of the factor vector space.
Remark 2.1.15 The comodule MIN, with the structure given as in the above proposition, is called the factor comodule of M with respect to the subcomodule N . 1 Proposition 2.1.16 Let M and N be two right C-comodules, and f : M -+ N a morphism of C-comodules. Then I m ( f ) is a C-subcomodule of N and K e r ( f ) is a C-subcomodule of &I.
-
-
Proof: We denote by p~ : M M 8 C and p~ : N N 8 C the maps giving the comodule structures. Since f is a morphism of comodules, we have ( f @ I)pM = pN f . Then
which shows that p ~ ( K e r ( f )C) Ker(f @ I ) = K e r ( f ) @ C , and hence K e r ( f ) is a C-subcomodule of M. Now
~ ~ ( I l n (=f () f )@ I ) P M ( C~ I)m ( f )@ C, which shows that I m ( f ) is a C-subcomodule of N .
1
Theorem 2.1.17 (The fundamental isomorphism theorem for comodules) Let f : M --+ N be a morphism of right C-comodules, p : M --+ M / K e r ( f ) the canonical projection, and i : I m ( f ) --+ N the inclusion. Then there exists a unique isomorphism 7 : M / K e r ( f ) I m ( f ) of C-comodules for which the diagram --+
is commutative.
CHAPTER 2. COMODULES
72
7
Proof: The existence of a unique morphism : M / K e r ( f ) I m ( f ) of vector spaces making the diagram commutative follows from the fundamental isomorphism theorem for k-vector spaces. We know that 7 is defined by 7(m)= f(m) for any 7 i i E M / K e r ( f ) . It remains to show that is a M / K e r (f ) @ C morphism of comodules. Denoting by w : M / K e r ( f ) and 8 : I m ( f ) -+ I m ( f ) @ C the maps giving the comodule structures, we have -+
7
-+
(7@ I ) w ( m )
=
C 7(mco,)
=
Cf(m(o))@m(l)
=
C f (m)(o) f
@ m(1)
@
(m)(l)
= O(f(m)) =
which shows that
eT(m)
7 is a morphism of comodules.
I
Proposition 2.1.18 Let C be a coalgebra. Then the category M~ has coproducts. Proof: Let (Mi)iEIbe a family of right C-comodules, with structure maps pi : Mi 4 Mi @ C. Let $iE~&libe the direct sum of this family in k M , and qj : M j -+ cBiEIMi the canonical injections. Then there exists a unique morphism p in k M such that the diagram
p ) is a right C-comodule, is commutative. It is easy to check that (@iErMi, and moreover that this comodule is the coproduct of the family (Mi)iE1in I the category M C . Corollary 2.1.19 The category M'
2.2
I
is abelian.
Rational modules
Let C be a coalgebra, and C* the dual algebra. If M is a k-vector space, and w : M -+ M @ C is a k-linear map, we define : C* @ M -+ M by
+,
2.2. RATIONAL MODULES
73
= 4(y @ IM)(Ic* @ T ) ( I c - @ w), where IMand Ic. are 'the identity maps, T : M @ C -+ C @M i s d e f i n e d b y T ( m @ c ) = c 8 m , y : C * @ C - +k is defined by y(c* @ c) = c*(c), and 4 : k 8 M -+ M is the canonical mi @ ci, then $, (c* @ m) = C* (ci)mi. isomorphism. If w(m) =
$u
xi
xi
Proposition 2.2.1 (M, w) is a right C-comodule if and only if (M, $,) a left C*-module.
is
Proof: Assume that (M, w) is a right C-comodule. Denoting by C* . m = $,(c* @ m ) , we have c* . m = C ~ * ( m ( ~ ) ) r for n ( ~any ) c* E C*, m E M . First, we have that lC. . m = E m =
~(m(,))m(,)= m
from the definition of a comodule. Then, for c*, d* E C* and m E M c* (d* . m) = c* . ( x d * ( m ( l ) ) m ( o ) ) a
=
C d*(m(1))(c* . m(o) C d*(mil))~*((m(o))(l))(m(~))(~) C d*(m(2))c*(m(l))m(o)
=
C ( c * d * ) ( m ( l)mjo) )
=
(c*d*) . m
= =
which shows that (M, $), is a left C*-module. ), is a left C*-module. We denote w(m) = rn(O)@ Assume now that (M, $ m ( ~ )From . E . m = m it follows that ~(m(~))m =(m, ~ )hence the second condition from the definition of a comodule is checked. If c*,d* E C*, m E M , then
C
x
where 4' : &I @ Ic @ k -+ M is the canonical isomorphism, and I stands everywhere for the suitable identity map. Also c* . (d* . m) = =
=
C*
.
(x
d*(n2(i))m(o))
C d* )(c* . m(0) C d*(m(l))c*((m(o))(l))(m(o))(o) b(l)
= # ( I @ c* @ d*)(w @ I)w(m)
CHAPTER 2. COMODULES
74 Denoting
we have ( I €3 c* @ d * ) ( y )= 0 for any c*,d* 6 C * . This shows that y = 0. Indeed, if we denote by (ei)i a basis of C , we can write y = m;j@ei@ej for some mij E M . Fix io and j o and consider the maps e,t E C* defined by e f ( e j ) = 6 i j for any j. Then miojn= ( I €3 e& €3 e;o)(y)= 0 , and from this we get that y = 0. I Let now M be a left C*-module, and t/jM : C* 8 M the module structure of M . We define PM : M
Let j
:C
f~
-+
-+
+M
the map giving
Hom(C*,M ) , pM ( m ) ( c * )= c*m.
C**,j(c)(c*)= c*(c)be the canonical embedding, and
:M
8 C** -+ H o m ( C * ,M ) , fM(m@ c**)(c*)= c * * ( ~ * ) ~ ,
which is an injective morphism. It follows that the map
is injective. It is clear from the definition that p M ( m 8 c ) ( c * )= c * ( c ) mfor C E C,c* E C * , r n E M .
Definition 2.2.2 The left C*-module M is called rational i f
Remark 2.2.3 1) M is a rational C*-module if and only if for any m E M there exist two finite families of elements (mi)iC M and (ci)i G C such that c*m = Cic*(ci)mi for any c* E C * . 2) If M is a rational C*-module, and for an element m E M there exist two pairs of families (mi)i,(ci)i and (m;)j,( c O j as i n I ) , then mi €3 ci = m; €3 c;, since p,(Ci mi 8 ci) = pM mi €3 c;) and P M is injective.
cj
I
(kj
xi
Example 2.2.4 If C is a finite dimensional coalgebra, then the left C * module C* is rational. Indeed, in this situation j : C 4 C** is an isomorphism of vector spaces from Proposition 1.3.14, and fc. : C* €3 C** --+
H o m ( C * ,C * ) is an isomorphism of vector spaces by Lemma 1.3.2.2). It follows that pc=(C*@ C ) = H o m ( C * ,C * ) , and hence pc* ( C * )G PC- (C*@ C ) , which shows that C* is rational. I
2.2. RATIONAL MODULES
75
We will denote by R a t ( c * M ) the full subcategory of c.M having as objects all rational C*-modules.
Theorem 2.2.5 The categories Mc and Rat (c.M) are isomorphic. Proof: Let (M, w) be a right C-comodule. We show that (M, $), is a rational C*-module. Let m E M . Then from the definition of ?,I it follows ( ~ )then , M is a rational module by the that c* . m = C ~ * ( m ( ~ ) ) mand preceding remark. N a morphism Let now M and N be two right C-comodules, and f : M of comodules. We show that if we regard M and N with the left C*-module structures, f is a morphism of C*-modules. Indeed, for m E M and c* E C* we have -+
f (c* . m)
=
f (Cc*(mc1, h c o ,
=
C c'(mc1,)f (mco,)
= =
C c * ( f(m)ci))f (m)cor c* . f (m)
In this way we have defined a functor T : M~ 4 Rat(c.M), such that T(M, w) = (M, ?),I for any C-comodule ( M ,w), and T ( f ) = f for any N. morphism of C-comodules f : M Let now (M,$) be a rational left C*-module. Since p~ : M @ C -+ Hom(C*,M ) is an injective map, it follows that iiM: M @ C p M ( M @ C ) , the corestriction of p ~ is ,an isomorphism of vector spaces. We define -+
-+
xi
mi @ci, where (rni)i, (ci)i We remark that for m E M we have w+(m) = are two families of elements in M , respectively C , such that
for any c* E C*. We show that (M, w+) is a right C-comodule. For this we = ?I. Then consider the map w+ : M -+ M @ C , and we show that $J(,,) (M, w*) will be a C-comodule by Proposition 2.2.1 and due to the fact that (M, $) is a C*-module. Let then W+ (m) = Czmi @ c,. We have fi;' ( p (m)) ~ = C ,m, @ c,, hence
CHAPTER 2. COMODULES
76
Ci
By evaluation in c* we obtain that c* . m = c*(ci)mi. On the other hand, C,(c* 8 m ) = c*(c,)mi = c* . m i
and thus we have showed that (M, w+) is a C-comodule. Let now (M,$JM) and ( N , @ N )be two rational left C*-modules, and f : M -+ N a morphism of C*-modules. We show that f is a morphism of C-comodules from (M, w*,) to (N, w,~,,). Let m E M and w ~ ,(m) = Cimi €3 ci. Then p ~ ( (@ f I ) ~ + M ( ~ ) ) ( c= * ) P N ( f ~(mi) 8 ci)(ca) i
=
f (c* . m)
and P N ( w +(m))(c*) ~~ = ( P N P NP~ N (m))(c*) ~ = P N (m)(c*) ~ = c* . f (m) We have obtained that p ~ (f (@ I)w+, (m)) = pN(w+, f (m)), and since p~ is injective, it follows that (f 8 I)w+, = w*, f , showing that f is a morphism of C-comodules. We have thus defined a new functor
by S((M, $J)) = (M,w+) for any rational C*-module M, and S(f) = f , for any morphism f : M -+ N of rational C*-modules. We show that ST = I d , the identity functor. Let (M, w ) E M C . The comodule structure of S ( T ( M ) ) is given by w(+w
:M
-+
M
@ C,
(m) = fi;'
(PM(m))
Then ( ~ M W ( + ~ ) ( ~ )= ) (PCM* ()~ ) ( c *= ) C* . rn
2.2. RATIONAL MODULES and
= w. We show now that and since p~ is injective, it follows that w(+~) T S = Id. Indeed, if ( M , $ ) E R a t ( p M ) , then ( T S ) ( M )= (Ad,$(,,)) = I ( M ,$), which ends the proof.
The following result presents some of the basic properties of rational modules.
Theorem 2.2.6 Let C be a coalgebra. Then: i ) A cyclic submodule of a rational C*-module is finzte dimensional. ii) If M is a rational C*-module, and N is a C*-submodule of M , then N and M / N are rational C*-modules. iii) If (Mi)iEIis a family of rational C*-modules, then e i E I M i , the direct sum as a C*-module, is a rational C*-module. iv) Any C*-module L has a biggest rational C*-submodule LTat. More precisely, LTat is the sum of all rational submodules of L. Morevoer, the correspondence L H LTat defines a left exact functor Rat :
CUM -, c*M. Proof: i) Let M be a rational C*-module, and C* . m the cyclic submodule generated by m E M . Since M is rational, there exist two finite families of elements (ci)iEFC C and (mi)iEF C M for which c* . m = CiEF c*(ci)mi for any c* E C * . It follows that C* . m is contained in the vector subspace spanned by the family ( m i ) i E F and , so it is finite dimensional. ii) If N is a submodule of M , then Remark 2.2.3.1) shows immediately that N is rational. If we denote by E the coset of an element m E M modulo N , then with the notation from i) we have that for any c* E C*
and using again the same remark it follows that M / N is rational too. iii) Let qj : M j -+ e i E r M i be the canonical inclusion of Mj in the direct qi(mi) be an element of @1,1Mi, where F is a sum, and let m = CiEF finite subset of I . For any i E F, the module Mi is rational, hence there j G ~ iC Mi, Fi finite exist two families of elements (cij)yEF, C C and (mij)
CHAPTER 2. COMODULES
78
sets, such that c* . mi = CjGFi c*(cij)mij for any c* E C*. Then
and now eiEI Mi is rational by Remark 2.2.3 1). iv) If L is a left C*-module, let
rat =
x{
N / N is a rational C* - submodule of L }
Assertions ii) and iii) show that. LTatis a rational C*-submodule of L. From the definition, it is clear that any rational submodule of L is contained in L~at Now let us check that the correspondence L H Lrat defines a functor. If f : L -+ L' is a morphism of C*-modules, define : LTat -+ LITat t o be the restriction and corestriction of f . The definition is correct due to ii). Let now 0-L1-Lz--+L3-0
rat
be an exact sequence in p M . Since we can assume that the two maps are the inclusion and the canonical projection, respectively, the fact that the sequence 0, L y t --+ 4 LYt
List
is exact amounts to the fact that L;at = Ll nL y t , which is clear by ii) and the definition of the rational part. I Corollary 2.2.7 The category R U ~ ( ~ * isMa) Grothendieck category. Proof: In view of Corollary 2.1.19 and Theorem 2.2.6, the' only thing that remains t o show is that Rat(c* M ) has a family of generators. Any rational C*-module M is a sum of finite dimensional C*-submodules (since any cyclic submodule is finite dimensional), so M is a homomorphic image of a direct sum of finite dimensional rational left C*-modules. This shows that the family of all finite dimensional rational left C*-modules is a family of generators of R U ~ ( ~ = M and ) , the proof is finished. I Corollary 2.2.8 The category M'
is a Grothendieck category.
2.2. RATIONAL MODULES
79
I Proof: It follows from Theorem 2.2.5 and the preceding corollary. Another consequence is a new proof for the fundamental theorem of comodules (Theorem 2.1.7). Corollary 2.2.9 Let M be a right C-comodule. Then: i ) The subcon~odulegenerated by an element of M is finite dimensional. ii) M is the sum of its finite dimensional subcomodules. Proof: i) The subcomodule generated by an element m E M is the cyclic left C*-submodule generated by m of the rational C*-module &I, and this is finite dimensional from Theorem 2.2.6. ii) We regard M as a rational left C*-module. It is clear that M is the sum of its cyclic submodules, and all of these are finite dimensional subcomodules I of M . Remark 2.2.10 If C is a coalgebra, then C * is a left C*-module, and hence it makes sense to talk about the biggest rational left submodule of C*. W e will denote this submodule by Cfrat. Similarly, regarding C* as a right C*module, we denote by C,"rat the biggest rational right submodule of C * . I n case C is finite dimensional, Remark 2.2.4 shows that CfTat= CPTat = C*.
I Remark 2.2.11 If C is a coalgebra, since C is a right C-comodule via A, we may regard C as a (rational) left C*-module. W e denote this left c = Cc*(cZ)cl for any c* E C * action of C* o n C by -. Thus C* and c E C . Similarly, C is a rational right C*-module with the action C C* = Cc*(c1)c2. I
-
Lemma 2.2.12 Let M be a rational right C*-module which is finite dimensional. Then M * , with the induced left C*-module structure, is rational.
Proof: Since M is a rational right C*-module, we can regard M as a left C-comodule too. Let p : M -t C @ M, p(m) = C m(-l) @ m(o) be the map giving this comodule structure. Since M is finite dimensional, Exercise 1.3.3 shows that there exists a linear isomorphism !
xi
Let z = mf @ ii @ mi E M * @ C @ M for which B ( z ) = p. This means that for any m E M we have m: (m)ci @ mi
p(m) = O(r) (m) = i
CHAPTER 2. COMODULES
80
The left C*-module structure of M* = Hom(M, k) is given by (c* m*)(m) = m*(m c*) = m*
(x
C* (ci)my (m)mi)
=
i
c*(ci)m*(mi)m: (m)
= 2
This shows that c* - m* =
1c*(ci)m*(m,)mi 2
for all c* E C*, hence M * is rational as a left C*-module. I Let C be a coalgebra and M E MC, i.e. M E Rat(c.M). We consider the dual space M * = Homk(M, k), which is a right C*-module with the action given by (uc*)(m) = u(c*m) for any u E M*, c* E C* and rn E M . In general M* is not a rational right C*-module. By an opposite version of Lemma 2.2.12, if dim(M) is finite, then M* is a rational right C*-module. Following P. Gabriel [85] a right C-module M is called pseudocompact if it is a topological C*-module which is Hausdorff separated, complete and it satisfies the following axiom
(MPC) M has a basis of neighbourhoods of 0 consisting of submodules N such that M / N is finite dimensional. Similar to Theorem 1.5.33, we have the following.
Corollary 2.2.13 If M E M', pact module.
then M * is a right topological pseudocorn-
Proof: We first show that the multiplication p : M * x C* --+ M * , p(u, c*) = UC*,is continuous. Let u E M*, c* E C* and uc* + WL an open neighbourhood of uc* in the finite topology of M*, where W is a finite dimensional subspace of M. If p : M -+ M 63 C is the comodule structure map of M , then there exist two finite dimensional subspaces Wl I M and W2 I C such that p(W) 2 Wl 63 W2.In this case u + W: is an open neighbourhood of u E M * and c* f W$ is an open neighbourhood of c* E C*. If f E W t and g E W$ we have (u
+ f)(c* + g) = uc* + fc* + ug + f g
For any m E W we have that (fc*)(m) =
f (c*m)
=
C f (moc'(m1))
=
0 (since f (Wl) = 0)
2.2. RATIONAL MODULES
81
Thus fc* E W'-, and similarly we have ug E WL and' f g E WL, showing that p(u c* C uc* w'-
+ w:, + w;)
+
which means that p is continuous. Clearly M * is Hausdorff separated in the finite topology. On the other hand, M is the union of all its finite dimensional subcomodules N , so
-
=
lim Homk(N, k)
=
lim N* +---.
where in all limits N ranges over the set of all finite dimensional subcomodules of M . Since N* I IM * / N L we obtain that M * E lirn,~*/N'. Thus M* is complete and the set {N'IN
is a finite dimensional subcomodule of M )
is a basis of open neighbourhoods of 0 in M*. Since for any finite dimensional N the space M * / N L has finite dimension, we obtain that M* is I pseudocompact. Theorem 2.2.14 Let C be a coalgebra and M a left C*-module. The following assertions are equivalent. (2) M E R a t ( c * M ) . (ii) annc*(x) = {c* 6 C*lc*x = 0) is a closed (and open) left ideal of finite codimension for any x E M . (iii) For any x E M there exists a closed (open) two-sided ideal I C C* of finite codimension such that I x = 0. Proof: (i) + (ii) Any element of M belongs to a finite dimensional submodule of M , so we can reduce to the case where M is finite dimensional. Denote N = M * , which is a rational right C*-module with dim(N) = dim(M) and N * N M. By Corollary 2.2.13, N* is a topological left C*-module. Since N has finite dimension, the finite topology M 21 N*, on N* is discrete. On the other hand, the map 4 : C* 4(c*) = C*X, is continuous, and since {x) is open and closed in M , we have that annc*(x) = Ker(4) is open and closed in C*. Clearly C*/annc* (x) is finite dimensional, so (ii) holds. (ii) + (iii) As in (2) + (ii) we may assume that M is finite dimensional, say M = kxl . . . + kx,. In this case
-
+
a n n c - ( M ) = n ~ s i ~ ~ a n(xi) nc-
CHAPTER 2. COMODULES
82
so I = annc*(M) is open and closed in C*. Clearly I is a two-sided ideal of finite codimension. (iii) + (i) If I x = 0, then C*x is a quotient of the left C*-module C*/I. Since I is closed, then I = (I-'-)", so
Then I' is a finite dimensional subcoalgebra of C. Since I' is a left and right C-comodule, then so is (IL)*. In particular C*/I is a rational left C*-module, so C*x is a rational left C*-module. We conclude that M is a I rational left C*-module.
Lemma 2.2.15 Let A be a k-algebra and M a right A-module. Then (2) If M is projective, then M*= Homk(M, k) is an injective left A-module. (ii) If A has finite dimension and M is finitely generated and injective as a right A-module, then M * is a projective left A-module. Proof: We recall that the left A-module structure of M* is given by (af)(m) = f ( m a ) for any a E A, f E M * and m E M . (i) Let u : N' -+ N be an injective morphism of left A-modules, and f : N' + M * a morphism of left A-modules. The dual morphism u* : N * + N'* is a surjective morphism of right A-modules. Let a~ : M + M" be the natural injection, aM(m)(v) = v(m) for any m E M , v E M*. Since M is projective as a right A-module, there exists a morphism of right Amodules g : M + N * such that f*aM= u*g. Denoting by a~ : N + N** , --, (N1)** the natural injections, we have that and a ~ : N'
But if m* E M * we have ( a L a ~ - ) ( m * ) = a b ( a ~(m*)) * = ab(m*)aM For any m E M we have
2.2. RATIONAL MODULES
83
u Thus a k ( m * ) a M = rn*,showing that akaM= IdM*. Hence ( g * a ~ ) = f , so M * is injective. (ii) Clearly M is finite dimensional. Since M is finitely generated, there M* 0 in the category of left A-modules. exists an epimorphism An M** Apply the functor Horn(-, k) and obtain an exact sequence 0 A* ". Since M** II M and M is injective, we have A* E M $ N for some right A-module N . Then An e (A* ")* EY M * @ N*, so M * is a projective I left A-module. The following gives in particular a description of the rational part of C*.
- -
- -
Corollary 2.2.16 Let C be a coalgebra and PI E ' M . Then the following three sets are equal. (i) Rat('* M * ) . (ii) The sum of all finite dimensional C*-submodules of M * . (iii) The set of all elements m* E M * such that Ker(m*) contains a left C-subcomodule of M of finite codimension. Proof: (i) C (ii) is obvious. (ii) C (i) Let m E (ii). Then there exists a finite dimensional C*-submodule X of M* such that m* E X , so annc-(m*) is a left ideal of C * of finite codimension. On the other hand, since the map from C* to M* obtained by right multiplication with m* is continuous, the kernel of this map is open and closed in C * (see Corollary 2.2.13), so m* E Rat('+M*) (Theorem 2.2.14). (i) C (iii) Let m* E Rat('.M*) and X a finite dimensional C*-submodule of M* such that m* E X . Then XIis a subcomodule of M and m*XL = 0, so X' C Ker(m*). Since X' has finite dimension we obtain that m* E (222). (iii) C (i) Let m* E M * such that Y C Ker(m*) for some subcomodule Y of M of finite codimension. Then we can regard m* as a linear map from M / Y to k , so m* E (M/Y)* 5 M*. Since (M/Y)* is a right Ccomodule of finite dimension, (M/Y)* is a rational left C*-submodule of I M * , so m* E R a t ( p M * ) . Exercise 2.2.17 Let C be a coalgebra and 4c injection. Then cjc(Rat('* C)) = Rat('* C**).
:
C
-+
C** the natural
Exercise 2.2.18 Let (Ci)iEI be a family of coalgebras and C = $iE1Ci the coproduct of this family in the category of coalgebras. Then the following assertions hold. C t . Moreover, this is an isomorphism of topological rings if (i) C * 2: Ct. we consider the finite topology on C* and the product topology on
niEI
niEI
CHAPTER 2. COMOD ULES
84
(ii) If M E M~ then for any m E M there exists a two-sided ideal I of C* such that I m = 0, I = I i , where Ii is a two-sided ideal of C,t which is closed and has finite codimension, and Ii = C: for all but a finite number of i p s . (iii) The category M C is equivalent to the direct product of categories nicI MCa. (iv) If A is a subcoalgebra of C = e i E r C i , then there exists a family (Ai)iG1 such that A; is a subcoalgebra of C; for any i E I , and A = $ i E ~ A i .
niEI
Exercise 2.2.19 Let (Ci)iEIbe a family of subcoalgebras of the coalgebra C and A a simple subcoalgebra of CicICi (i.e. A is a subcoalgebra which has precisely two subcoalgebras, 0 and A; more details will be given in Chapter 3). Then there exists i E I such that A C C;.
2.3 Bicomodules and the cotensor product Definition 2.3.1 Let C and D be two coalgebras. A k-vector space M is called a left-Dl right-C bzcomodule zf M has a left D-comodule structure given by p : M --, D @ M , p ( m ) = Cm[-l~ @ m[01,a right C-comodule structure given by p : M 4 M @ C , p ( m ) = C m ( o ) @ m ( l ) ,such that ( p 6 I ) p = ( I @ p)p. This compatibility may be written as
for any m E M . If M and N are two such bicomodules, then a morphism of bicomodules from M to N is a linear map f : M t N which is a morphism of left D-comodules, and in the same time a morphism of right C-comodules. I n this way we can define a category of left-Dl right-C bicomodules, which 1 we will denote by D M ~ . Example 2.3.2 Any coalgebra C is a left-C, right-C bicomodule, with the left and right comodule structures defined by the comultiplication of C . I If C and D are two coalgebras, then we consider the dual algebras C* and D*, and we denote by G*M D *the category of left-C*, right-D* bimodules. We recall that such an object is a vector space having a structure of a left C*-module and a right D*-module structure, which are compatible in thesense that c * . ( m . d * ) = ( c * . m ) . d * for any c* E C * , m € M , d * E D * . The morphisms between two objects in this category are linear maps which are also morphisms of left C*-modules, and in the same time morphisms of right D*-modules. We will denote by Rat(G. M D - )the full subcategory of
2.3. BICOMOD ULES
85
c-MD.consisting of al1,objects which are rational left C*-modules, and in the same time rational right D*-modules. Theorem 2.3.3 Let C and D be two coalgebras. Then the categories ~ . isomorphic. ) and R ~ t ( ~ - & tare
C
Proof: The proof relies essentially on the proof of Theorem 2.2.5. Let Then we know already from the cited theorem that M is a M E rational left C*-module with the action given by c* . m = C ~ * ( r n ( l ) ) m ( ~ ) , and that M is a rational right D*-module with the action given by m . d* = ~ d * ( m ~ - l l ) m l oItl remains to check that these two structures induce a bimodule. We have (c* - m) . d* = =
x
c*(m(l))m(o). d*
C ~*(m(l))d*((m(o))[-ll)(m(o))[o]
and c* . (m . d*) = =
C d'(ml-ll)c* - mpl C d*(m[-l])~*((m[ol)(~))(m[ol)(o)~
and the equality follows from the definition of the bicomodule. Conversely, if M is a left-C*, right-D* bimodule, which is rational on both sides, we know that M has a structure of a left D-comodule, which we ~ ) a structure of a right C-comodule, will denote by m +-+ C mi-lj @ m [ Oand which we will denote by m ++ @ m(l). From the fact that M is a bimodule, and from the above computations, we obtain that
for any c* E C* and d* E D*. This means that
for any c* E C*,d* E D*. But if for an element z E D @ M @ C we have (d* @ I @ c*)(z) = 0 for any c* E C*, d* E D*, then z = 0. Indeed, we choose a basis (di)i of D and a basis (cj)j of C . Then we can write z = 293. di @ mij @ cj for some mij E M. Fix r and s, and consider the Then maps d; E D*, cH E C* for which d:(di) = bTi 2i c,'(cj) = 6,
x.
86
CHAPTER 2. COMODULES
0 = (d; €4 I @c : ) ( z ) = m,,. From this we deduce that z = 0, and the proof I is now complete due to Theorem 2.2.5. We recall that for any subset X of a coalgebra C there exists a smallest subcoalgebra of C containing X , and this is called the subcoalgebra generated by X (Remark 1.5.30). If the set X has only one element c , then the subcoalgebra generated by { c ) is called simply the subcoalgebra generated by c. The following exercise illustrates the use of bicomodules for proving the fundamental theorem of coalgebras (Theorem 1.4.7).
.
Exercise 2.3.4 Let C be a coalgebra, and c E C . Show that the subcoalgebra of C generated by c is finite dimensional, using the bicomodule structure of c . Corollary 2.3.5 Let C be a coalgebra. Then the subcoalgebra generated by a finite family of elements of C is finite dimensional. Proof: It is clear that the subcoalgebra generated by { c l , . . . ,c, ) is the sum of the subcoalgebras generated by each of the ci, and since all of these are finite dimensional, it follows that their sum is finite dimensional too. We present now a construction dual to the tensor product of modules over an algebra. Let C be a coalgebra, M a right C-comodule with comodule structure map p~ : M -+ M @ C and N a left C-comodule with comodule structure map p~ : N -, C @I N . We denote by MOcN the kernel of the morphism
M a c N is a Ic-subspace of M @ N which is called the cotensor product of the comodules M and N. If f E Comc(M, M ' ) and g E Comc(N, N ' ) are two morphisms in the categories M C and G M ,then the map f @ g : M @ N -+ M' 8 N' induces a linear morphism fDcg : MDcN -, MtOcN'. It is easy to see that the correspondence (M, N ) MOcN defines an additive functor from M~ x C M -+k M , called the cotensor functor. Since the tensor functor of vector spaces is exact and the cotensor is a kernel, we have that the cotensor functor is left exact. Also, since the tensor functor commutes with direct sums and with filtered inductive limits, we obtain that the cotensor functor also commutes with direct sums and with filtered inductive limits. If C = k, then MOkN is just the tensor product M €4 N , since in this case PM@I-I@PN=O. Let C and D be two coalgebras and assume that M E D ~ C Denote . by p- : M --+ D @I M and p+ : M -, M @ C the comodule structure maps of M . We have that ( p - @ I)p+ = ( I @ p+)p-. Let N E with comodule
-
,
2.3. BICOMOD ULES
87
structuremapp: N - + C @ N . T h e n p - @ I : M @ N + D @ M @ N e n d o w s M @ N with a structure of a left D-comodule. The induced structure of a right rational D*-module of M @ N is given by (m @ n)d' = md* @ n for any m E M , n E N and d' E D*. On the other hand, if d' E D*, the map u : M -+ M , u(m) = md* for any m E M , is a morphism of right Ccomodules, since M E Hence by the above argument it follows that (u @ I)(MOcN) MOcN, thus MOcN is a D*-submodule of M @ N . Moreover, it is a rational D*-module, so MOcN is a D-subcomodule of M @ N , i.e. (p- @ I ) ( M O c N ) C D @ (MOcN). In this way, for any we obtain a left exact functor M E The following gives the main properties of the cotensor functor
Proposition 2.3.6 (i) If M E M~ and N E C M , then MOcC e M as right C-comodules and COcN 1: N as left C-comodules. (ii) If M E MC and N E C ~ then , MOcN NOccop M as linear spaces. In particular if C is cocommutative, then MOcN .v NOcM. (iii) If C and D are two coalgebras and M E C M D , L E C~ and N E D M , then we have a natural isomorphism (LOcM)ODN = L o c ( ~ o D h r ) .
--
Proof: (i) We define a linear map a : MOcC -+ Ad as follows. For z E M O c C C M @ C , say z = C , x , @ c , with (a,), M and (c,), E C , we set a ( z ) = C E(c~)x,. On the other hand, since (pM @ I)pM = ( I @ A ) ~ Mwhere , p~ is the comodule structure map of M and A is the comultiplication of C , then we have (pM @I I)pM = (I@I A)pM, showing that ~ M ( MC) MOcC. Thus it makes sense to define ,8 : M -+ MOcC by P(m) = p ~ ( m = ) mo @ m l for any m E M . If z = C, x, @ c, E MOcC, then we have p ~ ( x ,@ ) C, = x, @ A(c,), and applying I @I @ E we = C ,x, @ C, = Z. This shows that (Pa)(z) = z , i.e. obtain C, E(C~)PM(X,) pa = IdMncc. It is clear that for any m E M we have (cu,B)(m) = m, thus a/3 = I ~ M We . have obtained that CY and P are inverse each other. It is clear that p is a morphism of right C-comodules, and this ends the proof of (i). M @ C be the comodule structure map of M as a right (ii) Let PM : M C-comodule and p h : M 4 CcOp@M,the comodule structure map of M as a left CcOP-comodule.If p~ (m) = C mo @ m l , then p',l (m) = C m l @ mo. Similarly we denote by p~ : N C @ N and p h : N -+ N @ CCOPthe comodule structures map of N as a left C-comodule,respectively as a right CCOP-comodule.Let T : M @ N N @ M and ? : M @ C @ N-+ N@CcOP@M be the linear isomorphisms defined by r ( m @ n ) = n @ m and ~ ( m @ c @ n=) n @ c @I m. We have that
z,
z,
-+
-+
-+
CHAPTER 2. COMODULES
88
implying that MOcN cz NOCeopM. The second part is clear since for a cocommutative C we have C = CcOp. (iii) Since the tensor product functor over a field is exact, we have a natural isomorphism ( L 8 M)ODN 21 L 8 ( M O D N ) Now the exact sequence of right D-comodules
and the fact that the cotensor functor is left exact produces a commutative diagram
and then we have a natural isomorphism (LOcM)ODN .v LOc(MODN).
I Assume now that N is a finite dimensional left C-comodule with comodule structure map p~ : N -+ C 8 N , p ~ ( n = ) n ( - ~8 ) n ( ~ )Let . N* = H o m k ( N ,k). Since N is finite dimensional, the natural morphism
x
0 : N* 8 C -r H o m ( N ,C), B ( f 8c ) ( n )= f ( n ) c is a linear isomorphism. N* has a natural structure of a right C-comodule given by PN* : N* -+
N* @ C,
PN-
( n * )= &'((I
xi
@ n*)pN)
If n* E N* and p ~ * ( n *=) fi 8 ci with fi E N* and ci E C , then c*n* = c*(cZ) fi for any c* E C*. Therefore (c*n*)(n)= c*(ci)fi(n) for any n E N. On the other hand we have B(Cif i 8 ci) = ( I @ n * ) p N , so if p ~ ( n = ) n ( - l ) 8 yo)we have fi(n)ci = n * ( n ( o ) ) n ( - l )and , then fi(n)c*(ci)= C n * ( n ( o ) ) c * ( n ( - lfor ) ) any c* E C*. We obtain that (c*n*)(n)= C ~ * ( n ( - ~ ) ) n * ( nand ( ~ this ) ) , gives a structure of a rational left C*-module to N * , which is associated to the right C-comodule structure of N* defined by p ~ . .This left C*-module structure of N* is the same with the left C*-module structure of N* = H o m k ( N , k ) obtained by regarding
xi
xi
x
xi
xi
2.3. BICOMODULES
89
N as a right C*-module induced by the comodule map p~ (see also Lemma 2.2.12).
-
Proposition 2.3.7 Let N be a finite dimensional left C-comodule and M a right C-comodule. Then we have a natural linear isomorphism MOcN Comc(N* , M).
Proof: Since N has finite dimension, we have the natural linear isomorphism a : M 8 N + Hom(N*,M), a ( x 8 y)(f) = f(y)x Let z = Ci xi 8yi E MOcN 2 M 8 N , i.e. we have
For any c* E C* and n* E N* we have that
so a ( z ) is a morphism of left C*-modules, which is the same to a morphism of right C-comodules from N * to M . With the same computation we obtain that if a ( z ) E Comc(N*, M ) , then z E MOcN. I Let C and D be two coalgebras and 4 : C --+ D a coalgebra morphism. If M E M C , then the map (I8 4)pM : M --t M '8 D gives M a structure of a right D-comodule. We denote by M4 the space M regarded with this structure of a right D-comodule. In this way we construct an exact functor
In particular, since C is a left-C, right-C bicomodule, we can regard C as a left-D, right-C bicomodule via 4. Then 4 : C -+ D is a morphism of left D-comodules.
90
CHAPTER 2. COMODULES
If N E M ~we, can define the right C-comodule ~4 = NOD(?, and in this way we have a new functor
which is left exact. In particular if D = k and 4 = e, the counit of C , then the functor (-), is just the forgetful functor U : MC -11, M, and the functor (-)'#' is exactly the functor - @ C :k M -+M'.
Proposition 2.3.8 Let 4 : C --r D be a morphism of coalgebras. Then the functor (-)$ : M~ -+ M~ is a left adjoint of the functor (-)4 : MD MC. In particular the forgetful functor U : M' -+k M is a left adjoint of the functor - 8 C :k M -+ M C . Proof: Let M E M~ and N E M D , and define the natural maps
as follows. For u E Comc(M, N4) we put @(u)= ( I @ E)U,where E is the is regarded here as the restriction of I@& : N @ C -+N counit of C and I@E to NODC. In fact @(u)is the composition of the morphisms
If v E ComD(&l4, N ) , then we put Q(v) = (v @ I)pM. We prove that !P(v) E Comc(M, ~ 4 ) Indeed, . since v E ComD(M4,N ) we have
On the other hand we have that
This shows that for any m E M we have 9(v)(m) E NODC, so Q(v) can be regarded as a morphism from M to N4.
2.4. SIMPLE AND INJECTIVE COMODULES I f u E C o m c ( M , N 4 ) then we have
@ ( @ ( u ) )= ( I @ & @ I ) ( u @ I ) p M = ( I @ E @ I ) ( I @ A ) u ( u is a comodule map) = ( I@ I)u -
U
so 9@ = Id. Conversely, for v E C o m D ( M dN , ) we have @(9(v)) = ( I @ E ) ~ ( V ) = ( I @ E)(V @ I)PM = (I@ED)(I@~)(~@I)PM = (I@ED)(V@I)(I@@)PM = ( I 8 E D ) P ~ U (v is a comodule map) -
,
V
showing that @9 = Id, which ends the proof.
I
Remark 2.3.9 Let A be a k-algebra, M a right A-module with module structure map p : M @ A -+ M and N a left A-module with module structure map p : A @ N -+ N . By restricting scalars, M and N are k-vector spaces, and we can form the tensor product M @ N over k . Then the tensor product M @ A N of A-modules is precisely Colcer(p @ I - I @p) . Thus the cotensor product is a construction dual to the tensor product. Exercise 2.3.10 Let C and D be two coalgebras. Show that the categories D M C , M C @ D " ~ MD~O"@C, ~ D @ C C o Pand ~ C C o P @ Dare ~ isomorphic, 7
Exercise 2.3.11 Let C , D and L be cocommutative coalgebras and I$ : C -+ L, 11, : D -+ L coalgebra morphisms. Regard C and D as L-comodules via the morphisms 4 and 11,. Show that COLD is a (cocommutative) subcoalgebra of C @ D , and moreover, COLD is the fiber product in the category of all cocommutative coalgebras.
2.4
Simple comodules and injective comodules
Definition 2.4.1 A right C-comodule is said to be free if it is isomorphic to a comodule of the form X @ C , with X a vector space, with the right I comodule structure map I @ A : X @ C --+ X @ C @ C .
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C H A P T E R 2. COMODULES
Remark 2.4.2 It is clear that if X is a vector space having the basis (ei)iEr, then the comodule X 8 C is isomorphic to ~ ( ' 1 , the direct s u m I i n the category MC of a family of copies of C indexed by the set I . Proposition 2.4.3 Any right C-comodule is isomorphic to a subcomodule of a free C-comodule. Proof: Let M E MC. Then M 8 C is a right C-comodule via I 8 A : M 8 C -+ M 8 C 8 C . Let p : M -, M @ C be the morphism giving the comodule structure of M . Then the definition of the comodule shows that p is a morphism of right C-comodules from M to M @ C, where the second one is regarded with the above mentioned structure. Moreover, p is injective, since if for an m E M we have p(m) = 8 rn(11 = 0, then m = C ~ ( r n ( ~ ) ) = m (0.~ )It follows that M is isomorphic to the subcomodule I m ( p ) of the free comodule M 8 C . I Definition 2.4.4 A right C-comodule M is called injective if i t is an injec, for any injective morphism i : X -+ Y tive object i n the category M ~i.e. of right C-comodules, and for any morphisrn f : X --+ M of right C comodules, there exists a morphism of right C-comodules : Y -+ M for I which Ti = f .
7
Corollary 2.4.5 A free right C-comodule is injective. I n particular, C is an injective right C-comodule. Proof: Consider the adjunction from Proposition 2.3.8. Since U is an exact functor, and MC and kM are Grothendieck categories, it follows that X 8 C is an injective object in MC for any injective object X from kM. But in kM every obiect is injective, which ends the proof. I Remark 2.4.6 Since the category MC is Grothendieck, it follows that every object has an injective envelope. This means that for any M E MC there exists E ( M ) E MC such that M is a n essential subcomodule i n E ( M ) (i.e. for any nonzero subcomodule X of E ( M ) we have X n M # 0). I
The following result provides a characterization of injective comodules. Proposition 2.4.7 Let M be a right C-comodule. Then M is injective if and only if M is a direct summand in a free C-comodule. Proof: Assume that F = M $ X for some free right C-comodule F and some right C-comodule X. Since F is injective by Corollary 2.4.5, we obtain that both M and X are injective.
2.4. SIMPLE AND INJECTIVE COMODULES
93
Conversely, assume that M is injective. Proposition 2.4.3 yields the exisfor some set tence of an injective morphism of comodules i : M + I . The injectivity of M shows that the morphism i splits (i.e. there exists a morphism of comodules j : C(') + M with ji = I d M ) and then = i ( M )$ K e r ( j ) e M @ K e r ( j ) , hence M is a direct summand in a I free comodule. In a Grothendieck category a finite direct sum of injective objects is an injective object. In a category of comodules we have even more, as the following shows.
c(')
c(')
be a family of injective right C-comodules. Proposition 2.4.8 Let Then their direct sum @iEIMiis an injective object in the category M ~ . Proof: Proposition 2.4.7 shows that for each i E I there exists a set Jisuch that Mi is a direct summand in d J i ) . Then, denoting by J = Ui,, Ji the coproduct of the family (Ji)iE1in the category of sets (which is exactly the disjoint union of the family), we obtain that e i E r M i is a direct summand I , so it is injective. in C ( J ) and Definition 2.4.9 A right C-comodule is called simple if M # 0 and if the I only subcomodules of M are 0 and M .
Remark 2.4.10 1) From the isomorphism between the category M~ and the category of the rational left C*-modules it follows that a right C-comodule M is simple i j and only if it is a simple object when it is regarded as a rational left C*-module. Since any submodule of a rational module is rational, this is equivalent to the fact that M is simple as a left C*-module. We will therefore regard throughout a (simple) right C-comodule also as a (simple) left C*-module. 2) If M is a right C-comodule, then we regard M as a left C*-module, and thus it makes sense to consider the socle s ( M ) of M , which isfthe sum of the simple C*-submodules. This is a semisimple left C* -module, in particular it is a direct sum of simple submodules. From the above considerations it follows that s ( M ) is also the sum of the simple right C-subcomodules of M . A classical result from module theory says that zf M = &€'Mi, then s ( M )= $i€~s(Mi). I Proposition 2.4.11 Any nonzero comodule contains a simple subcomodule. Proof: Let M be a non-zero simple right C-comodule, and let m E M, m # 0. Then the subcomodule C* . m generated by m is finite dimensional (by Theorem 2.2.6.i)) and then there exists a subcomodule S of C* .m having the least possible dimension among all non-zero subcomodules of C* . m. It is clear that S is simple. I
94
CHAPTER 2. COMODULES
Corollary 2.4.12 Let M be a non-zero right C-comodule. Then its socle s(M) is an essential subcomodule in M . Proof: If s ( M ) would not be essential in M , then there would exist a nonzero subcomodule X of M with s(M) n X = 0. The preceding proposition shows that X contains a simple subcomodule S . But then S is also a simple I subcomodule of M, thus S 5 s ( M ) , a contradiction. Proposition 2.4.13 A simple C-comodule is finite dimensional. Proof: Let S be a simple right C-comodule, and let x E S , x # 0. Then S is generated by x as a comodule, and from Theorem 2.1.7 it follows that S is finite dimensional. 1. Proposition 2.4.14 A simple right C-comodule S is isomorphic to a right coideal of C (hence S may be embedded in C). Proof: Remark 2.4.2 and Proposition 2.4.3 show that there exists an injective morphism of comodules S --+ C n , where n = dim(S). Regarding this embedding as being an embedding of left C*-modules, we obtain that S s(Cn). Hence S is isomorphic to a submodule of s(Cn). Taking into account the formula for the socle of a direct sum of modules (which was recalled above), it follows that S is isomorphic to a simple submodule of C , hence to a right C-subcomodule of C , and the proof is finished. I Corollary 2.4.15 Let S be a simple C-comodule. Then there exists an injective envelope E ( S ) of S with the property that E ( S ) C . E ( S ) the Proof: Let E(S) be an injective envelope of S, and f : S canonical injection. We know that f (S)is essential in E ( S ) . The preceding proposition shows that there exists an embedding i : S + C of S in C , and since C is an injective object in the category M ~ we, obtain that there exists a morphism of C-comodules g : E ( S ) 4 C for which gf = i. Since i is injective, we obtain that Ker(g) n f (S) = 0. But f (S) is essential in E(S), hence Ker(g) = 0. Thus g is injective, and this shows that E(S) may be embedded in C. I
Theorem 2.4.16 Let C be a coalgebra, and s(C) = CBiEIMi the socle of C regarded as a right C-comodule (Mi are all simple right C-subcomodules of C). Then C = @iEIE(Mi), where E(Mi) is an injective envelope of Mi contained in C. Proof: Corollary 2.4.15 shows that for any i there exists an injective envelope E(Mi) C C of Mi. Since the (Mi)'s are simple C*-submodules of
2.4. SIMPLE AND INJECTIVE COMODULES
95
C, whose sum is direct, and every n/l, is essential in E(Mi), it follows that the sum of the subcomodules E(Mi) is also direct. Thus
Since ezElE(MZ)is an injective subobject of C by Proposition 2.4.8, it follows that it is a direct summand in C, hence C = ( e Z E ~ E ( M z@) X ) for a subcomodule X of C . But s(C) is essential in C by Corollary 2.4.12, so e Z E I E ( M z )is also essential in C. This shows that X = 0 and the proof is finished. I Let C be a coalgebra and Q E M ~ The . definition of injective comodules shows that Q is injective if and only if the functor Cornc(-, Q) : M~ -+ Ab is exact. A right C-comodule M is called coflat if the cotensor functor MUc- : C~ -+k M is exact. If M E MG, then an object Q E MC is called M-injective if for any subcomodule M ' of M, and for any f E Comc(Mf,Q) there exists g E Comc(M, Q) such that g 1 ~ = ' f , where by g 1 ~ we ' denote the restriction of g to M r . Clearly, an object Q E MC is injective if and only if Q is M-injective for any M E M C . Theorem 2.4.17 Let Q be a right C-comodule. The following assertions are equivalent. (i) Q is an injective comodule. (ii) Q is M-injective for any right C-comodule M of finite dimension. (iii) For any two-sided ideal I of C* which is closed and has finite codimension and for any morphism of left C*-modules f : I -, Q there exists q E Q such that f (A) = Xq for any X E I. (iv) Q is a right coflat comodule. Proof: (i) + (ii) is obvious. (ii) + (i) Let X be an arbitrary object of M C . Then there exist a family (Mi)iEI of finite dimensional C-comodules and an epimorphism q5 : @iEIMi -+ X . Denote Y = @iCrMi and Z = Ker(q5). If X' is a subcomodule of XI then there exists Y' 5 Y such that Y ' / Z CY ,Xf. The commutative diagram
O-Z-Y-
X-0
CHAPTER 2. COMODULES
96 produces a commutative diagram
\
If we assume that the natural morphism Comc(Y1,Q ) 4 Comc(Y, Q) is surjective, we obtain that the morphism Comc(X1,Q ) + C o m c ( X , Q ) is surjective. Therefore for proving (ii) + (2) it is enough to show that Q is Y-injective. For this, we will show that if Q is Mi-injective for any i E I, then Q is @iEIMi-injective. Denote M = eiE1Mi and take K 5 M and f E Cornc (K, Q). We consider the set
which can be ordered as follows. If (L, g) and (L', g') are two elements of P, the (L, g) L: (L', g') if and only if L & L' and g = g;. A standard argument shows that P is inductive, and by applying Zorn's Lemma we can find a maximal element (LO,fo). Lo for any i E I. If there were some io E I such We show that Mi that Mi, is not contained in Lo, let us denote by h the restriction of fo to Lo n Mi,, h : Lo n Mi, 4 Q. Since Q is Mi,-injective, there exists a morphism 'iE : Mi, 4 Q such that the restriction of 7;: to Lo n Mi, is h. Clearly Lo is strictly contained in Lo Mi,. We define : Lo Mi, 4 Q by x ( x y) = fo(x) for any x E Lo and y E Mi,. The morphism fo is correctly defined, since for x,xl E Lo and y, y' 6 Mi, such that ~ + y = x ' + ~ ' , w e h a v ye - y 1 = x ' - X E L o n M i o l ~ ~
+ x(y)
+
z
+
+
-
h(y - 9') = h(p - y') = fo(xl - X)
+
+
thus fo(x) L(y) = fo(xl) K(y1). Since the restriction of and Lo # Lo Mi,, we obtain a contradiction. (i) 3 (iii) is clear. (iii) =+ (i) We know that the set
+
-
t o Lo is fo
6 = {C*/III a closed two - sided ideal of finite codimension of C*) is a family of generators of the category R a t ( ~ e M ) MC (see Theorem 2.2.14). Let U be the direct sum of all the elements of 6 , which is a generator of M C . Since Q is M-injective for any M E G, we see as in the
2.4. SIMPLE A N D INJECTIVE COMODULES
97
proof of (ii) + (i) that Q is U-injective, and hence that Q is ~('1-injective for any non-empty set I. If M E MC,there exist a non-empty set I and an epimorphism u(') M. With the same argument as in the proof of (ii) (2) we see that Q is M injective. Thus Q is injective. EQ ) $ Q' as C-comodules for (i) + (iv) If Q is injective in MC,then c ( ~ some set J and for some C-comodule Q'. Let us consider an exact sequence of left C-comodules
+
-
By cotensoring with Q we obtain an exact sequence of abelian groups
We show that IOU is surjective. This is clear in the case where Q = ~ ( ~ 1 , since QQN Y ( c o ~ N ) ( ~E) N ( ~ ) In general, if
c(~ EQ ) @ Q', we have an exact sequence
We have the commutative diagram
which shows that IQOv is surjective, i.e. Q is a right coflat comodule. (iv) + (ii) If M is a finite dimensional right C-comodule, then M* = Homk(M,k ) is a finite dimensional left C-comodule. On the other hand
(where the first isomorphism comes from Proposition 2.3.7). Since Q is coflat we obtain that Q is M-injective. I
,
CHAPTER 2. COMODULES
98
Corollary 2.4.18 Let P be a projective left C-comodule. Then Rat(c* P*) is an injective right C-comodule. Proof: By Theorem 2.4.17 it is enough to show that Rat(C=P*)is Ninjective for any finite dimensional right C-comodule N . Let N be such an object, u : N t + N an injective morphism of comodules and f : N t + Rat(P&) a morphism of right C-comodules. Regard f as a morphism from N' to P*,and take the dual morphism f * : P**-+ N". If a : P -+ P** is the natural injection, a(p)(p*) = p*(p) for any p E P,p* E P*, then f * a : P -+ N'* and u* : N * -, N'* is a surjective morphism. Since N and N' are finite dimensional right C-comodules, we have that N * and N'* are left C-comodules, and then since P is a projective left C-comodule, there exists a comodule morphism g : P -+ N* such that u*g = f*a. Taking the dual, this implies that (f*a)* = g*u**. If a~ : N + N** and a ~ : N, t + N'** are the natural injections, we have that
where the last equality follows from the following easy computation.
for any n E N , p E P . Since a N g * is a morphism of left C*-modules and N is rational, we can regard aNg* as a morphism from N to R a t ( p P * ) , showing that Rat(c* P * ) is N-injective. I
Corollary 2.4.19 Let Q be a finite dimensional right C-comodule. Then Q is injective (projective) as a left C*-module if and only if it is injective (projective) as a right C-comodule. Proof: We only have to show the if part. Assume that Q is injective in MC. Since Q is finite dimensional, there exist a positive integer n and a right C-comodule Qt such that Cn Q @ Q'. Applying the functor Hornk(-, k) we obtain that C* " E Q* @ Q' * as right C*-modules, so Q* is a projective right (?*-module. Lemma 2.2.15 implies that Q cx (Q*)* is an injective left C*-module.
--
'
2.4. SIMPLE AND INJECTIVE COMODULES
99
Assume now that Q is a projective object in M C . Since the functor Hornk(-, k) defines a duality between the category of finite dimensional right C-comodules and the category of finite dimensional left C-comodules, we see that Q* is an injective object in the category of finite dimensional left C-comodules. Theorem 2.4.17 shows that Q* is an injective object in C ~ so ,there exist a positive integer n and a left C-comodule X such that Cn N Q* @I X , Then
I
showing that Q is a projective left C*-module.
Corollary 2.4.20 Let M be a finite dimensional right C-comodule. Then M is an injective right C-comodule if and only if M * is a projective left C-comodule. Proof: If M * is projective as a left C-comodule, then M cz (M*)* is an injective right C-comodule by Theorem 2.4.17. Conversely, assume that M is an injective right C-comodule. The proof of Corollary 2.4.18 shows that M * is a projective right C*-module, i.e. M* is a projective object in the category R a t ( M c * ) . Then M * is a projective I object in C ~ . We say that a Grothendieck category A has enough projectives if for any object M E A there exist a projective object P E A and an epimorphism P M in A.
-
Corollary 2.4.21 Let C be a coalgebra. The following assertions are equivalent. (i) The injective envelope of any finite dimensional right C-comodule i n the category M C is finite dimensional. (ii) The znjective envelope of any simple right C-comodule i n the category M C is finite dimensional. (iii) If N is a finite dimensional left C-comodule, then there exist a finite N in dimensional projective left C-comodule P and an epimorphism P the category C M . (iv) The category C~ has enough projective objects.
-
Proof: ( i )w (ii)is immediate. ( i )@ (iii)follows from Corollary 2.4.19. (iii) + (iv) follows from the fact that the family of all finite dimensional left C-comodules is a family of generators in the category C ~ . ( i v ) + (iii) Let N E ' M . By the hypothesis there exist a projective N. Since the family of all finite object P of C~ and an epimorphism P
-
CHAPTER 2. COMODULES
100
dimensional left C-comodules is a family of generators for the category C ~ we can find a family (Xi)iEI of finite dimensional left C-comodules and an P in the category C M . Since P is projective, it epimorphism eiEIXi is a direct summand of eiEIXi. We can clearly assume that all the Xi's are indecomposable. Then a result of Crawley, Jonsson and Warfield (see [3, Corollary 26.6, page 3001) shows that P BiEJXi for some subset J C I. Since N has finite dimension we can find a finite subset K of J and an epimorphism e i E ~ X i N . Any Xi, i E K , is projective, as a direct summand of P, and then so is $ i E ~ X i .Moreover $iGKXi is finite dimensional, which ends the proof.
-
-
-
a
Corollary 2.4.22 Let P be a projective object in the category MC. Then P is projective in the category c=M. Moreover, Rat(P$,) is dense in P*.
-
Proof: There exist a family (Mi)iEI of finite dimensional right C-comodules and an epimorphism eiErMi P of right C-comodules. Then P is a direct summand of BiEIMi, so there exists a right C-comodule N such that eiEIMi E P $ N. By the Theorem of Crawley-Jonsson-Warfield we obtain that P = $jEjPj, where Pj are finite dimensional projective left C-comodules (in fact we can assume more, that the Pj's are indecomposable). By Corollary 2.4.19 we have that Pj is a projective left C*-module for any j , so P is also a projective left C*-module. On the other hand P* is a direct summand in (eiEIMi)* z ZEI M:. Since @ i E ~ Misf dense in b+ and eiEIM? E R a t ( n i E I MT), we obtain that Rat(P5.) is dense in P*. I
niEI
n.
Exercise 2.4.23 Let C and D be two coalgebras and 4 : C -+ D a coalgebra morphism. Show that the following are equivalent. (i) C is an injective (coflat) lej? D-comodule. (22) Any injective (coflat) left C-comodule is also injective (coflat) as a left D-comodule. (ziz) The functor (-)+ = -ODC : MD -, MC is exact.
2.5
Some topics on torsion theories on M~
Let A be a Grothendieck category and C a full subcategory of A. Then C is called a closed subcategory if C is closed under subobjects, quotient objects and direct sums. If C is furthermore closed under extensions, then C is called a localizing subcategory of A. A closed subcategory of a Grothendieck category is also a Grothendieck category. Indeed, if U E A is a generator of A, then { U / K J KE A a n d U / K E C)
,
2.5. TORSION THEORIES
101
is a family of generators of C, and then the direct sum of this family is a generator of C. If C is closed, then for any M E A we denote by tc(M) the sum of all subobjects of M which belong to C. In this way we define a left exact functor tc : A --+ A, called the preradical functor associated to C. An object M E A is called a C-torsion object if tc(M) = M , and this is clearly equivalent to M E C. If tc(M) = 0, then M is called a C-torsionfree object. If C is a localizing subcategory, we have that tc(M/tc(M)) = 0 for any M E A, since C is closed under extensions. In this case tc is called a radical functor. Following [216, Chapter VI], a closed subcategory C of A is called a hereditary pretorsion theory, and a localizing subcategory of A is called a hereditary torsion theory in A. If A is a Grothendieck category and M E A is an object, we denote by ca[M] (or shortly a[M]) the class of all objects of A which are subgenerated by M , i.e. which are isomorphic to subobjects of quotient objects of direct sums of copies of M . Dually, we denote by aL[M] (or shortly ut[M]) the class of all objects of A which are isomorphic to quotient objects of subobjects of direct sums of copies of M. Proposition 2.5.1 With the above notation, the following assertions are true. (i) a [ M ] is the smallest closed subcategory of A containing M . (ii) a[M] = al[M]. (iiz) If C is a closed subcategory of A, there exists an object M E A such that C = a[M]. Proof: (i) Since the direct sum functor is exact, we obtain that a[M] is closed to direct sums. Let now
be an exact sequence in A such that Y E a[M]. The definition of u[M] shows immediately that Y1 E a[M]. Since Y E o[M], there exist an epimorphism f : M(') -+ X and a monomorphism u : Y -+ X . We have that Y" .v Y/Yt and Y/Yt 5 X/Y' = x",so X" is a quotient object of X , and then it is also a quotient object of Ad('), and then Y" E a[M]. Thus a [ M ] is a closed subcategory of A. Assume that C is a closed subcategory of A such that M E C. Then for Y, f and u as above, we have that M(') E C, so X E C, showing that Y E C. Therefore a[M] C C. (ii) The definition tells us that an object Y E A is in uh[M] if and only if there exist a set I, a subobject X of M('), and a morphism f : X + Y
CHAPTER 2. COMODULES
102
with I m ( f ) = Y. Since aA[M] is closed under direct sums, subobjects and quotient objects, we obviously have that a k [ M ] C ad[M]. Since M E a a [ M ] and ad[M] is the smallest closed subcategory containing M , in order to prove that ad[M] C a k [ M ] , it is enough t o show that uk[M] is closed. Clearly a h [ M ] is closed under direct sums and homomorphic images. Assume that Y E a h [ M ] (with I, f and X as above) and let Y' be a subobject of Y. Then X' = fW1(Y') is a subobject of M('), and Y' is a homomorphic image of X ' (via the restriction and the corestriction of f ) . Thus Y' E a h [ M ] , so uh[M] is also closed under subobjects. (iii) Since C is a closed subcategory of A, then C is a Grothendieck category. If U is a generator of A, the family {U/U'IU1 5 U and U/U1 E C) is a family of generators of C. Then the direct sum of this family
I
is a generator of the category C and clearly C = u[M].
If C is a coalgebra, then the category M' is a Grothendieck category which is isomorphic to the category Rat(C*M) of rational left C*-modules. Corollary 2.5.2 We have that Rat(c. M) = U [ ~ * C ] .
c
Proof: Clearly any object of a[c.C] is rational, so we have that U [ ~ - C ] Rat(c*M ) . Conversely, if M E Rat(c- M ) , then M is a right C-comodule, and then there exists a nonempty set I such that M is isomorphic as a C-comodule to a subobject of a direct sum c(')of copies of C . Regarding I this in the category M, it means that M E a[c*C]. Proposition 2.5.3 Let C be a coalgebra. Then the following assertions hold. (i) If I is a left ideal of C*, then IL= annc(I) = {c E C I I c = 0). (iz) If X is a left coideal of C , then X I = annc. (X), where
-
a n n c * ( X ) = {f E C*If
-x
= 0 for any
x
E X}.
(iii) If p : M -+ M @ C is the comodule structure map of the right Ccomodule M , and J is a two-sided ideal of C* such that J M = 0, then p(M) G M @ J L , i.e. M is a right comodule over the subcoalgebra J L of C. (iv) If M is a right C-comodule and A = (annc*(M))I, then A is the smallest subcoalgebra of C such that p(M) C M @ A. The subcoalgebra A is called the coalgebra associated to the comodule M .
2.5. TORSION THEORIES
Proof: (i) Let c E annc(I). Then f
103
-
c = 0 for any f E I. Then
so c E I L . Conversely, if c E IL,then f (c) = 0 for any f E I . Let A(c) = xi@ yi with (xi)lSiln linearly independent. If 1 t < n, there exists g E C* such that g(xt) = 1 and g(xi) = 0 for any i # t. Then g f E I and
<
-
so f(yt) = 0. Then f c = ClSzln f (yi)x, = 0, which shows that c E annc(I). Thus ILC annc(I). (ii) If f E X L then f ( X ) = 0. Let x E X . Then f x = C f (x2)xl = 0, thus x E annc*(X). Conversely, assume that f E annc- (X). Then for any x E X we have that
-
so f E X L . (iii) For m E M let p(m) = E r n 0 8 m l , and assume that the me's are linearly independent. If f E J we have that 0 = f m = C f(ml)mo, so f (ml) = 0 for any m l , thus m l E J L . We obtain that p(M) 5 M 8 J L . (iv) Denote J = annc*(M). Then J is a two-sided ideal of C* and by (iii) we have p(M) C M @ A, and A = JL is a subcoalgebra of C . Assume that B is a subcoalgebra of C such that p(M) C M @ B . If f E B L and m E M , then f m = 0, so B L a n n c * ( M ) = J . Thus J I C ( B L ) ' - = B , and we find that A B. I
Exercise 2.5.4 Let M be a finite dimensional right C-comodule with coa basis of M and module structure map p : M 4 M @ C ,
104
CHAPTER 2. COMODULES
( ~ i j ) l < i , j < nbe elements of C such that p(mi) = m j 8 cji for any i. Show that the coalgebra A associated to M is the &dipace of C spanned by the set (cij)lli,jln7 and that A(cji) = CILtln ~ j8 t Cti and &(cij)= 6ij for any i,j .
Theorem 2.5.5 Let C be a coalgebra and A a subcoalgebra of C . We denote by CA = {M E M C 1 p M ( ~G) M 8 A) where p~ stands for the comodule structure map of M . Then the following assertions hold. (i) M E CA if and only zf A I M = 0. (ii) CA is a closed subcategory of M ~ . (iii) The map A I+ CA is a bijective correspondence between the set of all subcoalgebras of C and the set of all closed subcategories of M C . Proof: (i) Assume that M E CA and f E AL. Then f(A) = 0. Since ~ M ( MC ) M 8 A we have f m = C f(ml)mo with mo E M and m l E A, so then f m = 0, and A I M = 0. Conversely, if A I M = 0, by Proposition 2.5.3 we have p M ( M ) 2 M @I ALL = M @I A. (ii) It is easy to see that (i) implies that CA is closed under subobjects, quotient objects and direct sums. (iii) Let C be a closed subcategory of M C . Since MC e Rat(C.M), which is a closed subcategory of c*M,we have that C is a closed subcategory of c.M. Since C is a left-C, r i g h t 4 bicomodule, we also have that C is a left-C*, right-C* bimodule. Let A = tc(c*C), where tc is the preradical associated to the closed subcategory C of c * M . Then A is a left C*submodule of c*C. For any g E C* the map u, : C -+ C,u,(c) = c g is a morphism of left C*-modules. C is closed under quotient objects, so g C A. We obtain that A is a left-C*, right-C* u(A) A, i.e. A subbimodule of c*Cc* . If we consider A as a comodule, then we have A(A) C A 63 C and A(A) 2 C 8 A, so A(A) (A 8 C ) n ( C 8 A) = A 8 A, showing that A is a subcoalgebra of C. It is clear that A, regarded as a right C-comodule, belongs to C. Let M E C. Then there exists a monomorphism
c
-
-
-0
d
M
c(')
for some non-empty set I. Since tc is an exact functor which commutes with direct sums, we obtain an exact sequence 0
M
tc(C)(')
i.e. an exact sequence of right C-comodules 0 + M ---+ A(')
2.5. TORSION THEORIES
105
Since A ~ ( A ( ' ) )= 0, then A L M = 0 and M E CA by assertion (i). Thus C & CA. Conversely, if M E CA, then pM(M) & M 8 A, so M is a right A-comodule. Thus there exists a right C-comodule morphism
for some non-empty set I . Since A E C we have M E C, therefore C = CA. We have obtained that the correspondence A H CA is surjective. Now if B is another subcoalgebra of C and CA = CB, then clearly A E CB and B E CA, SO AL B = 0 and BI A = 0. Proposition 2.5.3 shows that B 2 ALL = A, and A BLL = B. Thus A = B , which ends the proof. I
-
-
Corollary 2.5.6 If C is a closed subcategory of M e , then C is'closed to direct products. Proof: Let be a family of objects of C. Since C = C A for some subcoalgebra A of C , we have that AL(ni,, Mi) = 0, where Mi is considered as a left C*-module. Since R a t ( p M ) is a closed subcategory of e*M,it is easy to see that (n,,, Mi)Tat is the direct product of the family (Mi)iEI in the category Rat(c*M) M e . On the other hand (niEI C_ Mi) SO A L ( n i E IMi)Tat = 0, i.e. ( n i G IMi)Tat E C. Thus C is closed to direct products. I We introduce now some notation. For any subspaces X and Y of the coalgebra C we denote by X A Y the subspace
niEl
-
The subspace X A Y is called the wedge of the subspaces X and Y. We define recurrently AOX = 0, A'X = X , and A ~ = X (An-lX) A X for any n 2 2. With this notation we have the following. Lemma 2.5.7 For any subspaces X and Y of the coalgebra we have that X AY =(X~Y')~. In particular, if A is a subcoalgebra of C, then for any positive integer n we have that A ~ = A ( J n ) l , where J = AL.
+
Proof: Let f E X L and g E YL, then since A ( X A Y) G X 8 C C 8 Y we see that (fg)(X A Y) = 0, so X A Y C_ ( X L Y L ) l . Conversely, let c E ( x ~ Y ~ ) ' .We can write
where the set {xill 5 i 5 m} U {z,ll 5 j 5 n } is linearly independent, E X for any i and zj belongs to a complement of X for any j. Fix
xi
C H A P T E R 2. COMODULES
106
some j l . Then there exists f E C* such that f ( X ) = 0, f ( z j , ) = 1, and f ( z j ) = 0 for any j $ jl. For any g E YL we have that fg E XLy', so ( f g ) ( c ) = 0. But ( f g ) ( c )= g ( u j ) , and we obtain that uj E ( y L ) ' - = Y . Thus A ( c ) E X @ C + C @ Y , i.e. c E X A Y . We prove the second statement by induction on n . It is obvious for n = 1. Assume the assertion holds for some n. Then
A'"'A
( A ~ AA)A = ( J ~ ) ' A A (induction hypothesis) = ( ( J ~ J)' ) (by ~ ~the first part) =
=
(FJ)~
where J" is the closure of Jn. We clearly have the inclusion ( F J ) ~C_ ( J " + ' ) ~ .Let now take some c E ( J n f ' ) ' - . By Proposition 2.5.3(i) we have c = 0, so J n (J c ) = 0. Thus J n x = 0 for any that Jn+l x E J c, i.e. J n C annee (2). By Theorem 2.2.14, a n n c * ( x ) is closed, and then J n C annc.(x), so J n x = 0 for any x E J c. Then JrL ( J c ) = 0 and c E ( 7We ~obtain )~ that. A~+'A = (Jn+l)l.
--
- -
- -
-
-
-
Exercise 2.5.8 Let C be a coalgebra and X , Y ,Z three subspaces of C . Show that ( X A Y ) A Z = X A (Y A Z ) . Exercise 2.5.9 Let f : C --, D be a coalgebra morphism and X , Y subspaces of C . Show that f ( X A Y ) C f ( X ) A f ( Y ) .
A subcoalgebra A of C is called co-idempotent if A A A = A. Clearly, if J = A' is an idempotent two-sided ideal of C * , then A is a co-idempotent subcoalgebra. Keeping the same notation as in Theorem 2.5.5 we have the following. Theorem 2.5.10 Let C be a coalgebra and let A be a subcoalgebra of C . Then the following assertions hold. (i) If A is co-idempotent, then C A is a localizing subcategory of M ~ . (ii) The map A H C A defines a bijective correspondence between the set of all co-idempotent subcoalgebras of C and the set of all localizing subcategories of M ~ . Proof: (i) Taking into account Theorem 2.5.5, it remains to prove that C A is closed under extensions. Let
be an exact sequence in M~ with M ' , M" E C A . If we put J = A', then J M ' = J M " = 0 , and then J 2 M = 0. So p M ( M ) G M @ ( J ~ ) ' . Since A
2.5. TORSION THEORIES
107
is co-idempotent we have A = A A A = ( J ~ ) SO ~ ,p M ( M ) C M
@I A
and
M E CA. (ii) Let C be a localizing subcategory of M C . By Theorem 2.5.5 there exists a subcoalgebra A of C such that C = CA. It remains to prove that A = A A A. Clearly A A A A. Denote M = (A2A)/A, and we have an exact sequence of right C-comodules
Clearly A1
-
A = 0 and A E CA. On the other hand
-
But A2A = {c E CI J2 c = 0) by Proposition 2.5.3, so for any c E A2A we have J (J c ) = 0, which means that J c 2 JL = A. Then AL (A2A) = 0, SO AIM = 0. Since CA is closed under extensions, we have that A ~ EA CA. Finally, since A 2 A2A and A2A is a subcoalgebra, we obtain by Theorem 2.5.5 that A = A2A. I We present now some generalizations of the finite dual of an algebra. Let A be a k-algebra. We denote by Idf.,(A) the set of all two-sided ideals of A of finite codimension. Idf.,(A) is a filter, i.e. it is closed to finite intersections and if I J are two-sided ideals such that I E Idf.,(A), then J E Idf.,(A). Let y be a subfilter of Idf.,(A), i.e. y C Idf,,(A), if I , J E y then I n J E y, and for any two-sided ideals I C J with I E y,we also have that J E y. We denote by
-
- -
-
Since (A/I)* can be embedded in A* for any I, we have that = { f E A*l there exists I E y such that I C K e r ( f ) )
If y = Idf.,(A), then A">?= A', the finite dual of the algebra A defined in Section 1.5. Proposition 2.5.11 A''? has a natural structure of a coalgebra. Proof: If I E y, since I has finite codimension in A we have a natural monomorphism
If we apply the inductive limit functor we obtain a monomorphism
CHAPTER 2. COMODULES
108
On the other hand we have a natural inorphism
and by the universal property of the inductive limit we obtain a natural morphism A? : AO)Y-+ AO,Y €3 A',?' As in the case of A', a standard argument shows that AY is coassociative. The counit E : AO>r-+ k is defined by €(a*)= a * ( l ) for a* E A03r, i.e. a* E ( A / I ) *for some I E y. I
Remark 2.5.12 If y and y' are two subfilters of Idf,,(A) such that y C y', then we have AotYC A'?~'C A'. I Let 4 : A -+ B be a morphism of k-algebras, and let us consider two subfilters y C Idf.,(A) and y' C Idf.,(B) such that for any J E y' we have +-'(J) E y. The morphism induces a monomorphism
+
for any J E y'. Then we have a natural epimorphism
and taking the inductive limit we obtain a natural morphism
-
B'>" = lim ( B /J)* -+ 1 s( ~ 1 4 - ' ( J ) ) * JET'
JE-/
By the universal property of inductive limits we have a natural morphism
Therefore the algebra morphism q5 : A -+ B induces a natural morphism @'vryr' : --+ AOIY.By standard arguments one can show that @ ' V Yis ~Y' a coalgebra morphism. ~
'
1
7
'
We recall that a k-algebra A is called pseudocompact if A is a separated and complete topological space which has a basis of neighbourhoods of 0 which are two-sided ideals of finite codimension. We denote by APCk the category of all pseudocompact k-algebras. In this category morphisms are continuous algebra morphisms. If C is a k-coalgebra, then C* is a pseudocompact k-algebra, and in this way we can define a contravariant functor F : k - Cog --+ APCk, F ( C ) = C*.
2.5. TORSION THEORIES
109
Theorem 2.5.13 The functor F defined above is an equivalence between the dual category (k -.Cog)O and the category APCk. Proof: For A E APCk we take the subfilter y of Idf,,(A) consisting of all open two-sided ideals of A. We can consider the coalgebra AOlr, and so we can define now a contravariant functor G : APCk + k - Cog by G(A) = A02r. Moreover, for A E APCk we have that
-
thus F G IdAPCk. For the other composition, let C be a k-coalgebra. Then the set
B = {DIID is a finite dimensional subcoalgebra of C ) is a basis of neighbourhoods of 0 in the topological algebra C*, and then G(C*) =
l@(C*/DL)* DEB
E
l&l(D*)* DEB
-
N
l@D
=
C
DEB
so G F Idk-c,,. I Let A be a pseudocompact k-algebra. We denote by M P C ( A ) the category of pseudocompact right A-modules, whose objects are all right A-modules that are pseudocompact (see Section 2.2) and with morphisms A-linear continuous maps. If C is a coalgebra and M E M C , then M * is a pseudocompact right C*-module. The correspondence M M * defines a contravariant functor F : MC -+ M P C ( C * ) .
Theorem 2.5.14 With the above notation, the functor F defines an equivalence between the dual category ( M C ) " and the category M P C ( C * ) .
CHAPTER 2. COMODULES
110
Proof: Let M E M P C ( C * ) and N a right C*-submodule of M such that N is open and N has finite codimension in M . Then I = annc.(M/N) is a two-sided ideal of C* of finite codimension, and since the topology of MlN is discrete we see that I is open in C*. The multiplication morphism
defines a morphism
If we take M 0 = l@(M/N)* N
where the inductive limit is taken over the open submodules N of M of finite codimension, then we have a natural morphism fiMo : M 0 -+ C 8 M 0 and standard arguments show that M O is a right C-comodule. Then the correspondence M M M Odefines a contravariant functor G : M P C ( C * ) + Me, and as in Theorem 2.5.13 one can show that G F E IdMc and F G E Id~pc(c*). I
2.6
Solutions to exercises
Exercise 2.1.8 Use Theorem 2.1.7 to prove Theorem 1.4.7. Solution: Let C be a coalgebra, c E C and V a finite dimensional right subcomodule of C (i.e. A(V) 2 V 8 C ) such that c E V. If {vi) is a basis of V, we have A(vi) = 8 cji,
C
and so It follows that A(cki) = ckj 8 c j i , and so the subspace spanned by V and the cjils is a finite dimensional subcoalgebra containing c.
Exercise 2.2.17 Let C be a coalgebra and q5c : C -+ C** the natural injection. Then C)) = Rat(c*C**). Solution: Since 4c is injective we clearly have that C)) C Rat(c* C**). Conversely, let f E Rat(c. C**). Theorem 2.2.14 tells that there exists a two-sided ideal I of C*, which is closed, of finite codimension, and satisfies If = 0. Since for any c* E I we have (c*f)(C*) = 0, we see that f ( I ) = f ( C * I ) = 0, so f E (C*/I)*. But I = ( I L ) l since I is closed, SO
C * / I !2! L2*/(IL)l e ( I L ) *
-
2.6. SOLUTIONS TO EXERCISES
Thus f E ( I L ) * * Since . I L has finite dimension, then I-'- (IL)** via the restriction of the morphism 4 ~In. particular we can find some x E I L such that f = & ( x ) , so we have that Rat(C. C**)2 q5c(Rat(c*C)).
Exercise 2.2.18 Let (Ci)iEIbe a family of coalgebras and C = $ i E ~ C the i coproduct of this family in the category of coalgebras. Then the following assertions hold. (i) C* cz (2:. Moreover, this is an isomorphism of topological rings if we consider the finite topology on C* and the product topology on C:. (ii) If M E M C then for any m E M there exists a two-sided ideal I of C* such that I m = 0, I = Ii, where Ii is a two-sided ideal of C,f which is closed and has finite codimension, and Ii = C,f for all but a finite number of i's. (iii) The category M C is equivalent to the direct product of categories niErMCa. (iv) If A is a subcoalgebra of C = e i E ~ C ithen , there exists a family (Ai)iEI such that Ai is a subcoalgebra of Ci for any i E I , and A = $iEIA;. Solution: (i) Define the map 4 : C* --t C,t by $ ( f )= ( f i ) i E 1 ,where for f E C* = Hom(C, k), fi is the restriction of f to Ci for any i E I . It is easy to see that i f f , g E C * ,then ( f g ) i = figi (the convolution product), so 4 ( f g ) = 4 ( f ) d ( g ) . Also, it is clear that 4 is bijective. We prove now that $ is a morphism of topological rings. Let f E C* and c = Ct=,,, ci, E C with tit E C i t If C ( f ) = ( f i ) i ~then ~ , f +ci = ni+,,.,,,i, ci ~ n ~ = ~ , , ( f i , showing that 4 is a morphism of topological rings. (ii) By Theorem 2.2.14 there exists a closed (open) two-sided ideal I of C* of finite codimension such that I m = 0. Let A = I L , a finite dimensional subcoalgebra of C . Then there exist i l l . . ,in E I such that A Ci, $ . . . $ Ci,, and then AL is a two-sided ideal in C:,. Then there exist two-sided ideals litof CTt such that AL = - It. It is easy to see that
niEI
niEI
niEI
niEI
+c;),
nl,,
I = I J - I = A ~ =
tEI
where Dt = C,' for any t @ {zl,. . . ,z,) and Dt = It for t E { i l l . .., t n ) . Moreover Dt is closed (open) and has finite codimension in C,' for any t . (iii) Let M E M ~ We . have C* E C,*, and we denote by e, the element of this direct product having 1 (more precisely the identity of C:) on the z-th spot and 0 elsewhere. We obtain a family (e,),EIof orthogonal primitive central idempotents of C*. Since e, is central, we have that e,M is a C*-submodule of M. On the other hand
nZEI
cf
=
{ ( f t ) t ~ ~ l€f C: t
=
{flf
for any t E I and fi = 0 ) € C * a n d flci=O)
CHAPTER 2. COMODULES
112
and C: (ei M) = 0, so eiM is a rational C,t-module. We have M = @iE1eiM as C*-modules. Indeed, the sum is direct since the ei's are orthogonal. Also, if m E MI then by (ii) we see that etm = 0 for all but finitely many t E I. We obviously have that et(m - CiEI eim) = 0, and since et = E ~ c , , we eim) = 0, so m eim = 0, showing that obtain that E . (m - CiEl m E Xi,, eiM. We can define now a functor
xi,,
and it is easy to see that this defines an equivalence of categories. (iv) follows from (iii).
Exercise 2.2.19 Let (Ci)iEl be a family of subcoalgebras of the coalgebra C and A a simple subcoalgebra of CiE,Ci (i.e. A is a subcoalgebra which has precisely two subcoalgebras, 0 and A; more details will be given in Chapter 3). Then there exists i E I such that A G Ci. Solution: Since A has finite dimension, we can assume that the family (Ci)iG1is finite, say that A 5 C1 . . . C,. Moreover, if we prove for n = 2, then we can easily prove by induction for any positive integer n. Let A C D + El where D and E are subcoalgebras of C. If A n D # 0, the A n D = A, since A is simple, so then A C D. If A n D = 0, pick some a E A. Then
+ +
+
Thus A(a) = C a1 @ a2 C bl @ b2 with a l , a2 E D and bl, b2 E E. Since A n D = 0, there exists f E C* such that f l D = 0 and f l A = &,A.Then f a E A (note that A is a left and right C*-module) we have that f a =E a = a. On the other hand f -- a = C f (a2)a1+ f (b2)bl = C f (b2)bl E El so a E E . We obtain that A C E .
-- -
x
Exercise 2.3.4 Let C be a coalgebra, and c E C. Show that the subcoalgebra of C generated by c is finite dimensional, using the bicomodule structure of C. Solution: We know that C is an object of the category C ~ with C left and right comodule structures induced by A. A subspace V of C is a subobject in the category C ~ if Cand only if A(V) G C @ V and A(V) C V @ C . Since (V @ C ) n ( C @ V) = V @ V, this is equivalent to the fact that V is a subcoalgebra of C. Hence the subcoalgebra V generated by c is the smallest subbicomodule of C containing c. By Theorem 2.3.3 it follows that V is the subbimodule of C generated by c, when we regard C as a left-C*, right-C* c C*. Theorem 2.2.6 shows that C* bimodule, so V = C* c is
- -
-
2.6. SOLUTIONS TO EXERCISES
113
-
finite dimensional, hence C* c = ~,=,,,Icc, for some c, E C, and using C* the right hand version of the cited theorem, it follows that every c, is finite dimensional. We obtain that
-
is finite dimensional, which ends the proof.
Exercise 2.3.10 Let C and D be two coalgebras. Show that the categories DMC, MC@D'O~ M D ~ O ~ @ CD@CCop 7 M and C c o P @ D are ~ isomorphic. Solution: If M E D ~ with C comodule structure maps p- : M D @M , @ m101and p+ : M M @ C , pf (m) = m(o) @ m ( l ) , p-(m) = C then M becomes a right C @ DcOP-comoduleby y : M -t M @ C 63 DCop defined by I
-+
-+
In this way we obtain a functor F : DMC-t M ~ @ ~ ~ ~ ~ . Conversely, let M E M C@DcoP. If4:C@Dcop--+Cand11,:C@Dc0p+ DcOpare the coalgebra morphisms defined by +(c@d) = ~ ( dand ) Q(c@d) = ~ E ( c )then , we consider the comodules M4 E MG and M* E MDCop D M . These two structures make M an object of the category D ~ GThus . ~ 4 DMC, @ and ~ it is clear ~ that ~ ~ we can define another functor G : M the functors F and G define an isomorphism of categories.
-
Exercise 2.3.11 Let C, D and L be cocommutative coalgebras and 4 : C -+ L, 11, : D -t L coalgebra morphisms. Regard C and D as L-comodules via the morphisms 4 and 11,. Show that COLD is a (cocommutative) subcoalgebra of C @ D, and moreover, COLD is the fiber product i n the category of all cocommutative coalgebras. Solution: C is a right L-comodule via the morphism
and D is a left L-comodule via the morphism
and COLD = K e r ( ( ( I @ 4 ) A c ) @ I - I @ ((11, @ I ) A ~ ) ) ) Since C and D are cocommutative we have that COLD is a C - D-bicomodule, so it is also a left and right C @ D-comodule (since C and D are
I
114
C H A P T E R 2. COMODULES
cocommutative). This implies that C O L D is a subcoalgebra of C @D . The last part follows dirrectly from the definition.
Exercise 2.4.23 Let C and D be two coalgebras and 4 : C D a coalgebra morphism. Show that the following are equivalent. (i) C is a n injective (cofiat) left D-comodule. (ii) Any injective (cofiat) left C-comodule is also injective (coflat) as a left D-comodule. (iii) The functor (-)+ = -ODC : M D -+ M~ is exact. Solution: It follows from Proposition 2.3.8 and the properties of adjoint functors. Exercise 2.5.4 Let M be a finite dimensional right C-comodule with comodule structure map p : M --, M €3 C , (mi)i=i,, a basis of M and ( ~ i j ) l < i , j < nbe elements of C such that p ( m i ) = C,,j,n mj @ cji for any i. Show that the coalgebra A associated to M is the Luhpace of C spanned by the set (cij)l
and p x , p y are the natural projections. Then r x , induces ~ an injective linear morphism b X , y : C / ( X A Y ) + C / X €3 C / Y and we have that
But K e r ( ~ ~ , , y ,= ~ )( X A Y )A Z and
so ( X A Y ) A Z = K e r ( ( p x @ p y 8 p z ) o A,). Similarly X A (Y A Z ) = K e r ( ( p x 8 py @ p ~ 0 )Az), proving the required equality.
2.6. SOL UTTONS TO EXERCISES
115
Exercise 2.5.9 Let f : C -+ D be a coalgebra morphism and X , Y subspaces of C . Show that f ( X A Y ) f ( X ) A f ( Y ) . Solution: Let us consider the linear maps fx : C / X -4 D l f ( X ) and f y : C / Y -+ Dlf ( Y )induced by f . We know that ( f @ f ) o A c = AD o f and ( f x @ f Y )O ( P X @ P Y ) = ( P ~ (@x P) ~ ( Y )0) ( f @ f ) , so then
c
Since X A Y = K e r ( ( p x @ p y )
0
Ac), We have that
and then
Bibliographical notes Our sources of inspiration for this chapter include again the books of M. Sweedler (2181, E. Abe [1], and S. Montgomery [149], P. Gabriel's paper [85], B. Pareigis' notes [178], and F.W. Anderson and K. Fuller [3] for module theoretical aspects. Further references are R. Wisbauer [244], I. Kaplansky [104], Y. Doi [72]. We remark that the cotensor product appears for the first time in a paper by Milnor and Moore [146]. We have also used the papers by C. Niistikescu and B. Torrecillas [159], B. Lin [122]. The reader should also consult M. Takeuchi's paper [229], where the Morita theory for categories of comodules is studied. The contexts known as Morita-Takeuchi contexts are a valuable tool for studying Galois coextensions of coalgebras, in a dual manner to the one in Chapter 6.
Chapter 3
Special classes of coalgebras 3.1
Cosemisimple coalgebras
We recall that if A is a Grothendieck category and M is an object of A, the sum s ( M ) of all simple subobjects of M is called the socle of M . If M = 0, we have s ( M ) = 0. An object M is called semisimple (or completely reducible) if s ( M ) = M. It is a well known fact that M is a semisimple object if and only if it is a direct sum of simple subobjects. A category A with the property that every object of A is semisimple is called a semisimple (or completely reducible) category. If C is a semisimple Grothendieck category, then any rnonomorphism (epimorphism) splits. in particular we have that any object is injective and projective. Conversely, if C is a Grothendieck category such that any object is injective and projective (such a category is called a spectral category) we do not necessarily have that C is a semisimple category. We will be interested in the situations where A = MC,the category of right comodules over the coalgebra C, or A = C ~the ,category of left comodules over the coalgebra C. A coalgebra C is called simple if the only subcoalgebras of C are 0 and C. The Fundamental Theorem of Coalgebras (Theorem 1.4.7) shows that a simple coalgebra has finite dimension.
Exercise 3.1.1 Let C be a coalgebra. Show that C is a simple coalgebra if and only if the dual algebra C* is a simple artinian algebra. If this is the case, then C is a sum of isomorphic simple right C-subcomodules. Moreover, i f the field k is algebraically closed, then C is isomorphic to a matrix coalgebra over k .
118
CHAPTER 3. SPECIAL CLASSES O F COALGEBRAS
Exercise 3.1.2 Let M be a simple right comodule over the coalgebra C . Show that: (i) The coalgebra A associated to M is a simple coalgebra. Mc(n, k), where n = (ii) If the field k is algebraically closed, then A dim(M). In particular the family (cij)15i,jl, defined in Exercise 2.5.4 is a basis of A.
--
Exercise 3.1.3 Let M and N be two simple right comodules over a coalgebra C . Show that M and N are isomorphic if and only if they have the same associated coalgebra. For an arbitrary coalgebra C we denote by Co the sum of all simple subcoalgebras. Co is a subcoalgebra of C called the coradical of C . We also use the notation Co = Corad(C) for the coradical. Proposition 3.1.4 Let C be a coalgebra. Then Co = s ( c C ) = s(Cc), where s(Cc) is the socle of C as an object of M', and s ( c C ) is the socle of C as an object of CM. Proof: We will show that Co = s(Cc). The proof of the fact that Co = s ( c C ) is similar (or can bee seen directly by looking at the coopposite coalgebra and applying the result about the right socle). A simple subcoalgebra A of C is a right C-subcomodule of C. Since A is a finite direct sum of simple right coideals of A, we see that A is semisimple of finite length when regarded a s a right C-comodule. Thus A C s(Cc), and then COC s ( C c ) . Conversely, let S C s(Cc) be a simple right C-comodule, and let A be the coalgebra associated to S. By Exercise 3.1.2 A is a simple coalgebra, so A C Co. But S C A, since for c E S we have c = C E ( C ~ )ECA. ~ Thus I S G A C Co, so s(Cc) C Co. A coalgebra C is called a right cosemisimple coalgebra (or right completely reducible coalgebra) if the category MC is a semisimple category, i.e. if every right C-comodule is cosemisimple. Similarly, we define left cosemisimple coalgebras by the semisimplicity of the category of left comodules. In fact, the concept of cosemisimplicity is left-right symmetric, as the following shows. Theorem 3.1.5 Let C be a coalgebra. The following assertions are equivalent. (i) C is a right cosemisimple coalgebra. (ii) C is a left cosemisimple coalgebra. (iii) C = Co. (iv) Every left (right) rational C*-module is semisimple.
3.1. COSEMISIMPLE COALGEBRAS
119
(v) Every right (left) C-comodule is projective i n the category MC (resp. CM). (vi) Every right (left) C-comodule is injective i n the category MC (resp. C~
1.
(vii) There exists a topological isomorphism between C * and a direct topological product of finite dimensional simple artinian k-algebras (with discrete topology). A coalgebra satisfying the above equivalent conditions is called cosemisimple. Proof: ( i )@ (ii)@ (iii)follows from Proposition 3.1.4. ( i )e ( i v ) It follows from the fact that if M E MC, then M is a simple object in the category MC if and only if M is a simple left C*-module. ( i )+ ( v ) and ( i )+ ( v i ) follow from the general fact that in a semisimple Grothendieck category every object is projective and injective. ( v ) 3 ( i ) (and similarly ( v i ) + (i)) Assume that any object of MC is projective. Then if M E M C , any subobject N of M is a direct summand. This implies that any finite dimensional object is semisimple (of finite length). On the other hand, MC has a family of finite dimensional generators, which shows that every object of MC is semisimple. (iii)+ ( v i i ) follows from Exercise 2.2.18. ( v i i ) + (iii) Assume that C * 21 Ai, an isomorphism of topological rings, where (Ai)iEr is a family of simple artinian k-algebras of finite dimension, and each Ai is regarded as a topological ring with the discrete Ai. Let Ti : C* -+ Ai be topology. For simplicity assume that C* = the natural projection for any i E I. Denote Ki = K e r ( n i ) . Since the map ~i is continuous, Ki is a two-sided ideal of C* of finite codimension, which , is a simple subcoalgebra of is open and closed in C*. Let Ri = ~ kwhich C , and denote R = CiGI Ri. We have that
niEI
n,,,
showing that R = C . Thus C = Co.
I
Exercise 3.1.6 Let C be a cosemisimple coalgebra. Show that C is a direct sum of simple subcoalgebras. Moreover, show that any subcoalgebra of C is cosemisimple. Exercise 3.1.7 Let C be a finite dimensional coalgebra. Show that C is cosemisimple if and only if the dual algebra C* is semisimple.
CHAPTER 3. SPECIAL CLASSES O F COALGEBRAS
120
If M E C M , we denote by Cendc(M) the space of all endomorphisms of the C-comodule M. Clearly Cendc(M) is a k-algebra with map composition as multiplication. In particular, if we regard C as a left C-comodule, let us take the k-algebra A = Cendc(C). The following indicates the connection between the k-algebras A and C* and computes the Jacobson radical of C*.
Proposition 3.1.8 With the above notations we have that (2) The map 4 : A C*, $(f) = E o f for any f E A = Cendc(C), is a k-algebra isomorphism. (zi) The Jacobson radical of C* is J(C*)= C k . -+
Proof: (i) Let f , g E A. Since g is a morphism of left C-comodules, we have C c l 8 g(c2) = C g ( c ) l 8 g(c)2 for any c E C . If we apply I 8 E we find that Then
so
4 is an algebra morphism.
Define II, : C* -+ A by
-
denotes the left action of C* on C for any u E C* and c E C, where coming from the right C-comodule structure of C . It is clear that $(u) E A, since
-
@(u)(c- v) = u = (u =
-
(c-v) c) v
--L
-
$(U)(C)
v
where is the right action of C* coming from the left C-comodule structure of C. Note that we used the fact that C is a left-C*, right-C* bimodule
3.1. COSEMISIMPLE COALGEBRAS
121
-
with the left action and the right action -. We show that 1C, is an inverse of 4. Indeed, for any u E C* and c E C we have
and for any f E A, c E C
(ii) By Proposition 2.5.3(ii) we have C& = annc-(Co). Since J ( C * ) is the intersection of the annihilators of all simple left C*-modules and Co is a sum of simple right C-comodules, thus a sum of simple left C*-modules, we have J ( C * ) C C k . Let now u E C t and denote f = $-'(u). Thus f = E o u and ~ ((Co)) f = u(C0) = 0. Then for any c E Co
f cc,
=
C f (c)1r(l(c)2) C c l ~ ( f ( c 2 ) )(since / E A)
=
0
=
(since c2 E Co)
so f (Co) = 0. We show that f E J ( A ) . From the fact that 4 is an algebra isomorphism it will follow that u E J ( C * ) . Denote g = 1 - f . Then $ is injective. Indeed, for c E Ker(g) n Co we have 0 = g(c) = c - f (c) = c, so Ker(g) n Co = 0. This implies that Ker(g) = 0, since Co is an essential left C-subcomodule of C . Since C is an injective left C-comodule, Im(g) is a direct summand in C as a left C-subcomodulel so C = Im(g) $ Y for some left C-subcomodule Y of C . On the other hand Co C Im(g), and since Co is essential in C, we must have Y = 0. Thus Im(g) = C and g is an isomorphism. Then g = 1 - f is invertible in A, so f E J(A).
a
Let C be a coalgebra and M E Mc. We recall that the socle of M , denoted by s(M), is the sum of all simple subcomodules of M . Then s ( M ) is a semisimple subcomodule of M . Since any non-zero comodule contains a simple subcomodule, we see that s ( M ) is essential in M. We can define recurrently an ascending chain Mo C M1 G . . . G Mn C . . . of
122
CHAPTER 3. SPECIAL CLASSES O F COALGEBRAS
>
0 we subcomodules of M as follows. Let Mo = s(M), and for any n define Mn+l such that s(M/Mn) = Mn+l/Mn. This ascending chain of subcomodules is called the Loewy series of M . Since M is the union of all subcomodules of finite dimension, we have that M = Un>0Mn. If I is a two-sided ideal of C*, we denote by annM(I) =-{x E MlIx = 01, which is clearly a left C*-submodule of M . L e m m a 3.1.9 Let I = J ( C * ) = C k and M E M C . Then for any n we have Mn = annM(In+l).
20
Proof: We use induction on n. For n = 0, we have annM(I) = Mo = s(M). Indeed, I M o = J(C*)Mo = 0, since the Jacobson radical of C* annihilates all simple left C*-modules. Thus Mo annM(I). On the other hand C t a n n M ( I ) = I a n n M ( I ) = 0, so by Proposition 2.5.3, a n n M ( I ) is a right Co-comodule. Since Co is a cosemisimple coalgebra, annM(I) is a semisimple object of the category M ~ O and , then also of the category M ~ We . obtain that annM(I) C s ( M ) = Mo. Assume now that Mn-1 = annM(In) for some n 2 1. Since Mn/Mn-1 = S(M/M,-~) is semisimple, we have that I(Mn/Mn-l) = 0 , therefore I M n C_ Mn-l. Then rn+'Mn = In(IMn) InMn-1 = 0, so Mn annM(In+l). If we denote X = annM(In+l),we have I n + l X = 0, so I X a n n M ( I n )= Mn-l. Then I(X/Mn-l) = 0 and by the same argument as above X/Mn-l is a right Co-comodule, so X / M n - ~is a semisimple comodule. We have that s(M/Mn-1) = Mn/Mn-l, so we obtain that X Mn. Thus Mn = annM(In+l),which ends the proof. I
c
c
Corollary 3.1.10 Let C be a coalgebra and Co,C1,. . . the Loewy series of the right (or left) C-comodule C . Then Co is the coradical of C, Cn = An+lC0 and Cn is a subcoalgebra of C for any n 0.
>
Proof: We have seen in Proposition 3.1.4 that the coradical of C is just the socle of the right C-comodule C. Lemma 3.1.9 shows that Cn = a n n c ( ~ ( C * ) n + l ) L . By Proposition 2.5.3(i) we have Cn = (J(C*)"+')~, and by Lemmma 2.5.7 we see that Cn = An+lCo. By Lemma 1.5.23 Cn is a subcoalgebra. I If C is a coalgebra, the Loewy series of the right (or left) C-comodule C is an ascending chain of subcoalgebras
which can also be regarded as the chain obtained by starting with the ~ any C ~n 2 0. This coradical Co of C and then by taking Cn = A ~ + for
3.2. SEMIPERFECT COALGEBRAS
123
chain is also called the coradical filtration of C. Since C is the union of its finite dimensional coalgebras we have C = U , , ~ ~ C ,
Exercise 3.1.11 Show that the coradical filtration is a coalgebra filtration, i.e. that A((?,) c xi=o,nCi @ Cn-i for any n 2 0. Exercise 3.1.12 Let C be a coalgebra and D a subcoalgebra of C such that u , , ~ ~ ( A= ~C D .) Show that Corad(C)C D. Exercise 3.1.13 Let f : C -+ D be a surjective morphism of coalgebras. Show that Corad(D)G f (Corad(C))., Exercise 3.1.14 Let X , Y and D be subspaces of the linear space C . Show that
Exercise 3.1.15 Let C be a coalgebra and D a subcoalgebra of C . Show that D, = D U C, for any n.
3.2
Semiperfect coalgebras
Proposition 3.2.1 Let C be a coalgebra. The following assertions are equivalent. (i) R a t ( p C * ) is dense in C*. (zi) For any simple left C-comodule S, Rat(c-E(S)*)is dense in E ( S ) * , where E ( S ) denotes the injective envelope of S in the category (iii) For any injective left C-comodule Q, Rat(C.Q*) is dense in Q*. (zv) For any left C-comodule M , Rat(c*M * ) is dense in M* (in the finite topology). Proof: ( z ) + ( i i ) If S is a simple left C-comodule, then C = E ( S ) $ X for some left C-subcomodule X of C. Then C* P E ( S ) *@ X* as left C*modules, and Rat(c*C * )= Rat(c*E ( S ) * )@ Rat(C*X * ) . Since Rat(c*C*) is dense in C*,we obtain by Exercise 1.2.11 that Rat(c. E ( S ) * )is dense in E(S)*. (iz)+ (ziz)If Q is an injective object of C M ,we have that Q = @ z e ~ E ( S 2 ) for a family ( S z ) z EofI simple left C-comodules. Since Q* P E(S,)*, so @2EIE(S2)* is dense in Q* by Exercise 1.2.17. On the other hand Rat(@,,1E(5',)*)= @,,IRat(c*E(S,)*),which is dense in cB,~IE(S,)* by the assumption. Then Rat(@,EIE(S,)*) is dense in Q*, and the result fol2 Rat(c*Qe). lows from the fact that Rat(@,ErE(S,)*) (zzi) + (zv)Let i : M -+ E ( M ) the inclusion morphism, and i* : E ( M ) * --+
n,,,
,
CHAPTER 3. SPECIAL CLASSES O F COALGEBRAS
124
M * the dual morphism. Since Rat(C=M*) is dense in M*, we obtain by Exercise 1.2.18 that i*(Rat(c-M*)) is dense in M*. But i* is a morphism of left C*-modules, so i*(Rat(C*M*)) C_ Rat(C- M*), showing that Rat(C*M*) is dense in M*. (iv) (i) is obvious. I
*
Proposition 3.2.2 For any M E 'M the following assertions are equivalent. (2) R a t ( p M*) # 0. (ii) There exists a maximal subcomodule N of M , N # M. (iii) J ( M ) # M , where J ( M ) is the Jacobson radical of the object M (i.e. the intersection of all maximal subobjects of M). Moreover, if (i) - (iii) are true, then M / J ( M ) is a semisimple left Cocomodule. Proof: (i) + (ii) Since Rat(c*M*) # 0, there exists a simple C*-submodule X 5 Rat(c8M*). Since X has finite dimension, X is closed in the finite topology. Let N = X I , which is a subcomodule of M . Moreover, since X # 0 we have that N # M. If P is a subcomodule of M such that N P # M , then P L C NL = X L - ' - = X . Since P - ' - = ~ P # M , we have that P L # 0, so we must have P L = X . Then N = X I = pLL = P , which shows that N is a maximal subcomodule of M . (i) If N is a maximal subcomodule of M , then M I N is a simple (ii) object in the category M ~ in, particular it is finite dimensional. Then (M/N)* is a rational left C*-module and the projection p : M -+ M I N induces an injective morphism of left C*-modules p* : (M/N)* -+ M*. Then 0 # Im(p*) C Rat(c*M*), so Rat(c. M*) # 0. (iii) ++ (ii) is obvious. Assume now that (i) - (iii) hold. Since Ck = J(C*), we have
*
Then by Proposition 2.5.3 we obtain that M / J ( M ) is a left Co-comodule, which is semisimple since Co is cosemisimple. I We say that a right C-comodule M has a projective cover if there exist a projective object P in the category of right C-comodules and a surjective morphism of comodules f : P -+ M such that K e r ( f ) is a superfluous K e r ( f ) = P for some subobject X of P, we subobject of P, i.e. if X must have X = P .
+
Theorem 3.2.3 Let C be a coalgebra. The following statements are equivalent. (2) The category MC of right C-comodules has enough projectives.
3.2. SEMIPERFECT COALGEBRAS
125
(ii) Rat(c* C*) is a dense subset of C*. (iii) The injective envelope of any simple left C-comodule is finite dimensional. (iv) Any finite dimensional right C-comodule has a projective cover.
Proof: (i) + (ii) By Proposition 3.2.1 it is enough to show that for any simple left C-comodule S, Rat(E(S)*) is dense in E(S)*. We know that S* is a simple right C-comodule. By (i), there exist a projective object P E M~ and an epimorphism P --+ S*. Then we have a monomorphism S S** P*,therefore S C Rat(C.P*). By Corollary 2.4.18 Rat(c* P*)is injective as a left C-comodule, so we have an injective morphism of left C-comodules E ( S ) -+ Rat(C*P*). Let u : E ( S ) + P* be the injection obtained by composing this injective morphism with the inclusion Rat(P6,) -+ P * . Then u* : P** + E ( S ) * is surjective, and u*(Rat(c.P**)) C Rat(c*E(S)*). But P is dense in P**(through the natural embedding), and P E Rat(c.P**), so Rat(c. P**)is dense in P**. It follows that u*( R a t ( p P**)) is dense in E(S)*, and then Rat(c*E ( S ) * ) is dense in E(S)*. (ii) ==+ (iii) Let S be a simple left C-comodule. We may assume that S is a left subcomodule of Co, and then Co = S @ M for some left C-comodule M. Since C = E(Co) = E ( S ) @ E ( M )we obtain the following commutative diagram with exact rows
--
-
where 7r is the projection from C* to E(S)*. Clearly E ( S ) * = C*x, where . the other hand, since the diagram is commutative, we have x = T ( E ~ )On n ( C t ) C S'. Let f E E ( S ) * such that f (S) = 0. Then we can regard f as an element of C * for which f ( E ( M ) ) = 0. Then f (Co) = f ( S $ M) = 0, so f E C;. We have obtained that n(Ck) = S L . NOWn(C,f) = C ~ T ( E=~C) t x , SO C t x = S L . We have that Rat(E(S)*) is not contained in S', since Rat(E(S)*) is dense in E ( S ) * and SL is closed in E(S)*. Thus SL # S' + Rat(E(S)*), and since E(S)*/S' S* is simple, we must have S'- Rat(E(S)*) = E(S)* = C*x. Therefore
+
-
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C H A P T E R 3. SPECIAL CLASSES OF C O A L G E B R A S
Since C: = J ( C * ) , we have by Nakayama Lemma that E ( S ) * = C * x = R a t ( E ( S ) * ) . Thus E ( S ) * is a rational left C*-module, so it is finite dimensional as a cyclic C*-module. We obtain that E ( S ) is finite dimensional too. (iii) =. ( i v ) Let M be a finite dimensional right C-comodule. Since M * is a finite dimensional left C-comodule, the injective envelope E ( M * ) of M* has finite dimension. I t is easy to see that by taking the dual of the exact sequence 0 M* E ( M * ) , we find a projective cover E(M*)* M 0 of M . ( i v ) + ( i ) is obvious, since any comodule is a homomorphic image of a direct sum of finite dimensional comodules. I
--- -
Definition 3.2.4 A coalgebra C is called right semiperfect if one of the equivalent conditions of Theorem 3.2.3 is satisfied. A coalgebra C is called left semiperfect i f CcOp is right semiperfect, i.e. if C satisfies one of the conditions of Theorem 3.2.3 with the left and right hand sides switched. I Remark 3.2.5 The equivalence of assertions ( i ) ,(ii) and ( i v ) of Theorem 3.2.3 is also proved in Corollary 2.4.21. W e note that a finite dimensional coalgebra is left and right semiperfect since the category of right (left) C comodules is isomorphic to the category of left (right) C*-modules, which I has enough projectives.
Corollary 3.2.6 Let C be a right semiperfect coalgebra. Then any nonzero left C-comodule contains a maximal subcomodule. Proof: We know from Theorem 3.2.3 that R U ~ ( ~ . C *is ) dense in C*. Then Proposition 3.2.1 shows that Rat(c* M * ) is dense in M * for any nonzero left C-comodule M . In particular Rat(c* M * ) # 0 and we can use I Proposition 3.2.2. Example 3.2.7 Let H be a k-vector space with basis {c, I m 6 N ) . Then H is a coalgebra with comultiplication A and counit E defined by
for any m E N (see Example 1.1.4.2)). B y Example 1.3.8.2) we know that the dual algebra H* is isomorphic to the algebra k [ [ X ] of ] formal power series i n the indeterminate X . Also, the category M H is the class of all torsion left modules over Ic[[X]].But in this category the only projective object is 0, so H is not a right semiperfect coalgebra. Since H is cocommutative, H is not a left semiperfect coalgebra either. I
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127
Example 3.2.8 Let C be the coalgebra defined i n Example 1.1.4.6). C has a basis { gi, d, I i E N* ), and coalgebra structure defined by
W e define the elements g j , dj* E C* by
Clearly g5gj = Gijgf and E = Ci=, , g f , i.e. for any c E C g f ( c ) # 0 for g: only finitely many i 's, and ~ ( c=) c i = l , m g t ( c ) . Then C = @i>lC as left C*-modules, and C = @i21gf C as right C*-modules. It is straightforward to check that
-
-
-
for any i ,j . These relations show that gf C =< gi,diPl > (the vector C =< gl >. Similarly space spanned by gi and di-l) for i > 1, and g; C g5 =< gi,di >. The coradical of C is the space spanned by the set {gili 2 l), and then a simple (left or right) subcomodule of C is of the form kg; for some i. Indeed, let X be a simple right C-subcomodule of C , and pick a non-zero c = C jorjgj + C jPjdj E X . If there exists i such that Pi # 0 , the relation df c = Pigi shows that gi E X , and then X = k&, a contradiction. If Pj = 0 for any j , then pick some i with cui # 0 , and then
-
-
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CHAPTER 3. SPECIAL CLASSES OF COALGEBRAS
128
-
g$ c = aigi, so X = kgi. Thus any simple (left or right) C-comodule is isomorphic to some kgi. For any i we have that C g$ is the injective envelope of the right C comodule kgi. Indeed, since C = @i>lC g$ as right C-comodules and C is a n injective right C-comodule, we see that C gt is an injective right C-comodule. Moreover, kgi is an essential right C-subcomodule (or equivalently a left C*-submodule) of C gi+ =< gi, di >, since 0 # gi+ (agi ,Odi) = agi E kgi for a # 0 , and df di = gi. Similarly g$ C is the injective envelope of the left C-comodule kgi. Therefore C is a left and right semiperfect coalgebra. I
-
+
-
--
-
-
-
Example 3.2.9 W e modify Example 3.2.8 for obtaining a right semiperfect coalgebra which is not a left semiperfect coalgebra. Let C be a vector space with basis { gi,di I i E N* ). C becomes a coalgebra by defining the comultiplication and counit as follows.
-
W e define the elements gj*,dj" E C* as i n Exercise 3.2.8. Also, as i n Ezercise 3.2.8 we have that E = C i = l , w g f , g:g; = Sijgf, C = @i21C gf C as right C*-modules. W e see as left C*-modules, and C = @i21gt g$ = kgi for i > 1, and C g? =< gl, d l , da, . . . >. Also that C gf - C = < gi,diVl > f o r i > 1, a n d g ; C = < g l >. A s in Example 3.2.8 we can see that the coradical o f C is the space spanned by the set {gili 2 1 ) and a simple (left or right) subcomodule of C is of the g; (which is infinite dimensional) is the form kgi for some i. Also C injective envelope of the left C*-module k g l , and gt -- C (which has finite dimension) is the injective envelope of the right C*-module kgi. Therefore C is a right semiperfect coalgebra, but it is not a left semiperfect coalgebra.
--
-
-
-
I Corollary 3.2.10 Let C be a coalgebra. Then the following assertions are equivalent. (2) C is left and right semiperfect. (ii) Every right (or left) C-comodule has a projective cover in M~ (or
CM). Proof: (ii)+ ( i )is obvious from the definition o f a semiperfect coalgebra. ( i )+ (ii)Let M E MC. Corollary 3.2.6 shows that J ( M ) # M . Moreover,
3.2. SEMIPERFECT COALGEBRAS
129
M / J ( M ) is a direct sum of simple right comodules, say M / J ( M ) = $iEISi. If fi : Pi -+ Si is a projective cover of Si,let P = @iErPi, which is a fi : P + M / J ( M ) . If projective object of M', and denote f = n : M -t M / J ( M ) is the natural projection, then there exists a morphism g : P -+ M such that f = r g (use that P is projective). This implies that J ( M ) + I m ( g ) = M . Proposition 3.2.2 shows that I m ( g ) = M I thus g : P -+ M is surjective. Since K e r ( f ) is superfluous in P and Ker(g) 2 K e r ( f ) , we obtain that Ker(g) is superfluous in P, and then f : P M is a projective cover of M . I -+
Corollary 3.2.11 Let C be a right semiperfect coalgebra and A a subcoalgebra of C . Then A is also right semiperfect. Proof: Let M be a simple left A-comodule. Then M is simple when regarded as a left C-comodule. By Theorem 3.2.3 the injective cover E ( M ) of M in ' M is finite dimensional. If CA is the closed subcategory associated to the subcoalgebra A (see Theorem 2.5.5), and t A the preradical associated to C A I let E 1 ( M )= t A ( E ( M ) ) .Then E t ( M ) is the injective envelope of &I in the category A ~ Therefore . d i m ( E 1 ( M )< ) d i m ( E ( M ) )< oo. I Corollary 3.2.12 Let C be a coalgebra and Rat : '.M -+ Rat('* M ) the functor which takes every C*-module to its rational part. Then the following are equivalent. (i) C is right semiperfect. (ii) Rat is an exact functor. Moreover, if C is right semiperfect, then Rat(c* M ) is a localizing subcategory of c. M . Proof: ( i ) + ( i i ) Assume that C is right semiperfect. We already know that the functor Rat is left exact. Let
be an exact sequence in
c*M . Then we have the commutative digram
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CHAPTER 3. SPECIAL CLASSES OF COALGEBRAS
where i',i and i" are the inclusion maps. Since R U ~ ( ~ . M has) enough projectives, there exists an epimorphism f : P -+ Rat(M1') for some projective object P E R a t ( C * M ) . By Corollary 2.4.22, P is also projective in c = M ,and then there exists g : P t M such that vg = i" f . We have that I m ( g ) G R a t ( M ) , and denote by g' : P -, R a t ( M ) the corestriction of g . Then g = ig', and wig' = i" f . Since vi = il'v', we have il'v'g' = i" f , so v'g' = f . Since f is an epimorphism, we see that v' is an epimorphism, therefore Rat is exact. (ii) + (i)Let M E M C = Rat(C.M). We show that R a t ( M * ) is dense in M*. Let m E M , m # 0 and N a finite dimensional subcomodule of M such that m E N . If j : N -+ M is the inclusion map, then j* : M* -+ N* is a surjective morphism of C*-modules. The hypothesis implies that we have an exact sequence
But R a t ( N * ) = N* since N is finite dimensional. Pick f E N* such that f (m) # 0. The exactness of the above sequence shows that there exists g E R a t ( M r ) such that f = j*(g) = gj. Then g ( m ) = f(m) # 0, which shows that R a t ( M * ) is dense in M*. We prove now that if C is right semiperfect, then R a t ( C * M )is a localizing subcategory of c.M. Indeed, it is enough to prove that R a t ( p M ) is closed under extensions. If
is an exact sequence in c * M such that MI, M u E R a t ( c . M ) , we obtain a commutative diagram
The Serpent Lemma shows that Coker(i)= 0, so i is surjective.
I
Remark 3.2.13 If the category of rational C*-modules is a localizing subcategory of the category of modules over C* it does not necessarily follow that C is right semiperfect, as it can be seen by looking at the coalgebra from Example 3.2.7. I
3.2. SEMIPERFECT COALGEBRAS
131
Since M' is a Grothendieck category, it has arbitrary direct products, i.e. for any family of objects there exists a direct product of the family in M C . The direct product functor is left exact, but it is not always exact. However, for semiperfect coalgebras it is exact.
Corollary 3.2.14 Let C be a right semiperfect coalgebra. Then the direct product functor in the category M~ is an exact functor (i.e. the category M C has the property Ab4* of Grothendieck). Proof: It is enough to prove the fact for the subcategory R a t ( c - M ) of c * M . If (M,),,I is a family of rational left C*-modules, then the direct product of this family in the category R a t ( c * M )is R a t ( n L I M a ) ,where we denote here the direct product in the category c.M. Since by - is an exact functor in c . M , we obtain from Corollary 3.2.12 that I - is an exact functor in Rat(c.M).
n* n:,, n,,,
Lemma 3.2.15 Let M E M ~ X , a subspace of C*, and Y a subspace of M . Then YY = X Y , where Y is the closure of X in the finite topology. Proof: We obviously have X Y G XY. Let f E and y E Y. If p : M -, M @ C is the comodule structure map of M , p ( y ) = C yo @ y l , then f(y1)yo. Since X is dense in 7, there exists g E X such that fy = ~ ( Y I ) f (91) for all ~ 1 ' s .Then g y = C g ( y 1 ) y o = f y , so f y = g y E X Y . Thus X Y G X Y , which ends the proof. I We can give now some properties of coalgebras which are left and right semiperfect. Corollary 3.2.16 Let C be a right semiperfect coalgebra. Then Rat(C*C * ) is an idempotent two-sided ideal of C*. Moreover, if C is left and right semiperfect then Rat(C*C * )= Rat(C8. ) . Proof: Clearly I = Rat(C*C*)is a two-sided ideal of the ring C*. By Theorem 3.2.3 we have 'li = C*. Then Lemma 3.2.15 shows that
12 = I . Assume now that C is also left semiperfect and let J = Rat(C&), which is also a two-sided ideal of C*. We know that I E M~ and J E C M . Since C is left and right semiperfect we have ? = 7 = C * , and then by Lemma 3.2.15 we have JI =If1= C*I = I SO
On the other hand, by a version of Lemma 3.2.15 for left comodules we have JI=J~=JC*=J
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I
We obtain I = J.
Let C be a coalgebra. We know that the coradical Co = s ( c C ) = s(Cc), where S ( ~ C )respectively , s(Cc), is the socle of C as a left, respectively right, C-comodule. Let s(Cc) = $jGJMj and s ( c C ) = e p E P N p be representations of the socles as direct sum of simple comodules (the Mjls are right comodules and the N,'s are left comodules). Then C = BjEJE(Mj) as right C-comodules, where E(Mj) is the injective envelope of M j as a right C-comodule. Then
as right C*-modules. We will identify these two C*-modules. An element of the , ) restrictions ) ~ ~ ~ of c* to c* E C* is thus identified to the set ( ~ i ~ ( ~ the subspaces E(Mj)'s. Similarly we have C = e p E p E ( N p ) ,and
as left C*-modules, isomorphism which we also regard as an identification. Corollary 3.2.17 Let C be a left and right semiperfect coalgebra and keep the above notation. Then the following assertions are tme. 2) Rat(c-C*) = Rat(C&,) = e p E p E ( N p ) *= e j E ~ E ( M j ) * .In particular * )Rat(C&,) is left and right projective over the ideal C*Tat= R ~ t ( ~ - c = the ring C*. ii) C*Tatis a ring with local units, i.e. for any finite subset X of C*Tat there exists an idempotent e E C*Tatsuch that ex = xe = x for any x E X (we say that the set consisiting of all these idempotents e appearing from finite subsets X is a system of local units). Proof: i) Since C is left and right semiperfect we have that E(Mj) and E(Np) are finite dimensional for any j and p. Then E(Mj)* is a finite dimensional right ideal of C*, so we obtain from Corollary 2.2.16 that E(Mj)* G Rat(C&). Thus e j E j E ( M j ) * 5 Rat(C&.). Let h* E Rat(C.C*). Then there exist two families of elements hz E C* and hi E C such that g*h* = g*(hi)h;
C 2
for any g* E C*. We can find a finite subset F J such that hi E CBjE~E(Mj)for any 2. Then we define g* E C* such that g* = E on E(Mj)
3.3. (QUASI-)CO-FROBENIUS COALGEBRAS
133
if j $ F and g* = 0 on any E ( M J )with j E F. Then g*h* = 0 , since g*(h,) = 0 for any i. On the other hand, for any j $ F and any h E E ( M J )we have A ( h ) E E ( M J )8 C , therefore
Since h* = 0 on any E ( M j ) with j $ F, we have
Thus we have showed that
Similarly, working with simple left comodules we obtain
which ends the proof of the first part. Note that this provides a proof of the fact that Rat(C&) = Rat(c*C*) different from the one given in Corollary 3.2.16. The fact that C*'at is left and right projective over C* follows from the fact that E ( M j ) * are projective right ideals in C* and E(N,)* are projective left ideals of C * . ii) For any i E J we define ~i E C* to be E on E ( M i ) and 0 on any E ( M j ) with j # i. Since ( E ~ c * ) ( = c ) C E ~ ( c ~ ) c *and ( c ~A) ( E ( M j ) )C E ( M j )8 C , we see that ( E ~ c * ) ( = c )c*(c)for c E E ( M i ) and ( E ~ c * ) ( = c ) 0 for c E E ( M j ) , j # i. Thus ~ i c *= c* for c* E E ( M i ) * and cica = 0 for c* E E ( M j ) * ,j # i. In particular E: = ~i and ~ i e = j 0 for i # j . For a fixed i E I there exists a finite set F C J such that A ( E ( M i ) ) C E ( M i ) 8 ( e j E ~ E ( M j )Then ) . clearly for any c* E E ( M i ) * we have that c * ( z j E F ~=j )c*. It is obvious now that the set consisting of all finite sums of E ~ ' Sis a system of local units of C*Tat. I
3.3
Quasi-co-F'robeniusand co-F'robenius coalgebras
Definition 3.3.1 Let C be a coalgebra. Then C is called a lejl (right) quasi-co-fiobenius coalgebra (shortly QcF coalgebra) if there exists an injective morphism of left (right) C*-modules from C to a free lefi (right)
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C*-module. The coalgebra C is called left (right) co-l+obenius i f there exists a monomorphism of left (right) C*-modules from C to C * . I Clearly, i f C is a left co-Frobenius coalgebra, then it is also a left QcF coalgebra. Conversely, i f C is left QcF, then C is not necessarily co-F'robenius, as we will see in Remark 3.3.12. Let C be a coalgebra. T o any bilinear form b : C x C --, k we associate the linear maps bl,b, : C 4 C* defined by
b r ( x ) ( y )= b(x,Y ) for any x , y E C . Conversely, t o any linear map two bilinear forms b, b' : C x C --, k defined by
4 :C
--t
C* we associate
b(x,Y ) = ~ ( Y ) ( x ) b'(x,Y ) = ~ ( x ) ( Y ) for any x,y E C. In this case we clearly have q!~ = bl = b:.
Proposition 3.3.2 Let C be a coalgebra and b : C x C --t k a bilinear form. The following assertions are equivalent. (i) b is C* -balanced, i. e. b(x c*,y ) = b ( z ,c* y ) for any x , y E C and C* E C * . (ii) bl is a morphism of left C*-modules. (iii) b, is a morphism of right C*-modules.
-
Proof: Let x , y the other hand
E
C and c* E C*. Then bl(c* (c*
-
-
-y)(x)
= b(x,c*
- y). O n
b ( y ) ( x = bl(Y)( X c*) = b(xLc*,y)
showing the equivalence o f ( i ) and (ii).T h e equivalence o f ( i ) and ( i i i ) can be proved similarly. I Corollary 3.3.3 The correspondence b t+ bl (respectively b I-+ b,) defines a bijection between all C*-balanced bilinear forms and all morphisms of left (respectively right) C*-modules from C to C* . I
A bilinear form b : C x C 4 k is called left (respectively right) nondegenerate i f b(c,y ) = 0 for any c E C (respectively b(x,c ) = 0 for any c E C ) implies that y = 0 (respectively x = 0 ) . A left C*-module M is torsionless i f M embeds in a direct product o f copies o f C * . T h e following characterizes QcF coalgebras.
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135
Theorem 3.3.4 Let C be a coalgebra. Then the following assertions are equivalent. (i) C is left QcF. (ii) C is a torsionless left C*-module. bi : C x C (iii) There exists a family of C*-balanced bilinear forms (b;)iE~, k, such that for any non-zero x E C there exists i E I such that bi(C, x) # 0. (iv) Every injective right C-comodule is projective. (v) C is a projective right C-comodule. (vi) C is a projective left C*-module. -+
Proof: (i) =+ (ii) is obvious. (ii) + (iii) Let 0 : C + (C*)' be an injective morphism of left C*-modules, where by (C*)' we mean a direct product of copies of C*. For any i E I, let p, : (C*)' -+ C * be the natural projection on the i-th component of this direct product. Define 6, : C x C --+ k by
for any x, y E C . By Proposition 3.3.2, b, is a C*-balanced bilinear form. If y E C, y # 0, then O(y) # 0, so there exists i E I such that p,(O(y)) # 0, i.e. 0 # p,(O(y))(c) = b,(c, y) for some c E C. (iii) + (ii) Define 0 : C -t (C*)' by O(c) = (O,(C)),~I,where O,(c) : C + k is given by O,(c)(d) = b,(d, c). Since b, is C*-balanced, 8, is a morphism of left C*-modules. If O(c) = 0, then O,(c) = 0 for any i E I, so b,(d, c) = 0 for any i E I and d E C . This shows that c = 0, so 0 is a monomorphism and then C is a torsionless left C*-module. (12) and (iii) + (vi) We know that C = ezEIE(Sz)as right C-comodules, for some simple right C-comodules S, (E(S) denotes the injective envelope of S ) . Thus it is enough to show that for any simple right C-comodule S, the injective envelope E ( S ) of S in the category M~ is a projective left C*-module. We can assume that S is a minimal right coideal of C , say S = C* x for some non-zero x E C . Then there exist i E I and c E C such that b,(c, x) # 0. Denote by
-
U = {y E Clbi(z, y) = 0 for all z E c
- C*)
Since b, is C*-balanced, U is a left C*-submodule of C, i.e. a right coideal of C. If S n U # 0, since S is minimal we see that S C U. In particular x E U, so b,(c, x) = 0, a contradiction. Thus S n U = 0. Since S is essential in E ( S ) (we consider an injective envelope E ( S ) of S such that E ( S ) C ) , we must have E ( S ) n U = 0. Let (b,)l : C C* be the morphism of left C*-modules defined by b,, i.e. (b,)l(y)(z) = b,(z, y) for any z , y E C. Define the map ai : C -+ (c C*)* such that for any y E C, a,(y) is the -+
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136
-
-
restriction of (bi)l to c C*, i.e. ai(y)(z) = bi(z, y) for any z E c Clearly (c C*)* is a left C*-module. If c* E C* we have ai(c*
Y)(z)
- C*.
9) = bi(z-c*,Y) = ai(y) (z c*) =
bi (2, c*
-
=
(c*ai(~))(z)
so cui is a morphism of left C*-modules. On the other hand we clearly have K e r ( a i ) = U. Since dim(U) is finite, then dim(C/U) is finite. Since E ( S ) n U = 0, there exists a monomorphism E ( S ) C/U, in particular E ( S ) has finite dimension. We have an injective morphism of left C*-modules
-
Since E ( S ) is finite dimensional, it is an artinian left C*-module, so E ( S ) embeds in a finitely generated free left C*-module. But E ( S ) is injective as a left C*-module (see Corollary 2.4.19), so E ( S ) is isomorphic to a direct summand of a free left C*-module, and then it is a projective left C*module. (vi) + (v) is obvious. (v) 3 (iv) If Q is injective in M G , there exists a non-empty set I such that Q is isomorphic to a direct summand of c(')as a right C-comodule. The hypothesis tells that c(')is projective in M C , and then so is Q. (iv) =+ (i) Let C = eiEIE(Si) as right C-comodules, where Si are simple C-comodules. By hypothesis any E(Si) is projective in the category M C , and it is also indecomposable. Let us fix some i E I. There exist a family (Mj)jGJ of finite dimensional right C-comodules and an epimorphism
Since E(Si) is projective, then it is isomorphic to a direct summand of The Krull-Schmidt Theorem shows now that E(Si) is isomorphic to an indecomposable direct summand of some M j . Thus E(Si) is finite dimensional. Corollary 2.4.22 shows that E(Si) is a projective left C*module, thus isomorphic to a direct summand of a free left C*-module. Then C is isomorphic to a direct summand of a free left C*-module, i.e. C 1 is a left QcF coalgebra. The proof of Theorem 3.3.4 gives also the following characterization of co-F'robenius coalgebras. @ j E jMj.
Theorem 3.3.5 Let C be a coalgebra. Then the following assertions are equivalent.
.
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137
(2) C is a left co-Frobenius coalgebra. (ii) There exists a C*-balanced bilinear form which is left nondegenerate. Corollary 3.3.6 Let C be a left QcF coalgebra. Then (i) C is a left semiperfect coalgebra. (ii) R a t ( c * C * )# 0 Proof: ( i ) It follows from the proof of (ii) and (iii) + ( v i ) in Theorem 3.3.4 that the injective envelope of a simple right comodule is finite dimensional. (ii) Since C is a left QcF coalgebra, there exists a monomorphism of left C*-modules 0 --+ --t (c*)(')
c
for some set I. Then ~ a t ( ( C * ) ( ' )contains ) C . On the other hand so Rat(c* C * ) # 0.
I
Example 3.3.7 Let C be the coalgebra from Example 3.2.8. W e keep the same notation and define the linear map B : C -+ C * by B(gi) = df and B(di) = g>, for any i. It is easy to see that 9 is an injective morphism of left C*-modules, so C is a left co-Frobenius coalgebra. O n the other hand C is not right co-Frobenius. Indeed, if b : C x C + k is a C * -balanced bilinear form, then
b b l , g,)
-
b(g,, df di) = b(gl d f , di) = b(0,d i ) = 0 =
-
and
so b ( g l , c ) = 0 for anylc E C , i.e. b is not right nondegenerate. W e also show that C is left QcF (obviously, since C is left co-Frobenius), but C is not right QcF. Indeed, i f C were right QcF, then C would be a projective right C*-module. Then g; C is projective, too. If we denote C = k g l , then the left C*-module M * is indecomposable and M = g; injective. Thus &I* 2 C g: for some j . But d i m ( C g:) = 2 for any j , while d i m ( M * ) = 1, a contradiction. I
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Lemma 3.3.8 Let C be a coalgebra. Then the algebra C* is right selfinjective if and only if C is pat as a left C*-module. Proof: We know that C& is injective H the functor Hornc- (-, C;.) is C is exact. Now exact, and c-C* is flat if and only if the functor - @c* the result follows from the fact that for any right C*-module M we have that Homc*(M, C&) Homk(M @c* C, k)
-
This last isomorphism follows from the fact that the functor - @c* C : Mc. + kM is a left adjoint of the functor Homk(C, -) : kM -r M c - . I
Corollary 3.3.9 If C is a left QcF coalgebra, then C* is a right selfinjective algebra, i.e. C* is a n injective right C*-module. Proof: By Theorem 3.3.4, C is a projective left C*-module, in particular a flat C*-module. Now apply Lemma 3.3.8. I Corollary 3.3.10 If C is a left QcF coalgebra, then C is a generator of the category CM. Proof: I t is enough to show that any finite dimensional M E CM is generated by C . Since M is finite dimensional, there exists an epimorphism of right C*-modules (C*)" M 0
--
for some positive integer n. Since C is left semiperfect, the functor Rat Mc* + Mc. is exact. We obtain an exact sequence
:
By Corollary 3.3.9 the object Rat(C&) is injective in the category Rat(Mc.), i.e. Rat(C5.) is injective in the category 'M. Since any left comodule embeds in a direct sum of copies of C , Rat(CE,) is a direct summand (as a left C-comodule) of a direct sum c(')of copies of C. In particular there exists an epimorphism
in the category
C
~
This . shows that C generates M .
I
Corollary 3.3.11 The following conditions are equivalent for a coalgebra C. (i) C is left and right QcF. (ii) C is a projective generator i n the category C M . (iii) C is a projective generator i n the category M ~ . (iv) C* is a left and right seljinjective algebra and C is a generator for the categories C M and M ~ .
3.3. (QUASI-)CO-FROBENIUS COALGEBRAS Proof: ( i )+ (ii)follows from Theorem 3.3.4 and corollary 3.3.10. (ii)+ ( i )By Theorem 3.3.4 we have that C is a left QcF coalgebra. Moreover, C is right QcF if and only if Cc. is torsionless. But Cc* is an essential extension of its socle s ( C c * ) = s ( c C ) = Co, the coradical of C . Since C&. is injective by Corollary 3.3.9, we have that Cc- is torsionless if and only if so is every simple right C*-subcomodule of C. Let M be a rational simple right C*-module, i.e. a simple left C-comodule. Then M N N * for a simple right C-comodule N . Since C is generator in M', there exists an epimorphism 4 N -+ 0
c(')
for a finite set I . Then M 21 N* embeds in (c*)', so M is torsionless. ( i )H (iii)is proved similarly. ( i )+ ( i v ) follows from Corollaries 3.3.9 and 3.3.10. ( i v ) + ( i )is proved in the same way as (ii)+ (i).
I
Remark 3.3.12 A n algebra A is called quasi-Frobenius if A is left artinian and A is an injective left A-module. If A is so, then A is also right artinian and injective as a right A-module (see for instance 165, section 58]), thus the concept of quasi-Frobenius is left-right symmetric. A finite dimensional algebra A is called Frobenius if A and A* are isomorphic as left (or equivalently right) A-modules. It is known that a Frobenius algebra is necessarily quasi-Frobenius, while there exist finite dimensional algebras which are quasi-Frobenius but not FTObenius (such an example was given by T. Nakayama i n [I571 and [158]). Let C be a finite dimensional coalgebra. It follows immediatelly from the definition that C is left co-Frobenius if and only if the dual algebra C * is Frobenius, and this is also equivalent to the fact that C is right co-Probenius. Also, by Corollary 3.3.9, Theorem 3.3.4 and Corollary 2.4.20 we see that C is left QcF if and only if the dual algebra C * is quasi-Frobenius, and this is also equivalent to C being right QcF. Now we can see that there exist coalgebras which are left QcF but not left co-Frobenius. Indeed, i f we take A to be a finite dimensional algebra which is quasi-Frobenius but not Frobenius, then the dual coalgebra A* is QcF but not co-Frobenius. I Exercise 3.3.13 Let C be a right semiperfect coalgebra. Show that if the category M C (or C ~ has)finitely many isomorphism types of simple 06, jects, then C is finite dimensional. Exercise 3.3.14 Let C be a coalgebra. Show that the category M' is equivalent to the category of modules AM for some ring with identity A if and only if C has finite dimension.
CHAPTER 3. SPECIAL CLASSES O F COALGEBRAS
140
Exercise 3.3.15 Let (Ci)iEI be a family of coalgebras and C = @iEICi. Show that C is right semiperfect i f and only if Ci is right semiperfect for anyi E I. Exercise 3.3.16 Let C be a cocommutative coalgebra. Show that C is right (left) semiperfect if and only if C = $iE1Ci for some family (Ci)iE1 of finite dimensional coalgebras. Exercise 3.3.17 (a) Let (Ci)iEr be a family of left co-Frobenius (respectively left QcF) coalgebras and C = $iEICi. Show that C is left coFrobenius (respectively left QcF). (b) Show that a cosemisimple coalgebra is left (and right) co-Frobenius.
3.4 Solutions to exercises Exercise 3.1.1 Let C be a coalgebra. Show that C is a simple coalgebra if and only if the dual algebra C* is a simple artinian algebra. If this is the case, then C is a sum of isomorphic simple right C-subcomodules. Moreover, if the field k is algebraically closed, then C is isomorphic to a matrix coalgebra over k . Solution: Assume that C is a simple coalgebra. If I is a two-sided ideal of C*, then I L is a subcoalgebra of C, so either I-' = 0 or I L = C . In the first case we obtain I = C* and in the second case we have I = 0. Thus C* is a simple algebra. On the other hand C is finite dimensional since it is simple, and then C* is finite dimensional, in particular artinian. Conversely, assume that C* is simple artinian and let J be a subcoalgebra of C. Then J-' is a two-sided ideal of C*, so either J-' = 0 or J' = C*. This shows that J is either 0 or the whole of C. If C is simple, then C* is simple artinian, thus any left C*-module is semisimple. In particular, C is a semisimple left C*-module, so C is a sum of simple left C*-submodules, i.e. a sum of simple right C-subcomodules. For the last assertion, we have that C* is isomorphic to a matrix algebra over a finite extension of k. Since k is algebraically closed, this implies that the extension must be k, and then C* = Mn(k), where n = d i m ( M ) . This shows that C M c ( n ,k).
-
Exercise 3.1.2 Let M be a simple right comodule over the coalgebra C . Show that: (i) The coalgebra A associated to M is a simple coalgebra. (ii) If the field k is algebraically closed, then A II M c ( n , k ) , where n = d i m ( M ) . I n particular the family ( ~ i ~ ) l < i , ~defined
3.4. SOLUTIONS TO EXERCISES
141
Solution: (i) We know from Proposition 2.5.3 that A is the smallest subcoalgebra of C such that M is a right A-comodule. We regard M as an object in the category M*. Since A is the coalgebra associated to M , we have that A = (annA-( M ) ) I , so then a n n ~ (.M ) = 0. Thus M is a simple faithful left A*-module, so A* is a simple artinian algebra. By Exercise 3. i.1 we see that A is a simple coalgebra. (ii) follows from ( i )and Exercise 3.1.1. Exercise 3.1.3 Let M and N be two simple right comodules over a coalgebra C . Show that M and N are isomorphic if and only 2f they have the same assoczated coalgebra. Solution: If M and N are isomorphic as right C-comodules, they are also isomorphic as left C*-modules, so then a n n c . (M) = a n n c . ( N ) , and the associated coalgebras (annc* (M))'- and (annc. (N))'- are clearly equal. Conversely, if M and N have the same associated coalgebra A, then A is a simple subcoalgebra and any two simple A-comodules are isomorphic. In particular M and N are isomorphic. Exercise 3.1.6 Let C be a cosemisimple coalgebra. Show that C is a direct sum of simple subcoalgebras. Moreover, show that any subcoalgebra of C is cosemisimple. Solution: Let (Ci)iE1 be the family of all simple subcoalgebras of C . Then the sum CiG1 Ci is direct. Indeed, if for some j E I we have C j C CiGI-fjl Ci, then by Exercise 2.2.19 we have that Cj C Ct for some t E I - { j ) . Since Ct is simple we see that Ct = Cj, a contradiction. If D is a subcoalgebra of C = $iEICi, then by Exercise 2.2.18(iv) we have that D = eiEIAi, where Ai is a subcoalgebra of Ci for any i. Then either Ai = 0 or Ai = Ci, and we are done. Exercise 3.1.7 Let C be a finite dimensional coalgebra. Show that C is cosemisimple if and only if the dual algebra C* is semisimple. Solution: We know that C is cosemisimple if and only if the category of right C-comodules M C is semisimple. Also, C * is semisimple if and only if the category of left C*-modules c.M is semisimple. The result follows now from the fact that the categories M C and c * M are isomorphic. Exercise 3.1.11 Show that the coradical filtration is a coalgebra filtration, i.e. that A(C,,) Ci @ C,-i for any n 2 0. Solution: Exercise 2.5.8 shows that C, = Ci~C,-i-l for any 0 5 i 5 n-1, Ci @ C C @ Cn-i-l. Then SO A(C,)
c
+
142
CHAPTER 3. SPECIAL CLASSES O F COALGEBRAS
We prove that
< <
The right hand side is contained in the left hand side since for any 0 i n we have Ci €3 Cn-i C_ C j €3 C for i j, and Ci 8 Cn-i C C €3 Cn-'-j for jsz-1. For proving the other inclusion, let us take CiYi+'be a complement of Ci in Ci+l, and denote Ci,, = @i5jCj,j+l. Clearly
<
+
If z E (ni=o,n-l(Ci @ C C @ C,-i-l)) n (Cn €3 Cn) = S, write z = z o + z l + ...+ z , , w i t h z ~ ~ C o @ C , , €Co7l@Cn,...,zn€Cn-1,n@Cn. z~ We prove by induction on 0 i I n that zi E Ci @ Cn-i. This is clear for i = 0. Assume it is true for i < j (where 1 5 j). We show that zj E Cj @ Cn- Since z E S we have that
<
The induction hypothesis shows that zl + . . . + zj-1 E Cj-' @ C . We have that zj E Cj-1, j 8 Cn and obviously z j + l + . . . + zn E Cj,, @ Cn. Thus we must have
which ends the proof.
Exercise 3.1.12 Let C be a coalgebra and D a subcoalgebra of C such that U,>~(A~D =)C . Show that Corad(C) G D . Solution: Denote by I = D L , which is an ideal in C*. We know from Lemma 2.5.7 that An+lD = (In+')'-. Let f E I. We define g : C + k by g(c) = E(C)+ CnZl fn(c). The definition is correct since for any c E C , c E A ~ + ' Dfor some n, and then c E (In+')'-, showing that fm(c) = 0 for any m 2 n + 1. Moreover, on An+'D we have
Since U , , ~ ( A ~ D )= C , we have that E - f is invertible in C*. Thus D L J(C*), and then Corad(C) = J(C*)'- G D.
c
Exercise 3.1.13 Let f : C -+ D be a surjective morphism of coalgebras. Show that Corad(D) C_ f (Corad(C)).
3.4. SOLUTIONS T O EXERCISES
143
Solution: We know from Exercise 2.5.9 that f (AnCo) C n. Then we have
f (Co) for any
and since f (Co)is a subcoalgebra we get by Exercise 3.1.12 that Corad(D) C f (Corad(C)).
Exercise 3.1.14 Let X , Y and D be subspaces of the linear space C . Show that
+
Solution: Let D' and X' be complements of D f l X in D X , and U a complement of D X in C . Also let D" and Y' complements of D and Y in D Y , and V a complement of D Y in C . We have that ,
+
+
+
and
X @ C + C @ Y= = ((DnX)@(DnY))@((DnX)@D")@ @((Dnx)@y1)@((Dnx)@V)@(x'@(DnY))@ @(XI @ D") CB(x' Y ' )@ (x'@ V )@ (D' @ ( D n Y ) )@ @(U€4 ( D n Y ) )@ (Dl @ Y ' ) @ (XI@ Y ' ) $ (U @ Y ' ) These show that
( D @ D ) n ( X @ C + C @ Y=) =(Dc~X)@(D~Y))@((D~X)@D")@(D'@(D~Y)) and this is clearly equal to ( D n X ) @ D
+ D @ ( Dn Y ) .
Exercise 3.1.15 Let C be a coalgebra and D a subcoalgebra of C . Shovr that D, = D U C,, for any n. Solution: We prove by induction on n. For n = 0, the inclusion Do C D n Co is clear, since any simple subcoalgebra of D is also a simple subcoalgebra of C . On the other hand D n Co is a subcoalgebra of Co, so by Exercise 3.1.6, D fl Co is a cosemisimple coalgebra. Thus D i l Co is a sum of simple subcoalgebras, which are obviously subcoalgebras of D. This shows that D n C o c Do. Assume now that D, = DnC, for some n. We prove that D,+I = Since A(D,+,) c D n @ D + D @ D oC C n @ C + C @ C o
CHAPTER 3. SPECIAL CLASSES OF COALGEBRAS
144
we see that D,+1 D n C , + l , then
A(d)
C
Cn+1, so D,+l
C D
n Cn+l. conversely,
if d E
(D@D)n(C,@C+C@Co) ( D n Cn) @J D D ( D n Co) (by Exercise 3.1.14) = D, @ D D @J Do (by the induction hypothesis) E
=
+
+
thus d E D,+l, which ends the proof.
Exercise 3.3.13 Let C be a right semiperfect coalgebra. Show that if the categoy M~ (or C ~ has) finitely many isomorphism types of simple objects, then C is finite dimensional. Solution: If S is a simple object in the category M ~ then , Cornc ( S ,C ) N H o w (S, k)
S*
Since S is finite dimensional, the above isomorphism shows that in the representation of Co as a direct sum of simple right C-comodules there exist only finitely many objects isomorphic to S. By the hypothesis we obtain that Co is finite dimensional. Since C is right semiperfect we have that the injective envelope E ( S ) of S is finite dimensional. We conclude that C is finite dimensional since C is the injective envelope of Co in CM.
Exercise 3.3.14 Let C be a coalgebra. Show that the category MC is equivalent to the category of modules AMfor some ring with identity A if and only if C has finite dimension. Solution: If C has finite dimension, then the category MC is even isomorphic to the category c*M of modules over the dual algebra of C. Conversely, assume that MC is equivalent to AM for a ring with identity A. Let F : AM -+ M~ be an equivalence functor. Then P = F(*A) is a finitely generated projective generator of the category MC since so is A A in the category AM. Since any object of M' is the sum of subobjects of finite dimension, we obtain that P has finite dimension. Since P is also a generator we have that C is a right semiperfect coalgebra. Since P has finite dimension, M~ has a finite number of isomorphism types of simple objects. Since C is semiperfect, this implies that C has finite dimension. Exercise 3.3.15 Let (Ci)iEI be a family of coalgebras and C = eiErCi. Show that C is right semiperfect i f and only if Ci is right semiperfect for any i E I . Solution: The claim follows immediately from the equivalence of categories MC MCi,proved in Exercise 2.2.18.
- niE,
Exercise 3.3.16 Let C be a cocommutative coalgebra. Show that C is right (left) semiperfect zf and only i f C = @i,~Cifor some family ( C i ) i E ~of finite
3.4. SOLUTIONS TO EXERCISES
145
dimensional coalgebrcis. Solution: Assume that C is right semiperfect. Then C = $iE=E(Si) for some simple left C-subcomodules ( S i ) i E of ~ C (as usual E(Si) denotes the injective envelope of Si inside C). Since C is right semiperfect, any E(Si) is finite dimensional, and since C is cocommutative, any Ci = E(Si) is with all Ci's finite a subcoalgebra of C . Thus we can write C = $i&i dimensional. Conversely, if C = eiEICiwith any Ci finite dimensional, then any Ci is right semiperfect, and Exercise 3.3.15 shows that C is right semiperfect. Exercise 3.3.17 (a) Let (Ci)iEr be a family of left co-Frobenius (respectively left QcF) coalgebras and C = $ i e ~ C i . Show that C is left coFrobenius (respectively left QcF). (b) Show that a cosemisimple coalgebra is left (and right) co-Frobenius. Solution: (a) Assume that each Ci is left co-Frobenius. Then there exists a C:-balanced bilinear form bi : Ci x Ci -+ k which is left nondegenerate. We construct a bilinear form b : C x C -+ k by b(c,d ) = bi(c, d ) for any i E I and c, d E Ci, and b(c,d ) = 0 for any c E Ci, d E Cj with i # j . Then clearly b is C*-balanced and left nondegenerate, so C is left co-Frobenius. One proceeds similarly for the QcF property. (b) A cosemisimple coalgebra is a direct sum of simple subcoalgebras. On the other hand a simple coalgebra is left and right co-Frobenius since it is finite dimensional and its dual is a simple artinian algebra, thus a Frobenius algebra. By (a) we conclude that a cosemisimple coalgebra is left and right co-Frobenius. Bibliographical notes Besides the books of M. Sweedler [218], E. Abe [I], and S. Montgomery [149], we have used for the presentation of this chapter the papers by Y. Doi [72], C. Ngst5sescu and J. Gomez Torrecillas 11621, B. Lin [123], H. Allen and D. Trushin [2]. We have also used [28], R. Wisbauer's notes [244], and the paper by C. Menini, B. Torrecillas and R. Wisbauer [145], where coalgebras are over rings.
Chapter 4
Bialgebras and Hopf algebras 4.1
Bialgebras
Let H be a k-vector space which is simultaneously endowed with an algebra structure (H, M , u ) and a coalgebra structure (H,A, E). The following result describes the situation in which the two structures are compatible. We recall that on H @ H we have the structure of a tensor product of coalgebras, and the tensor product of algebras structure, while on k there exists a canonical structure of a coalgebra given in Example 1.1.4 4). Proposition 4.1.1 The following assertions are equivalent:
2) The maps M and u are morphisms of coalgebras. ii) The maps A and E are morphisms of algebras. Proof: M is a morphism of coalgebras if and only if the following diagrams are commutative M
H@H
*H
CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
H@H
M
pH
The map u is a morphism of coalgebras if and only if the following two diagrams are commutative
We note that A is a morphism of algebras if and only if the first and the third diagrams are commutative, and E is a morphism of algebras if and only if the second and the fourth diagrams are commutative. Therefore, I the equivalence of i) and ii) is clear.
Remark 4.1.2 I n the sigma notation, the conditions in which A and E are morphisms of algebras becomes
4.1. B I A L G E B R A S
149
Definition 4.1.3 A bialgebra is a k-vector space H , endowed with an algebra structure ( H ,M , u ) , and with a coalgebra structure ( H IA, E ) such that M and u are morphisms of coalgebras (and then by Proposition 4.1.1 zt follows that A and E are morphisms of algebras). I Remark 4.1.4 W e will say that a bialgebra has a property P , zf the underlying algebra or coalgebra has property P . Thus we will talk about commutative (or cocommutative) bialgebras, semisimple (or cosemisimple) bialgebras, etc. Note for example that i f the bialgebra H is commutative, then the multiplication is also a morphism of algebras, and if it is cocommutative, then the comultiplication of H is a morphism of coalgebras too. I Example 4.1.5 1) The field k , with its algebra structure, and with the canonical coalgebra structure, is a bialgebra. 2) If G is a monoid, then the semigroup algebra kG endowed with a coalgebra structure as i n Example 1.1.4.1) (in which A ( g ) = g 8 g and ~ ( g=) 1 for any g E G) is a bialgebra. 3) If H is a bialgebra, then Hop, Hcop and Hoplcop are bialgebras, where Hop has an algebra structure opposite to the one of H , and the same coalgebra structure as H , HcOphas the same algebra structure as H , and the coalgebra has the algebra structure structure co-opposite to the one of H , and HOP>COP opposite to the one of H , and the coalgebra structure co-opposite to the one of H . I
A possible way of constructing new bialgebras is to consider the dual of a finite dimensional bialgebra. Proposition 4.1.6 Let H be a finite dimensional bialgebra. Then H * , together with the algebra structure which is dual to the coalgebra structure of H , and with the coalgebra structure which is dual to the algebra structure of H is a bialgebra, which is called the dual bialgebra of H . Proof: We denote by A and E the comultiplication and the counit of H I and by S and E the comultiplication and the counit of H * . We recall that for h* E H" we have E ( h * ) = h * ( l ) and S(h*) = C h ; 8 h;, where h*(hg) = C h ; ( h ) h $ ( g )for any h , g E H . Let us show that S is a morphism of algebras. Indeed, if h*,g* E H* and S(h*) = C h ; h;, S(g*) = C g; 8 g; , then for any h , g E H we ha.ve
CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
which shows that
6(h*g*)=
C h;g; @ h;g; = 6(h*)6(ga)
Also ~ ( h g = ) ~ ( h ) e ( gfor ) any h , g E H , hence S ( E ) = E @ E , and so 6 is an algebra map. We show now that E is a morphism of algebras. This is clear, since
E(h*g*) = ( h * g * ) ( l )= h * ( l ) g * ( l )= E ( h * ) E ( g * ) and
E ( E )= ~ ( 1=) 1, which ends the proof. Numerous other examples of bialgebras will be given later in this chapter.
Definition 4.1.7 Let H and L be two k-bialgebras. A k-linear map f : H + L is called a morphism of bialgebras i f i t is a morphism of algebras and a morphism of coalgebras between the underlying algebras, respectively I coalgebras of the two bialgebras. We can now define a new category having as objects all k-bialgebras, and as morphisms the morphisms of bialgebras defined above. It is important to know how to obtain factor objects in this category.
Proposition 4.1.8 Let H be a bialgebra, and I a k-subspace of H which is a n ideal (in the underlying algebra of H ) and a coideal (in the underlying coalgebra of H ) . Then the structures of factor algebra and of factor coalgebra o n H / I define a bialgebra, and the canonical projection p : H + H / I is a bialgebra map. Moreover, if the bialgebra H is commutative (respectively cocommutative), then the factor bialgebra H / I is also commutative (respectively cocommutative). Proof: Denoting by 5;; the coset of an-element h-E H modulo I, the coalgebra structure on H / I is defined by A ( h ) = C hl @hz and 8(z) = ~ ( h ) for any h E H . Then it is clear that H and F are morphisms of algebras. I The rest is clear. Exercise 4.1.9 Let k be a field and n 2 2 a positive integer. Show that there is n o bialgebra structure on M,(k) such that the underlying algebra structure is the matrix algebra.
4.2. HOPFALGEBRAS
4.2
151
Hopf algebras
Let (C, A, E) be a coalgebra, and (A, M, u) an algebra. We define on the set Hom(C, A) an algebra structure in which the multiplication, denoted by * is given as follows: if f , g E Hom(C, A), then
for any c E C. The multiplication defined above is associative, since for f , g, h E Hom(C, A) and c E C we have ( ( f * 9) * h)(c) = C ( f *-g)(cl)h(cz) =
C f (cI)s(c~)~(c?)
=
Cf(c1)(g*h)(cz)
=
(f * (g * h))(c)
The identity element of the algebra Hom(C, A) is ue E Hom(C, A), since
hence f * (ue) = f . Similarly, (ue) * f = f . Let us note that if A = k , then * is the convolution product defined on the dual algebra of the coalgebra C . This is why in the case A is an arbitrary algebra we will also call * the convolution product. Let us consider a special case of the above construction. Let H be a bialgebra. We denote by HCthe underlying coalgebra H, and by H a the underlying algebra of H. Then we can define as above an algebra structure on Hom(HC,H a ) , in which the multiplication is defined by (f * g)(h) = C f(hl)g(ha) for any f , g E Hom(HC,H a ) and h E H, and the identity element is ue. We remark that the identity map I : H --, H is an element of Hom(HC,Ha). Definition 4.2.1 Let H be a bialgebra. A linear map S : H -+ H is called an antipode of the bialgebra H if S is the inverse of the identity map I : H --+ H with respect to the convolution product in Hom(Hc, H a ) . I Definition 4.2.2 A bialgebra H having an antipode is called a Hopf algeI bra.
Remarks 4.2.3 1. In a Hopf algebra, the antipode is unique, being the inverse of the element I in the algebra Hom(HC,H a ) . The fact that S :
152
CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
H --+ H is the antipode is written as S * I = I sigma notation
C ~ ( h 1 ) h 2=
* S = UE, and
using the
his(hz) = ~ ( h ) l
for any h E H Since a Hopf algebra is a bialgebra, we keep the convention from Remark 4.1.4, and we will say that a Hopf algebra has a property P, if the underlying algebra or coalgebra has property P. In order to define the category of Hopf algebras over the field Ic, we need the concept of morphism of Hopf algebras.
Definition 4.2.4 Let H and B be two Hopf algebras. A map f : H -+ B is called a morphism of Hopf algebras if it is a morphism of bialgebras. I It is natural to ask whether a morphism of Hopf algebras should preserve antipodes. The following result shows that this is indeed the case.
Proposition 4.2.5 Let H and B be two Hopf algebras with antipodes SH and SB. If f : H --+ B is a bialgebra map, then SBf = f SH. Proof: Consider the algebra Hom(H, B) with the convolution product, and the elements SBf and f SHfrom this algebra. We show that they are both invertible, and that they have the same inverse f , and so it will follow that they are equal. Indeed,
so
SBf is a left inverse for f . Also
hence f SH is also a right inverse for f . It follows that f is (convolution) 1 invertible, and that the left and right inverses are equal.
4.2. HOPF ALGEBRAS
153
We can now define the category of k-Hopf algebras, in which the objects are Hopf algebras over the field k, and the morphisms are the ones defined in Definition 4.2.4. This category will be denoted by k - Hopf Alg. We will provide examples of Hopf algebras in the following section. For the time being, we give some basic properties of the antipode.
Proposition 4.2.6 Let H be a Hopf algebra with antipode S. Then: i) S(hg) = S(g)S(h) for any g, h E H. ii) S(1) = 1, iii) A(S(h)) = C S(h2) 8 S(h1). iv) &(S(h))= ~ ( h ) . Properties i) and ii) mean that S is an antimorphism of algebras, and iii) and iv) that S is an antimorphism of coalgebras. Proof: i) Consider H 4 H with the tensor product of coalgebras structure, and H with the algebra structure. Then it makes sense to talk about the algebra H o m ( H 4 H, H ) , with the multiplication given by convolution, defined a s in the beginning of this section. The identity element of this algebra is UH&H@H : H 8 H -+ H . Consider the maps F, G, M : H 4 H -,H defined by
for any h , g E H . We show that M is a left inverse (with respect to convolution) for F, and a right inverse for G. Indeed, for h, g E H we have
(M * F ) ( h 8 g) = =
= =
C ~ ( ( 8hg ) l ) ~ ( ( 4h g)2) C w h l 4 g 1 ) ~ ( h82 g2) C hlgIs(g2)~(h2) C h l ~ ( g ) l S ( h 2 ) (definition of S for g)
&(h)&(g)l (definition of S for h) = E I I B H (4 ~ ~11 = U H E H @ H8( S) ~
=
which shows that M * F =
[email protected]
CHAPTER 4. BJALGEBRAS AND HOPF ALGEBRAS =
1s((hg)l)(hg)n
=
&(hg)l (definition of S for hg)
= UHEH~H 8 S) (~
and thus G * M = UHEH~H.Hence M is a left inverse for F and a right inverse for G in an algebra, and therefore F = G. This means that i) holds. ii) We apply the definition of the antipode for the element 1 E H . We get S ( 1 ) l = ~ ( 1 ) l Applying . E it follows that ~ ( s ( 1 ) = ) e(1) = 1. iii) Consider now H with the coalgebra structure, and H €3H with the structure of tensor product of algebras, and define the algebra Hom(H, H B H), with the multiplication given by the convolution product. The identity element of this algebra is UHBHEH. Consider the maps F, G : H -+ H @I H defined by F ( h ) = A(S(h)) and G(h) =
x
S(h2) 8 S(h1)
for any h E H . We show that A is a left inverse for F and a right inverse for G with respect to convolution. Indeed, for any h E H we have
and
= = =
=
x x
~ ( h 2 ) h@ 3 S(hi)h4
~ ( ( h 2 ) l ) ( h 2 )€3 2 S(hl)h3
&(h2)l@ S(hl)h3 (definition of S )
x
18 S(hl)h2 (the counit property)
=
18E(h)l
=
UH@HEH(~)
4.2. HOPF ALGEBRAS
155
which ends the proof. iv) We apply E to the relation 1 hlS(h2) = ~ ( h )to l get ' C & ( h l ) ~ ( S ( h 2 = )) ~ ( h ) .Since E and S are linear maps, we obtain & ( S ( E & ( h l ) h 2 )= ) ~(h). I Using the counit property we get €(S(h)) = ~ ( h ) . Proposition 4.2.7 Let H be a Hopf algebra with antipode S . Then the following assertions are equivalent: i) C S(h2)hl = ~ ( h )for l any h E H. ii) h2S(hl) = ~ ( h )for l any h E H. iiz) S2= I (by S2 we mean the composition of S with itselfl.
C
Proof: i)=+iii) We know that I is the inverse of S with respect to convolution. We show that S2 is a right convolution inyerse of S, and by the uniqueness of the inverse it will follow that S2= I. We have
=
S(S(h2)hl) ( S is an antimorphism of algebras)
= S(c(I1)l) =
E(h)l
which shows that indeed S * S2= UE. iii)+ii) We know that E hlS(h2) = ~ ( h ) l Applying . the antimorphism of algebras S we obtain C S2(h2)S(hl)= &(h)l. Since S2= I, this becomes C h2S(hl) = e(h)l. ii)+iii) We proceed as in i)+iii), and we show that S2is a left convolution inverse for S . Indeed,
iii)+i) We apply S to the equality C S ( h l ) h 2 = ~ ( h ) land , using S2 = I we obtain S(h2)hl = e(h)l. I
C
Corollary 4.2.8 Let H be a commutative or cocommutative Hopf algebra. Then S2= I . Proof: If H is commutative, then by C S(hl)h2 = e ( h ) l it follows that C h2S(hl) = ~ ( h ) li.e. , ii) from the preceding proposition. If H is cocommutative, then C h l @ h 2= C h 2 @ h l , and then by C S(hl)h2 = ~ ( h ) itl follows that C S ( h 2 ) h l = ~ ( h ) li.e. , i) I from the preceding proposition.
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CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
Remark 4.2.9 If H is a Hopf algebra, then the set G ( H ) of grouplike elements of H is a group with the multiplication induced by the one of H . Indeed, we first see that 1 E G ( H ) (this will be the identity element of G ( H ) ) . If g, h E G ( H ) , then gh E G ( H ) by the fact that A is an algebra map. Finally, if g is a grouplike element, then S ( g ) is also a grouplike element since the antipode is an antimorphism of coalgebras, and the property of the antipode shows that g is invertible with inverse g-l = S ( g ) . I Remark 4.2.10 Let H be a Hopf algebra with antipode S . Then the bialgebra HOp>cop is a Hopf algebra with the same antipode S . If moreover S is bzjective, then the bialgebras Hop and HCOpare Hopf algebras with antipode S-l. I In Proposition 4.1.6 we saw that if H is a finite dimensional bialgebra, then its dual is a bialgebra. The following result shows that if H is even a Hopf algebra, then its dual also has a Hopf algebra structure. Proposition 4.2.11 Let H be a finite dimensional Hopf algebra, with an-
tipode S . Then the bialgebra H* is a Hopf algebra, with antipode S * . Proof: We know already that H* is a bialgebra. It remains to prove that it has an antipode. Let h* E H* and 6 ( h * ) = C hr @J h;, where 6 is the comultiplication of H*. Then for h 6 H we have
where E is the counit of H*. We proved that C S * ( h ; ) h ' ; = E(h*)&.Sim, the proof is complete. ilarly, one can show that C h ; S * ( h l ) = E ( h * ) &and
I
Definition 4.2.12 Let H be a Hopf algebra. A subspace A of H is called a Hopf subalgebra i f A is a subalgebra, a subcoalgebra, and S ( A ) A. We note that if A is a Hopf subalgebra of H , then A is itself a Hopf algebra with the induced structures. This concept is just the concept of subobject i n the category k - Hop f Alg. I
4.2. HOPF ALGEBRAS
157
In order to define the concept of factor Hopf algebra of a Hopf algebra H , we need subspaces of H which are ideals (in order to define a natural algebra structure on the factor space), coideals (to be able to define a natural coalgebra structure on the factor space), and also to be stabilized by the antipode (so that the factor bialgebra which we obtain has an antipode). The following result is immediate. Proposition 4.2.13 Let H be a Hopf algebra, and I a Hopf ideal of H , i.e. I is an ideal of the algebra H , a coideal of the coalgebra H , and S ( I ) C I , where S is the antipode of H . Then on the factor space H I I we can introduce a natural structure of a Hopf algebra. When this structure is defined, the canonical projection p : H -+ H/I is a morphism of Hopf algebras. Proof: We saw in Proposition 4.1.8 that H / I has a bialgebra structure. Since S ( I ) C I , the morphism S : H -+ H induces a morphism 3 : H/I -+ H I I by 3 ( z ) = S(h),where denotes the coset of an element h E H in the factor space H / I . The 3 is an antipode in the factor bialgebra H / I , since
z
and similarly,
---
C hl S(hz) = ~ ( % ) i .
If A is an algebra, and M is a left A-module, then M has a right module structure over AOp,but in general M does not have a natural right module structure over A. Also, if C is a coalgebra, and M is a right C-comodule, then M has a natural left comodule structure over the co-opposite coalgebra CCOP,but not over C. The following result shows that if we work with a Hopf algebra, then all these are possibile. The role of the antipode is essential in this matter. Proposition 4.2.14 Let H be a Hopf algebra with antipode S . The following hold: i ) If M is a left H-module (with action denoted by hm for h E H , m E M), then M has a structure of a right H-module given by m . h = S ( h ) m for a n y mM ~ , ~ HE. ii) If M is a right H-comodule (with structure map p : M --t M @ H , p(m) = @ m ( l ) ) , then M has a left H-comodule structure with the morphism giving the structure p' : M -+ H @ M , pl(m) = S(m(l))@m(o). Proof: The checking is immediate, using the fact that S is an antimorI phism of algebras, and an antimorphism of coalgebras.
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158
Example 4.2.15 If H and L are two bialgebras, then it is easy to check that we have a bialgebra structure on H @ L if we consider the tensor product of algebras and the tensor product of coalgebras structures. Moreover, if H and L are Hopf algebras with antipodes S H and S L , then H 8 L is a Hopf algebra with the antipode S H @ S L . This bialgebra (Hopf algebra) is called I the tensor product of the two bialgebras (Hopf algebras).
Let H be a Hopf algebra and G ( H ) the group o f grouplike elements o f H . I f g, h E G ( H ) , then an element x E H is called ( g ,h)-primitive i f A ( x ) = x 8 g + h @ x. T h e set o f all ( g ,h)-primitive elements o f H is . ( 1 ,1)-primitive is simply called a primitive element. denoted by P g Y h ( H )A W e denote P ( H ) = P l , l ( H ) . Exercise 4.2.16 Let H be a finite dimensional Hopf algebra over a field k of characteristic zero. Show that if x E H is a primitive element, i.e. A ( x ) = x 8 1 1 8 x , then x = 0.
+
Exercise 4.2.17 Let H be a Hopf algebra over the field k and let K be a field extension of k . Show that one can define on H = K @k H a natural
structure of a Hopf algebra by taking the extension of scalars algebra structure and the coalgebra structure as i n Proposition 1.4.25. Moreover, 2f 3 is the antipode of ?T, then for any positive integer n we have that S n = Id if and only if ? = Id.
4.3
Examples of Hopf algebras
In this section we give some relevant examples o f Hopf algebras. 1) The group algebra. Let G be a (multiplicative) group, and kG the associated group algebra. This is a k-vector space with basis {gl g E G ), so its elements are o f the form CgEG a g g with ( c Y ~ a )family ~ ~ o~f elements from k having only a finite number of non-zero elements. T h e multiplication is defined by the relation
for any a ,,B E k,g,h E G , and extended by linearity. On the group algebra k G we also have a coalgebra structure as in Example 1.1.4 l ) ,in which A ( g ) = g 8 g and e(g) = 1 for any g € G . W e already know that the group algebra becomes in this way a bialgebra. W e note that until now we only used the fact that G is a monoid. T h e existence o f the antipode is directly related t o the fact that the elements o f G are invertible.
4.3. E X A M P L E S OF HOPF A L G E B R A S
159
Indeed, the map S : kG -+ k G , defined by S ( g ) = g-l for any g E G , and then extended linearly, is an antipode of the bialgebra k G , since
and similarly, C g l S ( g 2 ) = ~ ( g )for l any g E G. It is clear that if G is a monoid which is not a group, then the bialgebra k G is not a Hopf algebra. If G is a finite group, then Proposition 4.2.11 shows that on ( k G ) * we also have a Hopf algebra structure, which is dual to the one on kG. We recall that the algebra ( k G ) * has a complete system of orthogonal idempotents where pg E ( k G ) * is defined by p g ( h ) = 6g,h for any g, h E G . Therefore,
The coalgebra structure of ( k G ) * can be described using Remark 1.3.10, and is given by
The antipode of (kG)* is defined by S ( p g )= pg-1 for any g E G.
2) The tensor algebra. We recall first the categorical definition of the tensor algebra, since it will help us construct some maps enriching its structure.
Definition 4.3.1 Let M be a k-vector space. A tensor algebra of M is a pair ( X , i ) , where X is a k-algebra, and i : M -, X is a morphism of k-vector spaces such that the following universal property is satisfied: for any k-algebra A, and any k-linear map f : M -+ A , there exists a unique morphism of algebras 7 : X -+ A such that Ti = f , i.e. the following diagram is commutative.
160
CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
The tensor algebra of a vector space exists and is unique up to isomorphism. We briefly present its construction. Denote by T O ( M ) = k , T1(M) = M , and for n 2 2 by Tn(M) = M @ M @ .. . @ M , the tensor product of n copies of the vector space M . Now denote by T ( M ) = CB~>~T"(M), and i : M + T ( M ) , defined by i(m) = m E T1(M) for any m 2 M. On T ( M ) we define a multiplication as follows: if x = m l 8 . . . a m n E T n ( M ) , and y = h 1 8 . . . 8 h, E TT(M),then define the product of the elements x and y by
The multiplication of two arbitrary elements from T ( M ) is obtained by extending the above formula by linearity. In this way, T ( M ) becomes an , the pair (T(M), i) is a tensor algebra, with identity element 1 E T O ( M ) and algebra of M .
Remark 4.3.2 The existence of the tensor algebra shows that the forgetful functor U : k-Alg + k M has a left adjoint, namely the functor associating to a k-vector space its tensor algebra. I We define now a coalgebra structure on T ( M ) . To avoid any possible confusion we introduce the following notation: if a and P are tensor monomials from T ( M ) (i.e. each of them lies in a component Tn(M)), then the tensor monomial T ( M ) 8 T(M) having a on the first tensor position, and ,B on the second tensor position will be denoted by aBj3. Without this notation, for example for a = m 8 m E T ~ ( M )and ,Ll = m E T1(M), the elements a 8 0 and ,O 8 a from T ( M ) 8 T ( M ) would be both written as m m 8 m, causing confusion. In our notation, a 8 p = m 8 m g m , and ,B@a=rngm8m. Consider the linear map f : M -+ T ( M ) €9 T ( M ) defined by f (m) = m g l + l g m for any m E M . Applying the universal property of the tensor algebra, it follows that there exists a morphism of algebras A : T ( M ) + T ( M ) 8 T ( M ) for which Ai = f . Let us show that A is coassociative, i.e. ( A 8 I ) A = ( I 8 A)A. Since both sides of the equality we want to prove are morphisms of algebras, it is enough to check the equality on a system of generators (as an algebra) of T ( M ) , thus on i(M). Indeed, if m E M , then
and
4.3. EXAMPLES OF HOPF ALGEBRAS which shows that A is coassociative. We now define the counit, using the universal property of the tensor algebra for the null morphism 0 : M k. We obtain a morphism of algebras E : T ( M ) 4 k with the property that ~ ( m = ) 0 for any m E i ( M ) . To show that ( E @ I ) A = 4, where 4 : T ( M ) -+ k @ T ( M ) , 4(z) = 1 @ z is the canonical isomorphism, it is enough to check the equality on i ( M ) , and here it is clear that -+
Similarly, one can show that (I@ E)A = $', where 4' : T ( M ) 4 T ( M ) @ k is the canonical isomorphism. So far, we know that T ( M ) is a bialgebra. We construct an antipode. Coni sider the opposite algebra T(M)OP of T ( M ) , and let g : M -+ T(M)OP be the linear map defined by g(m)= -m for any m M. The universal property of the tensor algebra shows that there exists a morphism of algebras S : T ( M ) -+ T(M)OP such that S(m) = -m for any m E i(h/l). For an arbitrary element ml @ . . . @ mn E T n ( M ) we have S ( m l @ . . . @ m,) = (-l)nmn@. . .@ml. We regard now S : T ( M ) T ( M ) as an antimorphism of algebras. We show that for any m E T1(M) we have -+
Indeed, since A(m) = m B l
+l
m we have
and similarly the other equality. Therefore, the property the antipode should satisfy is checked for S on a system of (algebra) generators of T ( M ) . The fact that S verifies the property for any element in T ( M ) will follow from the next lemma.
Lemma 4.3.3 Let H be a bialgebra, and S : H -+ H an antimorphism of algebras. If for a , b E H we have (S * I)(a) = ( I * S)(a) = u&(a) and ( S * I)(b) = ( I * S)(b) = u&(b), then also ( S * I)(ab) = ( I * S)(ab) = u ~ ( a b ) . Proof: We know that
and
CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
162 Then
( S * I)(ab) = = =
~((ab)l)(ab)2
C s(aibl)az$ S ( b l ) S ( a l ) a 2 b 2 ( S is an antimorphism of algebras)
= E & ( a ) s ( b l ) b 2 ( from the property of a ) =
x ~ ( a ) ~ ( b ) (l from the property of b)
I Similarly, one can prove the second equality, and therefore we know now that T ( M ) is a Hopf algebra with antipode S. We show that T ( M ) is cocommutative, i.e. r A = A, where 7 : T ( M )8 T ( M ) -+ T ( M )8 T ( M ) is defined by ~ ( 8zv ) = v 8 z for any z, v E T ( M ) . Indeed, it is enough to check this on a system of algebra generators of T ( M ) , hence on i ( M ) (because r A and A are both morphisms of algebras), but on i ( M ) the equality is clear. 3) The symmetric algebra. We recall the definition of the symmetric algebra of a vector space.
Definition 4.3.4 Let M be a k-vector space. A symmetric algebra of M is a pair ( X ,i ) , where X is a commutative k-algebra, and i : M -, X is a k-linear map such that the following universal property holds: for any commutative k-algebra A, and any k-linear map f : M 4 A, there exists a unique morphism of algebras 7 : X --+ A such that Ti = f, i.e. the following diagram is commutative.
The symmetric algebra of a k-vector space M exists and is unique up to isomorphism. It is constructed as follows: consider the ideal I of the tensor algebra T ( M ) generated by all elements of the form x @J y - y @ x
4.3. EXAMPLES OF HOPF ALGEBRAS
163
with x, y E M . Then S ( M ) = T ( M ) / I , together with the map pi, where i : M -+ T ( M ) is the canonical inclusion, and p : T ( M ) -+ T ( M ) / I is the canonical projection, is a symmetric algebra of M. Remark 4.3.5 The existence of the symmetric algebra shows that the forgetful functor from the category of commutative k-algebras to the category I of k-vector spaces has a left adjoint. We show that the symmetric algebra M has a Hopf algebra structure. By Proposition 4.2.13, this will follow if we show that I is a Hopf ideal of the Hopf algebra T ( M ) . Since A and E are morphisms of algebras, and S is an antimorphism of algebras, it is enough to show that
~ ( x g y - y @ x ) = O a n d S ( x @ y - y @ x )I~ for any x, y E M . Indeed,
and this is clearly an element of I @ T(M)
+ T ( M ) @ I . Moreover,
and S ( x @ y - y @ x) = S(y)S(x) - S(x)S(y) = = (-9) @ (-x) - (-x) @ (-9) € I.
We obtained that S ( M ) has a Hopf algebra structure, it is a factor Hopf algebra of T ( M ) modulo the Hopf ideal I . It is clear that S ( M ) is a commutative Hopf algebra, and also cocommutative, since it is a factor of a cocommutative Hopf algebra. 4) The enveloping algebra of a Lie algebra. Let L be a Lie kalgebra, with bracket [ , 1. The enveloping algebra of the Lie algebra L is the factor algebra U(L) = T ( L ) / I , where T(L) is the tensor algebra of the k-vector space L, and I is the ideal of T(L) generated by the elements of the form [x,y] - x @ y y @ x with x, y E L. A computation similar to the one performed for the symmetric algebra shows that
+
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CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS = ([x,y] - x 8 y
which is in I @ T ( M )
+ y 8x)%1 + 1G([x,y] - x 8y + y 82)
+ T ( M ) @ I,
and S([x,y ] - x 63 y 3- y 8 x) = -([x,Y ] - @ Y
+ Y 8x) E I ,
so I is a Hopf ideal in T(L). It follows that U(L) has a Hopf algebra structure, the factor Hopf algebra of T(L) modulo the Hopf ideal I. Since T(L) is cocommutative, U ( L ) is also cocommutative. 5) Divided power Hopf algebras Let H be a k-vector space with basis {cili E N ) on which we consider the coalgebra structure defined in Example 1.1.4 2). Hence
for any m E N . We define on H an algebra structure as follows. We put
for any n , m E N , and then extend it by linearity on H. We note first that co is the identity element, so we will write co = 1. In order to show that the multiplication is associative it is enough to check that (cncm)cp= c, (cmcp) for any m, n , p E N. This is true because
4.3. EXAMPLES OF HOPF ALGEBRAS
165
We show now that H is a bialgebra with the above coalgebra and algebra structures. Since the counit is obviously an algebra map, it is enough to show that A(cncm) = A(cn)A(cm)for any n, m E N. We have
It remains to prove that the bialgebra H has an antipode. Since H is cocommutative, it suffices to show that there exists a linear map S : H -.. H such that C S ( h l ) h a = ~ ( h ) for l any h in a basis of H. We define S(cn) recurrently. For n = 0 we take S(co) = S(l) = 1. We assume that S(co), . . . ,S(cn-1) were defined such that the property of the antipode checks for h = ci with 0 5 i 5 n - 1. Then we define
and it is clear that the property of the antipode is then verified for h = cn too. In conclusion, H is a Hopf algebra, which is clearly commutative and cocommutative. 6. Sweedler's 4-dimensional,Hopf algebra.
Assume that char(k) # 2. Let H be the algebra given by generators and relations as follows: H is generated as a k-algebra by c and x satisfying the relations c2 = 1, x2 =o, x c = -CX Then H has dimension 4 as a k-vector space, with basis { 1,c, x, cx ). The coalgebra structure is induced by
CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
166
In this way, H becomes a bialgebra, which also has an antipode S given by S(c) = c-I, S(x) = -CX. This was the first example of a non-commutative and non-cocommutative Hopf algebra.
7. The Taft algebras. Let n 2 2 be an integer, and X a primitive n-th root of unity. Consider the algebra Hn2 (A) defined by the generators c and x with the relations
On this algebra we can introduce a coalgebra structure induced by
In this way, Hnz (A) becomes a bialgebra of dimension n2, having the basis { cixj I 0 5 i, j 5 n - 1 ). The antipode is defined by S(c) = c-' and S(x) = -c-'x. We note that for n = 2 and X = -1 we obtain Sweedler's 4-dimensional Hopf algebra. 8. On the polynomial algebra k[X] we introduce a coalgebra structure as follows: using the universal property of the polynomial algebra we find a unique morphism of algebras A : k(X] -+ k[X] C3 k[X] for which A(X) = X @I 1 1@ X . It is clear that
+
and then again using the universal property of the polynomial algebra it follows that A is coassociative. Similarly, there is a unique morphism of algebras E : k[X] -+ k with E(X) = 0. It is clear that together with A and E , the algebra k[X] becomes a bialgebra. This is even a Hopf algebra, with antipode S : k[X] -+ k[X] constructed again by the universal property of the polynomial algebra, such that S(X) = -X. This Hopf algebra is in fact isomorphic to the tensor (or symmetric, or universal enveloping) algebra of a one dimensional vector space (or Lie algebra). We take this opportunity to justify the use of the name convolution. The polynomial ring R[X] is a coalgebra as above, and hence its dual,
167
4.3. EXAMPLES O F HOPF ALGEBRAS
U = R[X]* = Hom(R[X],R ) is an algebra with the convolution product. If f is a continuous function with compact support, then f * E U, where f * is given by
and P is the polynomial function associated to P E R[X]. We have that A ( P ) E R [ X ] @3 R[X] R[X, Y],
--
A(P) =
C
PI@P2= P ( X
+ Y).
If g is another continuous function with compact support, the convolution product of f * and g* is given by
where h(t) = f(x)g(t - x)dx is what is usually called the convolution product (see [199]). 9. Let k be a field of characteristic p > 0. On the polynomial algebra k[X] we consider the Hopf algebra structure described in example 8, in which A ( X ) = X @ 1 + 1 €4 X ,E(X) = 0 and S ( X ) = -X. Since A(Xp) = XP @ 1 1 8Xp (we are in characteristic p, and all the binomial coefficients (p) with 1 5 i 5 p - 1 are divisible by p, hence zero), E ( X P )= 0 and S(X" = -XP (remark: if p = 2, then 1 = -I), it follows that the ideal generated by XP is a Hopf ideal, and it makes sense to construct the factor Hopf algebra H = k[X]/(XP). This has dimension p, and denoting by x the coset of X , we have A(x) = x @ 1 + 1@ x and x p = 0. This is the restricted enveloping algebra of the 1-dimensional p L i e algebra.
+
10. The cocommutative cofree coalgebra over a vector space.
Let V be a vector space, and (C, T) a cocommutative cofree coalgebra over V. We show that C has a natural structure of a Hopf algebra. Let ~(c)) p : C @ C -+ V @ Vbe the map defined by p ( c 8 d ) = ( ~ ( c ) ~ ( d ) , ~ ( d )for
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CHAPTER 4. BlALGEBRAS AND HOPF ALGEBRAS
any c, d E C. Proposition 1.6.14 shows that (C @ C,p) is a cocommutative cofree coalgebra over V $ V . The same result shows that if we denote by y : ( C @ C ) @ C - +( V $ V ) $ V the map defined by
we have that (C @ C @ C, y) is a cocommutative cofree coalgebra over V@V$V. Let m : V @ V V be the map defined by m ( x ,y) = x + y. Then V induces a morphism of coalgebras M : the linear map m : V $ V C €3 C -+ C between the cocommutative cofree coalgebra over these spaces. Also the linear map m $ I : V $ V @ V --, V $ V induces the morphism M @ I : C @ C @ C -t C @ C of coalgebras between the cocommutative cofree coalgebras over the two spaces (this follows from the relation p(M @ I ) = ( m $ I ) y , which checks immediately). By composition it follows that M(M €3 I ) : C €3 C @ C -+ C is the morphism of coalgebras associated to the linear map m(m @ I) : V @ V @ V + V (using the universal property of the cocommutative cofree coalgebra). Consider now the map y' : C €3 C @J C -+ V $ V $ V as in Proposition 1.6.14, for which (C @ (C @I C), 7') is a cocommutative cofree coalgebra over V &, (V @ V ) . Similar to the above procedure, one can show that M(I @ M ) is the morphism of coalgebras associated to the linear map m ( I $ m ) . But it is easy to see that y = y', and that m ( I $ m ) = m ( m @ I ) , hence M(M @J I ) = M(I €4 M ) , i.e. M is associative. Using Proposition 1.6.13, the zero morphism between the null space and V induces a morphism of coalgebras u : k -+ C. Also the linear map s : V -+ V, s(x) = -x, induces a map S : C -+ C , using again the universal property. As in the verification of the associativity of M , one can check that u is a unit for C , which thus becomes a bialgebra, and that S is an antipode for this bialgebra. In conclusion, the cocommutative cofree coalgebra over V has a Hopf algebra structure. -+
-+
Exercise 4.3.6 (i) Let k be a field which contains a primitive n - t h root of 1 (in particular this requires that the characteristic of k does not divide n ) and let Cn be the cyclic group of order n. Show that the Hopf algebra kcn is selfdual, i.e. the dual Hopf algebra (kc,)' is isomorphic to k c n . (iz) Show that for any finite abelian group C of order n and any field k which contains a primitive n - t h root of order n, the Hopf algebra kC is selfdual.
4.4. HOPF MODULES
169
Exercise 4.3.7 Let k be a field. Show that (i) If c h a r ( k ) # 2, then any Hopf algebra of dimenszon 2 is isomorphic to k c 2 , the group algebra of the cyclic group with two elements. (ii) If c h a r ( k ) = 2, then there exist precisely three isomorphism types of Hopf algebras of dimension 2 over k , and these are k c 2 , ( k c 2 ) * , and a certain selfdual Hopf algebra. Exercise 4.3.8 Let H be a Hopf algebra over the field k , such that there k x k x . . . x k (k appears n times). exists an algebra isomorphism H Then H is isomorphic to ( k G ) * , the dual of a group algebra of a group G with n elements.
--
Two more examples of Hopf algebras, the finite dual of a Hopf algebra, and the representative Hopf algebra of a group, are treated in the next exercises.
Exercise 4.3.9 Let H be a bialgebra. Then the finite dual coalgebra H O is a subalgebra of the dual algebra H * , and together with this algebra structure it is a bialgebra. Moreover, if H is a Hopf algebra, then H O is a Hopf algebra. Exercise 4.3.10 Let G be a monoid. Then the representative coalgebra R k ( G ) is a subalgebra of kGJ and even a bialgebra. If G is a group, then R k ( G ) is a Hopf algebra. If G is a topological group, then
R ~ ( G=) { f
E
R R ( G ) I f continuous)
is a Hopf subalgebra of R R ( G ) .
4.4
Hopf modules
Throughout this section H will be a Hopf algebra.
Definition 4.4.1 A k-vector space M is called a right H-Hopf module if H has a right H-module structure (the action of an element h E H on an element m E M will be denoted by m h ) , and a right H-comodule structure, given by the map p : M -+ M 8 H , p ( m ) = C m ( o ) 8 m ( ~ such ) , that for anym€M,h€H
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CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
Remark 4.4.2 It is easy to check that M B H has a right module structure over H 8 H (with the tensor product of algebras structure) defined by ( m@ h ) ( g B p )= m g @ h p f o r a n y m g h ~M 8 H , g B p E H B H . Considering then the morphism of algebras A : H H B H , we obtain that M 8 H becomes a right H-module by restriction of scalars via A. This structure is given by ( m @I h)g = C mgl 8 hg2 for any m 8 h E M 8 H , g E H . With this structure i n hand, we remark that the compatibility relation from the preceding definition means that p is a morphism of right H-modules. There is a dual interpretation of this relation. Consider H 8 H with the tensor product of coalgebras structure. Then M @ H has a natural structure of a right comodule over H 8 H , defined by m B h H C m ( ~ 8 h) 1 8 m ( l ) 8 h 2 . The multipliction p : H 8 H ---, H of the algebra H is a morphism of coalgebras, and then by corestriction of scalars M 8 H becomes a right H comodule, with m 8 h H E m(o)8 hl 8 m(l)h2. Then the compatibility relation from the preceding definition may be expressed by the fact that the map 4 : M 8 H -+ H , giving the right H-module structure of M , is a I morphism of H-comodules. -+
We can define a category having as objects the right H-Hopf modules, and as morphisms between two such objects all linear maps which are also morphisms of right H-modules and morphisms of right H-comodules. This category is denoted by M S , and will be called the category of right H-Hopf modules. It is clear that in this category a morphism is an isomorphism if and only if it is bijective.
Example 4.4.3 Let V be a k-vector space. Then we define on V 8 H a right H-module structure by ( v 8 h)g = v @hg for any v E V , h , g E H , and a right H-comodule structure given by the map p : V 8 H -+ V B H 8 H , p(v 8 h ) = C v @I hl B h2 for any v E V, h E H . Then V 8 H becomes a right H-Hopf module with these two structures. Indeed
p((v @ h ) g ) = p(v @ hg) =
Cv
@
( h g ) l @(hg)2
=
B hlgi B h2g2
=
C((v@hl)gl)Bh292
=
x ( v 8 h)(o)giB ( v 8 h)(l)gz
proving the compatibility relation.
I
We will show that the examples of H-Hopf modules from the preceding example are (up to isomorphism) all H-Hopf modules. We need first a definition.
4.4. HOPF MODULES
171
Definition 4.4.4 Let M be a right H-comodule, with comodule structure given b y the map p : M -+ M @ H . The set
is a vector subspace of M which is called the subspace of coinvariants of M. I
Example 4.4.5 Let H be given the rzght H-comodule structure induced b y A : H -, H @ H . Then HCoH= k l (where 1 is the identity element of H ) . , A ( h ) = C hl @ h2 = h @ 1. Applying E on the Indeed, if h E H " " ~ then first position we obtain h = ~ ( h )El k1. Conversely, if h = a1 for a scalar I a , then A ( h ) = a1 @ 1 = h @ 1. Theorem 4.4.6 (The fundamental theorem of Hopf modules) Let H be a Hopf algebra, and M a right H-Hopf module. Then the map f : M~~~ @ H -t Mydefined b y f(m @ h ) = m h for any m E MCoH and h E H , is an isomorphism of Hopf modules (on MCoH@ H we consider the H-Hopf module structure defined as in Example 4.4.3 for the vector space MCoH). Proof: W e denote the map giving the comodule structure o f M by p : 8 m ( l ) . Consider the map g : M -+ M , M + M @ H , p(m) = defined by g ( m ) = C m ( o ) S ( m ( l )for ) any m E M . I f m E M , we have
P(Cm ( o ) ~ ( m ( l ) ) ) C ( m ( o , ) ( o ) ( s ( m ( l , )@) l( m ( o ) ) , l ) ( s ( m ( l , ) ) 2 (definitiono f Hopf modules) C ( m ( o ; ) ( o , s ( ( m ( l , ) 28) ( m ( o , ) ( l ) s ( ( m ( l , ) l ) (the antipode is an antimorphism of coalgebras)
C m(o)s(m(s)) m(l)S(m(2)) @
(using the sigma notation for comodules) C m ( o ) s ( m ( 2 )8) ( m ( l ) ) i S ( ( m ( l ) ) n ) (using the sigma notation for comodules) m ( o ) S ( m ( 28 ) )~ ( m ( ~ (definition ))l o f the antipode)
C m ( O ) ~ ( m ( 2 ) & ( m ( l )1) ) @
C m ( o , ~ ( ( m ( l , ) 2 ~ ( ( m ( l )@ ) l 1) ) (using the sigma notation for comodules)
CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS m ( o ) ~ ( m c l@ ) )1 (the counit property)
=
= g(m)@l,
which shows that g(m) E M~~~for any m E M . It makes then sense to define the map F : M --+ MCoH@ H by F ( m ) = C ~ ( m ( ~@ )m(l) ) for any m E M . We will show that F is the inverse of f . Indeed, if m E M C O ~ and h E H we have
=
C g(m(o)hl)@ m(l)h2 (definition of Hopf modules)
= = =
g(mh1) €3 h
(since m E M ~ O ~ )
~ ( m h 1 ) ( o ) ~ ( ( m h l )@ ( l h2 )) m ( o ) ( h l ) i ~ ( m(h1)2) ( ~ ) @ h2 (definition of Hopf modules)
=
x m ( h l ) l ~ ( ( h l ) 2@) hz (since m E M " " ~ )
=
x m ~ ( h@~h2) (by the antipode property)
= m @ h (by the counit property)
hence Ff = I d . Conversely, if m E M , then
=
C m(o)s((mil,)l)(m,l))2 (using the sigma notation for comodules)
=
m ( o ) ~ ( m ( l )(by ) the antipode property)
= m (by the counit property)
which shows that f F = I d too. It remains to show that f is a morphism of H-Hopf modules, i.e. it is a morphism of right H-modules and a morphism of right H-comodules. The first assertion is clear, since f ( ( m 8 h)hl) = f ( m 8 hh') = mhh' = f (m 8 h)h'. In order to show that f is a morphism of right H-comodules, we have to prove that the diagram
4.5. SOLUTIONS TO EXERCISES
is commutative. This is immediate, since
( p f ) ( m @ h ) = p(mh) = m h l 8 h2 (since m E M ~ O ~ )
x
which ends the proof.
a
Exercise 4.4.7 Let H be a Hopf algebra. Show that for any right (left) H-comodule M , the injective dimension of M in the category M H is less than or equal the injective dimension of the trivial right H-comodule k . In particular, the global dimension of the category M* is equal to the injective dimension of the trivial right H-comodule k . Exercise 4.4.8 Let H be a Hopf algebra. Show that for any right (left) H-module M , the projective dimension of M i n the categoy M H is less than or equal the projective dimension of the trivial right H-module k (with action defined by a: +- h = ~ ( h ) for a any a: E k and h E H ) . I n particular the global dimension of the category M H is equal to the projective dimension of the right H-module k .
4.5
Solutions to exercises
Exercise 4.1.9 Let k be a field and n 2 2 a positive integer. Show that there is no bialgebra structure on M n ( k ) such that the underlying algebra structure is the matrix algebra. Solution: The argument is similar to the one that was used in Example 1.4.17. Suppose there is a bialgebra structure on M n ( k ) , then the counit E : M n ( k ) -+ k is an algebra morphism. Then the kernel of E is a two-sided ideal of M n ( k ) , so it is either 0 or the whole of M n ( k ) . Since ~ ( 1 =) 1, we have K e r ( ~ = ) 0 and we obtain a contradiction since dim(M,(k)) > d i m ( k ).
174
CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
Exercise 4.2.16 Let H be a finite dimensional Hopf algebra over a field k of characteristic zero. Show that if x E H is a primitive element, i.e. A(x) = x 8 1 + l a x , t h e n x = 0. Solution: If there were a non-zero primitive element x, we prove by induction that the 1,x, . . . ,xn are linearly independent for any positive integer n, and this will provide a contradiction, due to the finite dimension of H. The claim is clear for n = 1, since a1 bx = 0 implies by applying E that a = 0, and then, since x # 0, that b = 0. Assume the assertion true for n - 1 (where n 2), and let Cp=O,napxP = 0 for some scalars ao, . . . ,a p . Then by applying A we find that
+
>
+
Choose some 1 5 i , j 5 n - 1 such that i j = n, and let h;,h; E H* such that hf(xt) = biYt for any 0 5 t 5 n - 1 and hj*(xt) = 6j,t for any 0 5 t 5 n - 1 (this is possible since 1, x , . . . , xnP1 are linearly independent). Then by applying hf 8 h5 t o the above relation we obtain that a,(:) = 0, and since k has characteristic zero we have an = 0. Then again by the induction hypothesis we must have ao, . . . ,an-1 = 0.
Exercise 4.2.17Let H be a Hopf algebra over the field k and let K be a field extension of k. Show that one can define on ?? = K 8 k H a natural structure of a Hopf algebra by taking the extension of scalars algebra structure and the coalgebra structure as in Proposition 1.4.25. Moreover, if 3 is the antipode of X,then for any positive integer n we have that Sn = Id if and only if --n S =Id. Solution: The comultiplication and counit Z of & are given by
for any 6 E K and h E H. It is a straightforward check that 2 is a bialgebra over K , and moreover, the map 3 : -+ defined by S ( 6 ~ ~=h 6@kS(h) ) is an antipode of H. The last part is now obvious.
Exercise 4.3.6 (i) Let k be a field which contains a primitive n-th root of 1 ( i n particular this requires that the characteristic of k does not divide n ) and let Cn be the cyclic group of order n. Show that the Hopf algebra k c n is selfdual, i.e. the dual Hopf algebra (kcn)* is isomorphic to k c n . (ii) Show that for any finite abelian group C of order n and any field k which contains a primitive n-th root of order n, the Hopf algebra kC is
175
4.5. SOLUTIONS T O EXERCISES
selfdual. Solution: (i) Let Cn =< c > and let J be a primitive n-th root of 1 in k. Then k c n has the basis 1,c, c2,. . . ,cn-l, and let pl, p,, . . . ,pcn-l be the dual basis in (kcn)*. We determine G = G((kCn)*). We know that the elements of G are just the algebra morphisms from k c n to k. If f E G, for some 0 i 5 n - 1. Conversely, for then f ( c ) ~= 1, so f (c) = any such i there exists a unique algebra morphism fi : k c n -+ k such that fi(c) = t i . More precisely, fi(cj) = tij for any j (extended linearly). Thus fo, f l , . . . ,fn-1 are distinct grouplike elements of (kc,)*, and then a dimension argument shows that (kcn)* = kG. On the other hand fi = f: for any i, so G is cyclic, i.e. G E Cn. We conclude that (kcn)* = k c n . (ii) We write C as a direct product of finite cyclic groups. The assertion follows now from (i) and the fact that for any groups G and H we have that k(G x H) 2. kG @ kH.
<
Exercise 4.3.7 Let k be a field. Show that (i) If char(lc) # 2, then any Hopf algebra of dimension 2 is isomorphic to k c 2 , the group algebra of the cyclic group with two elements. (ii) If char(k) = 2, then there exist precisely three isomorphism types of Hopf algebras of dimension 2 over k, and these are kC2, (kC2)*, and a certain selfdual Hopf algebra. Solution: Let H be a Hopf algebra of dimension 2 and complete the set (1) to a basis with an element x such that E(X) = 0 (we can do this since H = k l $ K e r ( ~ ) ) .Since K e r ( ~ )is a one dimensional two-sided ideal of H, we have that x2 = ax for some a E k. We consider the basis B = { 1 @ 1 , x @ ~ 1 , 1 @ x , x @ xo)f H @ H . Write
for some scalars a , p, y, 6. If we write the counit property we obtain that x = c r l + y x and x = c r l f p x , so cu = 0 and ,f3 = y = 1. Thus A(x) = x @ 1 1@ x Sx @ x. If we write A(x2) = A(ax) and express both sides in terms of the basis t3,we obtain that a2S2 3aS 2 = 0, so either a6 = -1 or a6 = -2. On the other hand, if S is the antipode of H, then the relation C S(xl)xz = E ( x )= ~ 0 shows that S ( x ) ( l 6x) x = 0, and if we take S(x) = u l vx for some u, v E k, we find that u = 0 and v v6a = -1. Thus the situation 6a = -1 is impossible, and we must have Sa = -2. Now we distinguish two cases. (i) If char(k) # 2, then a f 0 and 6 = Then it is easy to determine that H has precisely two grouplike elements, namely 1 and 1 - i x . Then H = kG(H) k c 2 . (ii) If char(k) = 2, then a6 = 0, so either a = 0 or S = 0. If a = 0, it is
+
+
+
+
-
+
+
-2.
+
+
CHAPTER 4. BIALGEBRAS AND HOPE' ALGEBRAS
176
easy to see that if S # 0, then H has precisely two grouplike elements, 1 and 1 Sx, so again H 21 kC2. If S = 0 we obtain the Hopf algebra r with basis (1, x) such that x2 = 0 and A(x) = x 8 1 1 @ x. This has only one grouplike element, this is 1, so I? is not isomorphic to k c 2 . Finally, let us take the situation where S = 0 and a E k, a # 0. If we denote by H1 the Hopf algebra obtained in this way for a = 1, i.e. H1 has basis (1,x) such that x2 = x and A(%) = x @ 1 1 @ x, then it is clear that f : H --+ H1, f (1) = 1,f (x) = ix is an isomorphism of Hopf algebras. In fact H1 (kc2)*. Indeed, if we take C2 = (1,c}, then (1,c} is a basis of k c 2 and we consider the dual basis {pl,p,) of (kc2)*. Then
+
+
-
+
since char(k) = 2, so p, is a primitive element of (kCz)*. Also p: = p,, so (kc2)* HI. On the other hand r*cx I?. This can be seen again by taking in r*the dual basis {pl,p,) of { l , x ) , and checking that p, is a primitive element and p: = p,. Thus HI (kC2)*, r 21 r*,and HI is not isomorphic to k c 2 , and this ends the proof.
--
Exercise 4.3.8 Let H be a Hopf algebra over the field k, such that there exists an algebra isomorphism H E k x k x . . . x k (k appears n times). Then H is isomorphic to (kG)*, the dual of a group algebra of a group G with n elements. Solution: Since H k x k x . . . x k, we have that there exist precisely n algebra morphisms from H to k. But we know that G(H*), the set of grouplike elements of the algebra H* is precisely Hornk-,l,(H, k). Therefore H* has n grouplike elements, and since dim(H*) = n , we have that H* kG, where G = G(H*). It follows that H (kG)*.
-
-
Exercise 4.3.9 Let H be a bialgebra. Then the finite dual coalgebra HOis a subalgebra of the dual algebra H*, and together with this algebra structure it is a bialgebra. Moreover, if H is a Hopf algebra, then HOis a Hopf algebra. Solution: Let f , g E Ho, A(f) = C f l @ f 2 , A(g) = C g l @ g2. This means that
for all x, y E H . Then
4.5. SOLUTIONS TO EXERCISES hence f
* g E H 0 and
i.e. A is multiplicative. Now l H o = E , which is an algebra map. Then it is clear that A is an algebra map. It is also easy to see that E H (~ f ) = f (1) is an algebra map, so H 0 is a bialgebra. If H is a Hopf algebra with antipode S . We show that S* is an antipode for HO.Let f E H O , A(f) = f1 8 f2, i.e. f(xy) = C f l ( x ) f ~ ( y )for all x, y E H . First
=
C~ * ( f l ) ( ~ ) ~ * ( f 2 ) ( ~ ) ,
so S * ( f ) E HO.Then
The other equality is proved similarly, so HOis a Hopf algebra with antipode S*.
Exercise 4.3.10 Let G be a monoid. Then the representative coalgebra Rk(G) is a subalgebra of k G , and even a bialgebra. If G is a group, then Rk(G) is a Hopf algebra. If G is a topological group, then R ~ ( G= ) {f E RR(G) I f continuous] is a Hopf subalgebra of RR(G). Solution: The canonical isomorphism 4 : kG --+ (kg)* is also an algebra map, so its restriction and corestriction $ : Rk(G) -+ (kG)O is an isomorphism of both algebras and coalgebras. Thus Rk(G) is a bialgebra by Exercise 4.3.9. If G is a group, then kG is a Hopf algebra, so (kG)" is a Hopf algebra, and therefore Rk(G) is a Hopf algebra, with antipode given by S*(f ) ( x ) = f (x-l) for any f E Rk(G) and x E G. ) an Finally, if G is a topological group, then it is easy to see that R ~ ( G is RG-subbimodule of RR(G), and therefore it is a Hopf subalgebra. Exercise 4.4.7 Let H be a Hopf algebra. Show that for any right (left) H-comodule M , the injective dimension of M i n the category M H is less than or equal the injective dimension of the trivial right H-comodule k.
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CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS
In particular, the global dimension of the category MH is equal to the injective dimension of the trivial right H-comodule k . Solution: Let us take a (minimal) injective resolution
of the trivial right H-comodule k . Let M be a right H-comodule. For any right H-comodule Q the tensor product M @ Q is a right H-comodule with for any m € M , q E Q. the coaction given by m@q I+ m(0)@q(O)@m(l)q(l) Tensoring (4.1) with M we obtain an exact sequence of right H-comodules
*.
If Q is injective in M H , then M @ Q is injective in M Indeed, Q is a direct summand as an H-comodule in H(') for some set I, say H(') E Q @ X . Then (M @ H)(') ( M @ Q ) @ ( M @ X ) . Everything follows now if we show that M @ H is an injective H-comodule. But it is easy t o check that M 8 H is a right H-Hopf module with the coaction given as above by m 8 h I+ C m p ) 8 hl 8 m ( l ) h a ,and the action ( m8 h)g = m @ h g for any m E M , h , g E H . This shows that M @H is an injective right H-comodule. This implies that (4.2) is an injective resolution of M , thus i n j . d i m ( M ) inj.dim(k).
--
<
Exercise 4.4.8 Let H be a Hopf algebra. Show that for any right (left) H-module M , the projective dimension of M i n the category MH is less than or equal the projective dimension of the trivial right H-module k (with action defined by cr +- h = ~ ( h ) afor ! any a E k and h E H ) . In particular the global dimension of the category MH is equal to the projective dimension of the right H-module k . Solution: Let us consider a projective resolution of k in MH,
If M is a right H-module, then for any right H-module P we have a right Hmodule structure on M @ P with the action given by ( m @ p ) h= mhl@phz for any m E M , p E P, h E H. In this way we obtain an exact sequence of right H-modules
z
We show that this is a projective resolution of M , and this will end the proof. Indeed, if P is a projective right H-module, then P is a direct summand in a free right H-module, thus P @ X H(') as right H-modules for some right H-module X and some set I . Then ( M @ P) @ ( M @ X ) N
--
4.5. SOLUTIONS TO EXERCISES (M 8 H)('), so it is enough to show that M 8 H is projective. But this is true since M 8 H has a right H-Hopf module structure if we take the module structure and the right H-comodule structure given by m 8 h +-+ C m 8 hl 8 ha, and we are done.
Bibliographical notes Our main sources of inspiration for this chapter were the books of M. Sweedler [218], E. Abe (11, and S. Montgomery [149]. Exercises 4.4.7 and 4.4.8 are taken from [68] and [127], respectively. The Hopf modules are structures admitting a series of generalizations. One of these, the so called relative Hopf modules, introduced by Y. Doi [73], will be presented in Chapter 6. They generalize categories such as categories of graded modules over a ring graded by a group. An even more general structure was introduced independently by M. Koppinen [lo71 and Y. Doi [76]. These extend categories such as categories of modules graded by a G-set, categories of YetterDrinfel'd modules, etc. But things do not end here, because it is possible to find even more general structures, as some recent papers of Brzeziriski show [38, 391. We also recommend the book [51].
Chapter 5
Integrals 5.1
The definition of integrals for a bialgebra
Let H be a bialgebra. Then H* has an algebra structure which is dual to the coalgebra structure on H . The multiplication is given by the convolution product. To simplify notation, if h*,g* E H* we will denote the product of h* and g* in H* by h*g* instead of h* * g*.
Definition 5.1.1 A map T H* is called a left integral of the bialgebra H if h*T = h * ( l ) T for any h* E H*. The set of left integrals of H is denoted by S,. Left integrals of HCO"are called right integrals for H , and their set is denoted by S,. I
Remark 5.1.2 It is clear that T E H* is a left integral if and only zf C T ( x 2 ) x l = T ( x ) l V x L. H , and it is a right integral if and only if E T ( x l ) x z= T ( x ) l V x E H . I We discuss briefly the name given to the above notion. Let G be a compact group. A Haar integral on G is a linear functional X on the space of continuous functions from G to R, which is translation invariant, i.e.
for any continuous f : G -+ R, and any x E G. Then the restriction of X to the Hopf algebra R ~ ( G of ) continuous representative functions on G is an integral in the sense of the above definition. Indeed,
CHAPTER 5. INTEGRALS
and this explains the use of the name "integral" for bialgebras.
h
Remark 5.1.3 1) is clearly a vector subspace of H * . Moreover, is an ideal i n the algebra H * . That it is a right ideal is clear, since if g* E H * , and T E then for any h* E H* we have
A,
and so Tg* E J . To show that we have
is also a left ideal, with the same notation
showing that g*T E h. 2) Consider the rational part of H* to the left, hence H; rat is the sum of rational left ideals of the algebra H * . Then & E H; T a t . This follows immediately from Remark 2.2.3. In particular, if for a bialgebra we have H; Tat = 0 , then also h = 0 , hence there are no nonzero left integrals. I Before looking at some more examples, we give a result which will be frequently used in Chapter 6.
Lemma 5.1.4 If t is a left integral, and x,y
E
H , then
Proof: We show that the two sides are equal by applying an arbitrary h* E H*.
I We give now some examples of bialgebras, some having nonzero integrals, and some not.
5.1. T H E DEFINITION OF I N T E G R A L S FOR A B I A L G E B R A
183
Example 5.1.5 1 ) Let G be a monoid, and H = k G the semigroup algebra with bialgebra structure described i n Section 4.3. W e denote by pl E H* the map defined by p l ( g ) = S1,, for any g E G , where 1 means here the identity element of the monoid. Then pl is a left and right integral for H . Indeed, if h* E H * , then for any g E G we have ( h * p l ) ( g )= h * ( g ) p l ( g )= 61,,h* ( g ) = h* ( l ) p l( g ) . The last equality is clear if g = 1, and if g # 1, then both sides are zero. Since H is cocommutative, pl is also a right integral. 2) Let H be the divided power Hopf algebra from example 5) i n Section 4.3. Hence H is a k-vector space with basis {c, li E N ), the coalgebra structure is defined by m
A(crn) =
C cZe cm-2, &(em)= 60,m 2=0
and the algebra structure by
with identity element 1 = co. W e know that there exists an isomorphism of algebras h*(cn)Xn q5 : H* + k [ [ X ] ]q5(h*) , = 7~20
for h* E H * . Suppose T is a left (i.e. also right) integral of H . Then for any h* E H* we have h*T = h * ( l ) T , and applying q5 we obtain q5(h*)q5(T)= h * ( l ) + ( T ) . Noting that h * ( l ) = $(h*)(O),it follows that $ ( T ) is a formal power series for which F 4 ( T ) = F(O)$(T) for any formal power series F . Choosing then F # 0 such that F ( 0 ) = 0 (e.g. F = X ) , we obtain that q5(T)= 0 (since k [ [ X ] is ] a domain), and since q5 is an isomorphism, this implies = 0. 3) Let k be a field of characteristic zero, and H = k [ X ] with the Hopf algebra structure described i n example 8) of Section 4.3. Then H does not have nonzero integrals. Indeed, if T E H* is an integral, then h*T = h * ( l ) T for any h* E H* . Since A ( X ) = X @ 1 + 1 @ X , applying the above equality for X we get h * ( X ) T ( l )= 0 , and choosing h* with h * ( X ) # 0 it follows that T ( l )= 0 . Then we prove by induction that T ( X n ) = 0 for any n 2 0 . To go from n - 1 to n we apply the equality h*T = h * ( l ) T to Xn+l and we use the formula
By the induction hypothesis we obtain that h * ( X ) T ( X n )= 0 , and choosing again h* with h * ( X ) # 0 we find T ( X n ) = 0. Consequently, T = 0 . I
CHAPTER 5. INTEGRALS
184
Exercise 5.1.6 Let H be a Hopf algebra over the field k, K a field extension of k, and = K @ k H the Hopf algebra over K defined in Exercise 4.2.17. If T E H* is a left integral of H , show that the map E g* defined by T(6 @k h) = 6T(h) is a left integral of %. Exercise 5.1.7 Let H and HI be two Hopf algebras with nonzero integrals. Then the tensor product Hopf algebra H @ HI has a nonzero integral.
5.2
The connection between integrals and the ideal H*rat
Let H be a Hopf algebra. Throughout this section, by H * Tat we mean the rational part of H* to the left. Later in this section we will show that the rational parts of H* to the left and to the right are in fact equal, which will justify our not writing the index 1. We saw in the preceding section that if H * Tat = 0, then = 0. The aim of this section is to find a more precise connection between H*Tat and J. We know that H * Tat is a rational left H*-module, and this induces a right H-comodule structure on H*Tat, defined by p : H * Tat --+ H*Tat @ H , p(h*) = C h* @ hi,) such that g*h* = C g*(hil))h*o for any g* E H*. of H on H * defined as ko\lows: if h E H and Consider ( tO i le action h* E H*, then h h* E H* is given by (h h*)(g) = h*(gh) for any g E H . In this way, H* becomes a left H-module, and this structure is in fact induced by the canonical right H-module structure of H , taking into account that H* = Hom(H,k). Using Proposition 4.2.14, this left H-module structure on H* induces a right H-module structure on H* as follows: if h E H and h* E H*, define h* h = S(h) h*. We then have (h* h)(g) = h*(gS(h)) for any g E H .
- -
-
-
-
-
Theorem 5.2.1 H * Tat is a right H-submodule of H* (with action -). This right H-module structure, and the right H-comodule structure given by p define on H*Tat a right H-Hopf module structure. Proof: Let h* E H*Tat and h E H . We show that for any g* E H* we have the relation
Let g E H . Then
5.2. INTEGRALS AND H* RAT =
C(h2
g*)(hil))hio)(gS(hl))
-g*)h*)(gS(hl)) p) x ( h 2 - g * ) ( g S ( h l ) l ) h (*( g S ( h l ) ) 2 ) C(h3- g*)(glS(h2))h*(92S(hl))
= x((h2
(by the definition of
=
(by the definition of convolution)
= = = = =
C s*(gls(hz)h3)h'(gzs(hl)) C g*(sl.(hz))h*(szs(hl)) C g*(g1)h*(g2~(h)) (g*(h* - h ) ) ( g )
proving the required relation. This shows that h* moreover, that
p(h*
-h)
=
- h E H*
Tat,
and
C h;b) - h i @ hil)hz
I i.e. H* rat is a right H-Hopf module. The following lemma shows that there is a close connection between H* rat and J. Lemma 5.2.2 The subspace of coinvariants ( H * r a t ) C 0 H is exactly
L.
Proof: Let h* E H*'at. Then h* E ( H * T a t ) c 0 H if and only if p(h*) = h* @ 1, and this is equivalent to g*h* = g*(l)h* for any g* E H*. But this is the definition of a left integral. I We can now prove the result showing the complete connection between H* T a t and J.
A
Theorem 5.2.3 The map f : @H --+ H* Tat defined by f ( t @ h) = t h E H , is an isomorphzsm of right H-Hopf modules. for any t E
A,
-h
Proof: Follows directly from the fundamental theorem of Hopf modules I applied to the Hopf module H* T a t . We already saw that if H* ' a t = 0, then shows that the converse also holds.
h
Corollary 5.2.4 H* rat = 0 i f and only if
= 0. The preceding theorem
h = 0.
I
Exercise 5.2.5 Let H be a Hopf algebra. Then the following assertions are equivalent:
CHAPTER 5. INTEGRALS
186
i) H has a nonzero left integral. ii) There exists a finite dimensional left ideal i n H*. iii) There exists h* E H* such that K e r ( h * ) contains a left coideal of finite codimension i n H .
Corollary 5.2.6 Let H be a Hopf algebra with antipode S , and having a nonzero integral. Then S is injective. If moreover H is finite dimensional, then Js has dimension 1 and S is bijective. Proof: From Theorem 5.2.3, if there would exist an h # 0 with S ( h ) = 0, then for a t E &, t # 0 we would have f ( t @ h ) = 0, contradicting the injectivity of f . If H is finite dimensional, Example 2.2.4 shows that H* 'at = H*. We obtain an isomorphism of Hopf modules f : & @H -+ H* defined by f (t 8 h ) = t h = S(h) t . In particular, this is an isomorphism of vector spaces, and so d i m ( H * ) = dim(& @ H ) . But d i m ( H * ) = d i m ( H ) and dim(& @ H ) = d i m ( & ) d i m ( H ) . Therefore, d i m ( & ) = 1. Moreover, since S is an injective endomorphism of the finite dimensional vector space H, it follows that S is an isomorphism, and so it is bijective. !
-
-
When H is a finite dimensional Hopf algebra, there is still another way to work with integrals. We recall that there is an isomorphism of algebras 4 : H -+ H** defined by 4 ( h ) ( h * )= h*(h) for any h E H , h* E H * . Then it makes sense t o talk about left integrals for the Hopf algebra H * , these being elements in H**. Since q5 is bijective, there exists a nonzero element h E H such that +(h) E H** is a left integral for H*. Since any element in H** is of the form q5(1) with 1 E H , this means that for any 1 E H we . $ ( l ) $ ( h )= q5(lh) ($ is a morphism of have $(l)q5(h)= $ ( l ) ( l H * ) $ ( h )But , the fact that 4 ( h ) is a left algebras) and q5(l)(lH-)= q 5 ( 1 ) ( ~ )= ~ ( l )hence h any 1 E H . This justifies the integral for H* is equivalent to l h = ~ ( 1 ) for following definition. Definition 5.2.7 Let H be a finite dimensional Hopf algebra. A left integral in H is an element t E H for which ht = i ( h ) t for any h E H .
i Remark 5.2.8 i ) If H is a finite dimensional Hopf algebra, there is some danger of confusion between left integrals for H (which are elements of H*), and left integrals i n H (which are elements of H ) . Left integrals i n H are in fact left integrals for H* when they are regarded via the isomorphism 4 : H -t H**. This is why we will have to specify every time which of the two kind of integrals we are using. ii) Corollary 5.2.6 shows that i n any finite dimensional Hopf algebra there
5.2. I N T E G R A L S A N D H* RAT
187
exist nonzero left integrals, and moreover, the subspace they span has dimension 1, and is therefore k t , where t is a nonzero left integral i n H .
I Example 5.2.9 1 ) Let G be a finite group, and k G the group algebra with g is the Hopf algebra structure described i n Section 4.3 1). Then t = CgEc a left (and right) integral i n k G . 2) If G is a finite group, then ( k G ) * , the dual of k G , is a Hopf algebra, and the map pl E ( k G ) * ,pl(g) = bl,,, is a left (and right) integral in ( k G ) * . 3) Let k be a field of characteristic p > 0, and H = k [ X ] / ( X P )the Hopf algebra described i n Section 4.3 9). Then t = xp-' is a left (and right) integral i n H . 4 ) Let H denote Sweedler's 4-dimensional Hopf algebra described in Section 4.3 6). Then x+cx is a left integral i n H , and x-cx is a right integral in H . 5) Let Hnz(X) be a Taft algebra described i n Section 4.3 7). Then t = I ( 1 + c + . . . cn-')xn-' is a left integral i n Hnz(X).
+
An important application of integrals in finite dimensional Hopf algebras is is the following result, known as Maschke's theorem. Here is the classical proof. For a different proof see Exercise 5.5.13.
Theorem 5.2.10 Let H be a finite dimensional Hopf algebra. Then H is a semisimple algebra if and only if ~ ( t#) 0 for a left integral t E H . Proof: Suppose first that H is semisimple. We know that K e r ( ~ is) an ideal of codimension 1 in H . Regarding K e r ( ~ as ) a left submodule of H , by the semisimplicity of H we have that K e r ( ~ is) a direct summand in H , hence there exists a left ideal I of H with H = k e r ( e ) @ I . Let 1 = z h , with z E K e r ( ~ )h , E H , be the representation of 1 as a sum of two elements from Ker(E) and I . Clearly h # 0, because 1 4 K e r ( ~ ) . Since I has dimension 1 (since K e r ( ~ has ) codimension I ) , it follows that I = kh. Let now 1 E H. Then l h E I ; and so the representation of l h in the direct sum H = K e r ( & )@ I is l h = 0 lh. On the other hand, , so l h = ( 1 - ~ ( 1 ) l ) h ~ ( 1 ) h .Since we have 1 = ( 1 - ~ ( 1 ) l )~ ( l ) land ( 1 - ~ ( 1 ) l ) Eh K e r ( E ) , ~ ( l ) Eh I , and the representation of an element in H as a sum of two elements in K e r ( ~ and ) I is unique, it follows that ( 1 - ~ ( 1 ) l )=h 0 and ~ ( 1 ) = h lh. The last relation shows that h is a left integral in H . Since I n K e r ( & )= 0, it follows that ~ ( h#) 0, and the first implication is proved. We assume now that ~ ( t#) 0 for a left integral t in H . We fix such an integral t with ~ ( t =) 1 (we can do this by replacing t by t / ~ ( t ) ) In . order to show that H is semisimple, we have to show that for any left
+
+
+
+
CHAPTER 5. INTEGRALS
188
H-module M , and any H-submodule N of M , N is a direct summand in : M -+ N be a linear map such that n(n) = n for any n E N (to construct such a map we write M as a direct sum of N and another subspace, and then take the projection on N). We define
M . Let n
P :M
-+
N, P(m) =
tln(S(t2)m) for any m E M.
We show first that P ( n ) = n for any n E N . Indeed, P(n)
=
xtln(~(t2)n)
=
1t1s(t2)n
(since S(t2)n t N )
= ~(t)ln -
n
(by the property of the antipode) (since ~ ( t = ) 1)
We show now that P is a morphism of left H-modules. Indeed, for M and h E H we have
rn E
hP(m)
=
htln(~(t2)m)
=
hltla(S(t2)~(h2)m)(by the counit property)
=
hltln(S(t2)S(h2)h3m) (by the property of the antipode)
=
1hltln(s(h2tz)hsm)
=
~(hit)in(s((hit)2)h,m)
=
1~ ( h ~ ) t l s ( ~ ( t 2 ) h 2(since m ) h ~ =t ~ ( h 1 ) t )
= =
x
tln(s(t2)hm) (by the counit property)
P(hm)
We have showed that there exists a morphism of left H-modules P : M - - N such that P ( n ) = n for any n E N . Then N is a direct summand in M as left H-modules, (in fact M = N $ K e r ( P ) ) and the proof is finished.
Remark 5.2.11 If G is a finite group, and H = kG, then we saw that g is a left integral in H . Then ~ ( t = ) (Gllk, where IG( is the order t = CgEG The preceding theorem shows that the Hopf algebra kG is of the group G. semisimple if and only if IGllk # 0, hence if and only if the characteristic of k does not divide the order of the group G. This is the well known Maschke I theorem for groups.
5.3. FINITENESS CONDITIONS
189
If H is a semisimple Hopf algebra, and t E H is a left integral with ~ ( t =) 1, then t is a central idempotent, because S ( t ) is clearly a right integral in H (i.e. S ( t ) h = ~ ( h ) S ( t ) and ) , we have
Recall that a k-algebra A is called separable (see [112])if there exists an element C ai D b, E A D A such that C a,bi = 1 and
for all x E A. Such an element is called a separability idempotent.
Exercise 5.2.12 A semisimple Hopf algebra is separable. Exercise 5.2.13 Let H be a finite dimensional Hopf algebra over the field k, K a field extension of k, and H = K 81,H the Hopf algebra over K defined i n Exercise 4.2.17. If t E H i s a left integral i n H , show that ? = l K 81,t E is a left integral in H . A s a consequence show that H is semisimple over k if and only if is semisimple over K .
5.3 Finiteness conditions for Hopf algebras with nonzero integrals Lemma 5.3.1 Let H be a Hopf algebra. If J is a nonzero right (left) ideal which is also a right (leftl coideal of H , then J = H . I n particular we obtain the following: (i) If H i s a Hopf algebra that contains a nonzero right (left) ideal of finite dimension, then H has finite dimension. (ii) A semisimple Hopf algebra (i.e. a Hopf algebra which is semisimple as a n algebra) i s finite dimensional. (iii) A Hopf algebra containing a left integral i n H (i.e. a n element t E H with h t = ~ ( h )for t all h E H ) is finite dimensional. Proof: If J is a right ideal and a right coideal, then A ( J ) G J 8 H and J H = J . If E ( J ) = 0,then for any h E J we have h = C € ( h l ) h 2 E E ( J ) H = 0, so J = 0,a contradiction. Thus e ( J ) # 0, and then there exists h E H with ~ ( h=) 1. Then 1 = ~ ( h )=l C h l S ( h 2 ) E J H C J , so 1 E J and J = H . We show that the assertion (i) can be deduced from the first part of the statement. Indeed, let J be a nonzero right ideal of finite dimension in a
,
CHAPTER 5. INTEGRALS
190
-
Hopf algebra H , and let I = H* J, which is a right ideal, a right coideal and has finite dimension. The fact that I is a right ideal follows from
for any h* E H * , x E J, y E H . Then I = H, so H is finite dimensional. For (ii), let us take a semisimple Hopf algebra H . Then K e r ( ~ is ) a left ideal of H . Since H is a semisimple left H-module, there exists a left ideal I of H such that H = I @ K e r ( & ) .Since K e r ( & )has codimension 1 in H , we have that I has dimension 1. Then H is finite dimensional by (i). Note that by Theorem 5.2.10, I is the ideal generated by an idempotent integral in H . I (iii) The subspace generated by t is a finite dimensional left ideal.
Theorem 5.3.2 (Lin, Larson, Sweedler, Sullivan) Let H be a Hopf algebra. Then the following assertions are equivalent. (i) H has a nonzero left integral. (ii) H is a left co-Frobenius coalgebra. (iii) H is a left QcF coalgebra. (iv) H is a left semiperfect coalgebra. (v) H has a nonzero right integral. (vi) H is a right co-Frobenius coalgebra. (vii) H is a right QcF coalgebra. (viii) H is a right semiperfect coalgebra. (ix) H is a generator i n the category H M (or i n M ~ ) . (x) H is a projective object in the category H~ (or in M H ) . Proof: ( i )+ (ii). Let t E H * be a nonzero left integral. We define the bilinear application b : H x H + Ic by b(x,y ) = t ( x S ( y ) )for any x , y E H . We show that b is H*-balanced. Let x , y E H and h* E H*. Then
=
x
h * ( x ~ ) t ( x z S ( y ~ ) ~ (by ( y ~the ) ) counit property) h * ( x 1 S ( y 2 ) y ~ ) t ( x 2 S ( y l )(the ) property of the antipode)
= =
C(Y3
- h*)(xlS(Y2))t(x2S(Yl))
5.3. FINITENESS CONDITIONS =
C(y2
=
x(yz
=
=
- h*)((~S(~l))l)t((~S(~l))2)
- h * ) ( l ) t ( x S ( y l ) ) (t Ch*(~2)t(xs(~l)) b ( x , h* - y )
is a left integral)
Now we show that b is left non-degenerate. Assume that for some y E H we have b(x,y ) = 0 for any x E H . Then t ( x S ( y ) )= 0 for any x E H . But t ( x S ( y ) )= ( t y ) ( x ) and we obtain t y = 0. Now Theorem 5.2.3 shows that y = 0. This implies that H is left co-F'robenius. (ii)+ (iii)+ ( i v ) are obvious (by Corollary 3.3.6). ( i v ) + ( v ) Since H is left semiperfect we have that Rat(H;I.) is dense in H * . Then obviously Rat(H;I,) # 0, and by Theorem 5.2.3 applied to HOP,COP we have that ST # 0. ( v ) + ( v i ) + ( v i i ) + ( v i i i ) + ( i ) are the right hand side versions of the facts proved above. (iii)and ( v i i ) + ( i x ) and ( x ) follow by Corollary 3.3.11. ( i x ) + ( i )If H is a generator of HM,since k l is a left H-comodule, there exist a nonzero morphism t : H -+ Ic of left H-comodules. Then t is a nonzero left integral. ( x ) + ( i v ) (or ( x ) ( v i i i ) ) Since H is projective in HM, we have that l ? a t ( ~ . H * )is dense in H* by Corollary 2.4.22. I
-
-
+
Corollary 5.3.3 Let H be a Hopf algebra with nonzero integrals. Then any Hopf subalgebra of H has nonzero integrals. Proof: Let K be a Hopf subalgebra. Since H is left semiperfect as a coalgebra, then by Corollary 3.2.11 K is also left semiperfect. I Remark 5.3.4 If H has a nonzero left integral t , and K is a Hopf subalgebra of H , then the above corollary tells us that K has a nonzero left integral. However, such a nonzero integral is not necessarily the restriction o f t to K . Indeed, it might be possible that the restriction o f t to K to be zero, as it happens for instance i n the situation where H is a co-Frobenius Hopf algebra which is not cosemisimple, and K = k l (this will be clear in view of Exercise 5.5.9, where a characterization of the cosemisimplicity is I given). Exercise 5.3.5 Let H be a finite dimensional Hopf algebra. Show that H is injective as a left (or right) H-module.
CHAPTER 5. INTEGRALS
192
5.4
The uniqueness of integrals and the bijectivity of the antipode
Lemma 5.4.1 Let H be a Hopf algebra with nonzero integrals, I a dense left ideal in H * , M a finite dimensional right H-comodule and f : I + M a morphism of left H*-modules. Then there exists a unique morphism h : H* -+ M of H*-modules extending f . Proof: Let E ( M ) be the injective envelope o f M as a right H-comodule. Then E ( M ) is also injective as a left H*-module (see Corollary 2.4.19). Regarding f as a H*-morphism from I t o E ( M ) , we get a morphism h : H* -+ E ( M ) o f left H*-modules extending f. I f x = h ( ~E) E ( M ) , then I x = I h ( & )= ~ ( I E=)h ( I ) = f ( I ) C M . Since I is dense in H * , we have that x E I x . Indeed, there exists h* E I such that h* agrees with E on a11 the xi's from x H C xO@xl (the comodule structure m a p of E ( M ) ) . Then x = C E ( x ~ ) x = ~ C h*(xl)xo= h*x E I x . W e obtain x E M , and hence I m ( h ) C M . Thus h is exactly the required morphism. If h' is another morphism with the same property, then h - h' is 0 on I . Then I ( h - h') ( E ) = ( h - h') ( I E )= ( h - h l )( I ) = o and again since I is dense in H* we have that ( h - h ' ) ( ~E) I ( h - h l ) ( ~ SO ), ( h - h') ( E ) = 0. Then clearly h = h'. h
Theorem 5.4.2 Let H be a Hopf algebra with nonzero integral and M a finite dimensional right H-comodule. Then dim HornH-( H ,M ) = dim M . In particular dim ST = dim J = 1. Proof: Rat(H.H*) is a dense subspace o f H* by Theorem 3.2.3. Also by Theorem 5.3.2 there exists an injective morphism 9 : H --+ H* o f left H*-modules. Clearly 9 ( H ) E Rat(H.H*) and since H is an injective object in the category M H , we obtain that H is a direct summand as a left H*-module in Rat(H*H*).Thus there exists a surjective k-morphism Hemp ( H * r a t , M ) -+ HornH*( H ,M ) . By Lemma 5.4.1,
W e obtain an inequality dim HornH*( H ,M ) 5 dim HomH* ( H *T a t , M ) = dim M .
5.4. T H E UNIQUENESS OF INTEGRALS
193
If we take M = k , the trivial right H-comodule, we have that ST = HornH. (H, k ) has dimension a t most 1, so this has dimension precisely 1 since # 0. Similarly has dimension 1. Hence there exists even an isomorphism of left H*-modules B : H -+ HeTat, B(h) = t h, where t is a fixed non-zero left integral. Then the surjective morphism HornH- ( H * 'at, M ) -+ HornH*( H ,M ) obtained above from 6 is an isomorI phism, showing that in fact dim HornH*( H ,M ) = dim M .
h
ST
-
Remark 5.4.3 The proof of the result of the above theorem can be done i n a more general setting. If C is a left and right co-Frobenius coalgebra, and M is a finite dimensional right C-comodule, then dim Hornc. (C,M ) 5 dim M . I We will prove now that the antipode of a co-Frobenius Hopf algebra is bijective. The next lemma is the first step in this direction.
Lemma 5.4.4 1) Let H and K be two Hopf algebras and 4 : H -+ K an injective coalgebra morphism with 4 ( 1 ) = 1. If t E K* is a left integral, then t o 4 is a left integral i n H * . 2) Let H be a Hopf algebra with a nonzero left integral t and antipode S . Then t o S is a nonzero right integral of H . Proof: 1) Since t is a left integral for K , for any z E K we have that C t ( z 2 ) z 1= t ( z ) l . Take z = 4 ( h ) with h E H , and use the fact that 4 is a coalgebra morphism. We find that C t ( $ ( h 2 ) ) $ ( h l )= t ( d ( h ) ) l . This shows that 4 ( C t ( $ ( h 2 ) ) h l )= + ( t ( $ ( h ) ) l ) which , by the injectivity of q5 implies t(q5(hz))hl= t ( $ ( h ) ) l ,i.e. t o 4 is a left integral for H . 2) We consider S : H HOp~cOp, an injective coalgebra morphism with S ( 1 ) = 1. We use 1) and see that if t is a nonzero left integral for H , then t o S is a left integral for HOp~""p, i.e. a right integral for H. It remains to show that t o S # 0. Let J H be the injective envelope of k l , considered as a right H-comodule. Then J is finite dimensional and H = J $ X for a right coideal X of H . Let f E H* such that f ( X ) = 0 and f ( 1 ) # 0. Then K e r ( f ) contains a right coideal of finite codimension, thus f E H*rat. Hence f = t h for some h E H , and f ( 1 ) = ( t h)(l)= t(S(h)), I showing that t o S # 0. -+
c
-
-
Corollary 5.4.5 If H is a Hopf algebra with nonzero integrals, we have that S * ( h )= ST. Proof: It follows from the second assertion of Lemma 5.4.4 and the uniqueI ness of integrals. Corollary 5.4.6 Let H be a Hopf algebra with nonzero integral. Then the antipode S of H is bijective.
CHAPTER 5. INTEGRALS
194
Proof: We fix a nonzero left integral t . We know from Corollary 5.2.6 that S is injective. We prove that S is also surjective. Otherwise, if we assume that S ( H ) # H, since S ( H ) is a subcoalgebra of H, we have that S ( H ) is a left H-subcomodule of H, and by Corollary 3.2.6 there exists a maximal left H-subcomodule M of H such that S ( H ) G M . Let u E H* such that u # 0 and u ( M ) = 0. Since K e r ( u ) contains M, we have that u E R a t ( H . H*), thus there exists an h E H such that u = t h. Clearly h # 0 and for any x E M we have ( t h ) ( x ) = u ( x ) = 0. For any y E H we have S ( y ) E M, so
-
-
-
so (t o S ) ( h H ) = 0. Since t o S is a right integral, then t o S is a morphism of left H*-modules, so ( t o S ) ( H * ( h H ) ) = 0. Lemma 5.3.1 tells us that H* (hH) = H, and then (t 0 S ) ( H ) = 0. This is a contradiction with I Lemma 5.4.4.
-
Remark 5.4.7 If H is a Hopf algebra with nonzero integral and t is a non-zero left (or right) integral, we have that
-
Indeed, let t be a nonzero left integral. Then the relation t H = H*Tatfollows from Theorem 5.2.3 and the uniqueness of the integrals. Also H*Tat= t H = S(H) t =H t (we have used the bijectivity of the anH = H*ratfor Hop, we obtain tipode). If we write now the relation t H = H*rat, which also shows that H t = H*Tat. that t
-
-
-
-
-
-
The next exercise provides another proof for the uniqueness of integrals.
Exercise 5.4.8 Let spans S,.
5.5
x EA
and h
E
H be such that
x
0
S ( h ) = 1. Then x
Ideals in Hopf algebras with nonzero int egrals
We recall that for a coalgebra C we denote by G ( C ) the set of all grouplike elements of C (i.e. g E G ( C ) if A ( g ) = g 8 g and ~ ( g=) 1). It is possible to have G ( C ) = 0 (see Example 1.4.17).
5.5. IDEALS IN CO-FROBENIUS HOPF ALGEBRAS
195
Proposition 5.5.1 Let C be a coalgebra. Then (i) Every right (left) coideal of C of dimension 1 is spanned b y a grouplike element. (ii) There exists a bijective correspondence between G ( C ) and the set of all coalgebra morphisms from k (regarded as a coalgebra) to C . Thus any l-dimensional subcoalgebra of C is of the form kg for some g E G ( C ) . (iii) There exists a bijective correspondence between G ( C ) and the set of all continuous algebra morphisms from C* to k. Proof: (i) Let I be a right coideal of C of dimension 1. If c E I , c # 0, then I = kc, and since A ( I ) 2 I @ C there exists g E C such that A(c) = c @I g . The coassociativity of A shows that c@g@g= c@IA(g), thus A ( g ) = g @ g . Since c # 0 we must have g # 0, so g E G ( C ) . On the other hand the counit property shows that c = ~ ( c ) gso, I = kg. (ii) If g E G ( C ) ,then the map X -, Xg from k to C is a coalgebra morphism. Conversely, to any coalgebra morphism a : k -, C we associate the element g = a(1) E G ( C ) . In this way we define a bijective correspondence as required. (iii) Let g E G ( C ) and let cr : k --+ C be the coalgebra morphism associated in ( i i ) to g. Then a* : C* -, k* E k is a continuous algebra morphism. : C* -+ k is a continuous algebra morphism, then I = Conversely, if Ker(P) = P-l({O)) is closed, so II' = I . Also I' is a subcoalgebra of dimension 1 of C, so I L = kg for some g E G ( C ) . This implies that I = ( k g ) l . We show that P(c*) = c*(g)for any c* E C*. If c* E I we clearly have P(c*) = c*(g) = 0. For c* = E we have ,8(c*) = c*(g) = 1. These imply that P(c*) = c*(g) for any c* E I kc = C* (since I has codimension 1 and E 4 I ) . I
+
A coalgebra is called pointed if any simple subcoalgebra has dimension 1. By the previous proposition, we see that for a pointed coalgebra C , we have Corad(C) = kG(C). Exercise 5.5.2 Let f : C -+ D be a surjective morphism of coalgebras. Show that if C is pointed, then D is pointed and Corad(D) = f (Corad(C)). Let C be a coalgebra such that G ( C )# 0. For any g E G ( C ) we define the sets Lg = {c* E C*ld*c*= d*(g)c*for any d* E C * )
Rg = {c* E C*lc*d*= d*(g)c*for any d* E C * } I f f E C* and c* E Lg we have
I
CHAPTER 5. INTEGRALS
so fc* E L,. Also d*(c*f ) = d*c*f = d*(g)c*f , so c*f E L,, showing that L, is a two-sided ideal of C*. Similarly, R, is a two-sided ideal of C*. For any g E G(C), Proposition 5.5.1 tells that there exists a coalgebra morphism a, : k C , a,(X) = Xg. Then we can regard k as a left C , right C-bicomodule via a,, and we denote by ,k, respectively kg, the field k regarded as a left C-comodule, respectively as a right C-comodule. A morphism of left C-comodules (or equivalently of right C*-modules) from C to ,k is called a left g-integral. The space Hornc* (C,, k) is called the space of the left g-integrals. Similarly we define right g-integrals and the space Hornc- (C, kg) of right g-integrals. -+
Proposition 5.5.3 With the above notation we have L, = Hornc. (C, gk) and R, = Hornc* (C, kg). In particular if C is a left and right co-Frobenius coalgebra, then L, and R, are two-sided ideals of dimension 1. Conversely, if I is a 1-dimensional left (right) ideal of C*, then there exists g E G(C) such that I = L, (respectively I = Rg). In particular, 1-dimensional left (right) ideals are two-sided ideals. Proof: We prove that L, = Homc.(C, ,k). Let c* E C*. Then c* E Hornc* (C,g k) if and only if a,c* = (I@c*)A, where cr, : k -+ C@ k is the comodule structure map of ,k. This is the same with C c*(cz)cl = c*(c)g for any c E C , which is equivalent to the fact that u ( C c*(cz)cl)= u(c*(c)g) for any u E C* and any c E C . But u(Cc*(c2)cl) = Cu(cl)c*(c2) = (uc*)(c), and u(c*(c)g) = u(g)c*(c) = (u(g)c*)(c). Thus c* E Hornc* (C, k) if and only if uc* = u(g)c* for any u E C*, which means exactly that UE L,. Assume now that C is left and right co-Frobenius. By Corollary 3.3.11 C is a projective generator in the categories MC and CM (when regarded as a right or left C-comodule). Then Hornc*(C,, k) # 0 and Hornc* (C, kg) # 0. Remark 5.4.3 shows that dim(Homc* (C,, k)) = 1 and dim(Homc* (C, kg)) = 1, so L, and R, have dimension 1. Let I be a 1-dimensional left ideal of C*, say I = kx. Since I is a left ideal, there exists a k-algebra morphism f : C* -+ k such that c*x = f (c*)x for any c* E C*. Since the map c* H C*X is continuous we see that K e r ( f ) is closed in C*. Then by Proposition 5.5.1 (iii) there exists a coalgebra morphism cr : k -+ C , &(A) = Xg, where g E G ( C ) ,such that f = a * . Then for any c* E C* we have
,
,
5.5. IDEALS IN CO-FROBENIUS HOPF ALGEBRAS
197
i.e. x E L,. This shows that I C L, and since L, has dimension 1 we I conclude that I = L,. Assume now that II is a Hopf algebra with nonzero integral. Then G ( H ) is a group and 1 E G ( H ) (here 1 means the identity element of the k-algebra H ) and for any g E G ( H ) we have g-l = S ( g ) , where S is the antipode of H . For any g E G ( H ) the maps u,,u$ : H -+ H defined by u,(x) = gx and u $ ( x ) = xg for any x E H , are coalgebra isomorphisms. Then the dual morphisms u;,u$* : H* + H* are algebra isomorphisms. For any a* E H* and x E H we have that
-
-
thus u,(a*) = a* g. Similarly u:(a*) = g a*. It is easy to see L,, Lh-1, = u ; ( L g ) = Lg h , Rgh-1 = that Lgh-1 = U;(L,) = h u;(R,) = h R , and R,,-1, = u;(R,) = R , h. In particular by Proposition 5.5.3 there exists a E G ( H ) such that R , = L1 = This element is called the distinguished grouplike element of H .
-
-
-
h.
Proposition 5.5.4 With the above notation we have that (2) Rag = Lg = Rga for any g E G ( H ) . In particular a lies in the center of
the group G ( H ) . (ii)a-L,=L,-a=R,. (iiz) For any nonzero left integral t we have t S = a
- t , and t S -
-
-
=t
- a.
Proof: (i) Since L1 = R , we have g-l L1 = L,. On the other hand g-' L1 = g-I R , = R a g ,so L, = Rag. Similarly Lg = R,,. Since Rag = R,, we obtain by Proposition 5.5.3 that ag = ga for any g E G ( H ) , and this means that a belongs to the center of G ( H ) . (ii) We know that a Lg = L,,-I and ag-' R , = R,. Since R , = L1 then ag-' R , = ag-l L1 = L,,-1. We obtain that a Lg = R,. The relation L, a = Rg follows similarly. (iii) If x E H is such that t ( x ) = 1, then a = x t , because for all h* E H* we have h*(a)= h * ( a ) t ( x )= ( t h * ) ( x )= h * ( x L t ) . We know from (i) that a t is a right integral. So is t S , so by uniqueness there is an a E k such that t S = a ( a t ) . We prove cr = 1 by applying both sides of the equality to S P 1 ( x ) .So we want (a t ) ( S - l ( x ) )= 1, and
-
-
- -
-
-
-
-
CHAPTER 5. INTEGRALS
198 we compute it (a
-t)(sPl(x)) Ct ( ~ - ~ ( x ) t ( x ~ ) x ~ ) =
= =
1
t ( ~ l t ( ~ - ~ ( ~ ) ~ 2 ) )
1t ( r z t ( s - ' ( x 1 ) r ) ) ( b y Lemma 5.1.4 for HOP)
t ( x ) t ( ~ - ' ( 1 ) x )(since C t(x2)xl = t ( x ) l ) = t ( ~=)1 ~
=
I and the proof is complete. I f H is finite dimensional, and 0 # t' E H is a left integral, then for all h E H , t'h is also a left integral and t'h = A'(h)tl, A' a grouplike element in H*. Corollary 5.5.5 For H , A', t' as above, A'
- t'
= S(tl).
Proof: Let G be the Hopf algebra H*. Identify G* with H and consider t' as an element o f G*;A' is itself a grouplike element o f G and corresponds t o the element g o f Proposition 5.5.4 (iii). T h e statement follows immediately.
I Exercise 5.5.6 Prove Corollary 5.5.5 directly. Exercise 5.5.7 Let H be a Hopf algebra such that the coradical Ho is a . .. Hopf subalgebra. Show that the coradical filtration Ho C H I C . . . H, is an algebra filtration, i.e. for any positive integers m,n we have that HmHn C Hm+n. Theorem 5.5.8 Let H be a co-Frobenius Hopf algebra, and Ho, H I , . . . the coradical filtration of H . If J J H is an injective envelope of k l , considered as a left H-comodule, then JHo = H . Moreover, if Ho is a Hopf subalgebra of H , then there exzsts n such that Hn = H . Proof: Since Ho is the socle o f the left H-comodule H , we can write Ho = k l @ M for some left H-subcomodule M o f H . W e know that H = E ( k 1 ) @ E ( M ) = J $ E ( M ) (where E ( M ) is the injective envelope o f M ) . Let t # 0 be a right integral o f H . Then for h E E ( M ) we have t ( h ) l = C t ( h l ) h z E E ( M ) , and also t ( h ) l E J . Therefore t ( h ) l E E ( M ) n J = 0, so t ( h ) = 0. Thus t ( E ( M ) )= 0, and then t ( J ) must be nonzero. =J a (see Proposition 5.5.4). I f J H o # H , Let a E G ( H ) such that then aJHo is a proper left H-subcomodule o f H , so there exists a maximal left H-subcomodule N o f H with aJHo C N . Then H / N is a simple left comodule, so N L = ( H / N ) * is a simple right H-comodule. This shows
ST
-
5.5. IDEALS I N CO-FROBENIUS HOPF ALGEBRAS
199
that N I is a simple right subcomodule of H*T"t. We use the isomorphism of right H-comodules q5 : H + H*Tat,$ ( h ) = (t a-l) h (note that t a-I is a left integral), and find that there exists a simple right Hsubcomodule V of H such that N' = d(V) = ( t a-l) V. Then t(a-'NS(V)) = 0. As a simple subcomodule, V is contained in the (right) socle of H, thus V C Ho. Also, since A(V) C V 8 H , the counit property shows that there is some v E V with E(V)# 0. Then
-
-
-
-
-
Now we have J = a-'aJ C a-'aJHoS(V) C a-'NS(V), so t ( J ) = 0, a contradiction. Thus we must have J H o = H. Assume that Ho is a Hopf subalgebra of H . Since ST# 0, J must be finite dimensional, and then there exists n such that J C H,. Then H = JHo I HnHo = H, (by Exercise 5.5.7), thus H = H,. The previous theorem shows that if H is a co-F'robenius Hopf algebra and the trivial left H-comodule k is injective, then J = k l and H = Ho, i.e. H is cosemisimple. The following exercise gives a different proof of this fact for an arbitrary Hopf algebra (not necessarily co-Frobenius).
Exercise 5.5.9 Let H be a Hopf algebra. Show that the following are equivalent. (1) H is cosemisimple. (2) k is an injective right (or left) H-comodule. (3) There exists a right (or left) integral t E H* such that t(1) = 1. Exercise 5.5.10 Show that i n a cosemisimple Hopf algebra H the spaces of left and right integrals are equal (such a Hopf algebra is called unimodular), and if t is a left integral with t(1) = 1, we have that t o S = t . Remark 5.5.11 It is easy to see that a Hopf algebra is unimodular i f and only if the distinguished grouplike is equal to 1. Exercise 5.5.12 Let H be a Hopf algebra over the field k , K a field extension of k , and I? = K 81,H the Hopf algebra over K defined i n Exercise 4.2.17. Show that i f H is cosemisimple over k , then H is cosemisimple over K . Moreover, i n the case where H has a nonzero integral, show that if is cosemisimple over K , then H is cosemisimple over k. The equivalence between (1) and (3) in the following exercise has been already proved in Theorem 5.2.10 (Maschke's theorem for Hopf algebras). We give here a different proof.
CHAPTER 5. INTEGRALS
200
Exercise 5.5.13 Let H be a finite dimensional Hopf algebra. Show that the following are equivalent. (1) H is semisimple. (2) k is a projective left (or right) H-module (with the left H-action on k defined by h - CY = ~ ( h ) a ) . (3) There exists a left (or right) integral t E H* such that ~ ( t#)0 .
5.6
Hopf algebras constructed by Ore extensions
Throughout this section, k will be an algebraically closed field of characteristic 0. In fact, for most of the results we only need that k contains enough roots of unity. We will denote by Z f = N* the set of positive integers (non-zero natural numbers). We construct now the quantum binomial coefficients and prove the quantum version of the binomial formula. Define by recurrence a family of polynomials (Pn,i),>l,o5iSn in the indeterminate X with integer coefficients a s follows. We start with = = 1. If we assume that we have defined the polynomials (Pn,i)05iln for some n 2 1, then define (Pn+l,i)o
Also, for any positive integer i we denote by [i]the following polynomial . . . X 1. In fact [i]= if we regard the [i]= X" polynomial ring Z [ X ] as a subring of the field Q ( X ) of rational fractions. We also define [0] to be the constant polynomial [0] = 1. Now for any positive integer n we define the polynomial [n]!by
+
+
+ +
which is also a polynomial with integer coefficients. We make the convention [O]! = 1, a constant polynomial. We can describe now the polynomials Pn,i explicitely.
Proposition 5.6.1 For any positive integer n and any 0 5 i 5 n we have P n ,= &, 21. where the polynomial ring Z [ X ] is regarded as a subring of the ring of rational fractions Q ( X ) . Proof: We prove by induction on n that for any 0 5 i 5 n the desired formula holds. For n = 1 it is obvious. For passing from n to n + 1, we first
5.6. HOPF ALGEBRAS CONSTRUCTED B Y ORE EXTENSIONS 201 note that for i = 0 and for i = n + 1 the formula holds by the definition. If 1 5 i 5 n, then we have
-
+
[n I]! [i]![n- i l]!
+
i If q is an element of the field k, we define the element (i), of k as being if the value of the polynomial [i] at q, i.e. (i), = [i](q). Thus (i), = q # 1, and (i), = i if q = 1. Also, we define (n),! as being the value of [n]! a t q, i.e. (n)q!= [n]!(q). Note that if q = 1, then (n)q!= n!, the usual factorial. Finally, we define the q-binomial coefficients (n) for any positive 2 9 integer n and any 0 5 i 5 n by (n) = Pn,i(q). By Proposition 5.6.1, 4
(7
(i)q\yAfi)q!
we have that P,, = and then = whenever the 2 4 denominator of the fraction in the right hand side is nonzero. Nevertheless, (n) ! in any case, but with the warning that we write (n) to be (i)q!(n(li)q! 2 4 this is a formal notation, and we have to understand in fact P,,i(q) by it. Obviously, if q = 1, then is the usual The recurrence relation for 4 the polynomials Pn,ishows that
(7
(y).
(y)q
for any n 2 1 and 1 5 i 5 n. The reason for which the are called q-binomial coefficients is that they appear in a version of the binomial formula where the two terms of the binomial do not commute, and instead they anticommute by q.
CHAPTER 5. INTEGRALS
202
Lemma 5.6.2 Let a and b two elements of a Ic-algebra such that ba = gab for some q E k . Then the following assertions hold. . . (i) (The quantum binomial formula) ( a ~ b=)ELo ~ for any positive integer n. (iz) If q is a primitive n-th root of unity, then ( a + b)n = an + bn . I Proof: ( i ) T h e proof goes by induction on n. I t is obvious for n = 1. T o pass from n t o n + 1 we have that
=
(2(:) i=O
an-'bi)(a
+ b)
(induction hypothesis)
4
+
which gives the desired formula for n 1. (ii) Since q is a primitive n-th root of unity we have that ( n ) , = 0 , and ( i ) ,# 0 for any i < n. The result follows now from the quantum binomial formula. I Remark 5.6.3 We have seen in the proof of (ii) of Lemma 5.6.2 that if q is a primitive n-th root of unity, then
The converse also holds, i.e. if
5.6. HOPF ALGEBRAS CONSTRUCTED B Y ORE EXTENSIONS 203
then q is a primitive n-th root of 1. Indeed, by (y)q = 0 we see that q is a n - t h root of 1. If q is not a primitive n-th root, then q is a primitive d-th root of 1 for some divisor d of n with d < n . Then ( : ) q # 0, since
+
and q is not a root of any of the polynomials [n- 11,. . . , [n- d I ] ,[d I ] , . . . , [ I ] , and also q i s a simple root of both [n]and [dl. Thus we obtain a I contradiction, and conclude that q must be a primitive n - t h root of 1. Recall that for a k-algebra A, an algebra endomorphism cp of A, and a 9-derivation 6 of A (i.e. a linear map S : A -, A such that S(ab) = S ( a )b + q ( a ) S ( b ) for all a , b E A ) , the Ore extension A [ X ,cp, 61 is A[X] as an abelian group, with multiplication induced by X a = S ( a ) cp(a)X for all a E A. The following is an obvious extension of the universal property for polynomial rings.
+
Lemma 5.6.4 Let A [ X ,cp, S] be a n Ore extension of A and
the inclusion morphism. Then for any algebra B, any algebra rnorphism f : A -+ B and every element b E B such that bf ( a ) = f ( & ( a ) ) f (cp(a))b for all a E A, there exists a unique algebra morphism 7 : A[X, cp, S] + B such that f ( X ) = b and the following diagram is commutative:
+
We constrlict pointed Hopf algebras by starting with the coradical, forming Ore extensions, and then factoring out a Hopf ideal. Let A = kC be the group algebra of an abelian group C with the usual Hopf algebra structure, and let C* be the character group of C , i.e. C* consists of all group morphisms from C to the multiplicative group k* = k - (01, and the multiplication of C* is given by ( u v ) ( g )= u ( g ) v ( g ) for any U , V E C * a n d g E C . Let cl E C and cT E C* and take the algebra automorphism cpl of A
CHAPTER 5. INTEGRALS
204
defined by cpl(g) = c;(g)g for all g E C . Consider the Ore extension Al = A[X1,cp1,61], where 61 = 0. Apply Lemma 5.6.4 first with B = Al@Al,f = ( i ~ i ) ~ A A , b = c 1 € 3 X 1 + X l ~ 1 a n d t h e n w i t h B = k , f = € A , b = 0, to define algebra homomorphisms A : A1 + Al €3 A1 and €:A1 -kby
+
A(x1) = ci €3 X i X i €3 1 and e(X1) = 0. (5.5) It is easily checked that A and E define a bialgebra structure on Al. The antipode S of A extends to an antipode on Al by S(X1) = - c l i x l . Next, let c; E C*,712 E k*, and let cp2 E Aut(A1) be defined by,
We seek a cp2-derivation 62 of Al, such that 62 is zero on kC and &(XI) E k c . We want the Ore extension A2 = A1 [X2,92, 621 t o have a Hopf algebra structure with X2 a (1, c2)-primitive for some c2 E C, i.e. A(X2) = ca @ X2 + X2 €3 1. Then
Applying A to both sides of (5.6), we see that n2 = c;(c2)-l
= c % ( c ~and ) A(62(X1)) = clc2 @62(X1)
+ b2(X1) €3 1.
Thus 62(X1) is a (1, clc2)-primitive in k C and so we must have
for some scalar b12. If clcz - 1 = 0, then we define b12 t o be 0. If b12 = 0,
then S2 is clearly a cp2-derivation. Suppose that S2 # 0. In this case it remains to check that 62 is a cp2-derivation of Al. In order that S2 be well defined we must have, for all g E C , 62 (gX1) =
9 2 (g)62( X I )
= 62(c;(g)-lxlg) = c;(g)-'62(~1)g
Thus cp2(g) = ca(g)g = c;(g)-lg and therefore c;cs = 1 and n2
Now we compute
= c;(c2)-'
= c;(c2) = c;(cl) = c;(cl)-l
5.6. HOPF ALGEBRAS CONSTRUCTED BY ORE EXTENSIONS 205 and, by induction, we see that for every positive integer r, we have
A straightforward but tedious computation now ensures that for g, g1 E C,
and our definition of A2 = A1[X2,92, 621 is complete. Summarizing, A2 is a Hopf algebra with generators g E C , X1,X2, such that the elements of C are commuting grouplikes, X j is a (1, cj)-primitive and the following relations hold: gXj = cj*(g)-lXjg and X2Xl - y12XlXz = ~ I ~ ( C-I l), C~ where 712 = c;(c2)-l = c;(cl);, and, if & ( X I ) # 0,
&
We continue forming Ore extensions. Define an iutomorphisrn of Aj-1 by pj (g) = cj'(g)g where cj' E C* , and pj(Xi) = cj'(ci)Xi where ci E C , and Xi is a (l,ci)-primitive. The derivation 6j of Aj-1 is 0 on k C and 6j(Xi) = b,,(cicj - 1). If cicj = 1, we define bij = 0. We write XP for Xr' . . . Xf' where p E Nt. After t steps, we have a Hopf algebra At. Definition 5.6.8 At is the Hopf algebra generated by the elements g E C a n d X j , j = 1,. . . ,t where i) the elements of C are commuting grouplikes; ii) the X j are (1, cj)-primitives; iiz) Xjg = cjt (g)gXj; iv) XjXi = c;(ci)XiXj bij(cicj - 1) for 1 5 i < j 5 t; V) cf (cj)c:(ci) = 1 for j # i; vi) If bij # 0 then cfcj = 1. vii) If cicj = 1, then bij = 0. The antzpode of At is given by S(g) = g-' for g E G and S(Xj) = -cjlxj. I
+
The relations show that At has basis {gXPlg E C, p E N t ) . Since for 9j = c;(cj), ( X j @ l ) ( c j @ X j ) = qj(cj @ Xj)(Xj @ I),
,
CHAPTER 5. INTEGRALS
206
then, for n E Z+ , A(Xjn) = A(Xj), = (cj @ X j+ X j 8 l),, and expansion of this power follows the rules in Lemma 5.6.2. For g E C, p = (pl,. . . ,pt) E Nt,
where d = (dl,. . . ,dt) E Z t , the j-th entry d j in the t-tuple d ranges from 0 to pj, and the ad are scalars resulting from the q-binomial expansion described in Lemma 5.6.2 and the commutation relations. In particular, for 15 j 5 t , n E Z+,
Proposition 5.6.11 The Hopf algebra At has the following properties: (2) The term, (At),, in the coradical filtration of At is spanned by gXP, g E C, p E N t , pl . . . + pt 5 n. In particular, At is pointed with coradical k c . (ii) The Hopf algebra At does not have nonzero integrals.
+
Proof: (i) An induction argument using Equation (5.9) shows that for all n, < gXP(gE C, p~ N t , p l + ... + p t 5 n >C ~ ( , + l ) k c . Thus, A ( ' ~ ) ~ C= At and by Exercise 3.1.12 Corad(At) k c . Since k C is a cosemisimple coalgebra, it is exactly the coradical of At. (ii) This follows from Theorem 5.5.8, since (At)o = k C is a Hopf subala gebra and the coradical filtration is infinite.
Exercise 5.6.12 Give a different proof for the fact that At does not have nonzero integrals, by showing that the injective envelope of the simple right At-comodule kg, g E C , is infinite dimensional. In order to obtain a Hopf algebra with nonzero integral, we factor At by a Hopf ideal.
Lemma 5.6.13 Let n l , nz, . . . , nt ideal J ( a ) of At generated by
> 2 and a = (al, . . . ,at) E {O,lIt.
The
is a Hopf ideal if and only if qj = cj*(cj) is a primitive nj-th root of unity for any 1 5 j s t .
5.6. HOPF ALGEBRAS CONSTRUCTED B Y ORE EXTENSIONS 207 Proof: Since c y -1 is a (1, ~7~)-prirnitive, it follows that X,") - a j ( c y 1) is a (1, ~ y ) - ~ r i m i t i vif eand only if so is By (5.10) and Remark 5.6.3, this occurs if and only if = 0 for every 0 < k < n j , i.e. if and only
(7)qi
x?.
if qj is a primitive nj-th root of unity. Moreover, since S(Xj) = - c y l x j , induction on n shows that
Now, since qTi = 1, checking the cases n j even and n j odd, we see that (-l)"lq~n~(n'-1)12 = -1 and hence
for 1 5 j 5 t , so that the ideal J ( a ) is invariant under the antipode S, and is thus a Hopf ideal. 1 By Lemma 5.6.13, H = At/ J ( a ) is a Hopf algebra. However the coradical may be affected by taking this quotient. Since we want H to be a pointed Hopf algebra with coradical kC, some additional restrictions are required. We denote by xi the image of Xi in H and write xp for xY1 . . . x y , p = (pl, . . . ,pt) E N t .
Proposition 5.6.14 Assume J ( a ) as in Lemma 5.6.13 is a Hopf ideal. Then J ( a ) n k C = 0 if and only if for each i either ai = 0 or ( ~ f =) 1.~ ~ If this is the case then {gxplg E C , p E Nt, 0 5 pj 5 n j - 1) is' a basis of AtlJ(a). Proof: By Lemma 5.6.13, we know that J ( a ) is a Hopf ideal if and only if qz = cf (ci) is a primitive ni-th root of unity for 1 5 i 5 t . Now suppose that J ( a ) n k C = 0. Since
is in J ( a ) for every g E C, it follows that
c l ~ and (1-cf (g)-nt)Xr' But then for every g E C , both ~ ~ ( l - c f ( g ) - ~ " ( -1) are in J ( a ) . If ai # 0, which by our convention implies that cli - 1 # 0, then we must have cf (g)ni = 1 for all g, and thus c:ni = 1. Conversely, assume that cfna = 1 whenever ai # 0. By Definition 5.6.8 (iii), XTag = ~ f ( g ) ~ i g X "In~ .particular, Xznig= gXTi if ai # 0.. Also, if
208 i
CHAPTER 5. INTEGRALS
< j then by (5.7),
So, if bij = 0, then XjXzna= cj*( c ~ ) ~ ~ where X ~ ~ c; (ci)"z X ~ ,= cf ( ~ ~ ) = - ~1 i if ai # 0. If bij # 0 then cfcj' = 1, hence cf(ci) is a primitive ni-th root of unity, so that XjXzni = XYiXj. A similar argument works for i > j . Thus, Xzni is a central element of At if ai # 0. It follows that
so that J ( a ) is equal to the left ideal generated by
{x? - aj(c,"'
j _< t), and At is a free left module with basis {XplO
- 1)(15 - 1)
5 pj 5 n j
over the subalgebra B generated by C and X;', . . . ,X r t . We now show that no nonzero linear combination of elements of the form gXP, p E N t , 0 5 p j 5 n j - 1 lies in J(a). Otherwise there exist f j E A,, 1 j t, not all zero, such that
< <
where in the second sum g E C , p E N t , 0 5 p j 5 n j - 1. Since At is a free left B-module with basis {XplO 5 p j 5 n j - 11, each f j can be expressed in terms of this basis, and we find that
for some Fj E B. Now, B is isomorphic to the algebra R obtained from IcC by a sequence of Ore extensions with zero derivations in the indeterminates Y , = X y i , SO that yig = cfni(g)gyi and Y,Y, = ~ j * ~ j ( c y ~ ) Y ,Thus, Y , . we have
for some G j E R. It follows from Lemma 5.6.4 by induction on the number of indeterminates that there exists a kc-algebra homomorphism 0 : R -+ k c such that 0(Y,) = cj"j - 1 if a j # 0 and B(Y,) = 0 otherwise. Then B(~,,,.,,(Y, - aj(c? - 1))Gj) = 0, a contradiction. I
5.6. HOPF ALGEBRAS CONSTRUCTED BY ORE EXTENSIONS 209 From now on, we assume that nj 2 2, qj = cj'(cj) is a primitive nj-th root of 1, and cj*"j = 1 whenever aj # 0 , and we study the new Hopf algebra H = A t / J ( a ) . We have shown that the following defines a Hopf algebra structure on H .
Definition 5.6.15 Let t 2 1, C an abelian group, n = (nl,. . . , nt) E N t , c = ( c j ) E C t , c * = (c;) E C * t , a E ( 0 , b = (bij)l
-
1)
+ bji(cjci - 1)
< i 5 t The coalgebra structure is given by
A ( x i ) = ci @ x i f
Xi
8 1, for 1 5 i 5 t
I Remark 5.6.16 i ) If ai = 0 for all i , we write a = 0 . Similarly if bij = 0 for all i < j , we write b = 0 . If t = 1 so that no nonzero derivation occurs, we also write b = 0. ii) If a = 0 and b = 0 , then we write H = H ( C , n , c , c*) instead of H ( C , n, c, c*,0,O). iii) If i n Definition 5.6.15, the ai 's were arbitraq elements of k , then a simple change of variables would reduce to the case where the ai 's are 0 or 1.
iv) Since the elements bti do not appear i n the defining relations for
we always take them to be zero.
CHAPTER 5. INTEGRALS
210
Remark 5.6.17 In order to construct H ( C ,n , c, c*, a, b), it sufices to have c* and c such that cf (ci) is a root of unity not equal to 1, and cz(cj)c$(ci)= 1 for i # j . Then ni is the order of cf ( c i ) , and we choose a and b such that ai = 0 whenever c l i = 1, ai = 0 whenever cbni # 1, bij = 0 whenever cicj = 1, and bij = 0 whenever c; cf # 1. The remaining ai 's and bij 's are arbitrary. I
<
By Proposition 5.6.14, {gxplg E C , p E N t , 0 pj 5 nj - 1) is a basis for H . As in Equation (5.9), comultiplication on a general basis element is given by
A ( g x p )=
1adgcfl
C$
. . . c$ xp-* 8 g x d ,
(5.18)
d
<
where d = ( d l , . . . ,d t ) E Zt with 0 5 d j p j . Here the scalars nonzero products of qj-binomial coefficients and powers of c;(ci). In particular, for n E Z+,
ad
are
Proposition 5.6.20 Let H = H ( C , n, c, c*,a, b). Then H is pointed and the ( r 1)st term in the coradical filtration of H is H, =< gxplg E C , p E N t , p l + ...+pt 5 r >. InparticularH= Hn w h e r e n = n l + ...+n t - t so that the coradical filtration has nl . . . nt - t 1 terms.
+
+ +
+
Proof. The proof is similar to the proof of Proposition 5.6.11. The second part follows from the fact that the ad are nonzero. I Unlike A t , the Hopf algebra H has nonzero integrals. We compute the left and right integrals in H* explicitely. For g E C , and w = ( w l ,. . . ,w t ) E Z t , let E,,, E H* be the map taking gxw to 1 and all other basis elements to 0. Proposition 5.6.21 The Hopf algebra H = H ( C , n, c, c*, a, b) has nonzero integrals. The space of left integrals i n H* is kEl,,-l, where 1 = C 1 - n ~1-na . . . ~ : - ~ ~ = n : =( ,n ~j- ,1 ), and where n - 1 is the t-tuple (nl -
.,
1,. . . , nt - I ) . The space of rzght integrals for H is kE1,,-l the identity in C .
where 1 denotes
Proof. We show that is a left integral by evaluating h*El,n-l for h* E H*. This is nonzero only on elements z €3 lxn-l and such an element
5.6. HOPF ALGEBRAS CONSTRUCTED BY ORE EXTENSIONS 211 ,
can only occur as a summand in t
where y E k*. Now h * ~ ~ , , - ~ ( l = x ~h -* ~( l)) E l , , - l ( l ~ ~ - ~ ) . Similarly xn-I @ z only occurs in A(xn-I). Since A ( x n - l ) = xnL1 8 1 . . , thus El,,-lh* = ElYn-lh*(l). I
+.
Corollary 5.6.22 H is unimodular if and only if 1 = 1.
I
I f G is a group and g = ( g l ,. . . ,gt) E G t , we write g-I for the t-tuple ( g ~ l ., .. , g t l ) . Example 5.6.23 ( i ) I f H = H ( C ,n, c, c*,a, b) then Hop and HCOpare also of this type. Indeed, Hop H ( C , n ,c, c*-l, a , b'), where b:j = -cj*(ci)bij for i < j. Also HCOpS H ( C ,n, c-l, c*,a, b"); the isomorphism is given by the map f taking g to g and xj to zj = - c j l x j . Then zj is a (1,~ ; ' ) - ~ r i m i t i vand, e using the fact that (-1)njq;n'(n~-1)'2 - -1 where qj is a primitive n j - t h root of 1 , we see that its n j - t h power is either 0 or c ~ - ~' 1. The last parameter, b", is given by b$ = -c;(ci)bij for i < j. (ii) In particular i f H = H ( C ,n , c, c*) then
Hop E H ( C ,n, c, c*-l) and HCopE H(C,n,c-l,c*). (iii)The Taft Hopf algebras, in particular Sweedler's 4-dimensional Hopf algebra, are o f this form. I Exercise 5.6.24 Let A be the algebra generated by an invertible element 0 and ab = Xba, where X is a primitive a and an element b such that "b' 2n-th root of unity. Show that A is a Hopf algebra with the comultiplication and counit defined by
Also show that A has nonzero integrals and it is not unimodular. Exercise 5.6.25 Let H be the Hopf algebra with generators c , x l , . . . , x t subject to relations
A(c)=c@c, A(xi)=c@xi+xi@1. Show that H is a pointed Hopf algebra of dimension 2t+1 with coradical of dimension 2.
212
CHAPTER 5. INTEGRALS
The following exercise shows that our assumption that the derivations are zero on k C is not unreasonable.
Exercise 5.6.26 Let # be an automorphism of k C of the form # ( g ) = c*(g)g for g E C , and assume that c*(g) # 1 for any g E C of infinite order. Show that if 6 is a #-derivation of k C such that the Ore extension ( k C ) [ Y+,6] , has a Hopf algebra structure extending that of k C with Y a ( 1 ,c)-primitive, then there is a Hopf algebra isomorphism ( k C ) [ Y#,6] , E ( k c )1-T#I. We now classify Hopf algebras of the form H ( C , n, c, c*,a, 0 ) , i.e. they are constructed by using Ore extensions with zero derivations. Suppose H = H ( C , n,c, c*, a, 0 ) E H' = H ( C 1 ,n', c', c*', a', 0 ) and write g, xi (g', x:) for the generators of H (H' respectively). Let f be a Hopf algebra isomorphism from H to HI. Since the coradicals must be isomorphic, we may assume that C = C', and the Hopf algebra isomorphism induces an automorphism of C . Also by Proposition 5.6.20, t = t'. If 7r is a permutation ) ( v , ( ~ ).,. . ,v,(~)). of ( 1 , . . . , t ) and v E Z t , we write ~ ( vfor
Theorem 5.6.27 Let
H = H ( C ,n,c, c*,a, 0 ) and H' = H(c', n', c', c*',a', 0 ) be Hopf algebras as described above. Then H S H' if and only if C = C' (in fact we should write C E C ' , but we take for simplicity C = C'), t = t' and there is an automorphism f of C and a permutation 7r of { I , . . . ,t ) such that for 1 I i 5 t
Proof. We have seen that C CY C' as the group of grouplike elements in the two Hopf algebras. Assume that C = C'. Let I = {ill 5 i I t , ci = cl,c: = c ; ) and let
Note that since c:(ci) is a primitive ni-th root of 1 and for i E I , c:(ci) = c;(c1),then ni = nl for i E I . Similarly, since for j E J , cj"(ci) = c;'(f ( c l ) ) = c ; ( c l ) , = nl for j E J . Let L be the Hopf subalgebra of H generated by C and { x i ( i E I) and L' the Hopf subalgebra of H' generated by C and {xi 1j E J ) .
ni
5.6. HOPI? ALGEBRAS CONSTRUCTED B Y ORE EXTENSIONS 213 Since xl is a (1,cl)-primitive, f (xl) is a (1, f (cl)-primitive and so
f (51) = ao(f (cl) - 1) +
xsix;,
with
ai
E k,ji E j.
i=l
Then, since gxl = cr(g)-'xlg for all g E C , we see that
a 0 = 0,
and
L'. The and thus ai = 0 for any i for which c; # c;: o f . Thus f(L) same argument using f-' shows that f-'(L') & L and so f ( L ) = L'. If L # H , we repeat the argument for M, the Hopf subalgebra of H generated by C and the set {xilci = cp,c; = c;) where x, is the first element in the list 22,. . . , xt which is not in L. Continuing in this way, we see that there exists a permutation o such that
It remains to find T such that ai = a' . First suppose nl > 2. Then a(%). I = (1). For if p E I, p # 1, then
~ 1, a contradiction. Similarly J = { ~ ( l ) ) .Hence f ( x l ) = and c ; ( c ~ )= ax&,) for some nonzero scalar a, and the relation xyl = al(cyl - 1) implies anlxl
n'"(1) - al(eb;;, -
I), so that a;(,) = a l . 41) Next suppose nl = 2. Let Il = {i E Ilai = 1) and J1= { j E Jlajt = 1). For any i E I, there exist aij E k such that f (xi) = Cj, aijx>. As above, for all i E I, cz (ci) = -1 (for all j E J, cjl(c$) = -1) and thus the xi (respectively the xi) anticommute. If i E 11, f applied to x: = c; - 1 yields CjEJl a:j = 1. On the other hand, comparing f (xisk) and f (xkxi) for i, k E 11, i # k, we see that
and thus Cj, Jl a i j a k j = 0. This implies that the vectors Bi E kJ1, defined by Bi = (aij)jEJ1 for i E 11, form an orthonormal set in kJ1 under the ordinary dot product. Thus
CHAPTER 5. INTEGRALS
214
the space k J 1 contains at least /I11 independent vectors and so 1 Jl 1 2 [ I l1. The reverse inequality is proved similarly. Now define n to be a refinement of the permutation a such that for i E 11,n(i)E J1 and then ai = for all i E I . Conversely, let f be an automorphism of C and let n be a permutation of { 1 , 2 , . . . , t ) such that for all 1 5 i 5 t , ni =
nici,,f (ci) = c : ( ~ )C:,
= c $ ~ )0
f , and ai = a&i).
Extend f to a Hopf algebra isomorphism from H to H' by f ( x i ) = x : ( ~ ) . If we note that c$(i,(c:(j)) = c:;i) ( f ( c j ) )= cT(cj) the rest of the verification that f induces a Hopf algebra isomorphism is I straightforward. Note that in the proof above, it was shown that if n k > 2, then 111 = IJI = 1 where I = {ill 5 i 5 t , ci = ck,c: = c i ) and J = { j l l 5 j 5 t , $ = f ( c k ) c;' , o f = c:). Thus we can also classify Hopf algebras of the form H ( C , n, c, c*,a, b) if all ni > 2. We revisit later the case where some ni = 2.
Theorem 5.6.28 Let
H = H ( C , n, c, c*, a, b) and
H' = H ( C 1 ,n', c', c*',a', b') be such that all ni and n: > 2. Then H H' zf and only if C = C' (in fact we should write again C E C ' , but we identify C and C'), t = t' and there is an automorphism f of C , nonzero scalars (ai)lli
ali = 1 for
0
f , and ai = a'(i) ,
any i such that ai = 1, and for any 1 5 i < j
< t,
Proof: The argument is similar to that in Theorem 5.6.27. An application of the isomorphism f to the equation xjxi = c;(ci)xixj+bij(cicj- l ) , i < j, yields the relationship between b and b'. I In [107],I. Kaplansky conjectured that there exist only finitely many isomorphism types of Hopf algebras of a fixed finite dimension over an algebraically closed field of characteristic zero. The following corollary answers in the negative this conjecture.
5.6. HOPE' ALGEBRAS CONSTRUCTED B Y O R E EXTENSIONS 215
Corollary 5.6.29 Suppose that C , c E C t ,c* E C * t , are such that c;(cl) = ci(cj)-' i f 1 # j , c;(ci) is a primitive root of unity of order ni > 2, and there exist i < j such that c,fni = cJn' = 1, cli # 1, c? # 1, cicj # 1, and c,tc; = 1. Then for any a with ai = a:, = 1 and satisfying the conditions of Remark 5.6.17, there exist infinitely many non-isomorphic Hopf algebras of the form H ( C ,n, c, c*,a , b). Proof: Let b and b' be such that H = H ( C , n, c, c*,a , b) and H' = H ( C , n , c, c*,a , b') are well defined. By Remark 5.6.17, infinitely many such b and b' exist. I f f : H -+ H' is a Hopf algebra isomorphism, then the permutation .ir in Theorem 5.6.28 is the identity and thus bij = aiajb:j for some n i t h and n j t h roots o f unity ai and c u j . Since there exist only finitely many such roots, and k is infinite, the result follows. a Example 5.6.30 T o find a concrete example o f a class consisting o f infinitely many types o f Hopf algebras o f the same finite dimension, we need some data ( C ,c, c*) as in Corollary 5.6.29, with C finite. T h e simplest such data are the following. ( i ) Let p be an odd prime, and p a primitive p t h root o f 1. Take C = Cp2 =< g >, the cyclic group o f order p 2 , t = 2, c = ( g ,g ) , c* = ( g * , g * - l ) where g*(g) = p and a = ( 1 , l ) . Then nl = n2 = p and by Corollary 5.6.29, H ( C , n , c, c*,a , b) 2 H ( C ,n , c, c*,a, b') i f and only i f b12 = ybi2 for y a primitive p-th root o f 1. Thus there are infinitely many types o f Hopf algebras o f dimension p4. (ii) Let C = Cp, =< g >, the cyclic group o f order pq where p is an odd prime, q > 1 , and t = 2, c = (g,g),c* = (g*,g*-l) where g*(g) = p, p a primitive p-th root o f 1. Let a1 = a:! = 1. Then again nl = n:! = p, and as in ( i ) , there are infinitely many types o f Hopf algebras H ( C ,n, c, c*,a , b) o f dimension p3q. I Exercise 5.6.31 (i) Let C = C4 =< g >,t = 2 , n = ( 2 , 2 ) , c= (g,g),c* = (g*,g*) where g*(g) = -1, b12 = 1 , a = ( 1 ,l ) , a l = ( 0 , l ) . Show that there exists a Hopf algebra isomorphism H ( C ,n, c, c*,a , b) H ( C ,n, c, c*,a', b). (ii) Let C = C4 =< g >, t = 2, n = ( 2 , 2 ) ,c = ( g ,g), c* = (g*,g*) where g*(g) = -1,a = ( 1 , l ) and bl2 = 2,a1 = ( 0 ,l ) ,bi2 = 0 . Show that the Hopf algebras H ( C ,n, c, c*,a', b') and H ( C , n, c, c*,a , b) are isomorphic.
--
Exercise 5.6.32 Let C be a finite abelian group, c E C t and c* E C*t such that we can define H ( C , n, c, c*). Show that H ( C , n, c, c*)* E H ( C * ,n,c*,c ) , where i n considering H ( C * ,n, c*,c ) we regard c E C** by identifying C and C**. Remark 5.6.33 The previous exercise shows that i f H = H ( C ,n, c, c*) where C is a finite abelian group, then H 2 H* zf and only if there is
-
CHAPTER 5. INTEGRALS
216 a n isomorphism f l L j L t , %(j)
= nj,
:
C
-+
C* and a permutation
IT
E St such that for all
f ( c j ) = c : ( ~ ) ,< f ( c j ) , g >=< f ( g ) ,c , z ( ~ )> for all g
E C.
Let us consider now the case where C =< g > will be a cyclic group, either of order m , or infinite cyclic. We first determine for which values of the parameters t and m, finite dimensional Hopf algebras H = H(C,, n, c, c*, a , b) exist. By Remark 5.6.17, for a given t , in order to construct H , we need c E C&, c* E (CA)tsuch that ca(ci) is a root of unity different from 1 , and cf(cj)cj*(ci)= 1 for i # j . Let C be a primitive mth root of unity, and then g* E C& defined by g * ( g ) = 5 generates C&. Thus we may write ci = gut and cf = g*di. To find suitable c and c*, we require u, d E Zt with ui, di E Z mod m such that,
( d i u j + d j u i ) = 0 if i # j and diui $0.
(5.34)
Then H will be the Hopf algebra with basis gixP, p E Z t , 0 5 pi 5 ni and 0 5 i 5 m - 1 , and such that
. x . - 5 d . u '. X i x j
3
z -
+ bij(gui+u'
-
1 ) for 1 5 i < j 5 t.
Proposition 5.6.35 Let m be a positive integer. i ) If m is even, then the system (5.34) has solutions for any t. ii) If m i s odd, then the system (5.34) has solutions if and only if t where s i s the number of distinct primes dividing m.
< 2s,
Proof: i) If m = 2r then di = r,ui = 1 , 1 5 i 5 t , is a solution of (5.34). ii) We first prove by induction on s that the system has solutions for t = 2 s and thus for any t 5 2s. If s = 1 then dl = ul = 1 = u 2 , d 2 = -1 is a solution of (5.34). Now suppose the assertion holds for s - 1 and let m = P, . . .pFs with the pi prime. Then m' = m/pF* has s - 1 distinct prime divisors, so by the induction hypothesis there exist d:, ui for 1 i 5 2 s - 2 , such that (d:ui d ' . ~ ! ) 0 mod m' for 1 5 i # j 5 2 s ? 2 and d:u: $ 0 mod m' for 1 5 z 5 2 s - 2. Now a solution of the system for t = 2 s is given by di = p?~d:,ui = p z ~ u :for 1 5 i 5 2 s - 2 and d2s = d2s-1 = ~ 2 =~m '-, ~ ~2 =~ -m'. Next we show that for m = pa and t = 3 the system has no solutions. Suppose d , u E Z3 is a solution, and suppose di = d:pai, ui = uLPPi where ( d : , p ) = (u',,p) = 1 for 1 5 i 5 3. For i # j , pa divides di% + d j u i =
+
<
5.6. HOPF ALGEBRAS CONSTRUCTED BY ORE EXTENSIONS 217
+
d'.ul. , 3 +pa3 +pi diu6, and so ai ,L$ = aj +Pi. Since pa does not divide diui for any i, then ai +Pi < a, so ai ,Bj aj +pi < 2 a for all i, j . Thus dju; E ;d;ui mod p for all i # j . Multiplying these three congruences, we Eu0~mod p, a contradiction. obtain d l d 2 d 3 u ~ u ~ Now suppose that m = pyl . . .p:s and 2s 1 t. If the system had a solution d, u, then for every i there would exist ji, 1 ji 5 s, such that does not divide diui. By the Pigeon Hole Principle we find i l , i2,i3 such that jil = jiz = jis;denote this integer by j . Then does not divide any dkuk,but divides dku, +druk for a11 distinct r, k E {il, i2,i3), and this contradicts what we proved in the case m = pa. I
p"i+flj
+ +
+ <
pzi
<
pF
Corollary 5.6.36 i) If m is even, then Hopf algebras of the form
exist for every t . ii) If m is odd, then H(Cm,n, c, c*, a , b) exist for any t the number of distinct prime factors of m.
< 2s , where s
is
I
Corollary 5.6.37 If C =< g > is an infinite cyclic group, then Hopf algebras H(C, n , c, c*,a , b) exist for all t.
<
Proof: Let t be a positive integer and choose m such that t 2s where s is the number of distinct prime divisors of m. Then by Proposition 5.6.35, there exist d,, u,, 1 5 i 5 t solutions for the system (5.34). Now let c, = gUa and c: = g*dafor g*(g) = C, a primitive m-th root of 1, as before. I The classification results presented in Theorem 5.6.27 and Theorem 5.6.28 depend upon knowledge of the automorphism group of C . In case C is cyclic, Aut(C) is well known, and Theorem 5.6.27 specializes to the following. Proposition 5.6.38 If C =< g > is cyclic, then H(C, n, c, c*,a , 0 ) % H(C1,n', c', c*', a', 0) if and only if C = C', t = t' and there is an automorphism f of C mapping g to gh and a permutation .rr of (1,. . . , t) such that ni = n&i,, c: = c'4%) . (i.e. hui E unci)), c: = ( c $ ~ , ) (i.e. ~ di = hd,(i)), and ai = a'~ (. 2 ) . If C is cyclic of order m , then (h, m) = 1; if C is infinite cyclic, then h = 1 or h = -1. I
If C is cyclic, then it is easy to see when H(C,, n, c, c*) is isomorphic to its dual, its opposite or co-opposite Hopf algebra.
C H A P T E R 5. INTEGRALS
218
,
Corollary 5.6.39 Let C = C , =< g >, finite, and H = H(C,, n, c, c*) where ci = gui, cf = ( g * ) d i and < g*, g >= C, a fixed primitive m t h root of I. (i) H H* if and only i f there exist h , .rr as i n Proposition 5.6.38 such that for all 1 5 j 5 t , n,(j) = n j ,
huj
= d,(j)
mod m, u , z ( j )
--
uj
mod m.
In particular a Tuft Hopf algebra is selfdual. (ii) H 2 HcOp i f and only if there exist h,.rr such that for all 1 < j 5 t ,
n,(j) = n j , h u j (iii) H
-u,(~) mod m, d j
hd,(j) mod m.
Hop if and only if there exist h, IT such that for all 1 5 j 5 t , n,(j) = nj, h u j
= u,(j)
mod m, d j
-hd,(j) mod m.
I We study now Hopf algebras of the form H ( C , n, c*, c, 0 , I ) , where b = 1 means that bij = 1 for all i < j. Thus, the skew-primitives xi are all nilpotent and for i # j , xixj - cf ( c j ) x j x iis a nonzero element of k c . It is easy to see that if a = 0 and all bij are nonzero, then a change of variables ensures that all bij equal 1. This class produces many interesting examples. The following two definitions are particular cases of Definition 5.6.15.
Definition 5.6.40 For t = 2, let n 2 2, c = ( c l ,cp) E C 2 , g * E C* with g*(cl) = g*(c2) a primitive n-th root of unity, and clc2 # 1. Denote the pair ( n , n ) by ( n ) , and, if cl = c2 = g, denote (cl,c2) by ( g ) . Then H ( C , ( n ) (, c l ,c 2 ) ,( g * , g*-l), 0 , l ) denotes the Hopf algebra generated by the commuting grouplike elements g 6 C , and the ( 1 ,cj)-primitives xj, j = 1,2, with multiplication relations
23 = 0 , x1g =< g*,g > gx1,
x2g
=< g*-l,g > gxp
Definition 5.6.41 Let t > 2 and let c E Ct,g* E C* such that g*(ci) = -1 for all i and cicj # 1 if i # j . We denote the t-tuple ( 2 , . . . , 2 ) by ( 2 ) , and the t-tuple (g*,. . . , g * ) by (g*). Then H ( C , ( 2 ) ,( c l ,. . . , c t ) , ( g * ) ,0 , l ) is the Hopf algebra generated by the commuting grouplike elements g E C , and the ( 1 ,c j )-primitives xj , with relations
x: = 0 ,
xig = g*(g)gxi, xixj
+ x j x i = cicj - 1 for i # j.
5.6. HOPF ALGEBRAS CONSTRUCTED BY ORE EXTENSIONS 219 Example 5.6.42 (i) Let C, =< g > be cyclic of finite order m 2 2, let n be an integer 2 2, and let cl = gul, c2 = gu2,g* 6 C* be such that g*(g) = X where Am = 1, ul + u2 $ 0 mod m, and Xu' = Xu2, a primitive nth root of 1. Then H = H(C,, (n), c, (g*, g*-'), 0 , l ) is a Hopf algebra of dimension mn2, with coradical kc, and generators g, X I ,x2 such that g is grouplike of order m , xi is a (l,gut)-primitive, and
, =< g >. (ii) Let m L 2 , t > 2 be integers, m even, and let C = C Let u l , . . . , ut be odd integers such that ui u j $ 0 mod m if i # j and let ci = gui,cf = g* where g*(g) = -1. Then the Hopf algebra H(Cm, (2),c, (g*),0 , l ) has dimension 2tm and has generators g, X I , .. . ,xt such that g is grouplike, xi is a (1,gut)-primitive, and
+
(iii) Suppose C =< g > is infinite cyclic, and n 2 2. Let ul,u2 be integers such that ul u2 # 0, and let X E k such that Xu' = Xu2 is a primitive nth root of 1. Let g* E C* with g*(g) = A. Then there is an infinite dimensional pointed Hopf algebra with nonzero integral
+
with generators g, x1,x2 such that g is grouplike of infinite order, xi is a (1, gu"-primitive, and
(iv) Let C =< g > be infinite cyclic, t > 2 and let 211,. . . , ut be odd integers such that u,+uJ # 0 for i # j . Then there is an infinite dimensional pointed Hopf algebra with nonzero integral H(C, (2), c, (g*),0, l ) , where c, = gut and g*(g) = -1. The generators are g , x l , . . . ,xt such that g is grouplike of infinite order, x, is a (1, gua)-primitive,and
By an argument similar to the proof of Theorem 5.6.27, we can classify the Hopf algebras from Definition 5.6.40.
220
CHAPTER 5. INTEGRALS
Theorem 5.6.43 There is a Hopf algebra isomorphism from
to H' = H ( C 1 ,(n'),c', (g*',(g*')-I),0,1) if and only if C = C ' , n = n' and there is an automorphism f of C such that (i) f ( c l ) = ci ,f (c2)= C: and g* = g*' o f ; or (ii) f ( c l ) = cb, f (c2)= C; and g* = (g*')-' o f.
Proof: I f H H', then exactly as in the proof o f Theorem 5.6.27, there exists an automorphism f o f C and a bijection n o f { 1 , 2 } such that f (ci) = c$) and cf = c S g o f . The conditions ( i ) and (ii) in the statement correspon t o n the identity and n the nonidentity permutation. Conversely, i f ( i ) holds, define an isomorphism from H t o H' by mapping g t o f ( g ) and xi t o xi. I f (ii) holds, define an isomorphism from H t o H' by mapping g t o f ( g ) ,X I t o x i and xz t o -g* ( c l ) x i . I Corollary 5.6.44 If C =< g > is cyclic, then the Hopf algebras H and H' above are isomorphic i f and only i f C = C',n = n', and there is an integer h such that the map taking g to g h is an automorphism of C and either (2) cf = C f ' h and c f = guih = gu: = c: for i = 1,2; or (ii) cf = ( C f ' ) - hand g"lh = g";, guZh = 9U''. 4 For the Hopf algebras o f Definition 5.6.41 there is a similar classification result.
Theorem 5.6.45 There is a Hopf algebra isomorphism from
to H' = H ( C ' , (2), c', (g*'),0 , l ) if and only if C = C 1 , t = t' and there is a permutation n E St and an I automorphism f of C such that f (ci) = c&, and g* = g*' o f .
Corollary 5.6.46 Suppose C =< g > is cyclic. Then H and H' as above are isomorphic if and only if C = C', t = t' and there exists a permutation n E St and an azltomorphism of C taking g to gh, such that cf = gUih = for all i. I
5.7. SOLUTIONS TO EXERCISES
221
In Exercise 5.6.31 we saw that if a # 0, Ore extension Hopf algebras with nonzero derivations may be isomorphic to Ore extension Hopf algebras with zero derivations. The following theorem shows that if a = 0, this is impossible.
Theorem 5.6.47 Hopf algebras of the form H ( C ,n,c, c*) = H ( C ,n, c, c*,0,O) cannot be isomorphic to either the Hopf algebras of Definition 5.6.40 or Definition 5.6.41. Proof: Suppose that f : H ( C 1 ,(n'),c', ( g * ' g*'-l), , 0 , l ) -r H ( C ,n, c, c*)
is an isomorphism of Hopf algebras. Then, as in the proof of Theorem 5.6.27, we see that C = C', f ( x i ) = Cicvixi and f ( x ; ) = Pixi for scalars ai, Pi. But f applied to the relation
xi
yields C , , u i P j ( z j z i - (9"')-'(c;)xixj) = 1 - 1 in H ( C , n, c, c*), where 1 # 1 is a grouplike element. The relations of an Ore extension with zero derivations show that this is impossible. Similarly, H ( C , n, c, c*) cannot be I isomorphic to a Hopf algebra as in Definition 5.6.41.
5.7
Solutions to exercises
Exercise 5.1.6 Let H be a Hopf algebra over the field k, K afield extension H the Hopf algebra over K defined in Exercise 4.2.17. of k, and H = K If T E H* is a left integral of H , show that the map T E p* defined by T ( 6 @ k h ) = S T ( h ) is a left integral of p. Solution: Let S @ k h E H . Then
showing that
T is a left integral of H .
CHAPTER 5. INTEGRALS
222
Exercise 5.1.7 Let H and H' be two Hopf algebras with nonzero integrals. Then the tensor product Hopf algebra H 8 H' has a nonzero integral. Solution: Assume that H and H' have nonzero integrals. Then we show that t 8 t' is a left integral for H 8 H ' , and this is obviously nonzero. Note that we regard H* 8 H" as a subspace of ( H 8 H1)*,in particulat t 8 t' E ( H 8 HI)* is the element working by ( t 8 t l ) ( h8 h') = t ( h ) t l ( h ' )for any h E H , h' E HI. If h E H , h' E H' we have that
showing that indeed t @ t' is a left integral.
Exercise 5.2.5 Let H be a Hopf algebra. Then the following assertions are equivalent: i) H has a nonzero left integral. ii) There exists a finite dimensional left ideal i n H * . iii) There exists h* E H* such that K e r ( h * ) contains a left coideal of finite codimension i n H . Solution: It follows from Corollary 5.2.4 and the characterization of H*Tat given in Corollary 2.2.16. Exercise 5.2.12 A semisimple Hopf algebra is separable. Solution: Let t E H be a left integral with ~ ( t=)1. We show that
is a separability idempotent. Since compute
C t l S ( t z ) = ~ ( t )=l 1, let x
E H and
5.7. SOLUTIONS TO EXERCISES
Exercise 5.2.13 Let H be a finite dimensional Hopf algebra over the field k , K a field extension of k , and H = K @ k H the Hopf algebra over K defined i n Exercise 4.2.17. If t E H is a left integral i n H , show that T = l K B~ t E H is a left integral i n H . A s a consequence show that H is semisimple over k i f and only if H is sen~isimpleover K . Solution: For any S 81,h E H we have that
which means that ? is a left integral in ?T. The second part follows immediately from Theorem 5.2.10. Exercise 5.3.5 Let H be a finite dimensional Hopf algebra. Show that H is injective as a left (or right) H-module. Solution: Since H* has nonzero integrals, we have that H* is a projective left H*-comodule. By Corollary 2.4.20 we see that H is injective as a right H*-comodule. Since the categories M ~ and * H M are isomorphic, we obtain that H is an injective left H-module.
h
and h E H be such that x o S ( h ) = 1. Then x Exercise 5.4.8 Let x E spans Solution: The existence of x and h as in the statement was proved in 5.4.4, only in that proof we have used the uniqueness. So we show first that this can be done directly. Let J. be the injective envelope of k l , considered as a left H-comodule. Then J is finite dimensional and H = J @I K for a left coideal K of H . Let f : H -+ k be a nonzero linear map such that f (K) = 0 and f ( l H )= 1. Since K K e r ( f ) we have that f E Rat(H;J.) = rat(^-H*). By Theorem 5.2.3 there exist h, E H and t , E such that f = C t, h,, so (Ct, h , ) ( l ) # 0. Therefore, one of the ( t , h , ) ( l ) is not zero, and we can take this element of H as our h and a suitable multiple of the corresponding left integral for X . We will denote the right integral x o S by xS. We show first that for any t E and g E H there exists an 1 E H such that
h.
I
c
-
h
h
-
-
CHAPTER 5. INTEGRALS
224 where (g
-
t)(x) = t(xg). Let x E H and compute
-
=
=
(9 t)(x) = xS(h)t(xg) (by Lemma 5.4.4 ii))
x
~ S ( x ~ g l h ) t ( x 2 g 2(t) left integral)
X ~ ( x g h l ) t ( ~ ( h 2(xS ) ) right integral)
=
C ghlt(~(h2))
=
xs(x
=
xS(xu) (where u = C ghlt(S(h2)))
= xS(xu)xS(h)
~ S ( x ~ u ~ ) ~ ( x 2 ~ 2(xS S ( right h ) ) integral)
=
xS(h2)x(xuS(hl)) (X left integral) = ~ ( x l )(where 1 = C u S ( h l ) x ( S ( h ~ ) ) ) =
=
(1
-
x)(x)
so (5.35) is proved, and we can choose r E H such that
Now t(x> = xS(h)t(x) = xS(hl)t(xh2) (xS right integral) = ~S(S(~l)~2hl)t(~3h2) = X ~ 2 ( x l ) t ( x 2 h(t) left integral) =
xS2(x1>x(x2r) (by (5.36))
=
xS(r)x(x),
where the last equality follows by reversing the previous five equalities. It follows that t = xS(r)x, i.e. x spans and the proof is complete.
-
Exercise 5.5.6 Prove Corollary 5.5.5 directly. Solution: We know that 4 : H * H, q5(h*) = C t i h * ( t h ) is a bijection. Hence there exists a T E H* such that C t i T ( t h ) = 1. Applying E to this equality we get T(tl) = 1. For h E H we have
5.7. SOLUTIONS T O EXERCISES
In particular, we have S ( t ) = C t ' , T ( t l t L ) = C t i T ( t 1 A ' ( t L ) ) = (t' A1)T(t')= t' A'.
-
-
Exercise 5.5.2 Let f : C -t D be a surjective morphism of coalgebras. Show that if C is pointed, then D is pointed and C o r a d ( D ) = f ( C o r a d ( C ) ) . Solution: It follows from Exercise 3.1.13 and from the fact that for any grouplike element g E G ( C ) , the element f ( g ) is a grouplike element of D . Exercise 5.5.7 Let H be a Hopf algebra such that the coradical Ho is a Hopf subalgebra. Show that the coradzcal filtration Ho C H1 C . . . H , G . . . zs an algebra filtration, 2.e. for any positzve integers m , n we have that HmHn C Hm+n. Solution: We remind from Exercise 3.1.11 that the coradical filtration is a coalgebra filtration, i.e. A(H,) C Cz=o,nHz @ Hn-, for any n. In Ho @ Hn. particular this shows that A ( H n ) 2 Hn 8 Hn-l We first show by induction on m that HmHo = Hm for any nz. For m = 0 this is clear since Ho is a subalgebra of H . Assume that Hm-l Ho = Hm-l. Then
+
where for the second inclusion we used the induction hypothesis. Thus H,,Ho C Ho A Hm-1 = H,,,. Clearly, Hm C HmHo since Ho contains 1. Similarly HoHm = Hm for any m. Now we prove by induction on p that for any m, n with m n = p we have that H, Hn C H,. It is clear for p = 0. Assume this is true for p - 1, where p 2 1, and let m, n with m + n = p. If m = 0 or n = 0, we already proved the desired relation. Assume that m , n > 0. Then we have that
+
which shows that HmHn C Ho A HPp1= H, = Hm+,.
Exercise 5.5.9 Let H be a Hopf algebra. Show that the following are equivalent.
226
CHAPTER 5. INTEGRALS
(1) H is cosemisimple. (2) k is an injective right (or left) H-comodule. (3) There exists a right (or left) integral t E H* such that t ( 1 ) = 1. Solution: ( 1 ) and ( 2 ) are clearly equivalent from Theorem 3.1.5 and Exercise 4.4.7. To see that ( 2 ) and ( 3 ) are equivalent, we consider the unit map u : k -, H, which is an injective morphism of right H-comodules. Then k is injective if and only if there exists a morphism t : H -+ k of right H-comodules with tu = I d k . But such a t is precisely a right integral with t ( 1 ) = 1. Exercise 5.5.10 Show that i n a cosemisimple Hopf algebra H the spaces of left and right integrals are equal, and i f t is a left integral with t ( 1 ) = 1, we have that t o S = t . Solution: By Exercise 5.5.9 we know that there exist a left integral t and a right integral T such that t ( 1 ) = T ( l )= 1. Then t = T ( 1 ) t = Tt = t ( 1 ) T = T , so = J,. We know that t o S is a right integral. Since (t o S ) ( l ) = 1, we see that t o S = t. Exercise 5.5.12 Let H be a Hopf algebra over the field k , K a field extension of k , and 2 = K 81,H the Hopf algebra over K defined i n Exercise
4.2.17. Show that i f H is cosemisimple over k , then
is cosemisimple over
K . Moreover, in the case where H has a nonzero integral, show that if is cosemisimple over K , then H is cosemisimple over k . Solution: We know from Exercise 5.1.6 that if T is a left integral of H ,
z.
then T E %* defined by T ( 6 8 k h ) = 6 T ( h ) is a left integral of Everything follows now from the characterization of cosemisimplicity given in Exercise 5.5.9. Exercise 5.5.13 Let H be a finite dimensional Hopf algebra. Show that the following are equivalent. (1) H is semisimple. (2) k is a projective left (or right) H-module (with the left H-action o n k defined by h . a = ~ ( h ) a ) . (3) There exists a left (or right) integral t E H such that ~ ( t#)0. Solution: It is clear that ( 1 ) and ( 2 ) are equivalent from Exercise 4.4.8. If the left H-module Ic is projective, since E : H -+ k is a surjective morphism of left H-modules, then there exists a morphism of left H-modules 4 : k -t H such that E o 4 = I d k . Denote t = 4 ( l k )E H . We have that
for any h E H , showing that t is a left integral in H. Clearly ~ ( t=) & ( + ( I ) )= 1. Conversely, if there exists a left (or right) integral t E H such that ~ ( t#) 0, we can obviously assume that ~ ( t=) 1 by multiplying
,
5.7. SOLUTIONS TO EXERCISES
227
with a scalar. Then the map 4 : k -+ H , 4(a) = a t is a morphism of left H-modules and E o 4 = I d , so k is isomorphic to a direct summand in the left H-module H . This shows that k is projective.
Exercise 5.6.12 Give a different proof for the fact that At does not have nonzero integrals, by showing that the injective envelope of the simple right At-comodule kg, g E C , is infinite dimensional. Solution: Let E, be the subspace of At spanned by all
Then by Equation (5.9), E, is a right At-subcomodule of At and kg is Thus the ES7sare essential in E,. On the other hand, At = $SEC&,. injective, and we obtain that &, is the injective envelope of kg.
Exercise 5.6.24 Let A be the algebra generated by an invertible element a and an element b such that 6" = 0 and ab = Xba, where X is a primitive 2n-th root of unity. Show that A is a Hopf algebra with the comultiplication and counit defined by
Also show that A has nonzero integrals and it is not unimodular. Solution: Let C =< a > be an infinite cyclic group and a* E C* such that a*(a) = It is easy to see that A e H(C, n, a2, a*). Everything else follows from the properties of Hopf algebras of the form H(C, n, c , c*,a , b).
a.
Exercise 5.6.25 Let H be the Hopf algebra with generators c, X I , .. . ,xt subject to relations
Show that H is a pointed Hopf algebra of dimension 2t+1 with coradical of dimension 2. Solution: Let C = C2 =< c >, the cyclic group of order 2, c;', . . . ,cf E C* defined by c,*( c ) = -1, and cj = c for all 1 5 j 5 t . Then H = H(C, n, c, c*) and all the requirements follow from the general facts about Hopf algebras defined by Ore extensions. Exercise 5.6.26 Let 4 be an automorphism of k C of the form 4 ( g ) = c*(g)g for g E C , and assume that c*(g) # 1 for any g E C of infinite order. Show that i f 6 is a 4-derivation of k C such that the Ore extension (kC)[Y,4, S ] has a Hopf algebra structure extending that of k C with Y a (1, c)-primitive,
CHAPTER 5. INTEGRALS
228
then there is a Hopf algebra isomorphism ( k c )[Y, $,6] 2 ( k c )[X,$1. Solution: Let U = {g E Clc*(g) # 1) and V = {g E Clc*(g) = 1). Thus, if g E V then by our assumption g has finite order. In this case, $(gn) = gn for all n , and induction on n 2 1 shows that S(gn) = ngn-lS(g). Then S(1) = mg-lS(g), where m is the order of g, and S(1) = 0 imply that 6(g) = 0. Now let g E U. Applying A to the relation Yg = c*(g)gY + S(g), we find that A(S(g)) = cg @ S(g) + S(g) 8 g. Thus S(g) is a (g, cg)-primitive, and so 6(g) = a,g(c - 1) for some scalar a,. Therefore, for any two elements g and h of U
and similarly
+ a,c*(h))(c - 1)gh Since C is abelian a, + a h < c*,g >= a h + a, < c*, h >, or S(hg) = ( a h
Denote by y the common value of the a,/(l - c*(g)) for g E U. We have a, - y c*(g)y = 0. Let Z = Y - y(c - 1). For any g E U we have that
+
Obviously, Zg = g Z if g E V, so (kC)[Y,4, S] (kC)[Z,41 as algebras. Since 2 is clearly a (1, c)-primitive, this is also a coalgebra morphism, which completes the solution.
Exercise 5.6.31 (i) Let C = C4 =< g > , t = 2 , n = (2,2),c = (g,g), C* = (g*,g*) where g*(g) = -1, b12 = 1, a = (1, I),a' = ( 0 , l ) . Show that there exists a Hopf algebra isomorphism H ( C , n , c, c*, a , b) E H(C,n , c, c*,a', b). (ii) Let C = C4 =< g > , t = 2,n = (2,2),c = (g,g),c* = (g*,g*) where g*(g) = -1, a = ( 1 , l ) and b12 = 2, a' = (0, l ) , bi2 = 0. Show that the Hopf algebras H(C, n , c, c * , a', b') and H(C, n , c, c*, a, b) are isomorphic. Solution: (i) The map f : H(C, n, c, c*, a, b) --+ H(C, n, c, c*, a', b) defined
5.7. SOLUTIONS TO EXERCISES
+
229
+
by f ( g ) = g, f ( x 1 ) = -(p2 P)xi Pxi, f ( x 2 ) = x i , where P E k is a primitive cube root of -1 is a Hopf algebra isomorphism. (ii) The map f from H ( C , n , c, c*,al,b') to H ( C , n, c, c*,a , b) defined by f ( g ) = g, f ( x l ) = 2 2 , f ( x 2 ) = x1 - 2 2 , is a Hopf algebra isomorphism. Note that one of the Hopf algebras is an Ore extension with nontrivial derivation while the other is an Ore extension with trivial derivation.
Exercise 5.6.32 Let C be a finite abelian group, c E C t and c* E C*t such that we can define H ( C , n , c, c*). Show that H ( C , n, c, c*)* H ( C * ,n, c*,c ) , where in considering H ( C *, n, c* , c ) we regard c E C** by identifying C and C**. Solution: Suppose C = C 1 x C2 x . . . x C , =< gl > x . . . x < g, > where Ci is cyclic of order mi. For i = 1, . . . , s, let Ci € k* be a primitive mi-th root of 1. The dual C* =< g: > x . . . x < g,* >, where g,t(gi) = Ci and gf ( g j ) = 1 for i # j,is then isomorphic to C . We identify C and C** using the natural isomorphism C C** where g**(g*)= g* ( g ) . First we determine the grouplikes in H*. Let hf E H* be the algebra map defined by h f ( g j ) = gf ( g j ) and h f ( x j ) = 0 for all i , j . Since the hf are algebra maps from H to k , H* contains a group of grouplikes generated by the g,', and so isomorphic to C * . Now, let yj E H* be defined by y j ( g x j ) = ~ j + - l ( ~and ) , y j ( g x w ) = 0 for xw # x j . We determine the nilpotency degree of yj. Clearly y; is nonzero only on basis elements gx;. Note that by (5.19) and the fact that qj = cj*(cj),
By induction, using the fact that
(T)
'?i
= (1
+ qj + . . . + q;-l),
we see that
for r l j = q:l, 3
Since 9,-, and thus r l j , is a primitive n j - t h root of 1, this expression is 0 if and only if r = nj. Thus the nilpotency degree of yj is nj. Let g* E H* be an element of the group of grouplikes generated by the gb above. We check how the yj multiply with g* and with each other. Clearly,
CHAPTER 5. INTEGRALS
230
both yjg* and g*yj are nonzero only on basis elements gxj. We compute
so that
g*yj = g*(cj)yjg*,or yjg* = ~
j**-~(~*)~*~~.
Let j < i. Then yjyi and yiyj are both nonzero only on basis elements gxixj = cf ( c j ) g x j x i . We compute
and
Therefore for j < i,
Finally, we confirm that the elements yj are ( e H ,~j*-l)-~rimitives and then we will be done. The maps cj*-' @ yj yj @ E H and m * ( y j )are both only nonzero on elements of H @ H which are sums of elements of the form g @ l x j or gxj @ 1, where m : H @I H -+ H is the multiplication of H and m* : H* -r ( H @ H ) * is regarded as the comultiplication of H * . We check
+
and
m * ( y j ) ( g@I Zxj) = y j ( g l x j )= ~ j * - l ( ~ l ) Similarly,
and
yj(gxj1) = yj(c;(l)glxj) = ~ j * ( l ) c j * - ~=( ~c l; )- ' ( ~ ) Thus the Hopf subalgebra of H* generated by the hr, yj is isomorphic to H(C*,n,c*-',c-') and by a dimension argument it is all of H*. Now we only need note that for any H = H ( C , n, c, c*), the group automorphism of C which maps every element to its inverse induces a Hopf algebra isomorphism from H to H ( C , n,c-', c*-'), and the solution is complete.
5.7. SOLUTIONS TO EXERCISES
Bibliographical notes Again we used the books of M. Sweedler [218], E. Abe [I],and S. Montgomery [149]. Integrals were introduced by M. Sweedler and R. Larson in [120]. The connection with H*Tat was given by M. Sweedler in [219]. Lemma 5.1.4 is also in this paper. In the solution of Exercise 5.2.12 (which we believe was first remarked by Kreimer), we have used a trick shown to us by D. Radford. In [218], M. Sweedler asked whether the dimension of the space of integrals is either 0 or 1 (the uniqueness of integrals). Uniqueness was proved by Sullivan in [217]. The study of integrals from a coalgebraic point of view has proved to be relevant, as shown in the papers by B. Lin [123], Y. Doi [72], or D. Radford [189]. The coalgebraic approach produced short proofs for the uniqueness of integrals, given in D. Stefan [211], M. Beattie, S. Diisciilescu, L. Grunenfelder, C. N5st6sescu, [28], C. Menini, B. Torrecillas, R. Wisbauer [145], S. Dbciilescu, C. N k t b e s c u , B. Torrecillas, [68]. The proof given here is a short version of the one in the last cited paper. The idea of the proof in Exercise 5.4.8 belongs to A. Van Daele [236] (this is actually the method used in the case of Haar measures), and we took it from [198]. The bijectivity of the antipode for co-Frobenius Hopf algebras was proved by D.E. Radford [189], where the structure of the 1-dimensional ideals of H* was also given. The proof given here uses a simplification due to C. Ciilinescu [52]. The method for constructing pointed Hopf algebras by Ore extensions from Section 5.6 was initiated by M. Beattie, S. Dkciilescu, L. Griinenfelder and C. Nkt6sescu in (281, and continued by M. Beattie, S. D6sc6lescu and L. Grunenfelder in [27]. A different approach for constructing these Hopf algebras is due to N. Andruskiewitsch and H.-J. Schneider [13], using a process of bosonization of a quantum linear space, followed by lifting. This class of Hopf algebras is large enough for answering in the negative Kaplansky's conjecture on the finiteness of the isomorphism types of Hopf algebras of a given finite dimension over an algebraically closed field of characteristic zero, as showed by N. Andruskiewitsch and H.-J. Schneider in [13], M. Beattie, S. Dbciilescu and L. Grunenfelder in [25, 271. The conjecture was also answered by S. Gelaki [86] and E. Muller [154]. A more general isomorphism theorem for Hopf algebras constructed by Ore extensions was proved by M. Beattie in [24]
Chapter 6
Actions and coactions of Hopf algebras 6.1 I11
Actions of Hopf algebras on algebras
this chapter k is a field, and H a Hopf k-algebra with comultiplication E . The antipode of H will be denoted by S.
A and counit
Definition 6.1.1 W e say that H acts on the k-algebra A (or that A is a (left) H-module algebra i f the following conditions hold: (MA1) A is a left H-module (with action of h E H on a E A denoted by h . a). (MA2) h . (ab) = C ( h l . a ) ( h a. b), V h E H , a , b E A . (MA3) h . lA= ~ ( h ) l V~h ,E H . Right H-module algebras are defined i n a similar way. I Let A be a lc-algebra which is also a left H-module with structure given by
By the adjunction property of the tensor product, we have the bijective natural correspondence
-
H o m ( H @ A , A)
H o m ( A ,H o m ( H , A ) ) .
If we denote by $ : A H o m ( H , A ) the map corresponding to v by the above bijection, we have the following Proposition 6.1.2 A is an H-module algebra if and only i f $ is a morphism of algebras ( H o m ( H ,A ) is an algebra with convolution: ( f * g ) ( h ) = C f (h1)dhz)).
234
CHAPTER 6. ACTIONS AND COACTIONS
Proof: Since $ corresponds to v , we have that v ( h 8 a ) = $ ( a ) ( h ) ,Qh E H , a E A. Hence ( M A 2 ) holds @ v ( h @ a b ) = ~ v ( h l @ a ) u ( h 2 8 bV) ,~ E Ha , b E A @ $(ab)(h)= C $ ( a ) ( h l ) $ ( b ) ( h a )= ( $ ( a ) * $ ( b ) ) ( h ) V h E H , a,b E A @ $(ab) = $(a) * $(b), Va,b E A @ $ is multiplicative. Moreover, ( M A 3 ) holds I @ v ( h @ 1 ~=)$ ( l ~ ) ( h=) ~ ( h )@ l $~ ( 1 ~ = ) IHO~(H,A). L e m m a 6.1.3 Let A be a k-algebra which is a left H-module such that ( M A 2 ) holds. Then i ) ( h - a)b = C hl . ( a ( S ( h 2 ). b)), Va,b E A, h E H. ii) If S is bijective, then a ( h . b) = C hZ . ( ( S W 1 ( h l.)a)b), Qa,b E A , h E H.
Proof: By ( M A 2 ) we have:
ii) is proved similarly.
-
I
Proposition 6.1.4 Let A be a k-algebra which is also a left H-module. Then A is an H-module algebra if and only if p : A @ A A , p(a@b) = ab, is a morphism of H-modules ( A C3 A is a left H-module with h . ( a C3 b) =
[email protected]). Proof: The assertion is clearly equivalent to ( M A 2 ) . To finish the proof it is enough to show that ( M A 3 ) may be deduced from ( M A 2 ) . We do this using Lemma 6.1.3. Indeed, taking in Lemma 6.1.3 a = b = l A ,we have
so ( M A 3 ) holds and the proof is complete.
I
Definition 6.1.5 Let A be an H-module algebra. We will call the algebra of invariants = { a E A I h . a = ~ ( h ) a V, h E H ) .
6.1. ACTIONS O F HOPF A L G E B R A S O N A L G E B R A S
235
A H is indeed a k-subalgebra of A : if a, b E A H ,then for any h E H we have h . (ab) = E ( h l a)(hz - b) =
E &(hi)a&(hz)b
=
Ec(hl)&(hz)ab
=
x
~ ( h l e ( h 2 ) ) a=b &(h)ab.
Another algebra associated to an action of the Hopf algebra H on the algebra A is given by the following
Definition 6.1.6 If A is an H-module algebra, the smash product of A and H , denoted A # H , is, as a vector space, A#H = A @I H , together with the following operation (we will denote the element a @ h by a#h):
-
Proposition 6.1.7 i ) A # H , together with the multiplication defined above, is a k-algebra. ii) The maps a a#lH and h l A # h are injective k-algebra maps from A , respectively H , to A # H . iii) A#H is free as a left A-module, and if {hi)iEl is a Ic-basis of H , then { l A # h i ) i E 1 is an A-basis of A # H as a left A-module. iv) If S is bijective (e.g. when H is finite dimensional, see Proposition 5.2.6, or, more general, when H is co-Frobenius see Proposition 5.4.6), then A # H is free as a right A-module, and for any basis {hi)iel of H over k , { l A # h i ) i E I is an A-basis of A # H as a right A-module.
-
Proof: i) We check associativity:
hence the multiplication is associative. The unit element is l A # l H :
CHAPTER 6. ACTIONS AND COACTIONS
ii) It is clear that (a#lH)(b#lH) = ab#lH, Qa, b E A. We also have (lA#h)(lA#g) = hl 1 ~ # h 2 g= l ~ # & ( h l ) h a= l ~ # h g . The injectivity of the two morphisms follows immediately from the fact that 1~ (resp. l A ) is linearly independent over k. iii) The map a#h a @ h is an isomorphism of left A-modules from A#H to A @ H , where the left A-module structure on A @ H is given by a(b @ h) = ab @I h. iv) will follow from
-
-
Lemma 6.1.8 If A is an H-module algebra, and S is bzjective, we have
Proof:
I We return to the proof of iv) and define
and
0 :H
@I A + A#H,
6(h @I a ) = ( l A # h ) ( a # l ~ ) .
By Lemma 6.1.8 it follows that 0 o 4 = lA#jy. Conversely,
hence also 4 o 0 = lHBA. Since 0 is a morphism of right A-modules, we I deduce that A#H is isomorphic to H @I A as right A-modules. We define now a new algebra, generalizing the smash product.
6.1. ACTIONS O F HOPF A L G E B R A S O N A L G E B R A S
237
Definition 6.1.9 Let H be a Hopf algebra which acts weakly on the algebra A (this means that A and H satisfy all conditions from Definition 6.1.1 with the exception of the associativity of multiplication with scalars from H : hence we do not necessarily have h . ( 1 . a ) = ( h l ) . a for V h ,1 E H , a E A. A s it will soon be seen, this condition will be replaced by a weaker one). Let a :H x H A be a k-bilinear map. W e denote by A#,H the k-vector space A @ H , together with a bilinear operation ( A @ H ) @ ( A @ H-4 ) (ABH), ( a # h ) @ (b#l) H (a#h)(b#l), given by the formula
-
where we denoted a @ h E A @ H by a#h. The object A#,H, introduced above, is called a crossed product if the operation is associative and l A # l ~ I is the unit element (i.e. if it is an algebra). Proposition 6.1.10 The following assertions hold: i ) A#,H is a crossed product if and only if the following conditions hold: The normality condition for a :
The cocycle condition: x ( h l . a(11,m l ) ) o ( h 2 ,hmz) = The twisted module condition:
x
~ ( h lll)a(h212, , m ) , V h . 1, m E H (6.3)
-
For the rest of the assertions we assume that A#,H is a crossed product. (ii) The map a a # l H , from A to A#,H, is an injective morphism of 12-algebras. iii) A#,H 21 A @ H as left A-modules. iv) If c is invertible (with respect to convolution), and S is bijective, then A#,H E H 18 A as right A-modules. In particular, i n this case we deduce that A#,H is free as a left and right A-module. Proof: i) We show that l A # l H is the unit element if and only if (6.2) holds. We compute:
Hence, if u ( 1 , h) = & ( h ) l A V , h E H , it follows that l A # l H is a left unit element. Conversely, if l A # l H is a left unit element, applying I @ E to the equality l A # h = C u ( 1 ,hl)#h2
238
CHAPTER 6. ACTIONS AND COACTIONS
we obtain ~ ( hlA ) = a(1, h). Similarly, one can show that lA#lHis a right unit element if and only if u(h, 1) = &(h)lA. We assume now that (6.2) holds, and that the multiplication defined in 6.1 is associative, and we prove (6.3) and (6.4). Let h, 1, m E H and a E A. From (l#h)((l#l)(l#m)) = ((l#h)(l#l))(l#m) we deduce (6.3) after writing both sides and applying I 8 e. From (l#h)((l#l)(a#m)) = ((l#h)(l#l))(a#m) we deduce (6.4) after writing both sides, using (6.2) and applying I @ E. Conversely, we assume that (6.3) and (6.4) hold. Let a , b, c E A and h, 1, m E H. We have:
where we used, (MA2) for the second equality, (6.3) for h3, 12,ml for the third one, and (6.4) for ha, 11,c for the fourth. On the other hand, ((a#h)(b#l))(c#m) = C a(hl . b)a(h2,11)((h312) + ~ ) ) 4 h 4 1 3m, l ) # h s b m , hence the multiplication of A#,H is associative. ii) and iii) are clear. iv) We define a:H@A-A#,H,
where a-' is the convolution inverse of u, and S-' is the composition inverse of S . We also define
We show that a and p are isomorphisms of right A-modules, inverse one to each other. We prove first the following
Lemma 6.1.11 If u is invertible, the following assertions hold for any h , l , m E H: a) h . 4 , m) = C d h l , ldu(h212, ml)~-'(h3,13m2). b) h , a-'(1, m) = C u(hl, llml)a-1(hz12, m2)u-'(h3, 13). C) C ( h l . o-'(S(h4), h5))o(h2, S(h3)) = ~ ( h ) l ~ .
6.1. ACTIONS OF HOPF ALGEBRAS ON ALGEBRAS
239
Proof: First, if up' is the convolution inverse of o , we have ~ o - ' ( l l , m l ) o ( 1 2 , m 2 ) = ~ o ( l l , m l ) o - ' ( ~ , m 2=) = €(l)€(m)lA, V1,m E
H.
a) We have
where we used (6.3) for the last equality. b) Multiplying by h E H both sides of the equality
we get
- -
h . 0-'(l,m), from H 8 H 8 H to We deduce that the map h 63 163 m A, is the convolution inverse of the map h 8 18 m h . a(1, m). To finish the proof of b), we show that the right hand sides of the equalities in a) and b) are each other's convolution inverse. Indeed,
=
o(h1, ll)o(h212,ml)~1(h313,m2)o-1(h4, 14) = ~ ( h ) ~ ( l ) ~ ( m ) l ~ .
c) The left hand side of the equality becomes, after applying b) for hl, S(h4),h5:
CHAPTER 6. ACTIONS AND COACTIONS
and the proof of the lemma is complete. I We go back to the proof of the proposition and show that a and ,B are each other's inverse. We compute
=
C hs €3 (SP1(h7)- o-'(h2,
S-'(hi)))
(s-l(he) . (h3 . a))o(S-l (h5), h4) (by (MA.2)) =
C hs B (S-'(hi)
. o-' (h2, S-l (hi)))
ff(s-l(h6)' h3)((S-'(hS)h4) . a ) (by (6.4)) =
hr 4 (S-'(he)
= =
x
. o-'(hz,
S-l(hi)))o(S-'(hs), ha)r(hr)a
he €3(S-'(he) . o-l(h2,
S-l
(hl)))o(s-'(h4), h3)a
C h2 4 E(S-' (hl))a (by Lemma 6.1.11, c) for S-' (hl)) =
hz 4€(hl)a = h 4 a ,
hence ,B o a = l , y @ ~Conversely, .
=
1op1(h5, S-'(h4))(h6 =
((SV1(h3). a ) o ( S - ' ( h ~ ) ,hl))#h7
o-'(h5, s-l(h4))(h6 (sW'(h3) a)) (h7. u(S-'(hz), hi))#hs (by (MA211
=
u-l(h8,
S-l
(h7))(hg - (S-l(h6). a))o(hlo, S-' (h5))o(hllS-'(h4), h l )
a-l (hi2,S-I (h3)h2)#h13 (by Lemma 6.1.11, a)) =
o-'(h6, s-l(h5))(h7 . (s-l(h4) . a)) o(h8, S - ' ( h 3 ) ) ~ ( h 9 S - ~ ( h ~hl)#hlo ),
=
o-l(h6, s 1 ( h 5 ) ) o ( h 7 ,S-'(h4))((h~S-'(h3)) . a )
6.1. ACTIONS OF HOPF ALGEBRAS ON ALGEBRAS
=
241
x
a&(h1)#h2 = a#h,
hence also a o p = lA#,H. Finally, we note that
= =
o-l(h2, s - ' ( h l ) ) ( h ~. ( a b ) ) # 4 a ( h 8 ab) = a ( ( h @ a)b),
hence a is a morphism of right A-modules. It follows that phism of right A-modules, and the proof is complete.
-
a!
is an isomor-
I
Remark 6.1.12 In case u : H 8 H A is trivial, i.e. o ( h , l ) = ~ ( h ) ~ ( 1 ) 1A * , is even an H-module algebra, and the crossed product A#,H is the smash product A#H. I We look now at some examples of actions:
Example 6.1.13 (Examples of Hopf algebras acting on algebras) 1) Let G be a finite group acting as automorphisms on the k-algebra A. If we put H = kG, with'^(^) = g 8 g, ~ ( g = ) 1, S(g) = g-l, Vg 6 G, and g . a = g(a) = ag, a E A, g E G, then A is an H-module algebra, as it may be easily seen. The smash product A#H is in this case the skew group ring A * G (we recall that this is the group ring, in which multiplication is altered as follows: (ag)(bh) = (abg)(gh), = is the subalgebra of the elements fixed Qa, b E A, g, h E G), and by G (which explains the name of algebra of the invariants, given to AH in general). The smash product A#H is sometimes called the semidirect product. Here is why. Let K be a group acting as automorphisms on the group H (i.e.
CHAPTER 6. ACTIONS AND COACTIONS
242
+
-
: K Aut(H)). Then K acts there exists a morphism of groups as automorphisms on the group ring kH, which becomes in this way a kK-module algebra. Since in kH#kK we have, by the definition of the multiplication,
we obtain that kH#kK 21 k ( H X + K ) , where H x 4 K is the semidirect product of the groups H and K. 2) Let G be a finite group, and A a graded k-algebra of type G. This means that A = @ A, (direct sum of k-vector spaces), such that AgAh Agh.
c
OEG
If 1 E G-is the unit element, Al is a subalgebra of A. Each element a E A writes uniquely as a = C a,. The elements a, E A, are called the sEG
homogeneous components of a. Let H = kG* = Homk(kG,k), with dual basis {p, I g E G, p,(h) = 6,,h). The elements p, are a family of orthogonal idempotents, whose sum is l H . We recall that H is a Hopf algebra with A(pg) = C ~ ~ h @ph, - 1 E(P,) = 6,,1, S(pg) = pg-1. For a E A we put hEG
p, . a = a,, the homogeneous component of degree g of a. In this way, A becomes an H-module algebra, since p, . (ab) = (ab), = agh-1bh = hEG
C (pgh-1 . a)(ph.b ) . The smash product A#kG* is the free left A-module ~ E G with basis { p , I g E G), in which multiplication is given by The subalgebra of the invariants is in this case Al, the homogeneous component of degree 1 of A. 3) Let L be a Lie algebra over k, and A a k-algebra such that L acts on A Derk(A) a morphism as derivations (this means that there exists cr : L of Lie algebras). For x E L and a E A, we denote by x . a = cr(x)(a). Let H = U(L), be the universal enveloping algebra of L (for x E L A(x) = x 63 1 163 x, E(X) = 0, S(x) = -x). Since H is generated by monomials of the form X I . . .a,, xi E L, we put
-
+
XI..
.x, . a
= XI
. (xz . (. . . (x, . a ) . . .),
a E A.
In this way, A becomes an H-module algebra, and AH = {a E A ( x . a 0, Vx E L). 4) Any Hopf algebra H acts on itself by the adjoint action, defined by h . I = (ad h)1 =
hlls(h2).
=
6.2. COACTIONS O F HOPF ALGEBRAS ON ALGEBRAS
243
This action extends the usual ones from the case H = kG, where (ad x)y = xYx-l, x , y E G, or from the case H = U ( L ) , where (ad x)h = xh - hx, x E L, h E H (the second case shows the origin of the name of this action). We have then HH = Z ( H ) (center of H ) . Indeed, if g E HH, then Qh E H
The reverse inclusion is obvious. 5) If H is a Hopf algebra, then H* is a left (and right) H-module algebra h*)(g) = h*(gh) (and (h* h)(g) = h*(hg)) with actions defined by (h I for all h , g H, ~ h* E H*.
-
6.2
-
Coactions of Hopf algebras on algebras
We have seen in Example 6.1.13 2) that a grading by an finite group G on an algebra is an example of an action of a Hopf algebra. To study the case when G is infinite requires the notion of a coaction of a Hopf algebra on an algebra. Definition 6.2.1 Let H be a Hopf algebra, and A a k-algebra. We say
that H coacts to the right on A (or that A is a right H-comodule algebra) if the following coditions are fulfilled: (CAI) A is a right H-comodule, with structure map
(CA2) C(ab)o @ (ab)l = Caobo 8 albl, Va, b E A. (CA3) p(1) = 1~@ 1 ~ . The notion of a left H-comodule algebra is defined similarly. If no mention of the contrary is made, we will understand by an H-comodule algebra a right H-comodule algebra. I The following result shows that, unlike condition (MA2) from the definition of H-module algebras, conditions (CA2) and (CA3) may be interpreted in both possibile ways.
-
Proposition 6.2.2 Let H be a Hopf algebra, and A a k-algebra which is a
right H-comodule with structural morphism p : A assertions are equivalent: i) A is an H-comodule algebra.
A@H. The following
CHAPTER 6. ACTIONS AND COACTIONS
244
-
ii) p is a morphism of algebras. iii) The multiplication of A is a morphism of comodules (the right comodule structure o n A @ A is given by a @ b C a0 @ bo @ albl), and the unit of A, u : k A is a morphism of comodules.
-
Proof: Obvious. I As in the case of actions, we can define a subalgebra of an H-comodule algebra using the coaction. Definition 6.2.3 Let A be an H-comodule algebra. The following subalgebra of A ~ c o H = { a € A I p(a) = a @ 1). is called the algebra of the coinvariants of A .
I
In case H is finite dimensional, we have the following natural connection between actions and coactions. Proposition 6.2.4 Let H be afinite dimensional Hopf algebra, and A a k-
algebra. Then A is a (right) H-comodule algebra if and only if A is a (left) H*-module algebra. Moreover, i n this case we also have that A ~ =* AcoH. Proof: Let n = d i m k ( H ) , and { e l , . . . , e n ) C H , {e;, . . . , e;) C H* be dual bases, i.e. e * ( e j ) = b i j . Assume that A is an H-comodule algebra. Then A becomes an H*-module algebra with
f -a= xaof(al), Vf
E H * , a € A.
Indeed, we know already that A is a left H*-module, and
Conversely, if A is a left H*-module algebra, A is a right H-comodule with
6.2. COACTIONS OF HOPF ALGEBRAS ON ALGEBRAS We have, for any f E H*
i=l
f . (ab) 8 1
n
( I @f ) (
C ( e f . a ) ( e ; . b) 8 e i e j )
i,j=l
( I 8 f )(p(a)p(b)), and so p(ab) = p(a)p(b). Finally,
We now have
Example 6.2.5 (Examples of coact.ions of Hopf algebras o n algebras)
245
CHAPTER 6. ACTIONS AND COACTIONS
246
1) Any Hopf algebra H is an H-comodule algebra (left and right) with ~ . we comodule structure given by A. Let us compute H ~ If h~ E HCoH, have A(h) = C hl 8 hp = h 8 1. Applying I 8 E to both sides, we obtain , HCoH2 k1. Since the reverse inclusion is clear, we have h = ~ ( h ) lhence HCoH = k1. 2) Let G be an arbitrary group, and A a graded k-algebra of type G (see Example 6.1.13, 2) ). Then A is a kG-comodule algebra with comodule structure given by
where a =
C a,,
a, E A, almost all of them zero. We also have AcOkG =
sEG
Al. 3) Let A#,H be a crossed product. This becomes an H-comodule algebra with P : A#uH A#uH 8 H , p(a#h) = x ( a # h l ) 8 h2.
-
We have (A#,H)COH = A#,l E A. Indeed, if a#h E ( A # , H ) " o ~ , then applying I @ I 8 E to the equality p(a#h) = (a#h) 8 1 we obtain a#h E A#,l, and the reverse inclusion is clear. Since the smash product is a particular case of a crossed product, the assertion also hold for a smash I product A# H. It is possible to associate different smash products to a right H-comodule algebra A. First, the smash product #(H, A) is the k-vector space Hom(H, A) with multiplication given by
-
Exercise 6.2.6 With the multiplication defined in (6.5), #(H, A) is an associative ring with multiplicative identity UHEH. Moreover, A is isomorphic ~(h)a. to a subalgebra of #(H, A) by identifying a E A with the map h Also H* = Hom(H, k) is a subalgebra of #(H, A).
Remark 6.2.7 If we take k with the H-comodule algebra structure given by UH, then the multiplication from (6.5) is just the convolution product. I We can also construct the (right) smash product of A with U, where U is any right H-module subring of H*, (i.e. possibly without a 1). This smash product, written A#U, is the tensor product A 8 U over k but with multiplication given by
6.2. COACTIONS O F HOPF ALGEBRAS ON ALGEBRAS
247
If H is co-Frobenius, H*Tatis a right H-module subring of H*,A#H*Tat makes sense and is an ideal (a proper ideal if H is infinite dimensional) of A#H*. In fact, A#H*Tat is the largest rational submodule of A#H* where A#H* has the usual left H*-action given by multiplication by l # H * . To see this, note that A#H* is isomorphic as a left H*-module to H* 8 A, where the left H*-action on H* @ A is given by multiplication by H* @ 1. The isomorphism is given by the H*-module map
with inverse q!J defined by q!J(a#h*) = C h* -- S-l(al) @ a0 . Since ( H *@A)Tat= H*Tat@A,(A#H*)Tat = 4(H*Tat @A) = A#H*Tat Thus we have
If H is finite-dimensional, then H is a ,left H*-module algebra, and these smash products are all equal (this is the usual smash product from the previous section). Note that the idea for the definition of (6.5) comes naturally by transporting the smash product structure from A#H* to Hom(H, A) via the isomorphism of vector spaces from Lemma 1.3.2. Exercise 6.2.8 I n general, A#H*Tat is properly contained i n #(H, A)Tat. Remark 6.2.9 Let us remark that for graded rings A over an infinite group G, A#(kG)*Tat is just Beattie's smash product (211. W e can adjoin a 1 to A#H*Tat in the standard way. Let (A#H*Tat)l = A#H*Tat x A with componentwise addition and multiplication given by
Then (A#H*Tat)l is an associative ring with multiplicative identity ( 0 , l ) and with A#H*Tat isomorphic to an ideal i n (A#H*Tat)l via i(x) = (x,O). Again, for graded rings A over an infinite group G , (A#(kG)*Tat)l is just Quinn's smash product (1841. I We define now the categories of relative Hopf modules (left and right). Definition 6.2.10 Let H be a Hopf algebra, A an H-comodule algebra. W e say that M is a left (A, H)-Hopf module if M is a left A-module and a right H-comodule (with m H C mo @I m l ) , such that the following relation holds.
248
CHAPTER 6. ACTIONS AND COACTIONS
We denote b y A ~ the H category whose objects are the left ( A ,H)-Hopf modules, and in which the morphisms are the maps which are A-linear and H - colinear. We say that M is a right ( A ,H)-Hopf module if M is a right A-module and a right H-comodule (with m H Em0 B m l ) , such that the following relation holds.
We denote by M z the category with objects the right ( A ,H)-Hopf modules, and morphisms linear maps which are A-linear and H-colinear. Similar definitions may be given for left H-comodule algebras. If A is such an algebra, the objects of the category ZM are left A-modules and left H comodules M satisfying the relation
for all a E A , m E M , and the objects of the category H M are ~ right A-modules and left H-comodules M satisfying the relation
for all a E A , m E M . I f M is a left H-module, we denote by
M~ = {
m M ~I h . m = ~ ( h ) m V , hE H).
I f A is a left H-module algebra, and M is also a left A#H-module, it may be easily checked that M~ is an AH-submodule o f M . If M is a right H-comodule with m H C mo @ ml, we denote by
I f A is a right H-comodule algebra, and M is also a right ( A ,H)-module, it may be checked that M~~~ is an ACoH-submoduleo f M . The following result characterizes the categories o f relative Hopf modules in case H is co-F'robenius. Proposition 6.2.11 Let H be a co-Frobenius Hopf algebra, and A a right H-comodule algebra. Then: i) The category A ~ isHisomorphic to the category of left unital A#H*Tatmodules (i.e. modules M such that M = (A#H*Tat). M ) , denoted by A#H*~atM U . iz) The category M : is isomorphic to the category of right unital A#H*Tatmodules, denoted b y M i # H . , , t .
6.2. COACTIONS OF HOPF ALGEBRAS ON ALGEBRAS
249
Proof: i) The reader is first invited to solve the following Exercise 6.2.12 Let H be co-Frobenius Hopf algebra and M a unital left A#H*T"t-module. Then for any m E M there exzsts an u* E H*Tatsuch u* . m = (l#u*) . m, so M is a unital left H*Tat-module, and that m therefore a rational left H*-module.
"
. Exercise shows that M is a rational left Let M E A # H * v a t M UThe H*-module, and therefore a right H-comodule. M also becomes a left A#H*-module via (a#h*) . m = (a#h*u*) . m
for a E A, h* E H*, m E M , u* E H*Tat, and m = u* . m . The definition is correct, because we can find a common left unit for finitely many elements in at . Now we turn M into a left A-module by putting a . m = (a#e).m. We have
so M E Conversely, if M E then M becomes a left H*-module with H*Tat. M = M , and a left A#H*-module via
Then
so M becomes a unital left A#H*Tat-module. It is clear that the above correspondences define functors (which are the identity on morphisms) establishing the desired category isomorphism. ii) The proof is along the same lines as the one above. It should be noted that H*Tatis stabilized by the antipode, which is an automorphism of H considered as a k-vector space, and so if h* E H* with Ker(h*) I, I a finite codimensional coideal, then Ker(h*S) 2 S - l ( I ) , which is also a
>
CHAPTER 6. ACTIONS AND COACTIONS
250
coideal o f finite codimension. W e also note that i f M E M?, then the right A#H*-module structure on M is given by
Exercise 6.2.13 Consider the right H-comodule algebra A with the left and right A#H*-module structures given by the fact that A E M ? and A E AMH:
Then A is a left A#H* and right AcoH-bimodule, and a left ACoHand right A# H* -bimodule. Consequently, the map
is a
ring
morphism.
Exercise 6.2.14 Let A be a right H-comodule algebra and consider A as a left or right A#H*Tat-module as in Exercise 6.2.13. Then: i ) ACoHE End(A#H*ratA) iz) AcoH End(AA#H*rat).
,
Example 6.2.15 1 ) If H is a Hopf algebra, H is a right comodule algebra as i n Example 6.2.5, I), then the categories H M H and M z are the usual categories of Hopf modules. 2) If G is a group, H = IcG, and A is a graded Ic-algebra of type G (see Example 6.2.5, 2) ), then the category (respectively M:) is the categorg I of left (resp. right) A-modules graded over G. Proposition 6.2.16 If H is a co-Frobenius Hopf algebra, and 0 # t E J, (resp. M E Then: let M E i ) t . M MCoH ii) If m E M~~~ and c E A , then t . ( c m ) = ( t . c ) m (resp. t . ( m c ) = m ( t . c)). In particular, the map M McoH, m ++ t . m is a morphism of AcoHbimodules.
c
MY).
-
Proof: W e prove only one o f the cases. i) I f h* E H * , then h* . (t . m) = ( h * t ). m = h * ( l ) t .m. ii) t . (cm)= C t ( c l m l ) c 0 m o = C t ( c 1 ) c o m = ( t . c)m.
6.3. THE MORlTA CONTEXT
251
Corollary 6.2.17 If H is a finite dimensional Hopf algebra, 0 # t E H is
a left integral, and A is a left H-module algebra, the map
I
is a morphism of AH-bimodules.
Definition 6.2.18 The map t r from Corollary 6.2.17 is called the trace
function. W e say that the the H-module algebra A has an element of trace I 1 if t r is surjective, i.e. there exists an a E A with t . a = 1. Example 6.2.19 1) Let G be a finite group acting on the k-algebra A as g is a left integral automorphisms (see Example 6.1.13, 1) ). Then t = sEG
i n H = k G , and the trace function is i n this case
I n case A is a field, a Galois extension with Galois group G , the trace function is then exactly the trace function defined e.g. in N. Jacobson [99, p.2841, which justifies the choice for the name. The connection with the trace of a matrix is the following: i n the Galois case, the trace of an element is the trace of the image of this element i n the matrix ring via the regular representation (cf. [99, p.4031). 2) If H is semisimple, then any H-module algebra has an element of trace 1. Indeed, if t is an integral with ~ ( t=) 1, then t . 1 = 1. I Exercise 6.2.20 (Maschke's Theorem for smash products) Let H be a
semisimple Hopf algebra, and A a left H-module algebra. Let V be a left A#H-module, and W an A#H-submodule of V . If W is a direct summand i n V as A-modules, then it is a direct summand i n V as A#H-modules.
6.3
The Morita context
Let H be a co-Frobenius Hopf algebra, t a nonzero left integral on H , and A a right H-comodule algebra. In this section, we construct a Morita context connecting A#H*Tat and ACoH. T h e n we will use the Morita context t o study the situation when A/ACoH is Galois. Recall from Exercise 6.2.13 that A is an A#H*rat - ACoH-bimoduleand an AcoH - A#H*Tat-bimodule with the usual modu module structure o n A, and for a , b E A, h* E H*Tat,the left and right A#H*Tat-module structures are given by: (a#h*) . b = C a b o h * ( b ~ ) ,
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252
-
and
b . (a#h*) = (h*sW1) (ba) =
1boaoh*(S-'(blal)).
If g is the grouplike element of H from Proposition 5.5.4 (iii) (which was denoted there by a), we can also define a (unital) right A#H*'at-module structure on A by
b ., (a#h*) = b . (a#g
-
h*) =
boaoh'(~-'(bla~)~).
-
Since g defines an automorphism of A#H*rat, a#h* I-+ a#g h*, it follows that with this structure A is also an ACoH- A#H*Tat-bimodule. We define now the Morita context. Let P = A # H I T ~ ~A A c o ~with the standard bimodule structure given above. Let Q = ~ c o H AAWH*ratwhere now the right A#H*Tat-module structure on A is defined using the grouplike from Proposition 5.5.4 (iii), which we will now denote by g, as above. Define bimodule maps [-, -1 and (-, -) by
I-, -](a
abo#t
and
(-,-)
(-, -)(a
-
-
€3 b) = [a, b] =
bl,
: Q 8 P = A @ ~ ~ # ~A * r a tA ' o ~ , @Q
-
b) = (a, b) = t
-
(ab) =
aobOt(alb1).
Note that since t A C A " " ~ ,the image of (-, -) lies in AcoH. Then, with the notation above, we have Proposition 6.3.1 For H with nonzero left integral t, A, P, Q, [-, -1, (-, -) as above, the sextuple
is a Morita context.
-
Proof: We have to check that: 1. The bracket [-,-I: A B A c o ~ A A#H*rat satisfies [ab, c] = [a, bc] for b E A ' o ~ , which is clear, and that it is a bimodule map. Left A#H*Tat-linearity: [(a#l) . b, c] = C[abol(bl),c] = C abocol(bl)#t cl and
-
=
aboco#l(bl)t
-
cl since t is a left integral.
6.3. THE MORITA CONTEXT
253
Right A#H*rat-linearity: [a,b.,(c#l)] = C [ a ,b o ~ o l ( S - ~ ( b l c l ) g=) ]C aboco#(t and,
[a,b](c#l) = x ( a b o # t ='
=
-
- b1~1)1(S-~(b2~2)g),
- bl)(c#l)
C aboco#(t -- blcl)l C ab0co#(t -- l ( ~ - ' ( b z c l ) ) -) blcl
-
since X(l h ) = ( 1 h ) ( g )= l ( h g ) . 2. The bracket (-,-) : A @ A # H * T O ~A + t A C - ACuHis obviously r the definition is correct by Exercise 6.2.16. left and right ~ " " ~ - 1 i n e aand Moreover, ( a ,(b#l) . c) = C ( a ,bcol(c1)) = C aobocot(aibici)l(c2) = C aoboco((t -- a i b i ) l ) ( c i ) = C aoboco((t(1 S-l (a2b2))) a l b l ) ( c l ) = Caoboco(t a l b l ) ( c l ) ( l-- S-'(a2b2))(g) by X(m)= m ( g ) and (a.,(b#l), c ) = C(aobol(S-l (a1b l ) g ) ,c ) = C aobocot(alblcl)l(S-'(a2b2)g). 3. Associativity of the brackets. First note that we will use (g t)S-' = t from Proposition 5.5.4 (iii). Now,
-
-
-
-
-
u ,. [b,C ] =
C a ., (bco#t - c l )
=
C aoboco(t - ~ 2 ) ( S - l ( a i b l c l ) ~ ) C a o b o c t ( ~ - l( a l b l ) g ) C aobo((g t ) s - ' ) ( a l b i ) c C aobot(albl)cby the above
=
( a ,b)c.
= = =
Also [a,b]. c = C ( a b o # t bl)c = C a b o c o ( t b l ) ( c l ) = Cabocot(b1cl) = c). I If any of the maps of the above Morita context is surjective, then it is an isomorphism. While for the map (-, -) this is well known, there is a little problem with the other map, since A#H*rat has no unit. Although the proof is almost the same as the usual one, we propose the following
-
-
Exercise 6.3.2 If the map [-, -1 from the Morita context i n Proposition 6.3.1 is surjective, then it is bijective.
CHAPTER 6. ACTIONS AND COACTIONS
254
Now we discuss the surjectivity of the Morita map to AcoH,leaving the discussion on the other map for the section on Galois extensions.
Definition 6.3.3 A total integral for the H-comodule algebra A is an H comodule map from H to A taking 1 to 1. Since an integral for the Hopf algebra H is a colinear map from H to k, the H-comodule algebra k has a total integral if and only if H is cosemisimple.
Exercise 6.3.4 Let H be a finite dimensional Hopf algebra. Then a right H-comodule algebra A has a total integral if and only if the corresponding left H*-module algebra A has an element of trace 1. We give now the characterization of the surjectivity of one of the Morita context maps.
-
Proposition 6.3.5 The Morita context map to ACoH is onto if and only A. if there exists a total integral Q, : H Proof: (+) Let Q, be a total integral, i.e. Q, is a morphism of right Hcomodules, and cP(1) = 1. Then cP is also a morphism of left H*-modules, @ ( h )for any h E H . so Q,(t h ) = t Suppose h E H is such that t h = 1, then for (-, -) the map from Proposition 6.3.1,
-
-
-
-
which shows that (-, -) is onto. Since t H & HCoH= k l H , to find an . h E H with t h = 1, it is enough to prove that t H # 0. But if t H = 0 , then for any h,g E H we have that:
-
-
= C t ( g 2 h 3 ) g l h 2 ~ - ' ( h l= ) x(t
- -
-
-
( g h ~ ) ) S - l ( h i= ) 0
- -
and so ( H t) H = 0. But H t = H*Tat, so H*Tat H = 0. Finally, since H*Tatis dense in H*, this implies that H* H = 0 which is clearly a contradiction. (+) Choose a E A such that t a = 1, and define Q, : H ---+ A by
-
Then @ ( I ) = 1 and Q, is a morphism of left H*-modules since for h* E H * , ~H E,
6.4. HOPF-GALOIS EXTENSIONS
6.4
Hopf- Galois extensions
Let H be a Hopf algebra over the field k, and A a right H-comodule algebra. We denote by p:A-ABH, p(a)=~ao8al the morphism giving the H-comodule structure on A, and by ACoHthe subalgebra of coinvariants. We define the following canonical map
can : A g
A
A c o ~
-
A 8 H, can(a 63 b) = ( a 8 l ) p ( b )=
x
abo 8 bl
Definition 6.4.1 W e say that A is right H-Galois, or that the extension I A/AcoH is Galois, i f can is bijective.
-
We can also define the map
can' : A @
~
~
Ac
o
~
A 8 H , can(a 63 b) = p(a)(b 8 1) =
x
aob 8 a l .
Exercise 6.4.2 If S is bijective, then can is bijective if and only i f can' is
bijective. We give two examples showing that this notion covers, on one hand, the classical definition of a Galois extension, and, on the other hand, in the case of gkadings (Example 6.2.5, 2) ) it comes down to another well known notion. We will give some more examples after proving a theorem containing various characterizations of Galois extensions.
Example 6.4.3 (Examples of Hopf-Galois extensions) 1) Let G be a finite group acting as automorphisms on the field E
> k. We
know from Example 6.1.13, 2) that E is a left kG-module algebra, hence a right kG*-comodule algebra. Let F = E ~ It. is known that E / F is Galois with Galois group G if and only if [E : F] =I G I (see N. Jacobson
CHAPTER 6. ACTIONS AND COACTIONS
256
[99, Artin's Lemma, p. 2291). Suppose that E/F is Galois. Let n =I G 1, G = {TI,.. . , r],), ( ~ 1 ,... ,u,) a basis of E / F . Let {pl, . . . ,p,) c kG* be the dual basis for {qi) c kG. E is a right kG*-comodule algebra with p : E E €3 kG*, p(a) = C ( Q . a)@pi. c a n : E @ F E E€3kG* isgiven by can(a8b) = Ca(vi.b)€3pi. If w = Cxj@ u j E Ker(can), it follows that
-
.
-
(because pi are linearly independent). As in the proof of Artin's Lemma, it may be shown that if the system (6.9) has a non-zero solution, then all the elements x j are in F, which contradicts the fact that {ui) is a basis. Hence all x j are 0, so w = 0. It follows that can is injective. But can is F-linear, and both E @F E and E @ kG* are F-vector spaces of dimension n2, and therefore can is a bijection. Conversely, we use dimF@ € 3 E~ ) = [E : FI2and dimF(E €3 kG*) = [E : F] I G I. If can is an isomorphism, it follows that [E : F] =I G 1, so E / F is Galois. 2) Let A = @ A, be a graded k-algebra of type G. We know from Example sEG
"
6.2.5, 2) that A is a right kG-comodule algebra, and that A"" = Al . We recall that A is said to be strongly graded if A,Ah = Aghl Vg, h E G , or, equivalently, if AgAg-l = Al, Vg E G. We have that A/A1 is right kGGalois if and only if A is strongly graded. Assume first that A is strongly graded. Let
where ai E A,-I, bi E A,, C a i b i = 1. It may be seen immediately that (can o P)(a €3 g) = a €3 9. Moreover,
257
6.4. HOPE'-GA LOIS EXTENSIONS
Conversely, if can is bijective, it is in particular surjective. For each g E G, let ai, bi E A be such that
It follows that all b, may be assumed homogeneous of degree g, and C a,b, = 1. Since the sum of homogeneous components is direct, it follows that the I a, may be also assumed homogeneous of degree g-I. We remark that in the last example it was enough to assume that can is surjective to get Galois. As we will see below, in the main result of this section, this is due to the fact that kG is cosemisimple, in particular co-Frobenius. For any M E AM^, consider the left A#H*Tat-module map
where the A#H*Tat-module structure of A McoH is induced by the usual left A#H*Tat action on A. Thus 4~ is also a morphism in the category AM^. ~f 4 M is an isomorphism for all M E A M H , we say the Weak Structure Theorem holds for AM^. Similarly, if for any M E M z , the map 4h : McoHBAcon A M, $ h ( m 18a ) = m a @ A C ~ H
-
is an isomorphism, the Weak Structure Theorem holds for M z We prove now the main result of this section.
Theorem 6.4.4 Let H be a Hopf algebra with non-zero left integral t, A a right H-comodule algebra. Then the following are equivalent: i) A/ACoH is a right H-Galois extension. ii) The map can: A BAcon A A @ H is surjective. iii) The Morita map [-, -1 is surjective. iv) The Weak Structure Theorem holds for *M H . v) The map 4M is surjective for all M E AM^. vi) A is a generator for the category A ~ HA#H*TatMU.
-
Proof: Since H is co-Frobenius, the map
is bijective. Then, since [-, -1 = (IB r) o can, it follows that ii) + iii). This also shows that i ) ii),~ using Exercise 6.3.2. In order to show that v) =+ iii), we consider A#H*Tat E AM^, which is @ A as in (6.6). Therefore, (A#H*Tat)cOH= (H*Tut@ isomorphic to H*Tat ~ ) c o H= t @ A, and so 4 ~ # ~ - r a=t [-, -1.
258
CHAPTER 6. ACTIONS AND COACTIONS
iii) + iv). Let M E A ~ Hm, E M . We show first that 4~ is one to one. Suppose m = 4 ~ ( C a 8i mi) for ai E A, mi E MCoH.Let e* be an element of H*T"tthat agrees with E on the finite set of elements ai,, m l in H . Suppose E[ck,dk]= I#(?*. Then
=
C
Ot
-
(dr
aimi)) since mi E M~~~
So if m = 0 it follows that C ai @ A c o ~mi = 0, and so 4M is injective. To show that q5M is surjective, note that for m and e* as above
We have thus proved that iii), iv), and v) are equivalent. iv)+ vi). Let M E AM^. Since ACoHis a generator in Aco~M, for some set I, there is a surjection from ( A ~ O ~ ) (to' )McoH. Thus there is a A @ A c o ~( A ~ O ~ ) to ( ' )A M~~~II M . surjection from A(') vi) =+ v). Let M E AM^. Since A a generator, given x E M , there is an index set I, (fi)iEI, fi E Hom;(A, M), ai E A, with C fi(ai) = x. I Then fi(l) E M ' o ~ ,and z = 4 M ( Ca; @I f i ( l ) .
-
Remark 6.4.5 A similar statement holds with can' replacing can, and the category M z replacing
AM^.
Corollary 6.4.6 If H is co-Frobenius and the equivalent conditions of Theorem 6.4.4 hold, then the map n in Exercise 6.2.13 induces a ring isomorphism A#H*Tat 2 E n d ( A A c o ~ ) T a t ,
where the rational part is taken with respect with the right H*-module structure given by (f . h*)(b) = C h*(bl)f (bo). Proof:We prove first that the map
is injective. Let z E A#H*Tat be such that z . a = 0, Va E A. Let g* E H*T"t be such that z(l#g*) = z. Since [-, -1 is surjective, there exist ai, bi E A
6.4. HOPF-GALOIS EXTENSIONS
259
such that 1#g* = C [ a i , bi]. Then we have z = z(l#g*) = C [ z . ai, bi] = 0. Thus it remains to show that the corestriction of .rr to E n d ( A A c o ~ ) Tisa t a t , let h* E H*Tat such that f = f sh*. surjective. Let f E E n d ( A A c o ~ ) Tand Let t' be a right integral and C ai @ bi E A B A c o ~A such that 2
18(h* 0 S-I) =
aibio 8 S(bil)
-
t'.
Then, for any b E A we have
and the proof is complete. I If H is finite dimensional, then from the Morita theory it follows that A/ACoHis an H-Galois extension if and only if A is projective finitely generated as a right ACoH-moduleand the map .rr is an isomorphism. The behaviour of .rr in the general co-Frobenius case was exhibited in the previous proposition. The next result investigates the structure of A as a right ACoH-modulein the Galois case.
Corollary 6.4.7 If H is co-fiobenius, then any H-Galois H-comodule algebra A is a fiat right AcoH-module. Proof: A well known criterion for flatness ([3, 19.191) says that A is flat over AcoH if and only if for every relation
there exist elements cl, . . . , c, 1, . . . ,n) such that
E A and cij E AcoH
(2 =
1 , . . . ,m,j =
CHAPTER 6. ACTIONS AND COACTIONS
260 and
SO let a l , . . . , a n E A and bl,. . ., bn E AcoH such that sider the morphism in
AM^
n
C ajbj = 0.
j=1
Con-
Since A is a generator in AM^, there exist a set X and a surjective morphism 4 : A ( ~ -+ ) Ker(<). Therefore, there exist elements cl, . . . ,c, E A H that and morphisms q51,. . . ,$, : A -+ Ker(<) in A ~ such (a,, . . ., an) =
C dz(ci) = C ~ i 4 i ( l ) . i=l i=l
Applying the canonical projection .rrj to bothe sides we get (6.10) for cij (nj o 4i)(l) E AcoH. As for (6.11), we have
=
4
and the proof is complete.
Example 6.4.8 (Other examples of Hopf-Galois extensions) 1) Let H be a Hopf algebra. Then we know that H is a right H-comodule algebra, H~~~E k. The extension H/k is clearly Galois, since
-
can: H @ H - H O H ,
~ a n ( h @ ~ ) = C h ~ ~ @ ~
has inverse h @ g C hS(gl) 63 g2. 2) Let A be a left H-module algebra. By Example 6.4.3, 2), A#H is a right H-comodule algebra, and ( A # H ) ' o ~ E A. It is easy to see that A#H/A is a Galois extension. Indeed, in this case we have can : A#H
@A
A#H
-+
(A#H)
@ H,
and (a#h) @ (b#g) = (a#h)(b#l) @ (l#g), so it is enough to define can on elements of the form (a#h) @ (l#g). But can((a#h) @ (l#g)) = C ( a # h g l ) @ g2. It follows that the inverse of can may be defined as in
6.4. HOPF-GALOIS EXTENSIONS
261
1). 3) The assertion in 2) also holds for crossed products with invertible cocycle. This will be proved a t the end of this section. Until then, we give a proof in the case H is finite dimensional. Let A#,H be a crossed product with invertible a. By 1) H/k is Galois, so by Theorem 6.4.4, iii), for 0 # t E H* left integral, the map
H 8H
-+
H#H*, x 8 Y
-
(x#t)(y#l)
is surjective. Let xi, yi E H such that
We compute
which ends the proof by Theorem 6.4.4, iii). Note also that A#,H has an element of trace 1: if h E H is such that 1 t ( h ) = 1, then l # h is such an element. Remarks 6.4.9 1) If H is co-Frobenius, A/AcoH is H-Galois, and A has a total integral, then AcoH is Morita equivalent to A#H*Tat. This follows from Theorem 6.4.4 iii), Exercise 6.3.2, and Proposition 6.3.5. 2) In particular, i f H is finzte dimensional, A/ACoHis H-Galois, and A has an element of trace 1, then 1) above and Exercise 6.3.4 show that ACoH zs Morita equivalent to A#H*. A n example is provided by a crossed product with invertible cocycle A#,H, as Example 6.4.8 3) shows.
262
CHAPTER 6. ACTIONS AND COACTIONS
3) As it will be mentioned in the notes at the end of this chapter, there exist much more general theorems about the equivalence of the categories M z (or AM^) and the category of right (or left) AcoH-modules. In this context, the fundamental theorem of Hopf modules may be seen as an example of such a situation, due to Example 6.4.8 1). 4 ) If H is finite dimensional, A is a left H-module algebra, A / A H is H * Galois, and M is a left A-module, we have the isomorphism
(where the last isomorphism is [-,
-1
@ I ) , which is even functorial.
I
W e can now prove a Maschke-type theorem for crossed products. Proposition 6.4.10 Let H be a semisimple Hopf algebra, and A#,H a crossed product with invertible a. Let V be a left A#,H-module, and W an A#,H-submodule of V . If W is a direct summand i n V as A-modules, then it is a direct summand i n V as A#,H-modules. Proof: If W is a direct summand in V as A-modules, then we have that (A#,H) @ A W is a direct summand in (A#,H) V as A#,Hmodules, since the functor (A#,H) @ A - is an equivalence. By Remark 6.4.9, 4 ) , it follows that (A#,H)#H* @A#,H W is a direct summand in (A#,H)#H* @A#,H V as (A#,H)#H*-modules. Since H is semisimple, from Exercise 6.2.20 we deduce that (A#,H)#H* @A#,H W is a direct summand in (A#,H)#H* @A#,H V as ( ( A # , H)#H*)#H-modules. But again the functor (A#,H)#H* @A#,H - is an equivalence, and so W is a direct summand in V as A#, H-modules. I T h e following exercise shows again how close co-Frobenius Hopf algebras are t o finite dimensional Hopf algebras.
Exercise 6.4.11 Let H be a co-fiobenius Hopf algebra, and A a right H comodule algebra with structure map
If A/AcoH is a separable Galois extension, then H is finite dimensional. W e close this section by a characterization o f crossed products which will be used later on. Theorem 6.4.12 Let A be a right H-comodule algebra, and B = ACoH. The following assertions are equivalent: B, and a weak action a) There exists an invertible cocycle a : B @ B of H on B , such that A = B#,H.
-
6.4. HOPF-GALOIS EXTENSIONS
263
b) The extension A / B is Galois, and A is isomorphic as left B-modules and right H-comodules to B 8 H , where the B-module and H-comodule structures on B 8 H are the trivial ones.
Proof: a)+b). It is clear that the identity map is the required isomorphism, so all we have to prove is that the extension B # , H / B is Galois. We have to show that the map
can : B#,H
=
8~ B#,H
-
B#,H
8H,
a(h1 . b)o(hz,ll)#h312 8 13
is bijective. We define
and show this is an inverse for can. We compute first
Note that the proof would be finished if H would be co-Frobenius, due to Theorem 6.4.4, so this gives another proof for Example 6.4.8 3). But since
CHAPTER 6. ACTIONS AND COACTlONS
264
H is arbitrary, we also have to compute
so p is the inverse of can, and the proof is complete.
b)+a). Let @ : B @ H -+ A be an isomorphism of left B-modules and right H-comodules. Much like in the case of group extensions, we define first a sort of a "set-splitting" map y : H -+ A, y ( h ) = cP(1 @ h ) , which is clearly H-colinear. We show first that y is convolution invertible. Since AIB is Galois, for each h E H there exists an element C ai(h) bi(h) E A @B A, the preimage of 1 @ h by can. We have
which may be checked by applying can to both sides. We also have
which may be seen after applying can @ I to both sides. Now we define : H -t A, p(h) = C ai(h)(I@ &)@-l(bi(h)), and we compute
,LL
(Y * I - L ) ( ~= ) = = =
G(hl)~(hZ)
C ~ ( h l ) a i ( h 2 ) ( 1~@) @ - l ( b i ( h 2 ) ) C r ( h ) o a i ( ~ ( h))(iI @ ~ ) @ - l ( b i ( ~ ( h ) l ) )
( I @ &)@-I ( ~ ( h )(by ) (6.12)) = ( I @ & ) @ - ' ( @ ( lh @) ) = =
€(h)l,
6.4. HOPF-GALOIS EXTENSIONS and
so y is a convolution invertible H-comodule map. Since 1 is grouplike, y(1) is invertible with inverse yP1(l), so replacing y by y', yt(h) = y(h)y-'(l), we can assume that y(1) = 1. From now on, the proof follows closely the proof of the fact that an extension of groups is a crossed product: 1. We define a weak action of H on B by putting for h E H and a E B: h.a=
C y(hl)a?-l(h2),
and a cocycle 0 : H x H -+ B by
In order to check that the definitions are correct, we note first that
for all h E H . To see this, note that the left hand side of (6.14) is the convolution inverse of
and if we denote by [(h) the right hand side of (6.14), then it is immediate 1, SO (6.14) holds. that ($ * J)(h) = ~ ( h ) l @ Now
CHAPTER 6. ACTIONS AND COACTIONS
266
so H . B C B. To check (MA2), we compute x ( h l . a)(h2 - 6) =
-
~(hl)ay-~(h2)7(h3)by-'(h4)= h (ab),
while (MA3) is obvious. Now
so the definition of u is also correct. 2. It is clear that u is a normal cocycle (it satisfies (6.2)) and is convolution invertible, with inverse given by:
We check that a satisfies the cocycle condition (6.3):
To check the twisted module condition (6.4) we compute C ( h l . (11 . a))ff(hz,12) =
=
x
y(hl)y(h)ay-' ( l z ) ~ ' ( h 2 ) 7 ( h 3 ) 7 ( 1 3 ) 7 - ~ ( h 4 4 )
=
7(h1)~(1l)a~-l(h212)
=
r(hl)r(l1)7-~(h2l2)r(h3~3)a~~~(h414)
=
x
u(hlrll)(h2h . a).
3. By 1 and 2 we can introduce a crossed product structure (with invertible cocycle) on B @ H, and we show that transporting this multiplication via @ I , @'(a @ h) = @(a@ h)@(18 1)-l, we get the multiplication of A. First recall that the connection between the map y (from 1 and 2) and @ is given by: y(h) = @ ( I@ h)@(l @ 1)-l.
6.5. SOME DUALITY THEOREMS Now we compute
= = =
ay(hi)by(11)y-'(h2~2)~(~1313)
a @ ( l @h ) @ ( l8 l ) - l b @ ( l @1 ) @ ( 1 8I)-' @'(a8 h)@'(b8 l ) ,
I
and the proof is complete.
Remark 6.4.13 A crossed product with trivial weak action is called a twisted product. From the above proof we obtain that a Galois extension AIB with A = B 8 H (as left A-modules and right H-comodules), and with the property that B is contained in the center of A, is isomorphic to a I twisted product of B and H .
6.5
Application to the duality theorems for co-Frobenius Hopf algebras
Throughout this section, H will be a co-Frobenius Hopf algebra over the field k. In this section we show that it is possible to derive the duality theorems for co-hobenius Hopf algebras from Corollary 6.4.6. Let M , N E M T . Then we can consider H O M A ( M ,N ) consisting of those f E HomA(M, N ) for which there exist fo E HornA(M, N ) and fi E H such that
C f (mo)o8 f ( m o ) l s ( m ~=) C f o ( m ) €9
fi.
(6.15)
We remark that H O M A ( M ,N ) is the rational part of H o m A ( M ,N ) with respect to the left H*-module structure defined by
If M = N , we denote
CHAPTER 6. ACTIONS AND COACTIONS
268
Definition 6.5.1 We will denote by H M the ~ category whose objects are right A-modules M which are also left H-modules and right H-comodules such that the following conditions hold: i) M is a left H , right A-bimodule, ii) M is a right ( A ,H)-Hopf module, iii) M is a left-right H-Hopf module. The morphisms i n this category are the left H-linear, right A-linear and right H-colinear maps. We will call H M ; the category of two-sided ( A ,H)-Hopf modules. I Example 6.5.2 Let M E M z and consider H @ M with the natural structures of a left H-module and right A-module, and with right H-comodule structure given by
Then H @ Dl is a two-sided ( A ,H)-Hopf module. In particular, U = H 8 A is a two-sided ( A ,H)-Hopf module. I Remark 6.5.3 Let N E M A . Then N @ H E
H ~ z
if we put
and
-
C ( n @ h ) o @ ( n ~ h ) ~ = C n ~' h ~ ~ h ~ , the left H-module structure being the natural one. If M E M z , then M @ H H @ M (as i n Example 6.5.2) i n the isomorphism (of right A-modules, left H-modules and right H-comodules) being given by
In particular, U = H @ A
=A @H
H~z, I
in H M ~ .
Theorem 6.5.4 Let A be a right H-comodule algebra, and M E H M ~ . Then E ~ ~ ; ( M ) # H ENDA(M), f @ h f .h
-
is an isomorphism of right H-comodule algebras.
-
6.5. SOME DUALITY THEOREMS
269
Proof: E N D A ( M ) is a right H-comodule with respect to the structure induced by the fact that it is a rational left H*-module, the structure being given by (6.15). It is also a right H-module, with action induced by the left H-module structure on M. If f E E N D A ( M ) , we denote by C fo @ f l , the image of f through the H-comodule structure map: (h* . f )(m) = C h* (f l)fo(m), Vh* E H*. Let f E E N D A ( M ) , h E H, and h* E H*. Then we first have for any m E M
because M E
H ~ Now z . we compute
=
C((h2
=
C(h2
- h*) . f ) ( h l m )
-
h*)(fl)fo(hlm)
= Ch*(flhz)(fohl(m)
which shows that E N D A ( M ) is an H-submodule of EndA(M), and also a Hopf module. Now f E E N D ~ ( M ) ~ Oif" and only if
It is clear that (6.18) is satisfied if f is H-colinear and A-linear. We assume that f E E n d A ( M ) and (6.18) holds, and we compute
where we used (6.18) for mo. In conclusion, END*(M)""~ = End;(M), the endomorphism ring of M in M z , and the map in the statement is an isomorphism by the fundamental theorem for Hopf modules 4.4.6. It remains to show that it is an isomorphism of right H-comodule algebras. First, E n d f ( M ) is a left H-module algebra via
and it is even a left H-module algebra, because
CHAPTER 6. ACTIONS AND COACTIONS
and everything is proved.
-
Exercise 6.5.5 If A is a right H-comodule algebra, then A
1ENDA(A), a
fa, fa(b) = ab
is an isomorphism of H-comodule algebras. From Corollary 6.4.6 we obtain immediately the duality theorem for actions of co-Frobenius Hopf algebras.
Corollary 6.5.6 If A1#,H is a crossed product with invertible a , then we have an isomorphism of algebras
MA(A')
denotes the ring of matrices with rows and columns indexed where by a basis of H, and with only finitely many non-zero entries in A'.
Proof: The right H*-module structure on End(A1#,HA,) is given by the left H*-module structure on A1#,H: h*.(a#h) = C a#hl h*(ha). We know from Proposition 6.1.10 iv) that A1#,H 3 H @A' as right A'-modules and left H*-modules via a#h
-
h4 @ (S-l(hl) . a)o(S-'(ha), hi),
and so A1#,HA, is free. Then End(A1#,HA,) is the ring of row-finite matrices with entries in A' and rows and columns indexed by a basis of H. In order to see who is End(A'#,HA,)Tat we choose the following basis of H : as before Corollary 3.2.17, write H = $E(Sx), where the Sx's are simple left H-comodules and E ( S x ) denotes the injective envelope of Sx. Then we denote by {hi)i the basis of H obtained by putting together bases A1#,H to write in the E(Sx)'s, and use the basis {hi @ lIi in H @I A' End(A1#,HAI) as a matrix ring. We denote by px the map in H*Tatequal to E on E(Sx) and 0 elsewhere. In order to prove the assertion in the statement it is enough to prove that any f with f (hi @ 1) # 0 and f (hj €3 1) = 0 for j # i is in the rational part. Say hi E E(Sx). Then it is easy to see that f = f .PA. is 0 outside The converse inclusion is clear, since for any f and any px, f E(Sx). 1 We move next to the second duality theorem. We need first some preparations.
---
6.5. SOME DUALITY THEOREMS
271
Lemma 6.5.7 Let R be a ring without unit and M R a right R-module with the property that there exists a common unit (in R ) for any finite number of elements in R and M . Then
where, for f E H o ~ R ( R RM,R ) and r , s E R7 ( Proof: The isomorphism sends m E M to p,(r) inverse sends C fi . ri to C fi(ri).
f . r ) ( s )= f
(rs).
= m r , for all r E R . The
I
Corollary 6.5.8 If R and M are as in Lemma 6.5.7 and M R 21 R R , then
I
(as rings.)
Lemma 6.5.9 Let H@A be the two-sided Hopf module from Example 6.5.2. Then H @ A 21 A # H * ~ " ~
as right A#H*rat -modules, via
where t is a right integral of H Proof: Recall that H @ A becomes a right A#H*rat-module via
If we denote by 19 the map in the statement, which is clearly a bijection, then we have to show that
i.e. that
If we apply the second parts to an element x , and denote y z = S - l ( y ) , then this gets down to
which is exactly Lemma 5.1.4 applied to Hop ' O p . As a consequence of the above we obtain the following
=
halbl and
I
272
CHAPTER 6. ACTIONS AND COACTIONS
Corollary 6.5.10 If H is co-Frobenius and A is a right H-comodule algebra, then
Let us describe now explicitely the left H*-module structure on H €3 A obtained via the isomorphism B of Lemma 6.5.9. This is given by
where h* E H * , h E H , a E A and g is the distinguished grouplike element (denoted there by a ) from Proposition 5.5.4 (iii). Indeed, we have
since we have, for all h , x E H
h*(b(hl)g-')t
- h2x
-
= x(h* x i ) ( t
- hxz).
The last equality may be checked as follows. Apply both sides to an element w E H and denote y = x w to get
Now denote u = S ( g h ) and t' = t
-
g-'
to obtain
which is exactly Lemma 5.1.4 applied to Hop, for which t' is still a left integral. This proves (6.19). We are ready to prove the second duality theorem for co-Frobenius Hopf algebras.
Theorem 6.5.11 Let H be a co-Frobenius Hopf algebra and A a right H comodule algebra. Then
where M ~ ( A denotes ) the ring of matrices with rows and columns indexed by a basis of H , and with only finitely many non-zero entries i n A.
6.5. SOME DUALITY THEOREMS
273
Proof: We write first Theorem 6.5.4 for the two-sided Hopf module H @ A from Example 6.5.2: ~ n d f @( A)#H ~
ci E N D A ( H@
A).
Now we define the right H*-module structure on E N D A(H@ A) as follows: @ A) and if f E E N D A ( H @ A) and f = C f,e, for some f, E ~ n d z ( H e, E H , then (6.20) f . h* = C ( f , - h*)ei We now take the rational part on both sides of the above isomorphism with respect to the H*-module structure in (6.20), and use Corollary 6.5.10 to get (A#H*'~~)#H N E N D A ( H 8 A) - H * ' ~ ~ . To finish the proof, we describe E N D A ( H@ A) . H*ratas a matrix ring.
More precisely, we show that
where the H*-module structure is the one given by (6.20). We define another H*-module structure on E N D A ( H@ A) and show that the rational parts with respect to this module structure and the one given by (6.20) are the same. Define, for h* E H*, f E E N D * ( H @ A)
(f
h*)(h @ a ) =
f (h*(s(hl)g-')hz
@ a).
We check that E N D A ( H@ A) . H*rat= E N D A ( H @ A) O H*rat. Let f E E N D A ( H @ A ) ,f = f,e,, f, E E n d z ( @ ~ A ) , ej E H . Then
x
(f
h*)(h @ a ) = C ( ( f j . h*)ej)(h @ a ) =
x ( f j h*)(ejh @ a )
=
x
=
C(s(gej,g-l)
h*( ~ ( e jhl)g-l) , f j (ej, h2 @ a ) (we used (6.19))
= C(fjej,)
- h*)(S(hi)g-')(fjej2)(h2
a (s(sej,g-l)
4a )
h*)(h a a ) ,
which shows that E N D A ( H@ A ) . H*'at G E N D A ( H@ A) O H*rat. Conversely, if f E E N D A ( H @ A ) , f = fjej, f j E end:(^ @ A), e j E H , then
x
CHAPTER 6. ACTIONS AND COACTIONS
=
( C ( f j (s2(gejlg-')
-
h*))ej2)(h8 a ) ,
showing that E N D A ( H@ A) O H*ratC E N D A ( H@ A) . H*rat. Now write H = @E(Sx),where the Sx's are simple right H-comodules, and take {hi} a basis in each E(Sx). Put them together and take {g-'hi €311, which is an A-basis for H @ A. Use this basis to view the elements of E N D A ( H @ A) as row finite matrices with entries in A. We show now that under this identification the elements of E N D A ( H @ A) 0 HTatare represented by finite matrices. As in the proof of Corollary 6.5.6, we denote by px the linear form on H equal to E on E(Sx) and 0 elsewhere. It is easy is represented by a finite to see that for every f E E n d A ( H@A), f matrix for all A. Conversely, if f(gP'hi €31) # 0 and f (g-'hj @ 1) = 0 for all j # i, then clearly f E E N D A ( H@ A). Moreover, if hi E E(Sx), then if hj E E(Sx), we have
Since for hj @ E(Sx), (f o pxS-l)(g-'hj @ 1) = f (g-'hj @ 1) = 0, we get that f = f o E E N D A ( HO A) O H*Tat, and the proof is c0m~lete.1 We end this section with some exercise concerning injective objects in the category of relative Hopf modules. Exercise 6.5.12 Show that is a Grothendieclc category.
A
~ =HOA#H*
[A @ HI. In particular,
A
~
Exercise 6.5.13 Let H be a co-Frobenius Hopf algebra. Then the following assertions hold: (i) If M E H * M , then MTat = H*TatM. (ii) OA#H- [A @ HI is a localizing subcategory of A#H. M . (zii) The radical associated to the localizing subcategory OA#H* [A @ HI, and
H
6.5. SOME DUALITY THEOREMS
275
produces an exact functor the identification of OA#H*[A@ H ] and t : A # H * M -+ AM^, given by t ( M ) = H*TatM. (iv) H 8 A is a projective generator of . Exercise 6.5.14 Let H be a co-Frobenius Hopf algebra, v* E H*Tat and h E H . Then: @* ( ~) *~h *=) ~ v;h* 8 v;. i) For h* E H * , C ( V * ~ ii) v* h E H*Tat, and C ( v * h)o 8 (v* h)l = C ( v ; h2)8 ( S ( h 1 ) v ; ) , (where by v* H v; €3 v; we denote the right H-comodule structure of H*Tat induced by the rational left H*-structure of H * T ~ ~ ) .
--
-
C
-
-
Exercise 6.5.15 Let H be a co-Frobenius Hopfalgebra and A an H-comodule algebra. Then the following assertions hold: Then HornH*(H*Tat,M ) is a left A#H*-module by i) Let M E
for any f E H o m ~ . ( H * ~ ~ ~ , na#h* / l ) , E A#H* and v* E . If cp : M -+ P is a morphism i n then the induced application : M ) -t HornH. (H*Tat,P ) is a morphism ofA#H*-modules. HornHii) The functor F : -+ A # H * M F , ( M ) = H o ~ H * ( H * ~ ~is~ a, M ) right adjoint of the radical functor t : A # H * M t ( N ) = H*TatN. -+
For the next exercises we will need the following definition: we say that a right H-comodule M has finite support if there exists a finite dimensional subspace X of H such that P M ( M )C M @ X , where p~ : M -+ M €3 H is the comodule structure map of M . An object M E AM^ has finite support if M has finite support as a H-comodule.
Exercise 6.5.16 Let H be a co-fiobenius Hopf algebra, and A a right H comodule algebra. Then the following assertions hold: i ) Let M E Then the map
y M ( m ) ( v * )= v * m = C v * ( m l ) m ofor any m E M , v * E H*Tat, is an injective morphism of A#H*-modules. ii) If M E A ~ has H finite support, then y~ is an isomorphism of A#H*modules. Exercise 6.5.17 Let H be a co-Frobenius Hopf algebra, and A an H comodule algebra. If M E A M is~ an injective object with finite support, then M is injective as an A-module.
CHAPTER 6. ACTIONS AND COACTIONS
276
6.6
Solutions to exercises
-
Exercise 6.2.6 With the multiplication defined in (6.5), # ( H , A ) is an associative ring with multiplicative identity U H E H . Moreover, A is isomorphic to a subalgebra of # ( H , A ) by identifying a E A with the map h ~(h)a. Also H* = H o m ( H , k ) is a subalgebra of # ( H , A). Solution: We first have
so the multiplication is associative. The other assertions are clear.
Exercise 6.2.8 In general, A#H*Tat is properly contained i n # ( H , A)Tat. Solution: Let H be any infinite dimensional Hopf algebra with bijective antipode, let A = H , and consider the map 3 E # ( H , H ) . For any h* E H* c Hom(H, H ) , x E H , by the multiplication in #(H, H ) we have
(h* . S ) ( X= )
h * ( $ ( ~ 2 ) 2 ~ i ) S ( ~=2h)*1( l ) S ( x )
so 3 E # ( H , H)Tat. But no bijection $ from H to H lies in H # H * . For if $ = C hi#hf E H # H * , choose x E H , x not in the finite dimensional subspace of H spanned by the hi. However x = $($-' ( x ) )= hi (hf , $-' ( x ) ), a contradiction.
Exercise 6.2.12 Let H be co-Frobenius Hopf algebra and M a unital left A#H*Tat-module. Then for any m E M there exists an u* E H*Tat such that m = u* . m = ( l # u * ) . m, so M is a unital left H*Tat-module, and therefore a rational left H*-module. Solution: Let m E M, m = C ( a i # h f ) . m i . We take u* E H*Tatequal to E on the finite dimensional subspace generated by ai, hfl , and zero elsewhere. Then (l#u*)(ai#hf = ai#hf for all i , so m = u* . m. Then we can define an action of H* on M by h* . m = h*u* . m, if h* E H * , m E M , U * E H*Tat , and m = u* - m. The definition is correct, because H*Tat is an ideal of H * , and H*Tat itself is a unital left H*Tat-module. Now h* . m = h*u* m = C h*(u;)u; . m, so M is rational. Exercise 6.2.13 Consider the right H-comodule algebra A with the left and right A#H*-module structures given by the fact that A E M z and
6.6. SOLUTIONS TO EXERCISES
277
Then A is a left A#H* and right AcoH-bimodule,and a left ACoHand right A#H* -bimodule. Consequently, the map
is a ring morphism. Solution: I f c E AcoH,then ((b#h*) a)c = =
=
x x
baoch*(al)
~baoc~h*(a~cl) b ( a ~ ) ~ h * ( ( a= c ) (b#h*) l) . ac,
and a similar computation shows the other assertion. Exercise 6.2.14 Let A be a right H-comodule algebra and consider A as a left or right A#H*Tat-module as in Exercise 6.2.13. Then: i) AcoH II End(A#H""tA) ii) ACoHII End(AA#H*rat). Solution: W e prove only the first assertion. W e have that for each X E A ~ NH M o ~ ~ ( A # H *H~ ~~ r~n) ~, # ~ * ~ ~ !x t (~Ao r, nx ;) ( A , X )II xCoH, which for X = A gives the desired ring isomorphism. Exercise 6.2.20 (Maschke's Theorem for smash products) Let H be a semisimple Hopf algebra, and A a left H-module algebra. Let V be a left A#H-module, and W an A#H-submodule of V . If W is a direct summand in V as A-modules, then it is a direct summand in V as A#H-modules. Solution: Let t E H be a left integral with ~ ( t=) 1. Then t is also a right W be the projection as A-modules. integral, and S ( t ) = t. Let A : V W e define
-
W e show that A' is a projection as A#H-modules. W e check first that A' is A#H-linear. Since S is bijective, we can use tensor monomials o f the form a#S(h):
CHAPTER 6. ACTIONS AND COACTIONS
=
~ ( a # S ( t 1 ) ) A ( ( l # t 2 ~ ( h ~ ) S ( h 2 ) ) v )right ( t integral)
=
C(l#s(t,)2)(S-l(s(t,),)
.a#l)
A((l#tzS(h))v) (by Lemma 6.1.8) = C ( l # ~ ( t l ) ) ( t 2. a#l)A((l#tsS(h))v)
so A' is A#H-linear. It remains to check that it is a projection. If w E W, then (l#t2)w E W , so A((1#t2)w) = (l#t2)w. We have thus
and the proof is complete.
Exercise 6.3.2 If the map [-, -1 from the Morita context in Proposition 6.3.1 is surjective, then it is bijective. Solution: Assume [-, -1 is surjective, and let C ai 8 bi be an element in the kernel. Choose u*E HXTat a common right unit for the elements bi (u* may be chosen to agree with E on the finite dimensional subspace generated by the bil's). Let C [ a i , bi] = u*. Then we have
Exercise 6.3.4 Let H be a finite dimensional Hopf algebra. Then a right
6.6. SOLUTIONS TO EXERCISES
279
H-comodule algebra A has a total integral i f and only if the corresponding left H*-module algebra A has an element of trace 1. Solution: Let Q, : H -+ A be a total integral, and t E H* a left integral. Let h E H be such that t ( h ) = 1, so C h l t ( h z ) = t ( h ) l = 1. Then Q,(h) is an element of trace 1. Conversely, if t is as above, and c is an element of trace 1, then Q, : H -+ A , @ ( h )= C c o t ( c l S ( h ) )is colinear by Lemma 5.1.4, and takes 1 to 1, hence it is a total integral. Exercise 6.4.2 If S is bijective, then can is bijective if and only if can' is bijective. Solution: We have
can(a @ b) = 4 o canl(a @ b), where 4 : A @ H -+A @ H is the map given by 4 ( a @ h )= ( 1 @ S - ' ( h ) ) p ( a ) . This is invertible, with inverse given by $-'(a @ h ) = p ( a ) ( l @ S ( h ) ) .
Exercise 6.4.11 Let H be a co-Frobenius Hopf algebra over the field k , and A a right H-comodule algebra with structure map
If A / A C o H is a separable Galois extension, then H is finite dimensional. Solution: H has a direct sum decomposition
where the Mx1s are simple left subcomodules of H I and E ( M x ) is the injective envelope. We know the E ( M A ) ' s (which we are going to denote by H A ) are finite dimensional subspaces of H. Therefore, if H is infinite dimensional it follows that I is an infinite set. Denote, for each X E I, by p~ the map in H*rat which is equal to E on H A and equal to zero elsewhere. We also know that for any f E H*Tat there exists a finite set F 5 I such that
Now the extension A/ACoH is a Galois extension, which means that the map can : A A A @ H I can(a @ b) = abo @ bl
-
is bijective. Let us denote, for each X E I,
CHAPTER 6. ACTIONS AND COACTIONS
280
We have clearly that A =
C Ax, since for any a E A we have a = C p A. a , XEI
where the px's are the ones corresponding to the finite dimensional subspace of H generated by the al's. Furthermore, we have
Indeed, for an a E A we have
Now PA . H = {Chlpx(hz) I h E H) 5 HA,since for any p # X we have that PA annihilates the left subcomodule H,. We claim that Ax # 0 for all X E I. Indeed, suppose that Ax, = 0. Then can(A O A c oA) ~G
can(A @ A*)
G
A O HA,
by the definition of can and (6.22), and this is a contradiction because the image of can has to be all of A 8 H. We use now the fact that A/ACoHis a separable extension. Then there is A, C ai 8 bi. This means that a separabilty idempotent in A O A c o ~
C aibi = 1, and C cai O bi =
x
ai 4 bit, YC E A.
Denote by W the finite dimensional subspace of H generated by the elebil are ments bil, and by pw E H*rata map equal to E on W. Then pw a finite number of elements in H*Tat.Since I is infinite, we can choose Xo which does not appear among the px's associated to all the pw bi, as in bil) are zero on HA,. (6.21). In other words, we assume that (pw Pick now c E Ax,, c # 0, and apply I d @I p to the equality
-
-
We get
C
B bio 8 bi1 =
C
-
B bi0co8 bil cl
If we apply now M o (I@I I @ p w ) (where M is the multiplication of H ) to both sides of the above equality, on the left-hand side we get C caibi = c, while on the other side we get
since cl E HA, by (6.22). Therefore we have obtained that c = 0, which is the desired contradiction.
6.6. SOLUTIONS TO EXERCISES Exercise 6.'5.5 If A is a right H-comodule algebra, then
is an isomorphism of H-comodule algebras. Solution: If h* E H* and a , b E A, then
which proves the desired isomorphism.
Exercise 6.5.12 Show that A ~ =HO A # H * [A @ HI. In particular, AM H is a Grothendieck category. Solution: By Proposition 6.2.11, if M E AM^, then M becomes a left A#H*-module via
Since M is an H-comodule, M embeds in H(') for some set I (see Corollary 2.5.2). Then A@M embeds in A@H(') 21 (A@H)('), so M E a A # H * [A@H].
Exercise 6.5.13 it Let H be a co-Frobenius Hopf algebra. Then the following assertions hold: (i) If M E H * M , then Mrat = H*ratM. (ii) ~ A # H *[A @ HI is a localizing subcategory of A # H - M . (iii) The radical associated to the localizing subcategory a A # H * [A @ HI, and the identification of O A # H * [A@H] and A ~ produces H an exact functor t : A # H = M-+ AM^, given by t(M) = H*TatM. (iv) H @ A is a projective generator of A ~ H . Solution: i) The proof is the same as the one in Exercise 6.2.12: we know that H*Tat M G MTat. Let now m E Mrat, and v(m) = C mo @ m l , where v is the right H-comodule structure map of MTat. Since H*ratis dense in acting as E on the finite subspace generated by H * , there exists h* E H*Tat the mi's. Then m = h*m E H*TatM. (ii) It remains to show that C A # H * [A @ H ] is closed under extensions. Let 0 + N + M + P + 0 be an exact sequence of A#H*-modules such that N, P E U A # H * [ A @HI. Then N = H*ratN and P = H*ratP, and it can be easily seen that these imply M = H*ratM, i.e. M E U A # H * [A @ HI. (iii) follows from Corollary 3.2.12. (iv) Let M 6 AM^. By Corollary 3.3.11, H is a projective generator in
CHAPTER 6. ACTIONS AND COACTIONS
282
the category M H . Then regarding M as an object in M H , we have a surjective morphism of H-comodules H(') -, M for some set I. The functor A @ - : M H -+ has a right adjoint, so it is right exact and commutes with direct sums. We obtain a surjection ( A @H)(') .v A @ H(') --+ A @ M . Composing this with the module structure map A 8 M -+ M of M , we obtain a surjective morphism ( A @ H)(') -+ M in AM H. Thus A @ H is a generator. Since the functor A @ - takes projectives to projectives (as a left adjoint of an exact functor), A 8 H is projective.
Exercise 6.5.14 Let H be a co-Robenius Hopf algebra, v* E H*Tat and h E H . Then: 2) For h* E H * , C ( v * h * ) o@ ( ~ * h *=) C ~ v,*h*@ v;. ii) v* h E H*Tat, and C ( v * h)o @ (v* h)l = C ( v : h 2 )8 ( S ( h l ) v ; ) , (where by v* H v,* @ v; we denote the right H-comodule structure of H*rat induced by the rational left H*-structure of H*Tat). Solution: i) If l* E H*, we have l*(v*h*)= (l*v*)h*= C l*(v;)v(;h*. ii) Let h* E H* and 1 E H . We have
--
Therefore h*(v* thing.
-
-h)
=
-
- v;)(v,*- h2),
C h*(S(hl)
-
which proves every-
Exercise 6.5.15 Let H be a co-Frobenius Hopf algebra and A an H comodule algebra. Then the following assertions hold: Then H o ~ ~ * ( H M * ') ~is ~a ,left A#H*-module by i) Let M E
for any f E Home* (H*Tat,M ) , a#h* E A#H* and v* E H*Tat. If p : M -+ P is a morphism i n AM^, then the induced application : HornHt (H*Tat, M ) -t HornH*( F T a P t) ,is a morphism of A#H*-modules. ii) The functor F : -, A#H* M , F ( M ) = Hemp (H*Tat, M ) is a right adjoint of the radical functor t : A # H * M --+ AM^, t ( N ) = H*TatN. Solution: i) We first show that (a#h*)f is a morphism of H*-modules. Indeed
6.6. SOLUTIONS TO EXERCISES =
=
x
283
1*(alf ((v*
- a2)hf)1)aof((v* - a2)hf)o
-
l*(al((v*
a2)h*)l)aof ((v*
-
a2)h*)0)
(since f is a morphism of right HTcomodules) -
l*(al(v*
a2)l)aof ((v*
(by Exercise 6.5.14(i)) =
x x
l*(a1s(a2)
.;)a0 f ((v;
-
a2)oh*)
-
as)h*)
(by Exercise 6.5.14(zz))
=
l*(v;)aof ( ( 4
-
=
C .of
=
((a#h*)f )(l*v*)
(((1'~')
- al)h*) a1)h*)
To see that the action defines a left A#H*-module, we have
Now, if f E HornH. (H*'at, M ) , we have
ii) The structure defined in i) defines a functor
Let N E A # H * M .Then the map
y ~ ( n ) ( v *= ) v*n, is a morphism of A#H*-modules. Indeed, we have that
284
CHAPTER 6. ACTIONS AND COACTIONS
and
Let now M € Then the map 6~ : t ( F ( M ) )-+ M , SM(v*f ) = f ( v * ) for any v* E H*Tat,f E F ( M ) = HornH*(H*'at, M ) , is an isomorphism in the category vf fi = vj*gj E Indeed, we first show that SM is well defined. Let t ( F ( M ) )= H * H O ~ ~ * ( H *M' ") ,~with , v:,u; E H * , fi,gj E F ( M ) . Since vf ,u; E H*Tat,there exits 1' E H*rat such that l*vt = v: for any i and l*u; = u; for any j (it is enough to take some l* such that l* acts as E H on ; is possible since H*Tatis dense in H * ) . Then all ( v , ' ) ~and ( u j ' ) ~this
zi
zj
thus the definition of 6~ is correct. f ) ) = SM(h*v*f ) = f (h*v*)= h*f (v*)= h * S ~ ( vf*) , 6~ is since SM(h*(v* a morphism of H*-modules, thus a morphism of H-comodules. Finally, we prove that bM is a morphism of A-modules. Let v* E H*rat, f E ~ o r n ~ . ( H * ' "M~),, and a E A. We want S ~ ( a ( vf *) ) = a 6 ~ ( vf *) . Since a(v*f ) € H*TatHornH-(H*rat, M ) , there exists w* E H*Tat and g E HornH. (H*'at, M ) , such that a(v*f ) = w*g. This means that a0 f ((d* a l ) v * ) = g(l*w*) for any 1* E H*Tat. Then bM(a(v*f ) ) = G M ( w * ~ ) = g(w*),and aSM(v*f ) = af ( v * ) .We know that H*rat = @ p ~ p E ( N p ) thus *, there exists a finite subset Po of P such that v* E E B ~ ~ P , , E ( N ~Denot)*. ing by E ( N p ) l the space spanned by all the hi's with h E E ( N p ) , and A ( h ) = C hl 8 h2, all the spaces a1 E ( N p ) l ,p E PO,are finite dimensional, thus there exits a finite subset J E P such that al E ( N p ) ,are jNp. contained in B p Ej N p , and all w; E apE
z
-
-
-
6.6. SOLUTIONS T O EXERCISES
285
-
Take 1' E H*rat,which is E H on any E(N,), p E J . Then for h E E(N,),p $! POand any al we have ((l* a l ) v * ) ( h )= C l * ( a l hl)v*(h2)= 0 (since v*(hz)E v * ( E ( N p ) = ) 0 ) and for any h E E(N,), p E Po and any a l l
-
Therefore (1' al)v* = E H ( U ~ ) V *for any a l , and x u of ( ( l * a. f (EH(al)v*)= a f ( v * ) . On the other hand
-
a l ) v * )=
thus
asM(v*.f) = a f ( v * ) =
aof ((1'
=
g(l*w*)
=
dw*)
=
~M(w*s)
=
b,(a(v*f))
-
~I)v*)
We show that bM is injective. Indeed, let x i v f fi E t ( F ( M ) )such that SM(CiV: fi)= 0. Then for any v* E H*Tatwe have
xi
vf fi = 0. To show that SM is surjective, let m E M , choose some thus 1* at such that l * m = m , and let f E H o r n ~ . ( H * ~M~ ") defined by f (h*) = h*m for any h* E H*rat. Then clearly m = l * m = f (1*) = bM(l*f ) , which proves that deltaM is an isomorphism. Now for N E A # H * M and M E AM^, we define the maps
and
p : HomA#H. ( N ,H O V L ~ *( H * ~ "M~), ) -t ~ o m z ( t ( M ~) , by &(p)= H o m ~ - ( H * ' " ~ , p ) ' yfor , v p E H o m z ( t ( N ) , M ) ,and P ( q ) = 6,149 for q E HornA# H* ( N ,HornH*(H*'"l, M ) ) . Thus
286
CHAPTER 6. ACTIONS AND COACTIONS
and for any n E N, v* E H*Tat,
We have that
and (aP)(q)(n)(v*)= a(P(q))(n)(v*)= P(q)(v*n) = q(n)(v*) showing that a and P are inverse to each other, and this ends the proof.
Exercise 6.5.16 Let H be a co-Frobenius Hopf algebra, and A a right Hcomodule algebra. Then the following assertions hold: . the map i) Let M E A ~ HThen 7h.i : M -, HornH. (H*Tat,M ) ,
yM(m)(v*) = v*m = C v * ( m l ) m o for any m E M , v * E H*rat, is an injective morphism of A#H* -modules. ii) If M E A ~ has H finite support, then YM is an isomorphism of A#H*modules. Solution: i) We already know from the solution of Exercise 6.5.15 that y~ is a morphism of modules. The fact that it is injective follows from the density of H*Tatin H*. ii) Let ~ M ( M ) M @I X with X a finite dimensional subspace of H, and K be a finite subset of J such that X 2 @ t G ~ E ( M t )If. ~j E C* is E on E(Mj) and zero on any E(Adt), t # j, then obviously cj(X) = 0 for j $ K , thus E ~ = M 0. Let cp E H O ~ ~ - ( H * ~ ~and ~ ,jM$ )K, . We have p(&j)= (p(€:) = E j ~ ( & j )E E ~ = MO. Thus cp($jgKC*&j) = 0. Let m = CtEK cp(~t).Then for j $ K , y ~ ( m ) ( ~ = , ) Ejm = 0 = cp(Ej), and for j E K , y ~ ( r n ) ( ~=j )Ejm = C t E K & j c p ( ~= t ) cp(~j),showing that yM(m) = cp. Therefore y~ is surjective.
Exercise 6.5.17 Let H be a co-Frobenius Hopf algebra, and A an Hcomodule algebra. If M E A M is~an injective object with finite support, then M is injective as an A-module. Solution: Since the functor F from Exercise 6.5.15 has an exact left adjoint, we have that F ( M ) is an injective A#H*-module. Exercise 6.5.16 shows that F ( M ) 2 M , thus M is an injective A#H*-module. On the other hand the restriction of scalars, Res, from A#H* M to AM has a left adjoint A#H* @A - : AM-4 A # ~ * M and, this is exact, since A#H* is a free right A-module. Thus Res takes injective objects to injective objects. In particular M is injective as an A-module.
6.6. SOLUTIONS TO EXERCISES
287
Bibliographical notes The main source of inspiration was S. Montgomery [149]. Lemma 6.1.3 and Proposition 6.1.4 belong to M. Cohen [58, 591. Crossed products and their relation to Galois extensions were studied by R. Blattner, M. Cohen, S. Montgomery, [37], and Y. Doi, M. Takeuchi [78]. The last cited paper contains Theorem 6.4.12. The second condition in point b) is called the normal basis condition (an explanation of the name may be found in [149, p.128]), and the map y in the proof is called a cleft map. The presentation of the Morita context is a shortened version of the one given in M. Beattie, S. D6sc5lescu, $. Raianu, 1291, extending the construction given by Cai and Chen in [42], [54] for the cosemisimple case. This extends the construction from the finite dimensional case due to M. Cohen, D. Fischman and S. Montgomery, [61], and M. Cohen, D. Fischman [62] for the semisimple case. This last paper contains the Maschke theorem for smash products. The Maschke theorem for crossed products belongs to R. Blattner and S. Montgomery [36]. The definition of Galois extensions appears for the first time in this form in Kreimer and Takeuchi [I101 (see [149, p.1231 or [150]). Corollary 6.4.7 belongs to C. Menini and M. Zuccoli [143]. Theorem 6.5.4 belongs to Ulbrich [233], as well as the characterization of strongly graded rings as Hopf-Galois extensions [234]. Conditions when the induction functor is an equivalence (the so-called Strong Structure Theorem) were given by H.-J. Schneider (see [203] and [202]). The duality theorems for coFrobenius Hopf algebras belong to Van Daele and Zhang [237], and the proof was taken from [144]. Exercices 6.5.13-6.5.17 are from [68].
Chapter 7
Finite dimensional Hopf algebras 7.1
The order of the antipode
Throughout H is a finite dimensional Hopf algebra. We recall the following actions (module structures): 1) H* is a left H-module via (h h*)(g) = h*(gh) for h, g E H, h* E H*. 2) H* is a right H-module via (h* h)(g) = h*(hg) for h , g E H, h* E H*. 3) H is a left H*-module via h* h = C h*(hz)hl for h* E H*, h E H . 4) H is a right H*-module via h h* = C h*(hl)hz for h* E H * , h E H . As in Chapter 5, if g E H is a grouplike element, we denote by
-
-
Lg = {m E H*lh*m = h*(g)m for any h* E H* ) and R, = {n E H'Inh* = h*(g)n for any h* E H* ), which are ideals of H * , L1 = &, R1 = ST. Also recall from Proposition 5.5.3 that Lg and R, are 1-dimensional, and there exists a grouplike element (called the distinguished grouplike) a such that R, = L1. We can perform the same constructions on the dual algebra H*. More precisely, for any q E G(H*) = Alg(H, k) we can define { x E H I hx
= v(h)x
for any h E H )
R , = { y E H 1 yh
= v(h)y
for any h E H )
L,
=
290290
CHAPTER CHAPTER 7. 7. FINITE FINITE DIMENSIONAL DIMENSIONAL HOPF HOPF ALGEBRASALGEBRAS
We remark We remark that that if ifwe keep we keep the the same definition same definition we gave we gave for for Lg Lg (g (g EE H),H), then then L, L, should should be be a asubspace subspace of of H**. The H**.set Theset L,~, L,~, as as defined defined above, above, is isjustjust the the preimage preimage of of this this subspace subspace via via the the canonical canonical isomorphism 0isomorphism 0: : H -~ H -~ H**.H**. Prom the Prom the above above it it follows follows that that the the subspaces subspaces Ln and Ln and Rn are Rn are ideals ideals of of di-dimension mension 1 1in inH, H, and and there there exists exists c~ c~ E EG(H*) G(H*) such such that that R~ = R~=L~. L~. ThisThis element element c~ c~ is isthe the distinguished distinguished grouplike grouplike element element inin Remark 7.1.1 Remark 7.1.1 Using Using Remark Remark 5.5.11 5.5.11 and and Exercise Exercise 5.5.10, 5.5.10, we we see see that that ifif H is H issemisimple semisimple and and cosemisimple, cosemisimple, then then the the distinguished distinguished grouplikes grouplikes in inHH and and H* are H* are equal equal to to1 1and and ~, ~,respectively. respectively. || Lemma 7.1.2 Lemma 7.1.2 Let Let ~ ~~ ~G(H*), G(H*), g g~ ~G(H), G(H), m,n m,n ~ ~H* H* and and x xv vsucsuc hh that that m ~m~x x= g= g= = x x"- "-n. n.Then Then mm E ELg Lg and and nn Proofi Proofi Let Let h*,g* h*,g* ~ ~H*. H*. ThenThen (g*h*m)(x)(g*h*m)(x) == -= -= (g*h*)(g)(g*h*)(g) = =g*(g)h*(g)g*(g)h*(g)
== = =(g*h*(g)m)(x)(g*h*(g)m)(x) which which shows shows that that (g*(h*m (g*(h*m - - h*(g)m))(x) h*(g)m))(x) = =0, 0,so so (h*m (h*m - - h*(g)m)(x h*(g)m)(x H*) H*) =0.=0. But But x x~ ~A* =A* =A, A, since since Ln Ln "-- "-r~ r~ = =Le by Le by the the remarks remarks preceding preceding PropositionProposition 5.5.4, 5.5.4, and and L~ ~ L~~H* =H*=H by H by Theorem 5.2.3 Theorem 5.2.3 (applied (applied for for the the dual dual of of H°P).H°P). This This shows shows that that h*m h*m = h*(g)m, = h*(g)m, and and so so m~ m~La. La. The The fact fact that that n n~ ~Rg isis Rg proved proved in in a asimilar similar way. way. || Corollary Corollary 7.1.3 7.1.3 If Ifm m ~ ~H*, H*, x x~ ~L~, L~, and and rn rn ~ ~x x= 1, = 1,then then mm ~ ~LI LI andand
Proof: Proof: Lemma7.1.2 Lemma 7.1.2 shows shows that that m ~m~L~. L~. If Ifh* h* ~ ~H*, H*, thenthen h* h* (x (x ~ ~m)m)==
== = =h*(a)m(x) h*(a)m(x) (since (since mE m EL~ L~ = =R~)R~) = =h*(m(x)a)h*(m(x)a) Applying Applying s sto tothe the relation relation ~rn(x~)xl ~rn(x~)xl = =1 1we get we get re(x) re(x) = =1. 1.This This showsshows that that x x~ ~m =m=a. a. ||
7.1. THE ORDER OF THE ANTIPODE
291
Lemma 7.1.4 Let x E L,, g E G ( H ) ,m E H* such that m Then for any h* E H* we have
-x
= g.
Proof: We compute
C g(g)h;(g-')h;(g)
v(g)h*(1) =
(A(h.1 =
C h;
l ~ ; ( g - ~ ) h ; ( m ( x z ) x l q ( g(9) )= m
=
=
+
@
h;)
x)
h;(g-')h;(m(gx~)gxl)( o ( ~ )=x gx)
=
1h*(g-1gxl)m(gx2)(convolution)
=
C h*
( X I )m(gx2.)
I Lemma 7.1.5 Let g E G ( H ) ,7 E G ( H * ) ,x E L,, and m E H* such that x = g . Then for any h E H we have
m
-
-
S(g-1(77 h ) ) = ( m
-h)-x
Proof: Let h* E H*. Then
h*(S(g-'(rl-, h ) ) )= = =
C h*(~(h1)g)rl(h2) 0(g)h*(s(hl)g)0(g-~h2) ( O algebra map)
=
17(g)( ( h * S ) ~ ) ( ~ - ' h (convolution) )
=
C((h;s)v)(g-lh)v(g)h;(l) (counit property for h*)
= C((h;s)s)(s-1h)h;(m(gx2)xl)
(by Lemma 7.1.4 for h;) =
C ( h ; S )(g-1hl)~(g-1h~)h;(m(gx2)x~) (convolution)
~ g-lhzx) (since q(g-l h 2 = = -
h*(~(hl)gg-~hz~lm(h3~z))
h*(xlm(hxz)) = h*((m h) x)
- -
292
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
I I f we write the formula from Lemma 7.1.5 for the Hopf algebras H , HcOp,Hop~copand Hop, we get that for any h E H the following relations hold:
I f x E L,,m
-x
-
- h) -x then S-'((v - h)g-') ( h - m) - x g, then S ( ( h - v)g-l) x - ( h -
= g, then ~ ( g - l ( v h ) ) = ( m
I f x E R,,m x =g, I f x R~, , x - n = I f x ~ L , , x - n = g , then ~ - ' ( g - ' ( h In particular
=
-
n)
=
77)) = x
- (n
h)
I f x ~ L , , m - x = 1 , thenS(h)=(m-h)-x I f x E R , = L,,m x = 1, then S-'(a h ) = ( h m) x I f x E R, = L E , x n = a, then S ( ( h a)a-') = x ( h n) I f x E L E ,x n = g, then ~ - ' ( g - l h , )= x ( n h)
-
-
-
-
- -
- - -
(7.1) (7.2) (7.3) (7.4)
(7.5) (7.6) (7.7) (7.8)
Theorem 7.1.6 For any h E H we have
s 4 ( h )= a-'(a
- h - a-')a
Proof: Let x E L, = R,, and m E H* with m -+ x = 1. Corollary 7.1.3 shows that m E L1 and x +- m = a. Moreover, we have
- -x
( s 4 ( h ) m)
- - -
S-'(a s 4 ( h ) ) (by (7.6)) S-l ( S 4( a h ) ) = s(s2(a h)) = (m s 2 ( a h ) ) x (by (7.5)) = =
Since the map from H* to H , sending h* E H* to h* we obtain s4(h) m = m s2(a h) On the other hand,
-
-
-
-
x E H is bijective,
x - ( m - s2(o- h ) ) = s-'(a-' s 2 ( a h ) ) (by (7.8)) = S-'(s2(a-l ( a h))) = ~ ( a - (la h)) = ~ ( a - ' ( a h)aa-') =
-
-
-
-
x
-
-
(since a a = a ( a ) a ) ( ( a P ' ( a h a-')a) m) (by (7.7))
- -
-
7.2. THE NICHOLS-ZOELLER THEOREM Since the map h* H x
-
-
293
h* from H* to H is bijective, we obtain that
- -
m = (a-'(a h a-')a) We got that S4(h) formula follo~&from the bijectivity of the map h H h
I
-- rn
m, and then the from H to H * .
Theorem 7.1.7 (Radford) Let H be a finite dimensional Hopf algebra. Then the antipode S has finite order. Proof: Using the formula from Theorem 7.1.6 we obtain by induction that
for any positive integer n. Since G ( H ) and G(H*) are finite groups, their elements have finite orders, so there exists p for which ap = 1 and a p = E . Then it follows that S 4 P = Id. I
Remark 7.1.8 The proof of the preceding theorem not only shows that the order of antipode is finite, but also provides a hint on how to estimate the I order.
7.2
The Nichols-Zoeller Theorem
In this section we prove a fundamental result about finite dimensional Hopf algebras, which extends to finite dimensional Hopf algebras the well known Lagrange Theorem for groups. Everywhere in this section H will be a finite dimensional Hopf algebra and B a Hopf subalgebra of H . If M and N are left H-modules, then M €4 N has a left H-module structure (via the comultiplication of H ) given by h(x €4 y) = C hlx 8 hny for any h E H , x E M , y E N. Hence M €4 N is a left B-module by restricting the scalars. Any tensor product of H-modules will be regarded as an H-module (and by restricting scalars as a B-module) in this way unless otherwise specified.
Proposition 7.2.1 If M E g M , then H €4 M modules.
-
M ( ~ ~ as ~ (left~ B) - )
Proof: Let U = H €4 M regarded as a B-module as above, and V = H €4 M with the left B-module structure given by b(h €4 m) = h €4 bm for any b E B , h E H , m E M. Since B acts only on the second tensor position of
294
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
as left B-modules. We show that the B-modules V, we have V 21 U and V are isomorphic. Indeed, let f : V --t U and g : U -+ V by
for any h E H, m E M , where we denote as usual by m the left H-comodule structure of M . We have that
++
C m(-l) @ m(o)
and
showing that f and g are inverse each other. On the other hand
I
so f and g provide an isomorphism of B-modules.
Proposition 7.2.2 Let W be a left B-module. Then B 8 W
21
W @ B 21
B ( ~ ~ as ~ left ( ~B-modules. ) )
Proof: We know that B@W is a left B-module. It is also a left B-comodule with the comodule strcture given by the association bBw w C bl@b2Bw. In this way B D W becomes a left B-Hopf module and the fundamental theorem of Hopf modules (a left hand side version) tells us that B@W N B ( ~ ~ ~ ( ~
7.2. THE NICHOLS-ZOELLER THEOREM
295
Let us consider now the Hopf algebra BcOp,with the same multiplication as B and with comultiplication given by c ++ C c(l) 8 ~ ( 2 )= C c2 8 cl. In particular W is a left BcoP-module, so applying the isomorphism proved W e ( ~ ~ ~ p ) as( left ~ ~BcoP-modules. ~ ( ~ ) ) The above we obtain that Bcop@ left BcOp-modulestructure of BcOp8 W is given by b(c % w) = C b(l)c 8 b ( 2 )= ~ C b2c 8 blw, therefore since B = Bcop as algebras, we see that BcOp8 W is isomorphic to W 8 B as left B-modules (the isomorphism is I the twist map), so W @ B = ( ~ ~ ~ p ) =( B~ ( ~~ ~~ (~ ~( ~) )) ) . Lemma 7.2.3 Let A be a finite dimensional k-algebra and M a finitely generated left A-module. Then M is A-faithful if and only if A embeds in M(") as an A-module for some positive integer n .
Proof: If A embeds in M(") an A-module for some positive integer n, then a n n A ( M ) = a n n A ( ~ ( ~G) )annA(A) = 0, so M is A-faithful. Conversely, if M is A-faithful, let {ml, . . . , m,) be a basis of the k-vector space M . Then the map f : A + M(") defined by f ( a ) = ( a m l , . . . , a m n ) for any a E A is an A-module morphism with K e r ( f ) = ni=l,n a n n ~ ( r n i )= annA(M) = 0. I Corollary 7.2.4 Let A be a finite dimensional k-algebra which is injective as a left A-module, and let P I , . . . ,Pt be the isomorphism types of principal indecomposable left A-modules. If M is a finitely generated left A-module, then M is A-faithful if and only if each P, is isomorphic to a direct summand of the A-module M . Proof: Lemma 7.2.3 tells that M is A-faithful if and only if A embeds in ~ ( ' for ~ 1 some positive integer n. Since A is selfinjective, this is equivalent to the fact that M(") E A $ X for some positive integer n and some Amodule X. Decomposing both sides as direct sums of indecomposables, the Krull-Schmidt theorem shows that this is equivalent to the fact that M I contains a direct summand isomorphic to Pi for any i . = 1,.. . , t. Proposition 7.2.5 Let A be a finite dimensional k-algebra such that A is injective as a lefl A-module. If W is a finitely generated left A-module, then there exists a positive integer r such that w(')e F$ E for a free A-module F and an A-module E which is not faithful.
-
Proof: If W is not faithful we simply take r = 1,F = 0 and E = W. Assume now that W is A-faithful. Let A p,("')$.. .$P,("", where 5 , . . . , Pt are the isomorphism types of principal indecomposable A-modules. Corrolary 7.2.4 shows that W P,("') $ . . .$ P,("" )$ Q for some m l , . . . ,m t > 0 and some A-module Q such that any direct summand of Q is not isomorphic
296
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
to any Pi. Let r be the least common multiple of the numbers n l , . . . ,nt. We have that W(T)2
Let a = m i n ( y , . . . ,
p,('"l)
$ . ..$
pirrnt)Q(') @
y), say a = m.Then ni
,
This ends the proof if we denote F = A(*), which is free, and E = . . , pt(rmt -ant) @Q(r)lwhich is not faithful since it does not p:rrnl-anl) contain any direct summand isomorphic to Pi (note that rmi - a n i = 0 and that we have used again the Krull-Schmidt theorem). I
Proposition 7.2.6 Let W be a finitely generated left B-module such that w(') is a free B-module for some positive integer r . Then W is a free B-module.
Proof: Take B = B1@.. .@B,, a direct sum of indecomposable B-modules. If t is a nonzero left integral of B and t = tl . . . t, is the representation of t in the above direct sum, we see that
+ +
btl
+ . . . + bt,
= bt = &(b)tl
+ . . . + €(b)t,
=
~(b)t
for any b E B , showing that t l , . . . , t , are left integrals of B. Since the space of left integrals has dimension 1, we have that there exists a unique j such that t j # 0, and then t = tj. Hence B j is not isomorphic to any other Bi, i # j , since B j contains a nonzero integral and Bi doesn't, and if f : Bj --+ Biwere an isomorphism of B-modules we would clearly have that f (t) is a nonzero left integral. Let us take now PI = Bj, P2,. . . , P, the isomorphism types of principal @ . . . $P,("")the representation indecomposable B-modules, and B E of B as a sum of such modules. Note that n l = 1. We know that w(') is free as a B-module, say w(')1,B(P) for some positive integer p. Since
pin1)
the Krull-Schmidt theorem shows us that the decomposition of W as a sum of indecomposables is of the form W E @ . . . @ P , ( ~ for " ) some r n l , . . . ,m,. Moreover, since w ( ~E)P:'~') @ . . . @ P,("~"),we must have pni = rmi. In particular p = r m l . Then for any i we have that pni = r m l n i = rmi, so mi = mini. We obtain that W E B("'), a free Bmodule. I
pirn')
7.2. THE NICHOLS-ZOELLER THEOREM
297
Proposition 7.2.7 Let W be a finitely generated left B-module such that ( ~ ) ) there exists a faithful B-module L with L @ W = w ( ~ ~ as~ B-modules. Then W is a free B-module. Proof: We know from Exercise 5.3.5 that B is an injective left B-module. Then we can apply Proposition 7.2.5 and find that w ( ~=)F $ E for some positive integer r, some free B-module F and some B-module E which is not faithful. Proposition 7.2.6 shows that it is enough to prove that w(') is free. We have that
so we can replace W by ~ ( ~ and 1 , thus assume that W E F $ E . Similarly there exists a positive integer s such that L(") 2 F' $ E' with F' free and E' not faithful. Since L is faithful, L ( ~is) also faithful, so F' # 0. Since
. have reduced to the case L and we can replace L by ~ ( " 1 We Denote t = dim(L). Since L 8 W = w ( ~we ) , obtain' that
Since F is free, say F
LBF
2
F'
$ E'.
we see that
E~ ( 4 1 ,
L@B(~) = (L @ ~ ) ( 4 ) - ( B ( * ' ~ ( ~ ) ) ) (by ( ~ )Proposition 7.2.2) E
-
~ ( t 4 )
-
~ ( t )
-
Equation (7.9) and the Krull-Schmidt theorem imply now that E ( ~ ) L ~ E . If E # 0 we have that
Proposition 7.2.2 tells then that F' @ E is nonzero and free, hence it is , This shows that E = 0 and faithful, and then so is ~ ( ~ a1 contradiction. then W is free. a Lemma 7.2.8 If any finite dimensional M E gM is free as a B-module, then any object M E EM is free as a B-module.
298
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
Proof: We first note that any nonzero M E EM contains a nonzero finite dimensional subobject in the category E M . Indeed, let N be a finite dimensional nonzero H-subcomodule of M (for example a simple Hsubcomodule). Then B N is a finite dimensional subobject of M in the category E M . For a nonzero M E EM we define the set 3 consisting of all non-empty subsets X of M with the property that B X is a subobject of M in the category EM, and B X is a free B-module with basis X . By the first remark, 3 is non-empty. We order F by inclusion. If (Xi)iEI is a totally ordered subset of F, then clearly X = UicIXi is a basis of BXi is a subobject of M in E M , the B-module B X , and B X = CiEI so X E 3. Thus Zorn's Lemma applies and we find a maximal element Y E 3. If B Y # M , then M I B Y is a nonzero object of EM, so it contains a nonzero subobject S/BY, where B Y S 5 M and S E Z M . Let Z be a basis of S/BY and Y' C S such that .rr(Y') = Z, where n : S -,S/BY is the natural projection. Then S is a free B-module with basis Y U Y', so Y U Y' E F , a contradiction with the maximality of Y. Therefore we must have BY = M, so M is a free B-module with basis Y. I Theorem 7.2.9 (Nichols-Zoeller Theorem) Let H be a finite dimensional is free as Hopf algebra and B a Hopf subalgebra. Then any M E a B-module. I n particular H is a free left B-module and dim(B) divides dim(H).
EM
Proof: Lemma 7.2.8 shows that it is enough to prove the statement for finite dimensional M E g M . Let M be such an object. Since H is finitely generated and faithful as a left B-module, and from Proposition ~ B-modules, ( ~ ) ) we obtain from Proposition 7.2.7 7.2.1 H @ M E M ( ~ ~ as that M is a free B-module. The last part of the statement follows by taking
M = H.
I
Corollary 7.2.10 If H is afinite dimensional Hopf algebra, then the order of G ( H ) divides dim(H). I As an application we give the following result which will be a fundamental tool in classification results for Hopf algebras. If H is a Hopf algebra, , is an ideal of H . A Hopf subalgebra we will denote by HS = K e r ( ~ )which K of H is called normal if K + H = H K + .
Theorem 7.2.11 Let A be a finite dimensional Hopf algebra, B a normal Hopf subalgebra of A, and A/B+A the associated factor Hopf algebra. Then A is isomorphic as an algebra to a certain crossed product B#,A/B+A. Proof: We prove the assertion in a series of steps. Step I. B+A is a Hopf ideal of A.
7.2. T H E NICHOLS-ZOELLER THEOREM
299
It is clearly an ideal of A. Since c = (E@ & ) A ,if b E K e r ( ~ )then , A(b) E Ker(@ ~ c) = B + @ B B @ B + , so B+A is also a coideal. Finally, since S ( B + ) B + , it follows that the antipode stabilizes B+A. Step 11. A becomes a right H-comodule algebra via the canonical projection T : A -+ H = A/(B+A), and B = ACoH. The canonical inclusion i : B -t A is a morphism of left B-modules, and since B B is injective ( B is a finite dimensional Hopf algebra), i splits, i.e. there exists a left B-module map O : A -+ B with O i = IB. Let
+
We show that Q(B+A) = 0. Indeed, if b E B and a E A, then
g
It follows that there exists : H -t A a linear map such that Q n = Q. Thus from C a l Q ( a 2 ) = iO(a) we deduce that C a l Q ~ ( a 2 )=,iO(a). Let . a E AcoH, i.e. C a l @ 7r(a2) = a @ ~ ( 1 )Then
so a = iO(a) E B. Conversely, if b E B , then from C bl @ b2 = b @ 1 + C b i @ (b2 - c(b2)l) we obtain that C bl @ 4 b 2 ) = b @ n ( l ) , i.e. b E AcoH. Step 111. The extension A/B is H-Galois. Recall from Example 6.4.8 1) that the canonical Galois map
is bijective with inverse P-l(a @ b) = C a S ( b l ) @ b2. If M denotes the multiplication of A, then if we denote
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
300
we have P ( K ) = A 8 B+A. Indeed, if a,.al E A, b E B , and x = ab @I a' a 8 bat E K, then
which is in A @ B+A, since &(b)l- b E B + for all b E B . Conversely, for a , a' E A, b E B + , we have
-
Now p induces an isomorphism from (A@A)/K E A@BA to (AOA)/(A@ B+A) A 8 H , which is exactly the Galois map for the extension AIB. Step IV. We have that A 21 B @ H as left B-modules and right H-comodules. Recall first the multiplication in the smash product A#H*:
-
where (h* x)(y) = h*(xy). It follows that B 8H* is a subring of A#H*. We know from Nichols-Zoeller that BA is free of finite rank. Applying the same t o AOp,we get that B ~ ~ A OisPfree, then using the algebra isomorphism S : Bop 4 B it follows that A ~ free, S and thus AB is free (with the same rank as BA). Denote by 1 the rank of AB. Since the extension A / B is Galois, the map
is an isomorphism of A - B-bimodules, and hence we have that A$) r (B:) B B A)B E (A OB A)B ( A @ H ) B 2 ~ g ) By . Krull-Schmidt we get 1 = n, i.e. AB is free of rank n = dimk(H). By the above, we have that BA is also free and r a n k ( ~ A )= n. Now, A has an element of trace 1. Indeed, if t E H * is a left integral, and h E H is such that t(h)l = n(1) = hlt(h2), then an element a E A with n(a) = h
7.2. THE NICHOLS-ZOELLER THEOREM is an element of trace 1: t .a
Ct(li(u2))a1
= =
t(r(a1))f (.(a2))
=
C t ( T ( ~ ) l )(.(.)2) f C t ( h l ) f (h2) f (Ct(hlIh2)
=
f(n(l))=l.
= =
(f is a left inverse for T)
The extension A/B is also Galois, so the categories and AM^ are equivalent via the induced functor A @B -. If PI, P2,...,P, are the only projective indecomposables in E M it follows that A@BPI,A@BP2,...,A@B P, are the only projective indecomposables in AM^ N A # H * M .Write
Now A#H* is projective in
A
~ SOH
--
in this category, and in particular as left A-modules. But A#H* An as left A-modules, and again from this and from (7.11) we get as above that li = nlci and so A#H* = An (7.12)
as left A#H*-modules, therefore as left B @I H*-modules. We have now that A@H*=A#H*
(7.13)
as left B @ H*-modules, via
4 is bijective
-
Now it is clear that
with inverse
4-' : A#H* A @ H*, 4-'(a#h*) and 4 is left B €3 H*-linear because
=
C a.
@
h*
-
~ - l ( ~ ~ ) ,
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CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
Since BA is free of rank n, we get from (7.13) that
as left B 8 H*-modules. Combining this with (7.12) we obtain that
as left B 8 H*-modules. Since B 8 H* is finite dimensional, we can write
an indecomposable decomposition. Now A#H* A is projective, A#H* is free over B 8 H*, so A is also projective as a left B 8 H*-module. Thus
as left B 8 H*-modules. By (7.14) we have now
as left B 8 H*-modules, so by Krull-Schmidt we obtain that ki = ti, i = 1 , . . . ,s, i.e. A E B 8 H* as left B 8 H*-modules. But H* E H as left H*-modules by Theorem 5.2.3, so A B 8 H as left B-modules and right H-comodules. The result follows now from Theorem 6.4.12. I
--
7.3 Matrix subcoalgebras of Hopf algebras We say that a k-coalgebra C is a matrix coalgebra if C E Mc(n,k) for some positive integer n. This is equivalent to the fact that C has a basis (cij)lli,jln, with comultiplication A and counit E defined by
<
n. A basis of C for which the comultiplication and for any 1 5 i, j the counit work as above is called a comatrix basis of C. Of course, a matrix coalgebra might have several comatrix bases. Let C be a matrix coalgebra, and we fix a comatrix basis (cij)lsi,jgn of C. We know (see Exercise 1.3.11) that C* Mn(k), the matrix algebra, where the matrix units of C* are the elements (Eij)lli,jsn C* defined by Eij(c,,) = bi,bj,. We identify C* and Mn(k), in particular we can consider the trace T r ( c * )
-
7.3. MATRIX SUBCOALGEBRAS OF HOPF ALGEBRAS
303
of an element c* E C * ,which is the sum of the coefficients of all Eii's in the representation of c* in the basis (Eij)lli,jlnof C*. We define a bilinear form ( I ) : C x C -+ k by (cijlcTs)= bisbjTfor any 1 5 i ,j,r, s 5 n. This form induces a linear morphism J : C -+ C* by J(c)(d)= (cld) for any c, d E C . We clearly have that J(cij)= Eji for any i, j , thus J is an isomorphism of vector spaces.
Lemma 7.3.1 For any c, d E C we have that: (2) = Tr(J(c)J(d)). (ii) E ( C ) = C i = l , n ( ~ I ~=i iT)r ( J ( c ) ) , Proof: (i) It is enough to check the formula for elements of a basis of C , i.e. we have that
(ii) Again we check on elements of the basis. We have that ~ ( c j , = ) bjT,
and T r ( J ( c j T )=) T r ( E T j = ) bjT,which ends the proof. 1 Since J is a linear isomorphism, we can transfer the k-algebra structure of C* = Mn(k) on the space C. Thus C becomes an algebra with the multiplication o defined by
for any c, d E C. In particular we have that
The identity element of the algebra (C,0 ) is <-l(Ci=l,n Eii) = Ci=l,nC i i . We denote xc = CiXl,, ci,. Note that (C,o ) is the unique algebra structure on the space C making J an algebra morphism. In order to avoid confusions (for eg. in the situation where C is a subcoalgebra of a Hopf algebra, and there exists already a multiplication on elements of C ) we will denote by c ( ~the ) element c o c o . . . o c (c appears r times), and by c(-') the inverse of an invertible c in (C,0). We need a set of formulas.
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CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
Lemma 7.3.2 Let c, dl e E C. Then the following formulas hold. (2) (cld) = ( 4 4 . (ii) c o d = C ( c l 1d)cz = C(cld2)dl. (iii) (cld) = E(Co d) . (iv) (xly o z) = (a: o ylz) = (co dle) for any cyclic permutation {x, y, z ) of the set {c, d, e). (v) (c 0 die) = C(cle1>(dle2).
Proof: We use for the proof the well known commutation formula for traces Tr(AB) = Tr(BA) for any matrices A, B E Mn(k). (i) (cld) = Tr(t(c)t(d)) = Tr(t(d)t(c)) = (dlc). (ii) It is enough to check for basis elements. Take c = cij and d = c,,. Then c o d = &iiscTj and
The second formula can be proved similarly. (iii) Apply E to (ii) and use the counit property. (iv) We have that (XIYO z )
Tr(t(x)t(y O 2)) = ~r(t(x)t(t-l(t(~)~(z))) = Tr(<(x)t(y)E(z)) = Tr(t(c)t(d)[(e)) (by the commutation formula for traces) =
=
( c o dle)
The other equality follows now from (i). (v) C(cled(dle2)
=
( 4 C(dlez)e1)
(cld 0 e) (by (ii)) = (codle) (by (iv)) =
I We prove now a Skolem-Noether type result for C. Proposition 7.3.3 Let (b : C -+ C be a linear morphism. Then g5 is a coalgebra morphism if and only if there exists t E C, invertible in the
7.3. MATRIX SUBCOALGEBRAS OF HOPF ALGEBRAS
305
algebra (C, o), such that 4(c) = t(-') o c o t for any c E C . Moreover, in this case we have that T r ( 4 ) = & ( t ) ~ ( t ( - ' ) ) .
Proof: We transfer the problem to the dual algebra of C (which we identified with n / l , ( k ) ) , then use Skolem-Noether theorem there and transfer it back. Thus we have that 4 : C -t C is a coalgebra morphism if and only if +* : C* --+ C* is an algebra morphism. By using Skolem-Noether theorem, this is equivalent to the existence of some invertible T E C * such that 4*(c*) = Tc*T-I for any c* E C*, which applied to an element c E C means that c*(4(c)) = ~(cl)c*(~2)~-'(~3)
C
for any c* E C*. This is equivalent to
Let t = <-l(T), which is an invertible element in the algebra (C, 0). Since T(d) = c(t)(d) = (tld) for any d E C , the proof of the equivalence in the statement is finished if we show that for an invertible t in (C,o) and an element c E C we have that
This is equivalent to
which applied to some d E C means that
But x(tlcl)(t(-"lcr)(c2ld)
=
x(t1cl)(dl~2)(t(-~)1~3)
=
x(t0 dlcl)(t(-l)~c2)(by Proposition 7.3.2(v))
=
(t o d o t(-l)lc) (by Proposition 7.3.2(v)) o c o tld) (by Proposition 7.3.2(iv))
-
which is what we wanted. For the second part, if 4 is a coalgebra morphism, we have showed that ,
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CHAPTER 7. FINITE DIMENSIONAL HOPE' ALGEBRAS
for any i and j , and this shows that
I
The bilinear form ( I ) was defined in terms of the comatrix basis ( c ~ ~of the ) matrix ~ ~ coalgebra ~ , ~C . ~At this ~ point we are able to show the following.
Corollary 7.3.4 The bilinear form ( I ) : C x C the choice of the comatrix basis (cij)lsi,jsn.
-+
k does not depend o n
Proof: Let (c:j)lsi,j5n be another comatrix basis of C , and let [ , ] be C the bilinear form defined by (c;)lli,jln. Then the linear map 4 : C defined by q5(cij) = cij for any i,j is a coalgebra morphism, so we can apply the previous proposition and find an element t invertible in (C, o ) , where o is the multiplication defined on C by the form ( I ) associated to the basis ( c ~ ~ such ) ~ that ~ ~$(c), =~ it(-') ~ o~ c o, t for any c E C. In particular we have that cij = t(-') o cij o t for any i ,j . Then -+
( c : ~ ~ c ; , )= (t(-')0 C i j 0 tit(-') 0 G, 0 t ) = (cijlc,,) (by Proposition 7.3.2(iv)) (cij crs )
= =
siss,,
=
[c:j lc;sl
which shows that ( I ) = [
I 1.
I
Remark 7.3.5 A n immediate consequence of the corollary is that the map
< and the algebra structure (C,o ) , in particular the identity element of this,
do not depend o n the choice of the comatrix basis (cij)lli,jln. Thus we can regard the element xc as a distinguished element of the matrix coalgebra I C, no matter what comatrix basis we choose i n C . Let H be a Hopf algebra which has a nonzero left integral X E H*. h, is an isomorphism We know that the map q5 : H -+ H * , + ( h ) = X of right H-Hopf modules (Theorem 4.4.6). Since q5 is a morphism of right H-comodules, it is also a morphism of left H*-modules, which means that for any h E H and p E H* we have
-
7.3. MATRIX SUBCOALGEBRAS OF HOPF ALGEBRAS
307
If we apply (7.15) to an 'element a E H we obtain that
which can be rewritten as
Lemma 7.3.6 Let H be a Hopf algebra with antipode S , X E H* a left integral, and C , D subcoalgebras of H such that C n D = 0 . Then X(cS(d)) = 0 for any elements c E C , d E D. Proof: Take X a linear subspace of H such that H = C @ D $ X , and pick p E H* such that the restriction of p to D is 0 and the restriction of p to C is E . Then for c E C,d E D we have that
--
X(cS(d)) = X((c p ) S ( d ) ) = X(cS(p d ) ) (by formula (7.16)) = 0
I We know that a cosemisimple Hopf algebra has nonzero integrals, so it has a bijective antipode. The next result gives more information about the antipode.
Theorem 7.3.7 Let H be a cosemisimple Hopf algebra with antipode S . Then S 2 ( C ) = C for any subcoalgebra C . Proof: Exercises 5.5.9 and 5.5.10 guarantee the existence of a left integral X E H* with X(1) = 1, and moreover AS = A. The existence of a nonzero integral shows that the antipode is bijective. Since any nonzero subcoalgebra is a sum of simple subcoalgebras (see Exercise 3.1.6), it is enough to prove that S 2 ( C ) = C for any simple subcoalgebra C . Indeed, if C is simple, then S 2 ( C ) is a simple subcoalgebra of H, since S 2 is an injective morphism of coalgebras. Then we have either S 2 ( C ) = C or C m 3 2 ( C ) = 0. If we were in the second situation, then for any d , c E C we would have X(S2(c)S(d)= ) 0 from Lemma 7.3.6. But
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CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
so we obtain that X(dS(c)) = 0 for any c, d E C. In particular for any c E C we have
which implies that C = 0 (from the counit property), a contradiction. It 1 remains that S2(C) = C . Proposition 7.3.8 Let H be a Hopf algebra with antipode S , X E H* a left integral, C C H a matrix subcoalgebra with comatrix basis (cij) lli,jl,. Then the following assertions hold. (1) If i # j , then X(ciTS(csj)) = 0 for any r , s . (2) X(ciTS(csi)) = X(cjrS(csj)) for any i , j , r, 3. Proof: Let X be a linear complement of C in H. We first note that for c , d E C we have that C(c11d)c2 = c p, where p E H* is defined by p(h) = (hld) for h E C and p(h) = 0 for h E X. Then by Lemma 7.3.2(ii) we obtain c o d = c p. Similarly c o d = q d, where q(h) = (hlc) for h E C and q(h) = 0 for h E X. Let now pick some 1 i,j, r, v, s 5 n , and take p E H* such that p(h) = (hlcj,) for h E C and p(h) = 0 for h E X. Then
-
-
Use now ciT o cj, = biVcj, and cj,
0
-
<
csj = csv and obtain
For v = j we obtain
which for i # j shows relation (1). For v = i we obtain relation (2). Theorem 7.3.9 (The orthogonality relations for matrix subcoalgebras) Let H be a Hopf algebra, X E H* a left integral, and C,D matrix subcoalgebras of H . Then X ( X C ~ ( X D = )) bc,~X(l).
7.3. MATRIX SUBCOALGEBRAS OF HOPF ALGEBRAS Proof: If C # D we apply Lemma 7.3.6 to the elements X D E D and find that X ( x c S ( x D ) ) = 0. If C = D, then
=
x x
xc
309 E
C and
h(ciiS(cii)) (by Proposition 7.3.8(1))
l
=
h ( c l i S ( c i l ) ) (by Proposition 7.3.8(2))
llisn
= X(~(~11)1)
= X(1)
I For further use we need the following. Lemma 7.3.10 Let H be a cosemisimple Hopf algebra, X E H* a left integral, and C a matrix subcoalgebra of H . Let t € C an invertible element i n ( C ,o) such that S 2 ( c ) = t(-') o c o t for any c E C . Then ~ ( t#) 0 and c ( t ( - l ) # 0.
Proof: Let X be a linear complement of C in H , and define #,+ E' H* by + ( h ) = X ( X ~ S ( for ~ ) any ) h E H , # ( h ) = (tlh) for any h 6 C and # ( h )= 0 for any h E X . Let c E C and p E H* such that p(h) = (hlc) for h E C and p ( h ) = O f o r h ~ XW . ehaveseenthatp-d=codandd-p=docfor any d E C. Then for any d E C we have
We have showed that for any c, d E C we have that
+(c o d ) = X(cS(d)) Then
+(cod)
=
= = = =
X(cS(d)) XS(cS(d)) X(s2(d)S(c)) X((t(-') o d o t ) S ( c ) ) +(t(-') o d o t o c ) ( b y (7.17))
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
310
This means that $J is zero on the space V spanned by all cod - t(-l) o d o t oc with c, d E C. On the other hand +(t(-') o d o t o c) =
(tit(-') o d o t o c)
=
(tic o d) (by Lemma 7.3.2(iv)) = $(cod) so 4(V) = 0. Let C* Mn(k). Since codim(< AB- BAIA, B E M,(k) >) = 1 in Mn(k), by transferring this fact via the isomorphism : (C, o) -4 Mn(k), we obtain that codim(< c o d - d o c(c,d € C >) = 1 in C (by < S > we denote the linear subspace spanned by the set S). Since t is invertible in (C, o), this implies that codim(< (t o c) o d - d o (t o c)lc, d E C >) = 1 in C , and then codim(< t(-')(t o c o d - d o t o c)lc, d E C >) = 1 in C , which means that the codimension of V in C is 1. Since t # 0 we have that 4 # 0 (otherwise the image o f t in C* through the isomorphism would be zero). Since both 4 and $J are zero on V, we obtain that there exists CY E k such that = cu4. The orthogonality relation shows that $(xc) = X(xcS(xc)) = 1. But
--
<
+
, implies that ~ ( t #) 0. To obtain that &(t(-l)) # 0 we so 1 = a ~ ( t )which where t is replaced by t(-'1. regard everything in the Hopf algebra HOPICOP,
I
7.4
Cosemisimplicity, semisimplicity, and the square of the antipode
-
In this section H will denote a finite dimensional Hopf algebra and X E H* a left integral, 4 : H --, H*, $(h) = X h, the isomorphism of right H-Hopf modules. We recall from the previous section that for any h E H and p E H* we have (formula (7.15))
and for any a, h E H and p E H* we have (formula (7.16)
7.4. SEMISIMPLE AND COSEMISIMPLE HOPF ALGEBRAS
311
Let A = ~ - I ( E ) For . any p E H * we have that
-
A for any p E H*. If we apply the equality q5-lq5 = I d to thus $-'(p) = p h) A, which in particular for h = 1 an h E H, we obtain that h = (A A = 1, and then if we apply E we obtain that X(A) = 1. If shows that X A = E. we apply the relation $4-I = I d to E we find that X On the other hand for any h E H we have that
- -
-
-
showing that Ah = E ( ~ ) AThus . A E H is a right integral For any q E H* we denote by R(q) : H* -+ H* the linear morphism induced by the right multiplication with q, i.e. R(q)(p) = pq for any p E H * . We also denote by 7 : H* @ H -+ End(H*), ~ ( @Iq h)(p) = p(h)q for any p,q E H * , h E H . Then 7 is a linear isomorphism (Lemma 1.3.2). With h)(p) = C q ( ( X this notation equation (7.15) can be written R(X h l ) @ h 2 ) ( ~ )or , (7.18) R(X h) = x r l ( ( X h l ) B hz)
-
-
-
-
If f : H -+ H is a linear morphism, denote by f * : H* -t H* the dual morphism of f . Then for any p, q E H*,a , h E H we have
so q(q @ h) o f * shows that
=
q(q
f (h)), which combined with the relation (7.18)
R(A -- h) 0 f * =
C ll((h - h l ) 8 f ( h d )
(7.19)
We recall that for a finite dimensional vector space V with basis ( u , ) ~ ~ ~ ~ , and dual basis ( v , ) ; ~ , ~in , V*, the trace of a linear endomorphism u : V --, V is
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CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
This does not depend on the choosen basis of V , and in fact it is just the In particular, the fact that trace of the matrix of u in the basis (vi)15iLn. T r ( A B )= T r ( B A )for any A, B E Mn(lc)shows that Tr(uv)= Tr(vu)for any u,v E Endk(V).For the dual morphism u* : V * --+ V * we obtain
In particular, if q E H* and h E H we have that
thus
Tr(rl(9@ h ) )= q(h) If we write equation (7.19) for h = A and use the fact that R ( E )= Id, we obtain (7.21) f* = V ( ( A Al) 8 f ( A d )
C
-
Equation (7.21)shows by using (7.20) that
We are able to prove now an important result characterizing semisimple cosemisimple Hopf algebras.
Theorem 7.4.1 Let H be a finite dimensional Hopf algebra with antipode S . Then H i s semisimple and cosemiszmple zf and only zf T r ( S 2 )# 0 .
7.4. SEMISIMPLE AND COSEMISIMPLE HOPF ALGEBRAS
313
Proof: We have that
T ~ ( s ~ =) =
X(S~(AZ)S(AI))
(by (7.22))
,
C~s(~ls(h2))
= x(&(A>s(l)) =
X(l)€(A)
This ends the proof if we use the facts that H is semisimple if and only if € ( A ) # 0 (Theorem 5.2.10) and H is cosemisimple if and only if X(1) # 0 (Exercise 5.5.9). I We define for any h E H and p E H* the linear morphisms l ( h ) : H -+ H and l ( p ) :H 4 H by 1 ( h )( a ) = ha,
1 ( p )( a ) = p
-a
for any a E H . We have that
T r ( l ( h )o
soo 1 ) ) )
=
h ( h S 2 ( p-- A z ) S ( A l ) ) (by (7.22))
=
A(hS(AlS(p
= =
A2)))
C~ ( h s ( ~ l s ( ~ 2 ) ~ ( ~ . 3 ) ) ) C~ ( ~ s ( E ( A ~ ) P ( A ~ ) ) )
= X(h)p(A)
We have obtained
Exercise 7.4.2 Show that l(p)* = R ( p ) for any p E H*. In particular T r ( l ( p ) )= T r ( R ( p ) ) . Let us consider the element x E H such that
for any p E H*. Such an element x exists and is unique. Indeed, if i : H --+ H** is the natural isomorphism, and h** E H** is defined by h**(p) = T r ( l ( p ) )for any p E H * , we just take x = i-'(h*').
Exercise 7.4.3 Show that i f S 2 = Id and H is cosemisimple, then x is a nonzero right integral i n H .
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
314
Lemma 7.4.4 We have that Tr(l(x) o S 2 ) = A(l)&(A)= T r ( S 2 ) .
Proof: If we write (7.19) for f = I d we obtain
Then for any h E H (A
- h)(x)
=
- h)) (definition of x) V((X- hi) B h2)) (by (7.19)) C ( h - hlI(h2)
=
CNh2~(hl))
=
=
We obtain (A
Tr(R(X
- h)(z)
=
x
X(h2S(h1))
(7.24)
For h = 1, this shows that X(x) = X(1). We use now (7.23) for h = x and p = E and obtain Tr(1(z) o s 2 ) = X(x)&(A) = A(l)&(A) = T~(s~)
Lemma 7.4.5 The following formulas hold: 8x2 =EXZBXI. (i) (zi) x2 = dim(H)x = E(X)X. (iii) S2(x) = x. (iv) Tr(S2) = d i m ( ~ ) ~ r ( S i ~ ) . Proof: (i) Let ?1, : H*@I H* -+ ( H @H ) * be the linear isomorphism defined by $(pBq)(gB h) = p(g)q(h) for any p, q E H*, g, h E H. Then for proving that C x l B x2 = C 2 2 B x l it is enough to show that
for any p, q E H * . This can be seen as follows $(P @ q)(c x i @ 52)
(PB)(x) = Tr(l(pq)) (by the definition of x)
=
= T N P )0 l(9))
7.4. SEMISIMPLE AND COSEMISIMPLE HOPF ALGEBRAS
(ii) For any h E H we have that
(A
-
h)(x2) = (A
-
315
hS-l(x))(x)
=
A(~~S-'(X,)S(~IS-~(X~)))
=
A(~~s-~(xL)z-~S(~~))
= =
x
~(h2S-'(xz)xlS(hl)) (by
.(x)
(2))
C X(h2S(h1))
-
E(x)(A h)(x) (by (7.24)) = (A h) (E(x)x) =
for any h E H . Since H* = {A the other hand
-
-
hlh E H} we obtain that x2 = E(X)X.On
which completes the proof of (ii). (iii) Let p E H* and h E H. We have that
thus l(p o S 2 ) = S-2 o l(p) o S2. Then for any p E H* we have p(S2(x)) = ( P o S2)(x) = Tr(l(p o S2)) (definition of x) = TT(S-' o l ( p ) o s 2 )
316
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
which shows that S2(x) = x. (iv) Let T = 1(x) o S2. We know from Lemma 7.4.4 that Tr(T) = Tr(S2). We have that T(xh) = ( I (x) o s2)(xh) = x~'(xh) = xs2(x)s2(h) = x2s2(h) = dim(H)xs2(h) = dim(H)s2( x ) s 2(h) = dim(H)s2(xh) which shows that qxH= dim(H)SkH. Since obviously I m ( T ) & x H , we can regard TixH as a linear endomorphism of the space xH, and then
But since Im(T)
x H , we have that Tr(T) = T r ( q X H ) ,so we obtain
Theorem 7.4.6 (Larson-Radford) Let k be a field of characteristic zero and H a finite dimensional Hopf algebra over k, with antipode S . The following assertions are equivalent. (i) H is cosemisimple. (ii) H is semisimple. (zii) S2= Id. Proof: (iii)+(i) and (iii)+(ii) follow directly from Theorem 7.4.1 since T ~ ( S= ~ Tr(Id) ) = dim(H). (i)=+-(ii)We first use Exercises 4.2.17, 5.5.12 and 5.2.13 to reduce to the case where k is algebraically closed. Let C be a matrix subcoalgebra of H . We know that S2(C) = C (Theorem 7.3.7) and that there exists an invertible t in the algebra (C, o) such that S2(c) = t(-') o c o t for any c E C (Proposition 7.3.3). Let r be the order of S2 (which is finite by Theorem 7.1.7). Then c = S2r(c) = t ( - T ) o c o d T ) for any c E C, so dT) is in the
7.4. SEMISIMPLE AND COSEMISIMPLE HOPF ALGEBRAS
317
center of the algebra (C,0). Taking account of the algebra isomorphism [ : C + Mn(k) we have that the center of (C, o) is k x c , thus dT)= a X c for some a E k. Since t is invertible we have that a # 0, and then replacing t by &t (which still verifies S2(c) = t(-l) o c o t), we can assume that t(') = ~ c Then . [(t)' = I, the identity matrix, so the minimal polynomial of [(t) divides XT- 1, and hence it has only simple roots. This implies that the matrix [(t) is diagonalizable with eigenvalues r-th roots of unity. Since k has characteristic zero we may assume that the field of rational numbers Q k, and then, since k is algebraically closed, that the field Q (regarded as a subfield of the complex numbers) is contained in k. Since the inverse of a complex root of unity is the conjugate of that root, we obtain that [(t(-')) is diagonalizable with eigenvalues the (complex) conjugates of the eigenvalues of <(t). In particular ~ r ( t ( t ' - l ' ) ) ~ r ( t ( t )=) Tr(E(t))Tr(t(t))E R+ Proposition 7.3.3 tells that T~(s&) = ~ ( t ) ~ ( t ( - ' = ) )Tr ( t ( t ' - l ' ) ) ~ r ( ~ ( t ) )
'
On the other hand ~ ( t ) , ~ ( t ( - ' )#) 0 by Lemma 7.3.10, so T r ( S b ) > 0. We conclude that T r ( S 2 ) is a sum of positive numbers, since H is a direct sum of matrix subcoalgebras, so Tr(S2) > 0. In particular Tr(S2) # 0, showing that H is semisimple. (ii)=+(i) If H is semisimple, then H* is cosemisimple, and by (i)=+(ii) we obtain that H* is semisimple, so H is cosemisimple. (i)+(iii) Let H be cosemisimple. So H is also semisimple and then the distinguished grouplike elements of H and H * are 1 and E (by Remark 7.1.1). Using the formula of Theorem 7.1.6 we obtain S4= Id. Then S2 is diagonalizable and any eigenvalue of S2 is either 1 or -1. It follows that T r ( S 2 ) 5 dim(H). We have already seen that Tr(S2) > 0. On the other hand T r ( S 2 ) = (dim(H))Tr(SiH) by Lemma 7.4.5, which shows that dim(H) divides Tr(S2). Since 0 < T r ( S 2 ) 5 dim(H), we must have T r ( S 2 ) = dim(H), and this forces all the eigenvalues of S2 to be equal to 1. We obtain S2= Id. I Exercise 7.4.7 Let k be a field of characteristic zero and H a semisimple Hopf algebra over k. Show that a right (or left) integral t in H is cocommutative, i.e. t l @ t2 = C t2 C3 tl . Exercise 7.4.8 Let k be a field of characteristic zero and H be a finite dimensional Hop! algebra over k . Show that: (i) If H is commutative, then H N (kG)*for some finite group G. (22) If H is cocommutative, then H N kG for a finite group G.
318
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
Exercise 7.4.9 Let H be a finite dimensional Hopf algebra over a field of characteristic zero, and let S be the antipode of H . Show that S has odd order if and only if H H kG,where G = Cz x C2 x . . . x C2, and in this case S = Id, so the order of S is 1.
7.5
Character theory for semisimple Hopf algebras
In this section H will be a semisimple Hopf algebra over an algebraically closed field k of characteristic 0. Let V E H M be a finite dimensional H-module with basis and dual basis (vi);li5n in V * . We can regard V as a representation-of the algebra H , i.e. an algebra morphism p : H --+ Endk(V),defined by p(h)(v)= hv for any h E H,v E V . Then the character x ( V ) of the H-module V is defined to be the character of the representation p. This means that x ( V ) E H*, x ( V ) ( h )= Tr(p(h))for any h E H. In terms of the choosen basis we have
Exercise 7.5.1 Let V ,W E H M be finite dimensional. Then x ( V @W ) = x ( V ) + x ( W ) and x(V @ W ) = x ( V ) x ( W ) . The antipode S of H can be used to construct a left H-module structure on the dual of a left H-module. If V E H M is finite dimensional, then V * = H o m k ( H ,k ) has an induced structure of a right H-module and then we transfer this action to the left by using the antipode. We obtain that V* E H M by hv*(v)= v*(S(h)v) for any h E H, v* E V * ,v E V .
Exercise 7.5.2 Show that for a finite dimensional V E x ( V * )= S * ( x ( V ) )where , S* is the dual map of S .
H M we have
Proposition 7.5.3 1) I f f : V -+ W is a morphism of finite dimensional left H-modules, then f* : W * -+ V * is a morphism of left H-modules. 2) If V is a finite dimensional left H-module, then I/** z V as left Hmodules. Proof: 1) Let h E H, w* E W * and v
E
V . Then
7.5. CHARACTER THEORY
2) We show that the natural linear isomorphism i : V 4 V** defined by i(v)(v*) = v*(v) for any v E V, v* E V*, is also a morphism of left Hmodules. Indeed, we have that (hi(v))(v*) = i(v)(S(h)v*) (S(h)v*)(v) = v*(S2(h)v) = v*(hv) (since s2= Id) = i(hv)(v*) =
Corollary 7.5.4 If V is a simple left H-module, then V* is also a simple left H-module. Proof: Let f : X 4 V* be the inclusion morphism of a left H-submodule of V*. Then f * : V** 4 X * is a surjective morphism of left H-modules. Since V** 2: V, thus it is a simple H-module, we obtain that either f * = 0, and then X = 0, or f * is an isomorphism, and then f = I d and X = V*. I If V is a simple left H-module, then we can also regard V as a simple right H*-comodule. The coalgebra C associated to V (i.e. the minimal subcoalgebra of H* such that V is a C-comodule) is a simple coalgebra, so it is a matrix coalgebra, since k is algebraically closed (see Exercise 2.5.4 and Exercise 3.1.2). The following lemma gives a description of the character of V in terms of the associated coalgebra C. Lemma 7.5.5 Let V be a simple left H-module and C be the coalgebra associated to the simple right H*-comodule V. Then x(V) = x c . Proof: We know that C has a comatrix basis ( c ~ ~where ) ~n =~ dim(V), such that the coaction on the elements of a basis (vi)lli5n works by p(vi) = C1ljlnvj Uj cij, where p : V -+ V Uj H* denotes the comodule structure map of V. This implies that the action of H on V works by cij(h)vj for any h E H and 1 5 i 5 n. In terms of this hvi = coaction, the character of the H-module V is given by
~
~
320
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
We obtain that x(V) = Cllilncii = XC. I Let H be a semisimple Hopf algebra, Vl, . . . , Vn the isomorphism types of simple left H-modules, and XI = x(V1), . . . , xn = x(V,) E H* their characters (usualy called the irreducible characters of H ) . We choose Vl to be the trivial H-module, i.e. Vl = k with the action given by h a = ~ ( h ) a for any h E H and a E k. We know from Corollary 7.5.4 that for any i, the left H-module I/,* is simple, so there exists a unique ? E (1,. . . ,n) such that V,' e y. Since V** e V as H-modules, the assignement i H ? provides an involutory permutation of the set (1,. . . , n). Moreover S*(xi) = XT and dim(y) = dim(V,). Let H e Md,(k) x . . . x Md,(k) be the representation of the algebra H as a product of matrix rings, such that V, is the unique isomorphism type of simple left H-module produced by Mdi(k), in particular dim(V,) = di. Moreover, the decomposition of the left H-module H as a sum of simple Hmodules is H r b l l i 4 n ~ ( dini )particular , x ( H ) = El
-
@ci,
<
Theorem 7.5.6 Let H be a semisimple Hopf algebra with antipode S , XI,. . . ,xn E H* the irreducible characters of H , and A* E H** an integral. I Then for any 1 5 i, j n we have A*(xiS*(xj)) = Si,jA*(E).
<
Let
C
CQ(H) =
Qxi
l
be the Q-subspace of H* spanned by XI,. . . ,x,. In fact CQ(H) is a Qsubalgebra of H*. Indeed, for any 1 i, j 5 n, xixj = x(V,)x(V,) = x(V,@ V,), which is a linear combination of X I , . . . , xn with nonnegative integer coefficients, since V , @ V,, as a left H-module, is a direct sum of simple H-modules. The Q-algebra CQ(H) is called the Q-algebra of characters of H. Similarly, the k-subspace of H* spanned by X I , .. . ,xn is a k-subalgebra of H*, denoted by Ck(H), and called the k-algebra of characters. An immediate consequence of the orthogonality relations is the following.
<
7.5. CHARACTER THEORY
32 1
P r o p o s i t i o n 7.5.7 The irreducible characters dependent over k (inside H*).
XI,
. . . , x,, are linearly in-
aixi = 0 Proof: Let A* E H**be an integral with A*(E) = 1. If El<,<,, for some scalars ai E k, we have that for any t
Corollary 7.5.8 There exists an isomorphism of k-algebras Ck(H) CQ(H) mapping Xi E Ck(H) to 1 @ Xi E k @Q CQ(H).
-
k @ ~
I
L e m m a 7.5.9 Let M E HM and M = X I @ . . . @ X, a representation of M as a sum of simple H-modules. Then M H = @{X,IX, N VI). Proof: Let m E M~ and m 1 5 i t . Then
<
=
ml
+ . . . + m,
with mi E Xi for any
showing that hm, = ~ ( h ) m for , any for any i and h E H, i.e. m, E X: for any i. Thus M H = X F $ . . . @ X.: The characterization of M H follows immediately from the fact that if X is an H-module, the subspace of invariants X H is an H-submodule, and if X is simple, then either xH= 0, I or X = X H N Vl. L e m m a 7.5.10 Let V and W be simple left H-modules. Then dim(V @ w * ) ~= 1 if V and W are isomorphic, and dim(V 63 w * ) ~= 0 if V and W are not isomorphic. Proof: Let 4 : V @W* -+ Homk(W, V) be the natural linear isomorphism, i.e. @(v@ w*)(w) = W * ( W ) V for any v E V,w* E W * , w E W. We can transfer via 4 the H-module structure of V @ W* to Homk(W, V). This means that if f = @(v@ w*) E Homk(W, V) (in fact we should take a sum
322
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
of tensor monomials in V 8 W*, but we write like this for simplicity), then for any h E H and w E W we have
Thus (hf )(w) =
x
hi f ( s ( h 2 ) ~ )
Now we have Hornk (W,V) = HornH (W, I/). Indeed, if f E HomH (W, V), then (hf)(w) = C hlf ( S ( h 2 ) ~ )= Cf ( h l S ( h 2 ) ~ )= 4 h ) f (w),SO f E Homk(W,v ) ~ . Conversely, let f E Homk(W, v ) ~ . Then for any h E H,wEW h f (w) =
hlf ( ~ ( h 2 ) ~ )
so f is a morphism of H-modules. Therefore (V 8 W*)H E H o r n ~ ( WV). , I The result follows now by using Schur's lemma. Proposition 7.5.11 Let x l , . . . ,X, be the irreducible characters of H . Then for any i and j we have XiX7 = Sijxl C2
+
Proof: We know that X ~ X T= x(K @ Vj),which is a linear combination of the irreducible characters with nonnegative integer coefficients. Moreover, the coefficient of XI is the number of appearances of Vl in the decomposition of V,@T/,* as a sum of simple modules. Lemma 7.5.9 shows that this number a is dim(V 8 W*)H, and this is bij by the previous lemma. Theorem 7.5.12 The Q-algebra of characters C Q ( H ) is semisimple.
7.5. CHARACTER THEORY
323
Proof: We first show that if u E CQ(H), u # 0, then uS*(u) # 0. Indeed, xixi, for some 21,. . . ,x , E Q, not all zero. Then let u =
xlSisn
By Lemma 7.5.11, the coefficient of ~1 in the representation of uS*(u) as x! # 0, so uS*(u) # 0. a rational linear combination of X I , .. . , xn is Since CQ(H) is finite dimensional, to end the proof we only need to show that CQ(H) does not have nonzero nilpotents. Indeed, if z were a nonzero nilpotent, let v = zS*(z). The above remark shows that v # 0, and clearly S*(v) = v. Then v2 = vS*(v) # 0, and continuing by induction we obtain that v2'" # 0 for any positive integer m. But z lies in the Jacobson radical of CQ(H), and then so does v. In particular v is itself a nilpotent, a contradiction. I
xi
--
Exercise 7.5.13 Let C MC(m,k) be a matrix coalgebra and x E C such that X I €3 2 2 = C 2 2 €3 X I . Then there exists a! E k such that x = a x 6
x
Proposition 7.5.14 If H is a semisimple Hopf algebra and X E H* an integral with X(1) = 1 , then x(H) = - dixi = dim(H)X.
xlsi,,
Proof: Let X = El,i,, Xi, with Xi E Ci for any i. Since A(X) = TL(X) (by Exercise 7.4.7), 2nd A(&) E Ci €3 Ci, we see that A(&) = TA(Xi) for any i. According t o Exercise 7.5.13, we must have Xi = aixc, = Xi for some ai E k. If A* is an integral in H**such that A*(€) # 0, then for any j we have that
A* ( E ) %
aiA*( x i s *(xi)) (orthogonality relations)
=
lsisn =
A*(XS*(x?)
= A* (S*(X)S*( x ~ )(since S*(A) = A)
=
djA*(A) (since d j = d?)
Since A*(E)# 0, we obtain that a!, = #dj, for some a E k. Evaluating a t 1 we see that
so then
-
dixi = ' a h
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
324
I
showing the required formula.
Corollary 7.5.15 Let X E H* be an integral such that X(1) = 1, and denote by Z ( H ) the center of the underlying algebra of H . Then Ck(H) = Z(H) A.
-
Proof: We know that H cx Mdl (k) x . . . x Md, (k). Denote by cl, . . . , cn the images in H of the identity matrices in Mdl(k), . . . ,Mdn(k) via this isomorphism. Then ci acts as identity on the simple H-module V,, and c.V. - 0 for any j # i. We obviously have Z(H) = @ l-< i < n k ~ iand , ci = 1. Then for any 1 5 j 5 n , and h E H we have (cj
-
X)(h) = X(hcj) =
dim(^))-I
dixi(hcj) 11isn
= =
(dim(H))-ldjXj(hcj) ( d i m ( ~ ) ) - ' d ~ x ~ ( (since h) cjx = x for x E 4)
-
so cj X = (dim(H))-ldjXj for any j, and this clearly ends the proof. I The following is just a reformulation of the result in Proposition 7.5.14 for the dual Hopf algebra, but we will also need it in this form.
Corollary 7.5.16 Let A* E H**be an integral such that A*( E ) = 1. Then I the character of the left H*-module H* is dim(H)A*.
7.6
The Class Equation and applications
Lemma 7.6.1 Let A be a semisimple Q-algebra with basis {al,. . . , a n ) such that aiaj E CIS,,, Zat for any i, j. Then for any field extension Q C IC, there exists an isomorphism of K-algebras
for some positive integers s, rl, . . . , r,, such that for any a E A, 4(1 @ a ) = Ca,i,j raijEaijwith all r,ij algebraic integers, where for any 1 5 a 5 s we denoted by (Eaij)lSijlra the matrix units in MTa(K).
Proof: Let K be a splitting field of the Q-algebra A such that Q C K 5 Q and Q C K is a finite field extension (see for example (112, page 1131). Any finite extension of Q will be considered inside Q. Then K @qA ci n,=,., Mra(K) for some positive integers s , r l , . . . , r n . This K-algebra
7.6. THE CLASS EQUATION AND APPLICATIONS
325
isomorphism can be also written in the form K @Q A N n a = l , s end^ (V,), where Vl, . . . ,V, are the isomorphism types of simple left K @Q A-modules. Since K is a splitting field of A, we have that E n d ~ ( v , ) ? K for any lIa<s. We show that for any simple left K @Q A-module F , say of dimension r over K , there exists a basis of the K-vector space K @K F such that for the^ associated matrix basis of Endrc(K @ K F ) N M,(K), any endomorphism ya E EndK(KB K F),ya(x) = ax, the left multiplication with the element a E A, is a linear combination of the matrix basis with coefficients algebraic integers. The required isomorphism 4 follows immediately from here. So let us take a simple K @Q A-module F. Then F is a K-vector space by a x = ( a @ 1)x, and a left A-module by ax = (1 @ a)x, for a E K, a E A, x E F. Moreover, a ( a x ) = a(ax) for such a,a , x. Let { h l , .. . , h,) be a basis of F over K . We denote by R the ring of algebraic integers of K . It is known that R is a Dedekind ring and K is its field of fractions (see for example [65, page 108)). Now we define
which is an R-submodule of F . Then U is a torsionfree finitely generated R-module. Indeed, for any u E U , we have u E F, and then annR(u) C annK(u) = 0. We prove now that the morphism
is an isomorphism of K-vector spaces. It is surjective, since
where the last equality follows from the fact that F is a simple K @Q Aai @ R xi) = Q i X i = 0, for module. To show that 4 is injective, let some ai E K , xi E U . Since K is the field of fractions of R , there exists a nonzero N E R (in fact even N E 2) such that N a i E R for any i. Then
$(xi
xi
326
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
Then U is a projective R-module, since it is finitely generated torsionfree over the Dedekind ring R (see [65, Theorem 22.51). In particular U is R-flat, so R @R U embeds naturally in K @R U. Obviously, $(R @R U) = U. The structure theorem of finitely generated torsionfree modules over a Dedekind ring ((65,Theorem 22.51) tells that there exist (wq)lSqlT C U and fractional ideals of R (i.e. finitely generated R-submodules of K ) such that R g R U = @llq5TIq(1@wq),an internal direct sum of R-submodules. Since $ is an isomorphism of R-modules, we have that an internal direct sum of R-submodules. Note that since I, is not necessarily contained in R, and U is just an R-module, the product Iqwq makes sense only if we regard everything inside F, which is a K-vector space. However, Iqwq E U for any q. A classical result for Dedekind rings ((65, Theorem 20.141) ensures us that there exists a finite extension K E K', with the ring of algebraic integers of K' denoted by R', such that all the extensions of the fractional ideals I, to K' are principal, i.e. there exist C K ' such that R'I, = R'a, for any q. K' is also a splitting field of A and the simple modules over K' @Q A are obtained from the simple modules over K' @Q A by extending scalars from K to K'. We extend all the construction from K to Kt. Thus we take F' = K' BK F , which is a simple K' @Q A-module by (a' 8 a)(Pt 8 x) = a'P' @ ax for any a', P' E K t , a E A, x E F. A basis We construct U' inside F' in a way of F' over K t is (1 @ hpll 5 p 5 I-). similar to the construction of U inside F, i.e.
On the other hand The above isomorphism $ : K'@R U 4 F' is given by $(al@R X) = a ' @ a:~ for any a' E K t and x E F. It is easily checked that $(R' @R U) = U'. This implies that
7.6. THE CLASS EQUATION AND APPLICATIONS
327
Since RIIq = R'o,, it is easy to see that Q(R' @R Iqwq) = R1(oq@K w,). We obtain that U' = @,R1(aq @K w,), a direct sum of R'-submodules. It ~ ~linearly ~ , independent over R' (note follows that the elements ( Z L , ) ~are that the annihilators of these elements are zero, since we work inside the Kt-vector space F'), and then they are obviously linearly independent over K' (inside F'), thus forming a basis of F' over K'. Moreover, for any 1<j n we have that ajU1 C U'. Indeed, if r' E R', 1 i 5 n and 1 p r, then ajai = ztat from the hypotesis on the basis (ai)i of A, and then
< < <
<
=
xr's
@K
ath,
t
thus ajU' C U'. Since U' = @,R1uq, we obtain that aju, = Ctr:ut for , K'-vector some (ri)t C R'. This means that for the basis ( u , ) ~ ~ ,of~ the space F', the endomorphisms y, E EndK)( F ' ) given by multiplication with elements a E A, can be expressed as linear combinations of the matrix basis ~ , algebraic integer coefficients. of EndK/(F1) associated to ( u , ) ~ ~ , with Obviously, this fact can be extended by scalars to any field extension of K', in particular to K (since Q C K C K' C_ Q IC). I We apply the previous lemma for A = CQ(H) and K = k, where H is a semisimple Hopf algebra over the algebraic closed field k. We find that there exist some positive integers s, rl , . . . , r, and an isomorphism of k-algebras 4 : k @Q A -+ n,=l,, hfTa(k) such that for any a E A, 4(1 @ a ) = Cff,Z,f. . raijEaijwith all r,ij algebraic integers, where for any 1 a 5 s we denoted by (Ecuij)lli,j
<
-
<
eaii. In particular (eaii),,i is a complete for any a, p, i, j , u, v, and E = system of orthogonal primitive idempotents of Ck(H). Also, for any 1 5 u n XU = r~,aije,ij
<
a,ij
for some algebraic integers (ru,,ij),,i,3. If we denote by t r : H* -t k the character of the left H*-module H*,we have
,
328
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
that tr(eaii) is a positive integer for any a and i. Indeed, since e,ii is an idempotent, the left multiplication by e,ii is an idempotent endomorphism of H*, so it is diagonalizable and has eigenvalues either 0 or 1, and not all of them 0 (since e,ii # 0). Then tr(eaii), which is the trace of this endomorphism is a positive integer. On the other hand, for any a and any i # j, tr(eaij) = 0, since e,ij = e,iie,ij - e,ije,ii, and tr(xy) = tr(gx) for any x, y E H*.
Exercise 7.6.2 Show that for any idempotent e E H * , tr(e) = dim(eH*). This provides another way to see that tr(eaii) is a positive integer. Theorem 7.6.3 For any 1 I a divides dim(H) .
Is
and 1
I i I r,,
the integer tr(eaii)
Proof: Let A* E H**be an integral such that A*(&)= 1. We know that t r = dim(H)A* (Corollary 7.5.16). This shows that for any a: and any i # q we have A*(eaiq) = &tr(eaiq) = 0. We use the orthogonality relations and obtain that for any 1 u, v I n
<
L e t I = { l , . . . ,n ) a n d J = { ( c r , i , j ) l l I a I s , l I i , j I r , ) . T h e n I and J are sets with n elements and the above relation can be written using matrices as X Y = Id, where
are matrices defined by
Thus X is an invertible n x n-matrix and Y is its inverse matrix. Then YX is also the identity matrix, in particular for any a: and i we have that
7.6. THE CLASS EQUATION AND APPLICATIONS 1 = C , y,ii,,x,,,ii.
This means that
x,
1 In particular 7 = r~,,~ir,,,iiis an algebraic integer. A*(enzi On the other hand, using again the formula t r = dim(H)A* we find
is a rational number. We obtain that is an integer, which ends the B proof. Thus for a semisimple Hopf algebra H over the algebraic closed field k, with the character algebra over k
We have seen that (e,ii)15,ss,l~i~Tn is a complete set of primitive orthogonal idempotents of C k ( H ) , and tr(eaii) = dim(eiH*) divides d i ~ ( H ) for any a and i. Then (e,ii)lsals,15il~Q is a complete set of orthogonal idempotents of H*, SO H * = $l
Theorem 7.6.4 (Kac-Zhu) Let H be a Hopf algebra of prime dimension p over the algebraic closed field k. Then H cx k c p .
--
Proof: If p = 2, the result follows from Exercise 4.3.7. So we assume that p is odd. If H has a non-trivial grouplike element g, then g has order p by the Nichols-Zoeller theorem, and then H k c p . Similarly, if H* has a non-trivial grouplike element, then H * kc,, and then H e (kc,)* kcp (see Exercise 4.3.6). If neither H nor H * has non-trivial grouplike elements, then the distinguished grouplike elements of H and H* are 1 and E, and then by Theorem 7.1.6 we have S4 = Id. Therefore S2 is diagonalizable ) 0, so by with eigenvalues 1 and -1. In particular, since p is odd, T T ( S ~ # Theorem 7.4.1, H is semisimple. Use the complete system of orthogonal idempotents (e,,,)15,1,,l~z~r, of H*, to see that p = t r ( ~ )= El,,,,, 12z5,n tr(eaZZ).Also, by Theorem 7.6.3, each tr(e,,,) divides p, so it is either 1 or p. If tr(e,,,) = p for
-
330
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
some ac and i, then s = 1 and rl = 1, since any tr(eaii) is nonzero. In particular dim(Ck(H)) = 1, i.e. H has only one simple module, so H is a matrix algebra. This is impossible since a non-trivial matrix algebra does not have a Hopf algebra structure. It follows that all tr(eaii) are 1, and that their number is p. This implies that p = El,,,s r,. But El_ca5, ri = dim(Ck(H)) dim(H) = p, so we must have d i m l ~ k ( ~=) ) p, i.e. Ck(H) = H * , and r, = 1 for any 1 5 cr 5 s. Then H* = Ck(H) k x k x . . . x k (p of k). Then H is isomorphic to a group algebra by Exercise 4.3.8. I We have used as ingredients in the proof of the previous theorem the of primitive orthogonal idemfacts that the complete set (eaii)llals,llilTa , l
<
-
Theorem 7.6.5 (The Class Equation). Let H be a semisimple algebra over the algebraic closed field k, and (ei)llilm be a complete set of primitive orthogonal idempotents of Ck(H). Then dim(H) = Cllilmdim(ei H * ) and dim(eiH*) divides dim(H) for any 1 < i 5 m. Moreover, dim(eiH*) = 1 for some i. Proof: Since a complete system (ei)lli<m of primitive orthogonal idempotents of C k ( H ) provides a decomposition Ck(H) = @l
-
Exercise 7.6.6 Let A be a k-subalgebra of the k-algebra B, and e, e' idempotents of A such that eA e'A as right A-modules. Then e B ? e'B as right B-modules.
7.6. THE CLASS EQUATION AND APPLICATIONS
33 1
Theorem 7.6.7 Let H be a semisimple Hopf algebra of dimension pn over the algebraic closed field k, where p is a prime and n a positive integer. Then H contains a non-trivial central grouplike element. Proof: Let X E H* an integral such that X(1) = 1. If we take a complete system (ei)1ci<, of primitive orthogonal idempotents of Ck(H) such that el = A, thenthe Class Equation can be written as
Since any dim(e,H*) divides dim(H) = pn, we see that there exists some i 2 2 such that dim(e,H*) = 1, thus e,H* = Ice,. The structure of the 1-dimensional ideals of H* shows that there exists a grouplike element g in H such that e, = a g X for some a E k. Since i 2 2, g is non-trivial (otherwise e, = X = el). On the other hand, Corollary 7.5.15 shows that g I must be central.
-
Corollary 7.6.8 Let p be a prime number. Then a semisimple Hopf algebra of dimension p2 over an algebraic closed field of characteristic zero is isomorphic as a Hopf algebra either to k[Cp x Cp] or to kCpa. Proof: Let H be semisimple of dimension p2. Theorem 7.6.7 guarantees the existence of a central grouplike element g E H. If g has order p2, then the group of grouplike elements of H has order p2, and then H = kCp2. If g has order p, then let K be the Hopf subalgebra generated by g, this is the group algebra of a group of order p. Since g is central in H , K is contained in the center of H, in particular it is a normal Hopf subalgebra, and then we can form the Hopf algebra H / K + H . Moreover, by Theorem 7.2.11, H = K # , H / K + H as algebras, for some crossed product of K and H / K + H . In particular dim(H/K+H) = p, and then by Theorem 7.6.4 H / K + H cz kcp. Since K is central in H, the weak action of H / K + H on K is trivial (see Remark 6.4.13), so H is a twisted product of K and H / K + H . By Exercise 7.6.9 below, we see that H must be commutative. Since H is semisimple and commutative, we see that as an algebra H is a product of copies of k. Now Exercise 4.3.8 shows that H ci (kG)* for some group G with p2 elements. Since such a G is either Cp x Cp or Cp2, and in 1 both cases kG is selfdual, we obtain the result. Exercise 7.6.9 Show that if the algebra A is commutative, then a twisted product of A and k c p is commutative.
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
332
7.7 The Taft-Wilson Theorem Lemma 7.7.1 Let D be a finite dimensional pointed coalgebra, E a subcoalgebra of D and n : E --+ Eo a coalgebra morphism such that n o i = Id, where i : Eo -t E is the inclusion map. Then n can be extended to a coalgebra morphism n' : D --+ Do such that n'i' = Id, where i' : Do D is the inclusion map. -+
Proof: Since n is a coalgebra morphism, Ker(n) is a coideal of El and then also a coideal of D l and we can consider the factor coalgebra F = D / K e r ( r ) . Let p : D -, F be the natural projection and q5 = p o i' : Do F . We have that -+
so q5 is injective. By Exercise 5.5.2, Im(q5) = p(Do) = F o Then q5* : F* -+ D,* is a surjective morphism of algebras and Ker(q5*) = F& = J ( F ) * . We have that F * / J ( F * ) = F*/F&E F;, which is a finite direct product of copies of k as an algebra, so it is a separable algebra. By WedderburnMalcev Principal Theorem [183, page 2091 there exists a subalgebra A of F* such that F* = J ( F * ) $ A as k-vector spaces. Then A E F * / J ( F * ) E D; as algebras, so there exists an algebra morphism y : D,* --+ F* such that 4* o y = IdD6. Since Do and F are finite dimensional, there exists a coalgebra morphism 11, : F -4 Do such that y = +*. Then q5* o 11,* = I ~ D ; , so ?l, o q5 = IdDo. Then n' = $ o p : D -+ Do is a coalgebra morphism I extending n , and T' o i' = 11, o p o i' = 11, o 4 = IdD,.
Theorem 7.7.2 Let C be a pointed Hopf algebra. Then there exists a coideal I of C such that C = I $ Co. Proof: Let 3 be the set of all pairs (D, n), where D is a subcoalgebra of --+ Do is a coalgebra morphism such that n o i = IdDo, where i : DO D is the inclusion map. 3 is non-empty since (Co, Idco) E 3. The set 3 is ordered by ( D ln ) 5 (Dl, n') if D D' and .rr is the restriction of n' to D . It is easy to check that the ordered set ( 3 , I)is inductive, and then by Zorn's Lemma one gets a maximal element (D, n ) of 3. If D # C , let c E C - D , and let B be the subcoalgebra generated by c. The restriction of n to B n D induces a surjective coaIgebra morphism nl : B n D -+ ( B n D ) , such that nl oil = Id(BnD),,where i : (BnD)O -+ B n D is the inclusion map. Apply Lemma 7.7.1 and extend .rrl to .rr' : D Do
C and n : D -+
-+
7.7. T H E TAFT- WILSON THEOREM
333
such that n' o 'i = IdDo,where i' : Do + D is the inclusion map. Then n and n' produce a coalgebra morphism y : D B + ( D B ) o = Do Bo such that yo j = IdDo+Bo,where j : Do Bo 4 D B is the inclusion map. Since ( D + B, y ) E F and ( D ,n ) < ( D + B, Y), we find a contradiction. I Thus D = C , which ends the proof. Let C be a coalgebra. For any p E C* we denote by l ( p ) ,r ( p ) : C -+ C the maps defined by l(p)(c)= p c = C p ( c 2 ) c l and r ( p ) ( c )= c -- p = C p ( c l ) c 2for any c E C. Clearly l ( p ) is a morphism of right C*-modules, since C is a C*-bimodule. Then l ( p ) is a morphism of left C-comodules, so
+
+
+
+
+
-
Similarly r ( p ) is a morphism of right C-comodules, so
For any p, q E C*,c E C we have that
and also
We have obtained that
Lemma 7.7.3 Let E C_ C* be a family of idempotents such that CeEE e= i.e. for any c E C the set {e E Ele(c) # 0) isfinite and CeEE e(c) = ~ ( c ) . Then a O l(p) r ( 9 ) = C ( ( r ( 9 ) l ( e ) )63 ( l ( p )0r ( e ) ) ) A
E,
eEE
for any p, q
E
C*.
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
334
Proof: The condition CeEE e = E shows that the identity map of C . We have that
CeEE r ( e ) = CeEE l(e) = I ,
x ( l ( e )8 r ( e ) )o A =
X ( I8 r ( e 2 ) ) A
eEE
e€E
0
=
(by (7.27))
( I 8 r ( e ) )o A (e is an idempotent) eEE
= =
(I@I)oA A
Then
4 Let C be a pointed coalgebra with coradical Co = IcG and I a coideal of C such that C = I @ Co. For any x E G we consider the element p, E C* such that p x ( I ) = 0 and px(y) = Sxy for any y E G. Then E = {p,lz E G) is a system of orthogonal idempotents with C x E G p 2= E . Let us denote
Lemma 7.7.3 shows that
for any c E C , x,y E G. If we denote "CY = {"cYIc E C ) , then C = since E is a complete system of orthogonal idempotents. For any subspace S of C we denote S+ = S n K e r ( ~ ) . @ 2 , y E ~xCY
Lemma 7.7.4 (i) If z # y, then ("CY)+ = =Cy. (22) If x E G , then " C x = ("CX)+@ Icx.
Proof: Let c E C and x , y E G. Then
7.7. THE TAFT- WILSON THEOREM
and this is 0 when x # y and p,(c) when x = y. This shows that and since x = " x X ,that ~ ( " x , )= px(x) = 1. Thus the sum kx is direct and it is contained in "C". If c E C , then E("c" - p,(c)x) = p,(c) - p,(c) = 0, so
c
X I --
px(c)x+ ("c"
- p,(c)x) E
and (ii) follows.
Lemma 7.7.5 (i) I = nxEGKer(px) and p, X , y E G. (iz) ("Cv)+ I for any x , y E G. (iii) I = @x,yEG(xCY)+.
(2)
holds,
+ ("Cx)+
kx + ("Cx)+
- I C I ,I -
a p,
5 I for any
Proof: (i) Clearly I L nXEGKer(px). Let c E n x E G K e r ( p x and ) , write c = d CxEG a,x for some d E I and scalars a , E k . Then for any g E G we have 0 = p,(c) = a,, so c = d E I . Thus I = nxEG Ker(p,). If c E I we have that A(c) E C @ I + I @ C ,and then c p, = Cp,(cl)cz E I . Thus I p, 2 I . Similarly p, I G I. (ii) Let x # y and c E "CY = ("CY)+. Write c = d z with d E I and z E Co. Then c = "cy = "d, -t- "zy = "dy E I . If x = y, let c E ("Cx)+and write c = d z with d E I and z E Co. Then d z = c = "cX = "dX " z X ,and since "z" = a x for some a E k and "d" E I , we obtain that z = a x and d = "d". Then 0 = ~ ( c=) ~ ( d ) + = a a,
+
-
+
-
-
+
+
+
soc=d~I. (iii) We have that
Since $x,,G~("CY)+5 I by ( i i ) , and C = I $ Co, we obtain that I = CB~,~EG("C~)+. I Denote In = Cn n I . Since C = I $ Co and Co C Cn for any n, we have that Cn = (C,, n I ) $ CO= In $ CO.
Lemma 7.7.6 For any n we have In = $,,,E~(Inn(xo)+) = @ , , , E G ( C ~ ~ ("CY)+).
--
Proof: We clearly have that Inn("Cy)+= Cnn("CY)+for any n. If c E In and c = Cx,,EG~,,y with c,,, E ("CY)+, then c,,, = p, c p, E Cn
C H A P T E R 7. FINITE DIMENSIONAL HOPF ALGEBRAS
336
since Cn is a subcoalgebra, so then c , , ~E Cn n ("CY)+ for any x,y, proving 1 the claim. Let g and h be two grouplike elements o f C . A n element c E C is called a ( 9 ,h)-primitive element i f A ( c ) = c@g+h@c. T h e set o f all (g, h)-primitive elements is denoted by Pg,h(C). Obviously g - h E Pg,h(C).
Theorem 7.7.7 (The Taft- Wilson Theorem) Let C be a pointed coalgebra with coradical Co = kG. Then the following assertions hold. 1) If n 2 1, then for any c E C n there exists a family ( c ~ , ~G )C ~ , with finitely many nonzero elements, such that c = Cg,hEG cg,h and for with any g, h E G there exists w E Cn-1 8
2) If for any g, h E G we choose a linear complement Pi,h( C ) of k(g - h ) i n Pg,h(C), then C1 = kG @ ( @ g , h E ~ P i , h ( C ) ) . Proof: 1) Since C , = Co @ ( @ z , y E ~ ( C nn( " C Y ) + ) ) and A(Co) G Co 8 Co, it is enough t o consider the case where c E Cn n ("CY)+ for some x,y E G. Since c E C , and the coradical filtration is a coalgebra filtration (Exercise 3.1.11) we see that
for some ( have that
c ~ ) C ~ In, ~ G( d h ) h EC~ In,
xi,,
(ai)i,(bi)i G Cn-1- Since c = "cY we
where y = " a t 8 'b: E Cn-l 8 Cn-1 since pg -- C,-1 Cn-l ph C Cn-1 for any g, h E G. Therefore
-
C Cn-1 and
7.7. THE TAFT-WILSON THEOREM
337
+
By the counit property and the fact that & ( I )= 0 we find that c = " c l u for some u E Cn-l and c = "d,Y v for some v E C,-1. Replacing in (7.29) we obtain
+
wherew=y-u@y-x@v~C~-1'8C~-~. 2) We know that CI = CO@ I1 and I I = CI n I = @ x , y E ~ (n C ("CY)+ 1 ). We prove that py,x(C)= k(y - X ) @ (CI fl("Cy)+) (7.30)
Let c E C1 n ("Cy)+. The proof of 1) shows that'
for some a,,,, E k . Since c =
"cY
we have that
+
Applying I @ E and using the fact that ~ ( c=) 0 we get c = c Sx,yax,yy, so 6x,yax,y= 0. Then A(c) = c @ y x @ c, i.e. c E Py,,(C), proving an inclusion in (7.30). For the converse inclusion let c E Py,,(C), i.e. A(c) = c @ y x @ c. For any g,h E G we have
+
+
If h # y and g # x we obtain A(gch) = 0, so gch = 0. If h = y and g # x, then A(gcY) = ~ c Y@ y, SO by applying E @ I we see that ~ c YE ky. If h # y and g = x , then A("ch) = x @ "ch,SO similarly "ch E kx. If h = y and g = x we find "cY E Py,,(C). Taking into account all these cases and the fact that c = gch, we find c = "cY+ax+py for some a,/3 E k. Since both c and "cy are in PY,,(C) we obtain that a x +,By E PY,,(C). But ax py = a ( x - y) ( a f P)y, so
xg,hEG
+
+
338
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
+
(P a ) y E P,,,(C). Since y is a grouplike element, this implies /3 + a = 0. Thus c = a ( x - y) "cY. Obviously ~ ( " c y= ) 0 since "cY E C1 n P,,,, so c E k ( x - y ) C1 fl ("CY)+,and equation (7.30) is proved. Then we have
+
+
("cY)+))
Ci = Co @ ( c B ~ , ~ ~nG ( C ~ = c o+
C py,x(C) x,YEG
If Pg,h(C)= k(g - h ) $ Pi,h(C),we have
+
We show that the sum Co Cg,hEG Pi,h(C)is direct. Indeed, let c E Co and dg,h E Pi,h(C)for any g , h E G , such that c Cg,hEG dg,h = 0. Since
+
+
we have dg,h = ag,h(g- h ) bg,h with bg,h E C1 n (9Ch)+and ag,h E k. Then we have c+ ag,h(g - h) bg,h = 0
C
+
g,hEG
C
g,hEG
and from the fact that C I = Co $ ($g,hEG(Cln ( g C h ) + )we ) see that bg,h = 0 for any g , h E G. Then we obtain dg,h = ag,h(g- h ) and since Pi,h(C)n k(g - h ) = 0 we must have dg,h= 0 for any g , h. Hence c = 0, so the sum is direct, and this ends the proof. I
Exercise 7.7.8 Let H be a finite dimensional pointed Hopf algebra with dim(H) > 1. Show that 1G(H)I > 1. The next exercise contains an application of the Taft-Wilson Theorem. Recall that if R is a subring of S , the extension R c S is called a finite subnormalizing (or triangular)extension if there exist elements X I , . . . ,x, E R
S such that S =
C Rxi and for any 1 5 j
i=l (see [241]or [121]).Then we have
5 n we have
j
j
i=l
i=l
C Rxi = C xiR
Exercise 7.7.9 Let H be a finite dimensional pointed Hopf algebra acting on the algebra A. Show that A#H is a finite subnormalizing extension of A.
7.8
Pointed Hopf algebras of dimension pn with large coradical
Let p be a prime integer and n 2 2 an integer. Let G be a group of order pn-l, g E Z ( G ) an element in the center of G, and assume that there exists
7.8. POINTED HOPF ALGEBRAS OF DIMENSION PN
339
a linear character a of G such that a ( g ) is a primitive p t h root of unity. The linear map 4 : k G -+ k G defined by $ ( h ) = a ( h ) h for any h E G , is an automorphism of the group algebra k G , thus we can form the Ore extension k G [ X , $ ] ,this is X h = $ ( h ) X = a ( h ) h X for any h E G . Then using the universal property for Ore extensions (Lemma 5.6.4), the usual Hopf algebra structure of k G can be extended to a Hopf algebra structure of k G [ X ,$1 by setting
As C Y ( ~ ) P= 1, it is easy to see that the ideals ( X p ) and ( X p - gp Hopf ideals of k G [ X ,$1. Hence we define the factor Hopf algebras
+ 1) are
As in Section 5.6, we see that H 1 ( G , g ,a ) is a pointed Hopf algebra of dimension pn, with coradical kG. Also, if we require the extracondition that a" = E, then H 2 ( G ,g, a ) is a pointed Hopf algebra of dimension pn, with coradical kG (see Proposition 5.6.14). The Hopf algebra H I ( G ,g, a ) can be presented by generators x and the grouplike elements h , h E G , subject to relations
For the presentation of H2(G,g,a ) we just replace the relation xp = O by X P = gp - 1. Obviously H l ( G , g, a ) N H 2 ( G ,g, a ) in the case where gp = 1. In both cases the coradical filtration is the degree filtration in x , and is not contained in kG if and only if h = g . The following result shows that the two types of Hopf algebras described above are essentially different.
Lemma 7.8.1 Assume that g'p # 1. Then the Hopf algebras H l ( G , g , a ) and H 2 ( G ,g', a') are not isomorphic. Proof: If the two Hopf algebras were isomorphic, let f : H2(G,g1,a') -+ H l ( G , g , a ) be an isomorphism. Then f ( g ' ) must be g, as the unique grouplike with non-trivial PI,,. Then f ( x ) E PI,,, thus f ( x ) = y(g - 1 ) 62 for some scalars y and 6. Apply f to xg' = a'(g1)g'x,and find that y must be 0. Then apply f to x p = g'p - 1, and find 6pxP = gp - 1. This shows that gP = 1, and thus g'P = 1, a contradiction. I
+
Theorem 7.8.2 Let H be a pointed Hopf algebra of dimension pn, p prime, such that G ( H ) = G is a group of order pn-l. Then there exist g E Z ( g ) and a linear character a E G* such that a ( g ) is a primitive p-th root of unity and H N H1 ( G ,g, a ) or H N H2(G, g, a ) (for the second type the condition a p = E must be satisfied).
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
340
Proof: The Taft-Wilson Theorem shows that there is a g E G such that dim(P1,,) 2. Since gpn-l = 1, the conjugation by g is an endomorphism of PI,, whose minimal polynomial has distinct roots. Therefore it has a basis of eigenvectors, and moreover, we may assume that this basis contains g - 1. Let x be another element of the basis, x an eigenvector corresponding to the eigenvalue A. If X = 1, then xg = gx and A(x) = g B x + x @ l , so then the subalgebra of H generated by g and x is a Hopf subalgebra, and this is commutative, hence cosemisimple. We find that x E Corad(H) = kG, a contradiction. Thus we must have X # 1, which implies that X is a primitive root of 1 of order pel for some e 2 1.
>
Now we prove by induction on 1 5 a 5 pe - 1 that the set Sa = { hxi ( h E G, 0 5 i ( a ) is linearly independent. For a = 1, this follows from the Taft-Wilson Theorem. Assume that Sa-1 is linearly independent, and say that ah,jhxi = 0
+
for some scalars a h , i . Since A(x) = g 8 x x 8 1, and (x 8 l)(g 8 x) X(g 8 x)(x B l ) , we can use the quantum binomial formula, and get
=
Apply A to (7.31), then using (7.32) we see that
Fix some ho E H. Since Sa-1 is linearly independent, there exist $,$I E H* such that d(hoga-lx) = 1, 4(hxi) = 0 for any (h,i) # (hoga-', 1), $(hoxa-') = 1, and $(hxi) = 0 for any (h, i) # (ho, a - 1). Applying 4 B 11, to (7.33), we find that ( y ) A ~ h o , a= 0. AS a 5 pe - 1, we have # 0, thus a h o , a = 0. Now the induction hypothesis shows that all a h , i are zero, therefore Sa is linearly independent. But ISpe-l1 = pn-l+e shows that e must be 1, and Sp-l is a basis of H.
+
We prove now that H1 = kG+ ChEF P h , h g and P h , h g = k(hg - h) khx x 'I . Then as in (7.33) we have for any h E G. Let z = C ~ E Gc ~ ~ , ~Eh H Osisa
7.8. POINTED HOPF ALGEBRAS OF DIMENSION PN
341
Note that (hgZPS,s, h, i-s) = (h'gi'-S', s', h', it-s') if and only if s = s', i = i' and h = h'. Then for i 2 2, take s = 1, and hgi-sxs @ hxi-" has the coefficient ah,i in A(z). On the other hand, since A(z) E Ho@ H H@Ho, this coefficient must be zero, thus a h , i = 0. Therefore z E kG ChEC khx, which is what we want.
+
+
In particular, the only P,,, not contained in kG are P h , h g , h E G. On the other hand, xh E Ph,gh,thus P h , g h is not contained in kG. Thus we must have gh = hg for any h E G, i.e. g E Z(G). Also, xh E k(hg-h)+khx. Thus there exist a ( h ) E k*, P(h) E k such that xh = a(h)hx+P(h)(gh - h). We have that xgh
=
Xgxh
= Xg(a(h)hx =
Xa(h)ghx
+ P(h)(gh - h))
+ XP(h)g(gh - h)
and
showing that P(h) = 0. Now xh = cr(h)hx for any h E H, thus a is a linear character of G. Finally, taking the ptl' powers in A(x) = g@x+x@l, we obtain A(xP) = gP @ xp xp 8 1. Since gp # g, this implies that xp E PI,^^ = k(gp - 1). If XP = 0, then H E H1(G, g, a). If xp = P(gp - 1) for some nonzero scalar P, then by the change of variables y = ,B1'px (k is algebraically closed) we I see that H 2 H1(G, g, a ) .
+
Corollary 7.8.3 Let k be an algebraically closed field of characteristic zero and p a prime number. Then a pointed Hopf algebra of dimension p2 over I k is isomorphic either to a group algebra or to a Taft algebra.
Let p be a prime. Then a group of order p or p2 is abelian, therefore in order to find examples of non-cosemisimple pointed Hopf algebras of dimension pn with non-abelian coradical, we need n 2 4. We first investigate dimension p4, and the possibility of the coradical to be the group algebra of a non-abelian group of order p3. Proposition 7.8.4 Let H be a pointed Hopf algebra of dimension p4, p prime. Then either H is a group algebra or G ( H ) is abelian. Proof: It is enough to show that there do not exist Hopf algebras of dimension p4 with coradical the group algebra of a non-abelian group of
342
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
order p3. If we assume that such a Hopf algebra H exists, then the structure result given in Theorem 7.8.2 shows that there exist g E Z(G(H)) and a linear character a of G ( H ) such that a ( g ) is a primitive p t h root of unity. There exist two types of non-abelian groups with p3 elements. The first one is G1, with generators a and b, subject to
In this case Z(G1) =< ap >, and if cr E G;, then the relation bab-l = al+p shows that a(ap) = 1, thus a(g) = 1 for any g E Z(G1). The second type is G2, which is generated by a, b, c, subject to ap = bp = cp = 1, ac = ca, bc = cb, ab = bac Then Z(Gz) =< c >, and again the relation ab = bac shows that a(c) = 1 for any a E G;. Thus a(g) = 1 for any g E Z(Gz), which ends the proof. It is easy to see that there exist pointed Hopf algebras of dimension p5 with non-commutative coradical of dimension p4. We can take for example H = lcGl 8 Hp2, where Hp2 is a Taft Hopf algebra, and G1 is the first type of non-abelian group of order p3. Then H is pointed with G ( H ) = G I x Cp. We can also give examples of such Hopf algebras that are not obtained by tensor products as above. Example 7.8.5 i) Let M be the group of order p4 generated by a and b, subject to relations
Then Z ( M ) =< ap >. Let X be a primitive root of unity of order p2. Then a ( a ) = X and a(b) = 1 define a linear character of M , and a(ap) is a primitive root of unity of order p. We thus have a Hopf algebra H ( M , ap, a) with the required conditions. ii) Let E be the group of order p4 generated by a and b, subject to relations
Then Z ( E ) =< ap,bp >. Take a E E* such that a ( a ) = 1 and a(b) is a primitive root of unity of order p2. Then H ( E , bp, a ) is another example as we want. Now we show that if C = (Cp)n-l =< cl > x then a result similar to Corollary 7.8.3 holds.
-
< c2 > x . . . x < en-1 >
Proposition 7.8.6 If C = (Cp)"-l and H is a pointed Hopf algebra of dimension pn with G ( H ) = C , then H k(Cp)n-2 8 T for some Tajl Hopf algebra T. Moreover, there are exactly p - 1 isomorphism classes of such Hopf algebras.
7.9. POINTED HOPF ALGEBRAS OF DIMENSION P3
343
Proof: We know from Theorem 7.8.2 that H e H1 (C, g, a ) for some g E C and a E C * such that a(g) # 1. Regard C as a Zp-vector space. Then there exists a basis gl = g, g2, . . . ,gn-1 of C. Since a(g) # 1 we can find a ) 1. Then basis cl = g, c2, . . . , cn- 1 of C such that a(c2) = . . . = a ( ~ , - ~ = xc, = c,x for any 2 i < n - 1, the Hopf subalgebra T generated by g and x is a Taft algebra and we clearly have H 2 k(Cp)n-2 8 T. The second I part follows from Proposition 5.6.38.
<
Exercise 7.8.7 Let k be an algebraically closed field of characteristic zero. Show that there exist 3 isomorphism types of Hopf algebras of dimension Q over k: the group algebras k c 4 and k(Cz x C2) and Sweedler's Hopf algebra H4.
7.9
Pointed Hopf algebras of dimension
p3
In this section we classify pointed Hopf algebras of dimension p3, p prime, over an algebraically closed field k. The most difficult is the case where the coradical has dimension p, which we will treat first. So let H be a Hopf algebra of dimension p3 with Corad(H) = kcp. Lemma 7.9.1 There exist c E G(H), x E H and X a primitive p-th root of 1 such that xc = Xcx and A(x) = c 8 x x @ 1. The Hopf subalgebra T generated by c and x is a Taft Hopf algebra.
+
Proof: The Taft-Wilson theorem ensures the existence of some c E G(H) such that PI,, # k ( l - c). If 4 : Ply,-+ PI,, is the map defined by 4(a) = c-lac for every a E H, then $P = Id, so PI,, has a basis of eigenvectors for 4. Let x be such an eigenvector which is not in k(1 - c), and X the corresponding eigenvalue. If X = 1, then the Hopf subalgebra T generated by c and x is commutative, and hence the square of its antipode is the identity. Now T is cosemisimple by Theorem 7.4.6, dim(T) > p and Corad(H) = k c p , a contradiction. We conclude that X # 1, which ends the proof of the first statement. Taking the p-th power of the relation A(x) = c 8 x x 8 1, and using the quantum binomial formula, we find that xp is (1, 1)-primitive. Thus xp = 0 by Exercise 4.2.16. This shows that T is a Taft Hopf algebra. fl From now on T will be the Hopf subalgebra generated by c and x. The (n + 1)-th term T, in the coradical filtration of T is the subspace spanned by all c2x3 with 3 5 n. In particular Tp-l = T . In the sequel, we will assume that the sum over an empty family is zero.
+
344
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
Lemma 7.9.2 Let a E H be such that
for some g E G ( H ) ,n
and for all 1
and
v;,j E
H . Then
Proof: We have that
and
Looking at the terms with 1 on the third tensor position we find
P-1 n-1
and now we obtain (7.35) since
E v i , j@ i=O j=O
cis' = A ( a ) - g @ a - a @ 1.
Looking at the terms with xneT on the third position we find (7.36). 1 The key step is to show that there exist (1, g)-primitives that are not in T.
7.9. POINTED HOPF ALGEBRAS OF DIMENSION P3
345
Lemma 7.9.3 dim(H1)> 2p.
Proof: Suppose dim(HI) 5 2p. Since Tl C H I , we have H1 = T I . In particular dim(P,,,) = 2 only for v = cu. Step 1. We prove by induction on n 5 p - 1 that Hn = Tn. Assume that Hn- 1 = Tn- 1 and Hn # Tn, and pick some h E Hn - T,, . Write h = h,.,, as in the Taft-Wilson Theorem and pick some h,,, E Hn - TT,. u,vEG(H)
Denoting g = u-lv we have that a = u-'h,,, E Hn - T,, and
with vi,j E Tn-1. Let b = a+vo,o E Hn -Tn. 7.9.2 shows that A(vo,,-1) = 9 €3 Vo,n-1 vo,n-I 8 cn-'. If g # cn we have V O , ~ -E~ Ho, and then A ( b ) E Ho €3 H H €3 Hn-2, which is a contradiction since b @ Hn-l. Hence g = cn and V C I , ~ = - ~a(cn - cn-l) + P c ~ - ~for x some a ,P E k , P # 0. We have that
+
+
-pCn-lX
€3 xn-l E H € 3 H n - 2 + H o @ H .
(7.37)
Since ( A ( x n - I ) - cn €3 xn-I - x 7 ~ - 1 €3 1) + (cn €3 xn-l - Cn-1 8 , p - 1 ) E H €3 Hn-2 Ho €3 H and ( A ( x n )- cn €3 xn - xn 8 1) - (;),cn-lx €3 X n - 1 E H 8 HnV2, relation 7.37 implies that b' = b oxn-' - n -1 Pxn satisfies A(bl)- cn 8 b' - b' €3 1 E H €3 Hn-2 HO€3 H . Therefore b' E HnP1 = Tn-l and b E T n Hn = Tn, providing a contradiction. Step 2. We have from Step 1 that Hp-1 = Tp-1 = T # H,. Using the Taft-Wilson Theorem and 7.9.2 as in Step 1, we find some b E H p - T with
+
+
+
P- 1
A(b) = 18 b
+ b €3 1 + X u , 8 xJ for some v, E T
(note that we need here ,=I CP = 1). We use induction to show that for any 1 5 m 5 p there exists b, E Hp - T such that
for some w, E T, oj E k. For m = 1, we see again as in Step 1 that some o,,B E k , ,B # 0. Observe that
up-1
= ~ ( 1 cp-I) -
+ pcp-lx
for
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CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
+
Applying (7.9.2) to a = b axp-' (in the case n = p - l),we obtain a bl as wanted. Assume that we have found b, for some 1 m 5 p - 1 satisfying (7.38). Applying relation (7.36) to bm and r = m we obtain
<
-
(y)
a
2 7 (P -
i=l
~ - ~ X ~ - '
X
i=O
+
. .
cp-'x' @ C
ap-m+i~-.n+iC~-m+ xim-i
+ i,
G3 p - m x i
X
and
which implies that
for every 1 5 i 5 m - 1. For r = m + 1 the relation (7.36) gives
On the other hand we have
a(?--1
x m + l ) = l @Cp-m-l
m+l +?-m-1
x m+l
We obtain the following identities after we apply (7.39) with i replaced by i - 1 (first equality) and some elementary computation with A-factorials (second equality).
7.9. POINTED HOPF ALGEBRAS OF DIMENSION P3
347
We obtain that
As cp-"
#
1 we find
for some a: E k . Since
+
we can apply (7.9.2) to b,, crxp-m-l and get a b,+l Step 3. Take a b = b, satisfying (7.38):
It follows easily that A(bc - cb) = c Q (bc - cb) Also
+ (bc - cb) C3 c, and bc = cb.
Applying (7.39) with i = 1, m = 2, we obtain
and
(A' - 1 ) ~ , --~ (Xp-'
satisfyirlg (7.38).
- 1)aP-l = 0
348
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
We see that the coefficient of x2@xp-I is 0 in (7.41) and bx-xb E Hp-l = T . Clearly 6 = bx - xb E T + = T fl K e r ( ~ ) . The relations bc = cb, bx xb = 6 and 7.40 show that the algebra generated by c, b and x is a Hopf subalgebra of H , so it is the whole of H by the Nichols-Zoeller Theorem. They also show that T + H = H T + , which means that T is a normal Hopf subalgebra. Hence by Theorem 7.2.11 we have an isomorphism of algebras H cz T#,H/T+H ( a certain crossed product). But in H / T + H we have 2. = i,i = 0 and 6 is (1, 1)-primitive, thus 0. We obtain H / T + H E k, and then dim(H) = dim(T), which provides a final contradiction. I At this point we know that there are two different cases: 3 5 dim(PI,,) or there exists g # c such that 2 dim(Pl,g). In the first case let us pick some y E PI,, - k x such that yc = pcy for some primitive p t h root p of 1 (recall that PI,, has a basis of eigenvectors for the conjugation by c). In the second case pick y E PI,, such that yg = pgy for some p # 1. Write g = cd (in the first case we will take d = 1).
<
Lemma 7.9.4 The set {cqxiyjI 0
< q, i, j 5 p - 1 ) is a basis of H .
Proof: We prove by induction on 1 5 n 5 2 p - 2 that the set B, = {cqxiyj(0 5 q , i, j 5 p - 1 and i + j n) is linearly independent. For n = 1 this follows from the Taft-Wilson Theorem. Suppose that B, is linearly P- 1
independent and take q=O
a,,i,jcqxiyj = 0. Applying A, we find i+j
+
+
Fix some triple (go,io,jo) with io jo = n 1 and assume io # 0 (otherwise jo # 0 and we proceed in a similar way). Take 4 E H* mapping ~ ~ ~ + ~ ~to- 1~and + ~any j oother x element of B , to 0, and E H* mapping c q ~ x ~ o - 'to ~ j1~and the rest of B , to 0. Applying 4 @ $, we find that I cr,,,io,j0 = 0. Now all the aq,i,jare zero by the induction hypothesis.
+
< n < 2p - 2. We prove by induction on 1 < n < 2p - 2 that B,
Corollary 7.9.5 B, spans H, for every 0
Proof: - BnP1C H, - HnP1. This is clear for n = 1. For the induction step, pick xiyj E Bn+l - B,. Then
7.9. POINTED HOPF ALGEBRAS OF DIMENSION P3
349
E Ho@H+H@Hn,
Assuming again that i # 0 , choose 4,+ E H* such and xiyj E H that q 5 ( ~ ~ + ~ j) - = ' x +(xi-lyj) = 1, $(Bo) = 0 and +(B,-l) = 0. Then (48 +)(Ho8 H H 8 Hn-1) = 0 , while ( 4 8 + ) ( A ( x i y j ) )= ( i ) X ~ d#j 0 , I and this shows that xiyj Hn. We define now some Hopf algebras. If X is a primitive p t h root of 1 and 1 < i 5 p - 1 an integer, we denote by H ( X ,i) the Hopf algebra with generators c, x , y defined by
+
6
cp = 1, xP = yP = 0 ,
XC
= Xcx, yc = r i c y , yx = X - ~ X ~
We also denote by Hs(X) the Hopf algebra with generators c, x , y defined by
cp = 1, xP = yP = 0 ,
XC
= Xcx, yc = A-Icy, yx = x-Ixy
+ c2 - I
Note that for p = 2 the only Hopf algebra of the second type, H 6 ( - I ) , is equal to H(-1, I ) .
Theorem 7.9.6 Let H be a Hopf algebra of dimension p h i t h Corad(H) = k c p . Then H is isomorphic either to some H6(X) or to some H ( X , i).
Proof: We first consider an odd prime p. We distinguish the following cases. Case 1. x , y E PI,,. Then yx E H2, and since B2 spans Hz we have
for some a i , p i , yi E k,6 E H I . Applying A and replacing everywhere yx by the right hand side in (7.42), we find
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CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
Since B2 is linearly independent we get ai = yi = 0 for all i (looking at the coefficients of ci+lx €3cix and ci+ly @ ciy), pi = 0 for any i # 0 (looking a t + ~ ,BOX= 1 (looking at the coefficient of cx@y), the coefficient of c ~ @cixy), p = Po (looking at the coefficient of cy @ x), and A(6) = c2 @ 6 + 6 @ 1. Thus p = A-' and 6 E = k(c2 - 1). If 6 = 0, then H 21 H(X, 1). If 6 # 0, then H E Hs(i). Case 2. x E PI,,, y E PI,^, where g = cd # c. Writing again yx a s in (7.42) and applying A, we find
Since B2 is linearly independent we obtain that cri = yi = 0 for any i,
pi = 0 for any i f 0, Po = p , X d ~ o= 1 and
A(6) = 6 @ 1 + cg @ 6. Thus p = XVd and 6 PI,++'= k(cd+' - 1). If 6 = 0, then H N H(X,i). If 6 # 0, then yxc = (Poxy
E
+ S)c = Popxcy + 6c = Xpopcxy + Sc,
and yxc = Xycx = Xpcyx = Xpc(p0xy
+ 6) = Xpp0cxy + Xpc6.
These show that Xp = 1, and this implies d = 1, which is impossible. If p = 2, then x2 = y2 = 0 and B2 = {cq, cqx, cqy, cQxy105 q p - 1), so in the equation (7.42) we consider from the beginning that oi = yi = 0, and the proof works with the same computations. I The number of isomorphism types of Hopf algebras of the form H6(X) and H(X, i) can be evaluated using the following.
<
7.9. POINTED HOPF ALGEBRAS O F DIMENSION P3
351
-
Proposition 7.9.7 1) H(X,i) E H ( p , j ) if and only if either (X,i) = ( p , j ) or p = XPi2 and i j 1 (modp). 2) I f p is odd, then H6(X) E H6(p) if and only if = p . 3) If p is odd, then no one of the H6(p) is isomorphic to any H(X, i ) . Proof: 1) We use the classification result given in Proposition 5.6.38, taking into account the fact that H(X, i) = H(C,, (p,p), (c, ci), (c*,c*-~)) where c* E C* is such that c*(c) = A, while
where d* E C* is such that d*(c) = p. The permutation T as in Proposition 5.6.38 is either the identity or the transposition. In the first case, H(X,i ) Y H ( p , j ) if and only if there exists h such that ch = c,chi = d , c* = d*h and c*-i = d*-hj . This implies that h l(modp), and then i = j and
-
X = c*(c) = d*(c) = p. In the second case H(X, i ) = H ( p , j ) if and only if there exists h such that ch = cj,chi = C, C* = d*-jh and c * - ~= d*h. Then h is the inverse of i modulo p, j = h, and d* = c*-~', and we obtain that p = d*(c) = C*-i2 (c) = x-~'. 2) It follows immediately from Corollary 5.6.44. I 3) It follows from Theorem 5.6.28.
Corollary 7.9.8 If p is odd, then there exist (p - 1)'/2 types of Hopf algebras of the form H(X, i ) and p - 1 types of the form Hs(X). If p = 2, then there is only one Hopf algebra of the form H(X, i), namely H(-1, I ) , and it is equal to H6(-1). Proof: Assume that p is odd. There exist (p - 1)2 Hopf algebras of the form H(X,i). The classification of these, given in the previous proposition, shows that any of them is isomorphic to precisely one other in this list. Indeed, if we take some 1 < i p - 1 and some primitive p t h root of unit .Y A, then H(X, i) E H ( p , j ) where j is the inverse of i modulo p and p = X-? Then (X,i) # (p, j ) . Otherwise we would have that i2 l(modp), so then p = X-I # A. We conclude that there exist (p - 1 ) ~ / 2isomorphism types I of Hopf algebras of the form H(X, i). The rest is obvious. Let H be a pointed Hopf algebra of dimension p3. We have seen (Exercise 7.7.8) that the coradical of H can not be of dimension 1. By the Nichols-Zoeller Theorem we have dim(Corad(H)) E {p,p2,p3). If dim(Corad(H)) = p, then Corad(H) = k c p , and this case was discussed. If dim(Corad(H)) = p2, then Corad(H) = kCp2 or Corad(H) = k(Cp x C,), and the classification in these cases was done in the previous section, where pointed Hopf algebras H of dimension pn with G(H) of order pn-l
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CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
352
are classified. For every X # 1 and i such that X p Z = 1 and Xi is a primitive p t h root of 1, denote by Hp2(X,i) is the Hopf algebra with generators c and x and
cpZ = 1, x p = 0,
-
XC
= Xcx
A(c) = c @ c , A ( x ) = c i @ x + x @ l
By Proposition 5.6.38 two such Hopf algebras Hpz (A,i ) and Hp2( p ,j ) are isomorphic if and only if there exists h not divisible by p such that p = X h and i h j (mod p 2 ) . If X is a primitive p t h root of 1, we denote by H ( X ) the Hopf algebra with generators c and x defined by
A(c) = c @ c , A(x) = c @ x + x @ l Then using again Proposition 5.6.38, we see that H ( X ) if X = p. These facts imply the following.
2
H ( ~if) and only
Proposition 7.9.9 Let H be a pointed Hopf algebra of dimension p3 with Corad(H) = kCpz. Then H is isomorphic either to some Hpz(X,i ) or t o a H ( x ) , and there are 3 ( p - 1 ) types of such Hopf algebras. I The case where G ( H ) = C, x C p follows from the more general Proposition 7.8.6.
Proposition 7.9.10 Let H be a pointed Hopf algebra of dimension p3 with Corad(H) = k ( C p x C p ) . Then H = TA@I k c p , where TAis one of the Tuft Hopf algebras, and there exist p - 1 types of such Hopf algebras. I Finally, if dim(Corad(H)) = p3, then H is the group algebra of one of the following groups: C p x C p x C p ,Cpz x C p ,Cph G I = Cp2 >a C p ,G 2 = C p >a Cp2,where G I ,G 2 are the two types of nontrivial semidirect products. We have now the complete classification of pointed Hopf algebras of dimension p3 -
Theorem 7.9.11 Let H be a pointed Hopf algebra of dimension p3. Then H is isomorphic to one of the following: Hs(X),H ( X , i ) ,H ( x ) ,TA@ k c p , where X is a primitive p-th root of 1 , Hp2 (A,i ) for some X # 1 and i such that X p 2 = 1 and X i is a primitive p-th root of 1, k ( C p x C p x C p ) , k(CPz x Cp), k(Cp3), k(G1), k(G2). If p is odd, then there are 5 such types. If p = 2, then there are 10 types. I
+
7.10. SOL UTIONS TO EXERCISES
7.10
353
Solutions to exercises
Exercise 7.4.2 Show that l(p)* = R ( p ) for any p E H * . I n particular T r ( l ( p ) )= T r ( R ( p ) ) . Solution: If q E H* and a E H we have
Exercise 7.4.3 Show that if S2 = Id and H is cosemisimple, then x is a nonzero right integral i n H . Solution: For any p E H* we have P(X)
= =
Tr(l(p)) T r ( l ( 1 )o s2o l ( p ) ) (since l ( 1 ) = s2= I d )
X ( l ) p ( A ) (by (7.23)) = ~(4l)A) =
so x = X(1)A. Since H is cosemisimple, X(1) # 0, and then x is a nonzero right integral.
Exercise 7.4.7 Let k be a jeld of characteristic zero and H a semisimple Hopf algebra over k . Show that a right (or left) integral t i n H is cocommutative, i.e. t l 8 t2 = t2 8 t l . Solution: We know that H is cosemisimple and S2 = I d , so the element x E H for which p ( x ) = T r ( l ( p ) ) for any p E H * , is a right integral in H by Exercise 7.4.3. By Lemma 7.4.5, x is cocommutative. Exercise 7.4.8 Let k be a field of characteristic zero and H be a finite dimensional Hopf algebra over k . Show that: (i) If H is commutative, then H ( k G ) * for some finite group G . (ii) If H is cocommutative, then H E k G for a finite group G . Solution: If H is either commutative or cocommutative, then S2 = I d , showing that H is semisimple and cosemisimple. For (i),if H is semisimple and commutative, then H k x k x . . . x k as an algebra, and now H 1.( k G ) * for some finite group G by Exercise 4.3.8. For (ii),we apply ( i ) for H * .
--
--
Exercise 7.4.9 Let H be a finite dimensional Hopf algebra over a field of characteristic zero, and let S be the antipode of H . Show that S has odd k G , where G = C2 x C2 x . . . x C 2 , and i n this order if and only if H
--
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CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
case S = I d , so the order of S is 1. Solution: We know that S is an antimorphism of coalgebras, so if it has odd order, we obtain that S is also an coalgebra morphism. Since S is bijective, this implies that H is cocommutative. Then necessarily S2 = I d , and the odd order must be 1. Also, by Exercise 7.4.8 we have that H E kG for some group G. Since S has order 1, we must have that g = g-I for any g E G, so then G = C 2 x C2 x ... x C2. Exercise 7.5.1 Let V,W E H M be finite dimensional. Then x(V $ W ) = x ( V ) + x ( W ) and x(V @J W ) = x ( V ) x ( W ) . Solution: If ( ~ ~is a basis ) ~of V ~and i(wj)l<j<m ~ ~ is a basis of W , then we take as a basis for V @ W the union of these two bases and obtain
,j. The dual proving the first formula. In V @W we take the basis (vi@wj)i basis in (V @I W ) * is (vf @I wj*)i,j,if we identify (V @I W ) *with V* @ W * via the natural isomorphism. Then
proving the second formula.
Exercise 7.5.2 Show that for a finite dimensional V EH M we have x(V*)= S * ( x ( V ) )where , S* is the dual map of S. Solution: If ( v ~is a )basis~of V~and~the dual ~ basis ~ of V* is (vi);
Exercise 7.5.13 Let C 2i Mc(m, k ) be a matrix coalgebra and x E C such that C X I C3 2 2 = C x2 18X I . Then there exists a E k such that z = axc. Solution: Let (cij)lli,Jlm be a comatrix basis of C , and
x=
C
aijcij
l
for some scalars aij. Then
C xl 63 x2 = C 2 2 €3X I
is equivalent to
Looking a t the coefficients of the elements of the basis (cij @ ~ u v ) l < ~ , j , ~ , ~ < m of C @ C ,we see that this is equivalent to aij = 0 for any i # j and aii = app for any 1 I: i , p 5 m. But this means that x = all Ellismcii = a11xc.
Exercise 7.6.2 Show that for any idempotent e E H * , t r ( e ) = dim(eH*). This provides another way to see that tr(eaii) is a positive integer. Solution: The endomorphism u : H* -, H * , u ( p ) = ep for any p E H * , satisfies I m ( u ) C eH*. This implies that T r ( u ) = T r ( u l e H . ) .On the other hand, for any p E eH* we see that u ( p ) = ep = p, thus UI,H. = Id, and then T r ( u l e H . )= dim(eH*).
-
Exercise 7.6.6 Let A be a k-subalgebra of the k-algebra B , and e, e' idempotents of A such that eA e l A as right A-modules. Then e B = e'B as right B-modules. Solution: Let f : eA -+ e l A be an isomorphism of right A-modules with Note that ex = x for any x E e A , and e'x' = x' for any inverse f-'. x' E e'A. Define g : e B --+ e'B by g(eb) = f(e)b for any b E B . This is correctly defined, since ebl = ebz implies that With the same argument we can define g' : e'B for any b E B. Then
-+
e B by gl(eb) = fel(e')b
356
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
thus g'g = Id, and similarly gg' = Id. Then g provides an isomorphism of right B-modules. Exercise 7.6.9 Show that if the algebra A is commutative, then a twisted product of A and kc, is commutative. Solution: If A#,H is a twisted product, the weak action of H on A is trivial, i.e. h . a = ~ ( h ) afor any h E H , a E A. The normality condition tells us that a(1,g) = ~ ( g1), = 1 for any g E C,, and the cocycle condition tells that a(1, m)u(h, lm) = a ( h , l)u(hl, m) for any h, 1, m E C,. For 1 = hi and m = h, this equation becomes
We prove by induction that u(hi, h) = a(h, hi) for any i. This is clear for i = 0. Also, equation (7.43) shows that if a(hi, h) = a(h, hi), then a(h, hi+') = a(hi+', h). Thus we have proved that a(hi, h) = a ( h , hi) for any h E C, and i 1 0. Since C, is cyclic, this implies that h) = a(h, g ) for any g, h E C,. Then the multiplication of A#,kC,, given by
for any a, b E A and h, 1 E C,, is commutative. Exercise 7.7.8 Let H be a finite dimensional pointed Hopf algebra with dim(H) > 1. Show that IG(H)I > 1. Solution: If G(H) is trivial, then the Taft-Wilson Theorem and the fact that Pl,l(H), the set of all primitive elements of H , consists only of 0 (by Exercise 4.2.16) show that HI = Ho. Then H, = Ho for any n, in particular H = Ho. Thus H = kG(H), which has dimension 1, a contradiction. Exercise 7.7.9 Let H be a finite dimensional pointed Hopf algebra acting on the algebra A. Show that A#H is a finite subnormalizing extension of
A. Solution: We construct by induction a basis {xl,. . . ,x,) of H such that
for 1 5 j 5 n (we denote ax = a#x = (a#l)(l#x) and xa = (l#x)(a#l)). We put G = G(H), and denote by
7.10. SOLUTIONS TO EXERCISES
357
the coradical filtration on H . Now the elements of G form a basis of COand they clearly normalize A in A#H. So we put G = { a l , . . . , x t ) , and assume that a basis { x l , . . . , x,.) j
(t 5 r ) of Ci-l has been constructed such that Since Ci = Ci-1
@
C Ki,,,,, where
Let x E Ki,g,7. By Lemma 6.1.8 we have a h = a E A, h E H. So we get ax = CT(S-' (x) a )
+ T(T-'
,
a)
j
C xiA = i=l C Axi, 1 5 j 5 r. i= 1
C h2(S-'(hl)
. a) for all
+ C v s ( ~ - ' ( u s ) . a).
On the other hand,
Thus if we put x,+l = x, we have Ax,+l E
r+l
r+l
C xjA, and x,.+lA E C Axj.
j=1 j=1 If Ci = Ci-l+kxr+l we are done. If not, take xr+2 E Ki,,,,\(Ci-l+kx,.+l), for some U , T E G, and continue as above. The process will stop when dim(H) is reached.
Exercise 7.8.7 Let k be an algebraically closed field of characterzstic zero. Show that there exist 3 isomorphism types of Hopf algebras of dimension 4 over k: the group algebras k c 4 and k(C2 x C2) and Sweedler's Hopf algebra H4. Solution: Let H be a Hopf algebra of dimension 4. If H has a simple subcoalgebra S of dimension greater than 1, then this must be a matrix coalgebra, so it has dimension at least 4. Thus H = S , and then the dual Hopf algebra H* is isomorphic as an algebra to M2(k), and this is impossible by Exercise 4.1.9. We obtain that any simple subcoalgebra of H has dimension 1, i.e. H is pointed. We know from Exercise 7.7.8 that G(H) is not trivial. Then G(H) has order either 4, and in this case H is a group algebra, or 2, in in this case H is isomorphic to Sweedler's Hopf algebra by Corollary 7.8.3. Bibliographical notes The fact that the antipode of a finite dimensional Hopf algebra has finite order was proved by D. Radford [187]. E. Taft and R. Wilson gave in [225]
358
'
CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS
examples of finite dimensional Hopf algebras over an arbitrary field with antipode having the order a given odd positive integer. The Nichols-Zoeller theorem was proved in [173], and answered a conjecture of I. Kaplansky 11041 concerning the freeness of a finite dimensional Hopf algebra as a module over a Hopf subalgebra. Examples of an infinite Hopf algebra which is not free over a Hopf subalgebra were given by U. Oberst and H.-J. Schneider [176], H.-J. Schneider [201], and M. Takeuchi [230] (see also [149]). D. Radford proved in [I881 that any pointed Hopf algebra is free over a Hopf subalgebra, and in [I901 that commutative Hopf algebras are free over finite dimensional Hopf subalgebras. For the proof of Theorem 7.2.11, which is from A. Masuoka [133], we have used an old argument of Nakayama (see [149, 8.3.6, p. 136]), adapted as in [197]. In Section 7.3 we used the papers of W. Nichols [I701 and R. Larson [115]. In Section 7.4 we used the papers [117, 1181 of R. Larson and D. Radford, and the paper [170] of W. Nichols. The last ten years showed an increasing interest for classification problems for finite dimensional Hopf algebras over an algebraically closed field of characteristic zero. A very nice and complete survey on this is the paper [6] of N. Andruskiewitsch. Small dimensions ( 5 11) were classified by R. Williams [243] (see also D. Stefan (2141). Dimension 12 was recently solved by S. Natale [167]. In prime dimension p, I. Kaplansky conjectured in [lo41 that any Hopf algebra is isomorphic to kc,.This was proved by Y. Zhu in [247], using an argument of Kac [102]. Y. Zhu also proved a result, called the class equation (for another proof see M. Lorenz [126]), which is very useful for other classification problems. We present in Sections 7.5 and 7.6 the class equation, the classification in dimension p, and also the classification of semisimple Hopf algebras of dimension p2. We have used the papers [102, 2471, the lecture notes of H.-J. Schneider [207], and A. Masuoka's paper [139]. The problem of classifying all Hopf algebras of a given finite dimension is wide open. Apart from prime dimension and several small dimensions, no other general result is known. In dimension p2, p prime, it is conjectured that any Hopf algebra is isomorphic either to a group algebra or to a Taft algebra (some partial answer is given by N. Andruskiewitsch and H.-J. Schneider in [12]). The classification efforts have focused on two important classes: semisimple Hopf algebras and pointed Hopf algebras. About these two classes, we should remark that the theory of semisimple Hopf algebras parallels in some sense the theory of finite groups, while pointed Hopf algebras have a geometric flavour. D. Stefan proved that there are only finitely many isomorphism types of semisimple Hopf algebras of a given finite dimension [213], while we have seen in Section 5.6 that the number of types of pointed Hopf algebras of a given finite dimension may be infinite. Several results on the classification of semisimple Hopf algebras have been obtained by A. Masuoka [136, 137, 138, 139, 1401, S. Gelaki and
7.10. SOLUTIONS TO EXERCISES
359
S. Westreich [87], P. Etingof and S. Gelaki [81], S. Natale [165, 1661, N. Fukuda [84] and Y. Kashina [105]. A survey paper is S. Montgomery [153]. For the classification of pointed Hopf algebras, a fundamental result is the Taft-Wilson theorem, given in [223]. We give the proof presented in S. Montgomery's book [149], which uses techniques of D. Radford [192, 1931. A new proof was given by N. Andruskiewitsch and H.-J. Schneider in [18]. Exercise 7.7.9 is taken from [64], and the solution was suggested by D. Quinn. In Section 7.8 we present the classification of pointed Hopf algebras of dimension pn, p prime, with coradical of dimension pn-l. We follow the paper [66] of S. D&cMescu, with techniques similar to the one in M. Beattie, S. D&ciilescu, L, Griinenfelder [26]. The same result was proved by N. Andruskiewitsch and H.-J. Schneider in [15]. A more general result about pointed Hopf algebras with coradical of prime index was proved by M. Grafia [91]. For the classification of pointed Hopf algebras of dimension p3 we used the paper [45] of S. Caenepeel and S. Dkc&lescu. The same result is proved in N. Andruskiewitsch, H.-J. Schneider [13], and D. Stefan, F. Van Oystaeyen 12151. Other classification results in the pointed case have been given by N. Andruskiewitsch and H.-J. Schneider in [14, 15, 161, S. Caenepeel and S. DSscSlescu [46], S. Caenepeel, S. DSscSlescu and 8. Raianu [47], M. Graiia [89, 901, I. Musson [155], D. Stefan [212], etc. If the field is not algebraically closed, the classification of Hopf algebras is much more difficult. Even in dimension 3 (where for an algebraically closed field of characteristic zero the only isomorphism type is kc3) it was shown in S. Caenepeel, S. D&cSlescu and L. le Bruyn [48] that there exist infinitely many isomorphism types of Hopf algebras of dimension 3 over certain fields of characteristic zero. Finally, we mention two aspects which we did not include in this book, since it was not our aim to go too deep in classification problems for finite dimensional Hopf algebras. For the classification of semisimple Hopf algebras a central role is played by the theory of extensions of Hopf algebras. The reader is referred for this to [208], [5], [7], [98], [135], [130], [205]. Among the approaches to classification of pointed Hopf algebras, the technique which has proved to be the most powerful is the lifting method invented by N. Andruskiewitsch and H.-J. Schneider. However, we did not include it in this book since several technical concepts are necessary. We refer for this to the papers [6] and [17].
Appendix A
The category theory language A. 1
Categories, special objects and special morphisms
A category C consists of a class of objects (which we simply call the objects of C; if A is an object of this class, we simply write A E C), such that: for any pair (A, B ) of objects of C, a set Homc(A, B ) is given (also denoted Horn(A, B ) if there is no danger of confusion), whose elements are called the morphisms from A to B, for any A E C there is a distinguished element l A (or I A ) of Hornc(A, A), and for any A, B, C E C there exists a map (composition) Hornc (A, B ) x Homc (B, C )
Home (A, C )
-+
associating to the pair ( f , g), where f E Homc(A, B) and g E Hornc(B, C), an element of Homc(A, C ) , denoted by g o f , such that the following properties are satisfied. i) Composition is associative, in the sense that if f E Homc(A, B), g E Homc(B, C ) and h E Homc(C, D), then h o (g o f ) = (h o g) o f . ii) f o l A = f and l g o f = f for any f E Homc(A, B). iii) The sets Hornc(A, B ) and Hoinc(A1,B1) are disjoint whenever ( A ,B ) # (A', B'). We also write f : A + B instead o f f E Homc(A, B). The morphism 1 ~ is uniquely determined for a given A, and it is called the identity morphism of A.
362
APPENDIX A. THE CATEGORY THEORY LANGUAGE
Example A . l . l 1) The category of sets, denoted by S e t , whose class of objects is the class of all sets, and for any sets A and B , Homset(A, B ) is the set of all maps from A to B . The composition is just the map composition, and l A is the usual identity map of a set A. 2) The category of topological spaces, denoted by Top. The objects are all topological spaces, and for any topological spaces X and Y , HomT,,(X, Y ) is the set of continuous functions from X to Y . The composition is again the map composition and the identity morphism is the usual identity map. 3) If R is a ring, then the category of left R-modules, denoted by R M , has the class of objects all left R-modules, and for any left R-modules M and N , H o m R M ( M ,N ) , which is usually denoted by H o m R ( M ,N ) , is the set of the morphisms of R-modules from M to N . The composition and the identity morphisms are as i n Set. Similarly, one can define the category M R of right R-modules. I n particular, if Ic is a field, kM is the category of k-vector spaces. Also, if Z is the ring of integers, then z M , the category of Z-modules, is just the categoy of abelian groups, which is also denoted by Ab. 4 ) The category G r of groups has as objects all the groups, and the morphisms are group morphisms. 5) The category Ring of rings, has as objects all the rings with identity, and the morphisms are ring morphisms which preserve the identity. The dual category Let C be an arbitrary category. We denote by C0 the category having the same class of objects as C, and such that Homco ( A ,B ) = Homc ( B ,A) for any objects A , B E C. The composition of the morphisms f E Homco(A, B ) and g E Homco ( B ,C ) is defined to be f o g E H o m c ( C , A ) = Homco ( A ,C ) . The category C0 is called the dual category of C. Clearly (CO)O= C. The introduction of the dual category is important since for any concept or statement in a category, there is a dual concept or statement, obtained by regarding the initial one in the dual category. Subcategory Let C be a category. By a subcategory C' of C we understand the following. a) The class of objects of C' is a subclass of the class of objects of C. b) If A , B E C', then H o m p ( A , B) C H o m c ( A , B ) . c) The composition of morphisms in C' is the same as in C. d) l Ais the same in C' as in C for any A E C'. If furthermore Homcl(A, B) = H o m c ( A , B ) for any A , B E C', then C' is called a full subcategory of C. Direct product of categories Let (Ci)i,zr be a family of categories indexed by a non-empty set I. We
A. 1. CATEGORIES, OBJECTS AND MORPHISMS
363
define a new category C as follows. The objects of C are families (Mi)iGI where Mi E Ci for any i E I . If M = (Mi)iEI and N = (Ni)iGI are two objects of C, then Hornc(M, N )
=
{(ui)iErlui E Hornci (Mi, Ni) for any i E I}
are three objects of C, If M = (Mi)iEI, N = (Ni)iEI and P = and u = (ui)iEI E Homc(M, N) and v = (vi)iGI E Hornc(N, P), then the composition of u and v is defined by v o u = (vi o ui)iGI. The category C defined in this way is called the direct product of the family (Ci)iEI, and Ci. If moreover we have Ci = 2) for any it is usually denoted by C = Ci is also denoted by D'. If I is finite, say I = {1,2,. . . ,n}, i E I , then then instead of Ci we also write C1 x C2 x . . . x C,.
niGI niEI
niGI
Monomorphisms, epimorphisms and isomorphisms in a category Let C be a category. A morphism f : A -+ B is called a monomorphism if for any object C and any morphisms h, g E Hornc(C, A) such that f o h = f o g , we have g = h. The morphism f is called an epimorphism if for any object D and any morphisms k, 1 E Hornc(B, D ) such that k o f = 1 o f , we have k = 1. The morphism f is called an isomorphism if there exists a morphism g E Hornc(B,A) such that g o f = lAand f o g = lB. It is easy to see that such a g is unique (when it exists), and it is called the inverse of f and denoted by g = f - l . It is easy to check that any isomorphism is a monomorphism and an epimorphism. The converse is not true. Indeed, in the category of rings with identity the inclusion map i : Z -+ Q is a monomorphism and an epimorphism, but not an isomorphism. The composition of any two monomorphisms (respectively epimorphisms, isomorphisms) is a monomorphism (respectively epimorphism, isomorphism). B is a morphism in a category C, then f is a monomorphism If f : A (epimorphism) in C if and only if f is an epimorphism (monomorphism) when regarded as a morphism from B to A in the dual category CO. Thus the notion of epimorphism is dual to the one of monomorphism. Let us fix an object A of the category C. If a1 : A1 --+ A and a 2 : A2 A are monomorphisms, we shall write a1 5 a 2 if there exists a morphism y : Al 4 A2 such that a s o y = a l . Clearly, if such a y exists, then it is unique, and it is also a monomorphism. If a1 5 a 2 and a 2 5 a l , then there exist morphisms y and 6 such that a 2 o y = ol and a1 0 6 = oz, and this implies that S o y = lA,and y o 6 = lA,, so y and 6 are invertible. The monomorphisms ol and crz are called equivalent if ol L a 2 and a 2 5 (2.1. The equivalence of monomorphisms is an equivalence relation, and by Zermelo's axiom we can choose a representant in any equivalence class. The resulting monomorphism is called a subobject of the object A. Similarly we can define the notion of a quotient object of A by using the concept of -+
-+
364
APPENDIX A. THE CATEGORY THEORY LANGUAGE
epimorphism.
Initial and final object. Zero object. If C is a category and I (respectively F) is an object with the property that Homc(I, A) (respectively Homc(A, F ) is a set with only one element for any A E C, then I (respectively F) is called an initial (respectively final) object of C. Clearly, any two initial (respectively final) objects are isomorphic. An object 0 E C is called a zero object if it is an initial object and a final object. When exist, a zero object is unique up to an isomorphism. In this case we call a morphism f : A -+ B a zero morphism if it factors through 0. Each set Homc(A, B) has precisely one zero morphism, which we denote by OAB or simply by 0. Equalizers Let a , ,B : A -+ B be two morphisms in a category C. We say that a morphism u : K -+ A is an equalizer for a and ,f3 if a o u = ,B o u, and whenever u' : K' -+ A is a morphism such that a o u' = ,8 o u', there exists a unique morphism y : K ' -+ K such that u' = u o y. Clearly, if u : K -+ A is an equalizer for a and ,B then u is a monomorphism, and two equalizers of a and ,B are isomorphic subobjects of A. Dually, we can define the notion of coequalizer of two morphisms a and ,B, which is just the equalizer in the dual category CO. Now assume that the category C0 has a zero object. If a : A 4 B is a morphism in CO, then the equalizer (respectively the coequalizer) of a and 0 is called the kernel (respectively the cokernel) of a . The associated subobject (respectively quotient object) is denoted by K e r ( a ) (respectively Coker(a)). Let C be a category with zero object. Assume that for any morphism a : A -+ B in C there exist the kernel and the cokernel of a, so we have the sequence of morphisms
where i and n are the natural morphisms. We have that i is a monomorphism and n is an epimorphism. If there exists a cokernel of i , then it is called the coimage of a , and it is denoted by Coim(a). Also, if a kernel of .rr exists, then it is called an image of a, and it is denoted by I m ( a ) . By the universal properties of the kernel and cokernel, there exists a unique morphism E : Coim(a) -+ Im(cr) such that a = p o 5 o A, where X and p are the natural morphisms.
Products and coproducts Let C be a category and ( M i ) i c l be a family of objects of C. A product of the
A.2. FUNCTORS AND FUNCTORIAL MORPHlSMS
365
niEI
family is an object which we denote by Mi, together a family of morphisms (ri)ieI, where 9 : Mi + M j for any j E I, such that for any object M E C, and any family of morphisms (fi)iEI, where f j : M -+ M j for any j E I, there exists a unique morphism f : M -, Mi such that .rri o f = fi for any i E I . If it exists, the product of the family is unique up to isomorphism. In the case where I is a finite set, say I = {1,2,. . . , n), the product is also denoted by MI x M2 x . . . x Mn. The categories Set, Gr, Ring, R M , Top have products. Dually, one defines the notion of coproduct of a family of objects (Mi)iEI, which (if it exists) is denoted by flierMi. This is exactly the product of the family in the dual category. In fact, the coproduct Ui,, Mi is an object together with a family (qi)iEI of morphisms such that qj : M j -+ UiEIMi for any j E I, and for any object M and any family (fi)ier of morphisms with fi : Mi -+ M for any i, there exists a unique morphism f : UiEIMi -+ M such that f o q; = fi for any i. Again, in the case where I is finite, say I = { 1 , 2 , . . . , n ) , the coproduct is also denoted by MI U M2 U . . . U M,,, or MI @ M2 @ . . . @ Mn.
niEI
n,,,
Fiber products Let S be an object of a category C. We define the category C I S in the following way: - the objects are pairs (A, a ) , where a : A -+ S is a morphism in C. - If ( A ,a ) and (B, p ) are objects of CIS, then a morphism between (A, a) and ( B , 0)is a morphism f : A -+ B in C such that p o f = a. If (A, a ) and (B, p) are two objects of CIS, the product of these two objects in the category CIS is called the fiber product(or pull-back) of the two objects, and it is denoted by A n s B or A x s B . The dual notion of fibred product is called a pushout.
A.2
Functors and functorial morphisms
Functors Let C and V be two categories. We say that we give a covariant functor (or simply a functor) from C to V if to any object A E C we associate an object F ( A ) E V, and to any morphism f E Homc(A, B ) we associate a for any morphism F ( f ) E H o m v ( F ( A ) , F ( B ) ) such that F ( l A ) = object A E C, and F ( g o f ) = F(g) o F ( f ) for any morphisms f and g in C for which it makes sense the composition g 0 f . A covariant functor from the dual category C0 to D (or from C to Do) is called a contravariant functor from C to D. If F : C -+ V is a covariant functor, then for any objects A, B E C we have Homv(F(A), F ( B ) ) defined by f ++ F ( f ) . If this a map Homc (A, B ) map is injective (surjective, bijective), then the functor F is called faithful
-
366
APPENDIX A. THE CATEGORY THEORY LANGUAGE
(full, full and faithful). If C is a category, then the identity functor lc : C + C is defined by lc(A) = A for any object A, and l c ( f ) = f for any morphism f .
Functorial morphisms (natural transformations) Let F , G : C -+ D be two functors. A functorial morphism from F to G, G, is a family {+(A)IA E C) of morphisms, such that denoted by : F +(A) : F(A) -+ G(A) for any A E C, and for any morphism f : A -+ B in C we have that 4(B) o F ( f ) = G(f) o +(A). If moreover 4(A) is an isomorphism for any A E C, then 4 is called a functorial isomorphism. If there exists such a functorial isomorphism we write F e G. Clearly if : F G and $ : G -+ H are two functorial morphisms, we can define the composition $ o 4 : F -+ H by ($ o q5)(A) = $(A) o 4(A) for any object A. We denote by Hom(F, G) the class of all functorial morphisms from F to G. If the category C is small, i.e. its class of objects is a set, then Hom(F, G) is also a set. For every functor F we can define the functorial morphism IF : F -4 F for any A E C. This is called the identity functorial by lF(A) = morphism. .
+
+
+
-+
-+
Equivalence of categories Let C and V be two categories. A covariant functor F : C -+ V is called an equivalence of categories if there exists a covariant functor G : D -+ C such t h a t G o F ~ l ~ a n d F o G = lI.f ~m.o r e o v e r G o F = l c a n d F o G = l v , then F is called an isomorphism of categories, and C and V are called isomorphic categories. An equivalence of categories is characterized by the following.
Theorem A.2.1 If F : C -+ D is a covariant functor, then F is an equivalence of categories if and only if the following conditions are satisfied. (i) F is a full and faithful functor. (ii) For any object Y E D there exists an object X E C such that Y F ( X ) .
--
A contravariant functor F : C -+ D such that F is an equivalence between C0 and V (or C and Do) is called a duality. Yoneda's Lemma Let C be a category and A E C. We can define the covariant functor hA : C -+ Set as follows: if X E C, then h A ( x ) = Homc(A,X), and if u : X -+ Y is a morphism in C, then hA(u) : h A ( x ) -+ h A ( y ) is defined by hA(u)(<)= h o for any E h A ( x ) . Similarly, we can define a contravariant functor hA : C Set as follows: if X E C, then ~ A ( X= ) Homc(X, A), and if u : X -+ Y is a morphism in C, then h ~ ( u ): h ~ ( y ) h ~ ( xis)
<
<
-+
-+
A.3. ABELIAN CATEGORIES defined by h~ (u)(q) = q 0 u for any q E ~ A ( Y ) .
Theorem A.2.2 (Yoneda's Lemma) Let F : C functor and A E C. Then the natural map
-, Set
be a contravariant
is a bijection, We indicate the construction of the inverse of a. We define the map P : F(A) --+ Hom(hA,F) as follows. If J E F(A), then for X E C we consider the map $(XI : ~ A ( X-)' F ( X ) , + ( X ) ( f ) = F ( f ) ( t ) Note that since the functor F is contravariant we have that F ( f ) : F ( A ) F ( X ) . It is easy to check that the family of morphisms {$(X)IX E C) defines a functorial morphism $ : hA 4 F . We define then P(<) = $. One consequence of Yoneda's Lemma is that the class Hom(hA,F) is a set. We also have the following important consequence. -+
Corollary A.2.3 If A and B are two objects of C, then A e B if and only ifhA ? h B . A contravariant functor F : C Set is called representable if there exists an object A E C such that F 2 hA. By the above Corollary we see that if such an object A exists, then it is unique up to an isomorphism. -+
A.3
Abelian categories
Preadditive categories A category C is called preadditive if it satisfies the following conditions. a) For any objects A, B E C, the set Homc(A, B) has a structure of an abelian group, with the operation denoted by +. The neutral (zero) element of the group (Homc(A, B), +) is denoted by O A , ~ ,or shortly by 0, and it is called the zero morphism. b) If A, B , C are arbitrary objects of C, then for any u, U I , u2 E Homc(A, B ) and v, ~ 1 , 2 1 2E Homc(B, C), we have that v o (u, + u2) = v o u l + v 0 u2 and (vl + 212) o u = vl o u + v2 o U. c) There exists an object X E C such that lx = 0. If A E C and f E Homc(A,X), then by conditions b) and c) we have that f = lx o 7 = 0 o f = 0. Also, if g E HomCalc(X,A), we have that g = 0. Therefore the object X is a zero object in the sense given in Section
368
APPENDIX A. THE CATEGORY THEORY LANGUAGE
1 of the Appendix. Since the zero object is unique up to an isomorphism, we denote it by the symbol 0. If A is any object, we denote by 0 -+ A (respectively A --+ 0) the unique morphism from 0 to A (respectively from A to 0). Clearly 0 --+ A is a monomorphism and A --+ 0 is an epimorphism. Clearly if a category C is preadditive, then so is the dual category CO. If C and 'D are two preadditive categories, then a functor F : C -+ V is called additive if for any two objects A, B E C and any f , g E Homc(A, B) we have that F ( f g) = F ( f ) F(g). Clearly, if F is an additive functor and 0 is the zero object of C, then F(0) is the zero object of 2). As in Section 1, in any preadditive category C we can define for any morphism f : A --+ B the notions of Ker( f ) , Coker(f ) , I m ( f ) and Coim(f ) . We say that a preadditive category C satisfies the axiom (AB1) if for any morphism f : A 4 B there exist a kernel and a cokernel of f . In this case, we have a decomposition
+
+
x , i, p are monomorphisms, and A, n are epimorphisms. where f = ~ o - ~ oand using the above decomposition of f , we say that the category C satisfies the axiom (AB2) if 7 is an isomorphism for any morphism f in C. Clearly, if C verifies (AB2), then a morphism f : A 4 B is an isomorphism if and only if f is a monomorphism and an epimorphism. Assume that C is a preadditive category which satisfies the axioms (AB1) and (AB2). Then a sequence of morphisms
is called exact if I m f = K e r g as subobjects of B. An arbitrary sequence of morphisms is called exact if every subsequence of two consecutive morphisms is exact. If C and 2) are two preadditive categories satisfying the axioms (AB1) and (AB2), then an additive functor F : C --+ 'D is called left (respectively right) exact if for any exact sequence of the form
A. 3. ABELIA N CATEGORIES
-
we have an exact sequence
0
F(A)
Fx) F ( B ) 9F ( C )
respectively F(A)
F(I) F(B) 2) F(C)
-
0
A functor is called exact if it is left and right exact.
Additive and abelian categories A preadditive category C with the property that there exists a coproduct (direct sum) of any two objects is called an addztzve category. An additive category which satisfies the axioms (AB1) and (AB2) is called an abelzun category. If C is an additive category and Al, A2 E C, let A1 $ A2 be a coproduct of the two objects. Bytheuniversal property we have natural morphisms ik : AI, + Al $ A2 and nk : Al $ A2 -+ AI, for k = 1,2, such that TI,OZI, = lAL for k = 1,2, 7rloik = 0 for k # 1, and ilo7rl +i207r2 = lAIeA2. These relations show that (A1 $ A2,T I , 7r2) is a product (also called direct product) of the objects A1 and A2. Therefore if C is an additive (respectively preabelian, abelian) category, then so is the dual category CO. Also, a functor F : C -+ D between two additive categories is additive if and only if F commutes with finite coproducts. Grothendieck categories In the famous paper "Sur quelques poznts d'algkbre homologzque (TohGku Math. J . 9(1957), 119-221), A. Grothendieck introduced the following axioms for an abelian category C. (AB3) C has coproducts, i.e. for any family of objects (A,),,g there exists a coproduct of the family in C. (AB3)* C has products, i.e. for any family of objects there exists a product of the family in C. Assume that the category C satisfies the axiom (AB3). Then for any nonempty set I we can define a functor etE': e(') C, by associating to any family of objects indexed by I the coproduct (direct sum) of the family. This functor is always right exact. We formulate a new axiom. (AB4) For any non-empty set I, the functor $ , € I is exact. We also have the dual axiom for a category which satisfies (AB3)*. (AB4)* For any non-empty set I, the direct product functor is exact. Assume now that C is an abelian category satisfying the axiom (AB3). If A E C and is a family of subobjects of A, then by using (AB3) we see that there exists a smallest subobject CtEI A, of A such that all AZ7sare subobjects of CzEI A,. The subobject CZEI A, is called the sum of the family (At)2EI. The dual situation is when C is an abelian category -+
370
APPENDIX A. THE CATEGORY THEORY LANGUAGE
satisfying the axiom (AB3)*. Then for A E C and (Ai)iEI a family of subobjects of A, then there exists a largest subobject niErAi of A, which is a subobject of each Ai. The subobject niEIAi is called the intersection of the family ( A i ) i E ~ . If C is an abelian category, since C has finite products, then for any subobjects B , C of an object A, there exists the intersection of the family of two subobjects. This is denoted by B n C and is called the intersection of the subobjects B and C. We can introduce now a new axiom. (AB5) Let C be a category satisfying (AB3). If A is an object of C, and (Ai)iEI and B are subobjects of A such that the family (Ai)iEI is right filtered, then ( C i E I Ai) n B = CiEI(Ai n B). The dual of this axiom is called (AB5)*. It is easy to see that a a category satisfying (AB5) also satisfies (AB4). Let C be an abelian category. A family (Ui)iGr of objects of C is called a family of generators of C if for any object A E C and any subobject B of A such that B # A (as a subobject), there exist some i E I and a morphism cr : Ui 4 A such that I,m(a) is not a subobject of B. We say that an object U of C is a generator if the singleton family { U ) is a family of generators. In the case where C is an abelian category with (AB3), then a family (Ui)iEI of objects of C is a family of generators if and only if the direct sum GiEIUi of the family is a generator of C. An abelian category C which verifies the axiom (AB5) and has a generator (or equivalently a family of generators) is called a Grothendieck category. In the same cited paper of Grothendieck it is proved that an abelian category which verifies both (AB5) and (AB5)' must be the zero category. In particular we see that if C is a non-zero Grothendieck category, then the dual category CO is not a Grothendieck category.
A.4
Adjoint functors
Let C and V be two categories and F : C -+ V , G : V -+ C be two functors. The functor F is called a left adjoint of G (or G is called a right adjoint of F ) if there exists a functorial morphism
where HomD(F, -) : CO x V -+ Set is the functor associating to the pair of ) ,) , and Hornc(-, G) : CO x V 4 Set objects (A, B ) the set H o ~ D ( F ( A B is the functor associating to (A, B ) the set Homc(A, G(B)). In the case where C and V are preadditive categories, and F and G are two additive functors, then we assume that 4(A, B) is an isomorphism of
A.4. ADJOINT FUNCTORS
371
abelian groups for any A E C and B E 2). We give now the main properties of adjoint functors.
Theorem A.4.1 With the above notation, assume that F is a left adjoint of G. Then the following assertions hold. 1) The functor F commutes with coproducts and the functor G commutes with products. 2) If C and D are abelian categories and the functors F and G are additive, then F is right exact and G is left exact. 3) Assume that the category V has enough injective objects, i.e. for any B 6 V there exist an injective object Q E V and a monomorphism 0 -+ B 4 Q i n V . Then F is exact i f and only if G commutes with injective objects (i.e. for any injective object Q E V ,the object G(Q) is injective i n
C.
4 ) Assume that C has enough projective objects, i.e. for any A
E C there exist a projective object P E C and an epimorphism P -+ A 0 i n C . Then G is exact if and only i f F commutes with projective objects.
The concept of adjoint functor is fundamental in mathematics, since many properties are in fact adjointness properties. A standard example of an adjoint pair of functors is provided by the Horn and @ functors, more precisely, if M E RMs, then M B R - : SM + RM is a left adjoint of H o r n ~ ( M-), : RM --+ S M .
,
Appendix B
C-groups and C-cogroups B. 1
Definitions
Let C be a category and A E C. We say that we have an internal composition on the object A if for any object X E C there exists an binary operation on the set h A ( X ) = Homc(X, A) such that for any morphism u : Y -+ X in C the map hA(u) : h A ( X ) -+ hA(Y) is a morphism for the operations on the two sets. This means that for any X E C there is a map "*x" (or simply "*"): h A ( X ) x h A ( X ) -+ h A ( X ) such that for any morphism u : Y -4 X the diagram
is commutative, i.e.
for any f , g E h~ (X). In case * x turns h A ( X ) into a group (abelian group, resp.) for any A E C, then A is called a C-group (C-abelian group, resp.). If A is a C0group (Co-abelian group, resp.), then A is called a C-cogroup (C-abelian cogroup, resp.).
374
APPENDIX B. C-GROUPS AND C-COGROUPS
If A, B E C are two C-groups, and u : A -+ B a morphism in C such that for any X E C the map hA(u) : hA(X) -' hA(Y) is a morphism of groups, i.e. u o ( f * x 9) = ( u o f ) *Y ( ~ o g )
(B.2)
then u is called a morphism of C-groups. It is clear that the class of all C-groups defines a category, denoted by Gr(C). Similarly, the class of all C-abelian groups defines a category, denoted by Ab(C).
Remark B . l . l If the category C has a "zero" object, then for any objects X and Y we have the zero morphism from Y to X . Then, putting u = 0 in equality (B.l), we have in that
in Hornc (Y, A ) .
If A is a C-group, then from (B.3) it follows that 0 = el where e is the unit I element of the group Homc(Y,A) for any Y E C. We assume now that C has finite direct products, and a final object, denoted by F . If A E C is a C-group, we denote by V A E Homc(F, A) the unit element of this group. Then for any X E C, the element
where EX is the unique element of Homc(X, F ) , is the unit element of the group Homc(X, A). Indeed, from (B.l) we have
and since Homc(X, A) is a group, it follows that VA O E X is its unit element. Now since C has finite direct products, we have the direct products A x A and A x A x A. By the Yoneda Lemma, the maps *x,X E C, yield a morphism m : A x A -+ A. Then the associativity and the existence of the unit element imply the commutativity of the following diagrams
B. 1. DEFINITIONS
AxAxA
I x m -AxA
where I is the identity morphism, and since F is a final object, we have the canonical isomorphisms A x F A and F x A E A. In the same way, the existence of the inverse shows that there exists an S E Homc(A,A) making the following diagram commutative
--
F - A - F
(i)(s)
(B.6)
Here and denote the canonical morphisms defined by the universal property of the direct product using the morphisms I : A -+ A and S : A -+ A. Example B.1.2 I . If C = S e t (the category of sets), then it has arbitrary
direct products and a final object. I n this case we have that G r ( S e t ) is just the category of groups. 2. If C = T o p (the category of topological spaces), then G r ( T o p ) is the
APPENDIX B. C-GROUPS A N D C-COGROUPS
376
category of topological groups. 3. Assume that C is an additive category. Let A E C, and consider the A x A, and 7rj : A x A --,A, j = 1,2, such canonical morphisms i j : A that 7rj o i l = 6 j l I ~ I. n fact, il = and i 2 = B y the commutativity of the diagram (B.4), it follows that m o il = m o iz = I A . O n the other hand, let f,g 6 H o m c ( X , A ) . W e denote by : A x A -,A the canonical morphism with the property that o i l = o i2 = I A (we used the fact that A x A is also a coproduct). The morphisms f and g also define a canonical morphism (:) : X -+ A x A, such that 7rl o (:) = f and 7r2 o = g. Since il o 7rl i2o ~2 = I A x A , we have -+
(i)
(y).
v
v
(i)
+
and = ~ o i ~ o g = g ,
(L)
and therefore g o = f + g. Since m o i l = m o i2 = I*, we get by uniqueness that m = v, so m o (9f ) = f + g . ~ u t m o ( : ) = f *X g,so f *X g = f + g . I n conclusion, if C is a n additive category, we have that every object A E C is an abelian C-group, with the multiplication of H o m c ( X , A) the sum of morphisms. S o G r ( C ) = Ab(C) is isomorphic to the category C. Since CO is also an additive category, i t follows that every object A E C is also an abelian C-cogroup. I
B.2 General properties of C-groups Proposition B.2.1 Assume that C is a ctegory which has finite direct products, and a final object. Then the following assertion hold: i ) G r ( C ) has direct products. ii) Ab(C) is a n abelian category. iii) If C also has fiber products (pull-backs), then in the category Ab(C) e v e y morphism has a kernel. Proof: i) If A, B E G r ( C ) ,since we have
H o m c ( X , A x B ) 2 H o m c ( X ,A ) x H o m c ( X , B ) , it follows that A x B is also a C-group. ii) Assume that A, B E Ab(C). Then we consider the operation "*" on HomAb(q(AB , ) as in the first section. From ( B . l ) and (B.2) we get the
B.3. FORMAL GROUPS AND AFFINE GROUPS
377
distributivity of the composition "on with respect to "*". Now if F denotes the final object of C, then F is the zero object of Ab(C), and from i) we get that Ab(C) is an abelian category. iii) Let f : A -+ B be a morphism in the category Ab(C). If F is the final object and, and q : F -+ B is the unit element of the group Homc(F, B ) , then for X E C and E : X --+ F the unique morphism, we saw that q o E is the unit element of the group Homc(X, B). We denote by A X B F the fiber product associated to the diagram
Then by the definition of the fiber product we have that Homc(X, A x~ F ) is identified to the set of all elements u E Homc(X, A) such that f ou = ~ O E . Since Homc(X, A) is an abelian group, and f : A -+ B is a morphism of groups, then Homc (X, A x B F ) is a subgroup of Homc(X, A). So A x B F is a C-abelian group. It is easy to see that A x~ F is the kernel o f f in the I category Ab(C).
Remark B.2.2 A category C with finite direct products and final object is a braided monoidal category (even symmetric), where the "tensor product" functor is the direct product.
a
B.3 Formal groups and affine groups Let k be a field. We consider two categories: k - CCog, i.e. the category of all cocommutative k-coalgebras, and the dual (k-CAlg)", where k-CAlg is the category of commutative k-algebras. These categories have finite direct products and final objects (the final object is in both cases the field k, regarded as a coalgebra or an algebra). It is easy to see that Gr(k - CCog) is exactly the category of cocommutative Hopf algebras over k. An object in this category is called a formal group. Similarly, Gr((lc - CAlg)") is exactly the category of commutative Hopf algebras over k. In fact, the objects of this category are cogroups in the category k - CAlg. When we consider only the category of all affine kalgebras (i.e. commutative k-algebras which are finitely generated as k-
APPENDIX B. C-GROUPS AND C-COGROUPS
378
algebras), then a group object in the dual of this category is called an afine group. We remark that Ab(k-CCog) = Ab(k-CAlg). The following important result concerns this category.
Theorem B.3.1 Ab(k - CCog) is a Grothendieck category. Proof: We will give a sketch of proof. To simplify the notation, put A = Ab(k - CCog). The main steps of the proof are the following: Step 1. If H , K E A, then Homd(H, K ) is the set of all Hopf algebra maps. Now if f , g € Homd(H, K ) , then the multiplication of f and g is given by the convolution product f * g. Since H and K are commutative and cocommutative, then f * g is also a Hopf algebra morphism. It is easy to see that r] o E is the unit element, where E : H -+ k and r] : k -+ K are the canonical maps. Moreover, the inverse of f is given by SK o f = f o S H . Hence Homd(H, K ) is an abelian group. It is also easy to see that
Step 2. We know from Proposition B.2.1 that A is an additive category. If H, K E A, then H €3 K is a commutative and cocommutative Hopf algebra. It is easy to see that H @ K is the coproduct and the product of H and K in the category A. The zero object in A is k. Step 3. A verifies the axiom AB1). Let f : H -+ K be a morphism in the category A. By Proposition B.2.1, f has a kernel. In fact, the kernel is exactly H X K k = HOKk (see Exercise 2.3.11). On the other hand, if we put L = I m ( f ) , then L is a Hopf subalgebra of K , and I = L + K is a Hopf ) . it is easy to see that K I I , together ideal (recall that L+ = K e r ( ~ ~Then with the canonical projection .rr : K -+ K I I , is a cokernel of f . Step 4. A verifies the axiom AB2). This is the main step, and it follows from the following result: if H is a commutative and cocommutative Hopf algebra, the map K H K + H establishes a bijective correpondence between the Hopf subalgebras of H and the Hopf ideals of H. A proof of a more general version of this result may be found in M. Takeuchi [228], A. Heller [94],or more recently in H.-J. Schneider [203]. Step 5. J1 verifies the axioms AB3) and AB5). Let (Hi)iEI be a family of objects in A. Consider the infinite tensor product Qg Hi, which is a iEI
k-algebra, and the subalgebra, denoted by
BH~, generated by all images r "..
ic
of morphisms
ai : Hi -+
@ H j , cri(x) = Qg x j , where xi = x, and x j = 1 j€I
for j
#
i. It is easy to see that
.l€I
BH~ is also a Hopf algebra, and it is iEI
the direct sum of the family (Hi)i,g in the category A. Now by Step 4, if
B.3. FORMAL GROUPS AND AFFINE GROUPS
379
f : H -+ K is a morphism in A, then f is a monomorphism in this category if and only if K e r ( f ) = k (k is the zero object in A), if and only if f is injective (since the Hopf ideal associated to the Hopf subalgebra k is zero as a vector space). So the subobjects of K are all Hopf subalgebras. From this it follows that A satisfies AB5). Step 6. A has a family of generators. Indeed, if we consider the category k - CCog, the forgetful functor
U : Ab(k - CCog) has a left adjoint, denoted by
F : k - CCog
-
-
k - CCog
Ab(k - CCog).
Then if we consider the set of all finite dimensional cocommutative coalgebras (Ci)iEI, from the above it follows that the family (F(Ci))iEr is a I family of generators for A.
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t
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Notices 1 (1994), 53-59.
Index (g, h)-primitive element, 158 M-injective comodule, 95 C-abelian cogroup, 373 C-abelian group, 373 C-cogroup, 373 C-group, 373
a [ M ] ,101 abelian category, 369 additive category, 369 adjoint action, 242 adjoint functors, 370 affine group, 378 algebra, 1 algebra of characters, 320 algebra of coinvariants, 244 algebra of invariants, 234 antipode, 151 bialgebra, 149 bicomodule, 84 category with enough projectives, 99 character of a module, 318 Class Equation, 330 co-idempotent subcoalgebra, 106 co-opposite coalgebra, 32 coalgebra, 2 coalgebra associated to a comodule, 102 cocommutative bialgebra (Hopf algebra), 149 cocommutative cofree coalgebra, 53
coequalizer, 364 coflat comodule, 95 cofree coalgebra, 50 coideal, 24 cokernel, 364 comatrix basis, 302 commutative algebra, 8 commutative bialgebra (Hopf algebra), 149 comodule, ti6 comodule algebra, 243 comodule of finite support, 275 contravariant functor, 365 convolution, 166 convolution product, 18 coproduct, 365 coradical filtration, 123 coreflexive coalgebra, 43 cosemisimple coalgebra, 119 covariant functor, 365 crossed product, 237 distinguished grouplike element of H, 197 divided power Hopf algebras, 164 duality, 366 enveloping algebra of a Lie algebra, 163 epimorphism, 363 equalizer, 364 equivalence of categories, 366 essential subcomodule, 92
INDEX family of generators in a category, 370 fiber product, 365 formal group, 377 free comodule, 91 Frobenius algebra, 139 full, faithful functor, 366 functorial morphism, 366 Fundamental Theorem of Coalgebras, 25 of Comodules, 68 of Hopf modules, 171
Nichols-Zoeller theorem, 298
generator in a category, 370 Grothendieck category, 370 group algebra, 158 grouplike element, 28
opposite algebra, 33 Ore extension, 203 orthogonality relations for matrix subcoalgebras, 308
Haar integral, 181 Hopf algebra, 151 Hopf module, 169 Hopf subalgebra, 156 Hopf-Galois extension, 255
pointed coalgebra, 195 preadditive category, 367 primitive element, 158 product, 364 projective cover, 124 proper (residually finite dimensional) algebra, 44 pseudocompact algebra, 48 pseudocompact module, 80
injective comodule, 92 injective envelope, 92 integral, 181 integral in a Hopf algebra, 186 internal composition, 373 irreducible characters, 320 isomorphism, 363 isomorphism of categories, 366 Kac-Zhu theorem, 329 kernel, 364 Larson-Radford theorem, 3 16 left (right) coideal, 24 Loewy series of a comodule, 122 Maschke's theorem, 187 for crossed products, 262 for smash products, 251 matrix coalgebra, 4
module, 65 monomorphism, 363 morphism, 361 morphism of algebras, 9 of bialgebras, 150 of coalgebras, 9 of comodules, 69 of Hopf algebras, 152 of modules, 68
QcF coalgebra, 133 quasi-Frobenius algebra, 139 quotient object, 363 radical functor, 101 rational module, 74 relative Hopf modules, 247 representable functor, 367 representative coalgebra, 41 selfdual Hopf algebra, 168 semiperfect coalgebra, 126 separable algebra, 189 sigma notation, 5 sigma notation for comodules, 66 simple coalgebra, 117
INDEX simple comodule, 93 smash product, 235 subcoalgebra, 23 subcomodule, 70 subobject, 363 subspace of coinvariants, 171 Sweedler's d-dimensional Hopf algebra, 165 symmetric algebra, 162 Taft algebras, 166 Taft-Wilson theorem, 336 tensor product of coalgebras, 30 tensor algebra, 159 tensor product of bialgebras (Hopf algebras), 158 torsion theory, 101 total integral, 254 trace function, 251 twisted product, 267 unimodular, 199 weak action, 237 Weak Structure Theorem, 257 wedge, 105 zero object, 364
