Hopf algebras and quantum groups : proceedings of the Brussels conference 9780824703950, 0824703952, 9781138442139

Table of contents :
Content: 1. Lifting of Nichols Algebras of Type A[subscript 2] and Pointed Hopf Algebras of Order p[superscript 4] / Nicolas Andruskiewitsch, Hans-Jurgen Schneider 1 --
2. Survey of Cross Product Bialgebras / Yuri Bespalov, Bernhard Drabant 15 --
3. A Morita-Takeuchi Context for Graded Coalgebras / Crina Boboc 35 --
4. Coalgebra-Galois Extensions from the Extension Theory Point of View / Tomasz Brzezinski 47 --
5. Separable Functors for the Category of Doi-Hopf Modules II / Stefaan Caenepeel, Bogdan Ion, Gigel Militaru, Shenglin Zhu 69 --
6. Cyclic Cohomology of Coalgebras, Coderivations and De Rham Cohomology / Marco A. Farinati, Andrea Solotar 105 --
7. Schur-Weyl Categories and Non-Quasiclassical Weyl Type Formula / Dimitri Gurevich, Zakaria Mriss 131 --
8. A Generalized Power Map for Hopf Algebras / Yevgenia Kashina 159 --
9. Associated Varieties for Classical Lie Superalgebras / Ian M. Musson 177 --
10. Algebraic Versions of a Finite-Dimensional Quantum Groupoid / Dmitri Nikshych, Leonid Vainerman 189 --
11. Quasi-Hopf Algebras and the Centre of a Tensor Category / Florin Panaite, Freddy van Oystaeyen 221 --
12. An Easy Proof for the Uniqueness of Integrals / Serban Raianu 237 --
13. Coquasitriangular Hopf Algebra Associated to a Rigid Yang-Baxter Coalgebra / Mitsuhiro Takeuchi 241 --
14. On Regularity of the Algebra of Covariants for Actions of Pointed Hopf Algebras on Regular Commutative Algebras / Andrzej Tyc 261 --
15. A Survey on Multiplier Hopf Algebras / Alfons van Daele, Yinhuo Zhang 269.

Citation preview

Hopf algebras and quantum groups

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft

Zuhair Nashed

Rutgers University New Brunswick, New Jersey

University o f Delaware Newark, Delaware

EDITORIAL BOARD M. S. Baouendi University o f California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute o f Technology

Anil Nerode Cornell University Donald Passman University o f Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi David L. Russell University o f California, Virginia Polytechnic Institute Berkeley and State University Marvin Marcus University o f California, Santa Barbara W. S. Massey Yale University

Walter Schempp Universitdt Siegen Mark Teply University o f Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2. 3. 4.

N. Jacobson, Exceptional Lie Algebras L .-A Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis /. Satake, Classification Theory of Sem i-Sim ple Algebraic Groups

F. Hirzebmch a t at., Differentiable Manifolds and Quadratic Forms 5. I. Chavel, Riemannian Symmetric Spaces of Rank One 6. R. B. Burvkel, Characterization of C (X) Among Its Subalgebras 7. B. R. M cDonald et al., Ring Theory 8. V.-T. Siu, Techniques of Extension on Analytic Objects 9. S. R. Caradus et al., Calkin Algebras and Algebras of Operators on Banach Spaces 10. E. O. Roxin et al., Differential Gam es and Control Theory 11. M. Orzech and C. Small, The Brauer Group of Commutative Rings 12. S. Thornier, Topology and Its Applications 13. J. M. Lopez and K. A. Ross, Sidon Sets 14. W. W. Comfort and S. Negrepontis, Continuous Pseudometrics 15. K. McKennon and J. M. Robertson, Locally Convex Spaces 16. M. Carm eli and S. Matin, Representations of the Rotation and Lorentz Groups 17. G. B. Seligman, Rational Methods in Lie Algebras 18. D. G. de Figueiredo, Functional Analysis 19. L. Cesari et al., Nonlinear Functional Analysis and Differential Equations 20. J. J. Schaffer, Geom etry of Spheres in Normed Spaces 21. K. Yano and M. Kon, Anti-Invariant Submanifolds 22. W. V. Vasconcelos, The Rings of Dimension Two 23. R. E. Chandler, Hausdorff Compactifications 24. S. P. Franklin andB . V. S. Thomas, Topology 25. S. K. Jain, Ring Theory 26. B. R. McDonald and R. A. Morris, Ring Theory II 27. R. B. Mura and A. Rhemtulla, Orderable Groups 28. J. R. Graef, Stability of Dynamical Systems 29. H.-C. Wang, Homogeneous Branch Algebras 30. E. O. Roxin et al., Differential Gam es and Control Theory II 31. R. D. Porter, Introduction to Fibre Bundles 32. M. Altman, Contractors and Contractor Directions Theory and Applications 33. J. S. Golan, Decomposition and Dimension in Module Categories 34. G. Fairweather, Finite Elem ent Galerkin Methods for Differential Equations 35. J. D. Sally, Numbers of Generators of Ideals in Local Rings 36. S. S. M iller, Complex Analysis 37. R. Gordon, Representation Theory of Algebras 38. M. Goto and F. D. Grosshans, Semisimple Lie Algebras 39. A. I. Arruda et al., Mathem atical Logic 40. F. Van Oystaeyen, Ring Theory 41. F. Van Oystaeyen and A. Verschoren, Reflectors and Localization 42. M. Satyanarayana, Positively Ordered Semigroups 43. D. L Russell, Mathem atics o f Finite-Dimensional Control Systems 44. P .-T. Liu and E. Roxin, Differential Gam es and Control Theory III 45. A. Geramita and J. Seberry, Orthogonal Designs 46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

P.-T. Liu and J. G. Sutinen, Control Theory in M athem atical Economics C. Byrnes, Partial Differential Equations and Geometry G. Klambauer; Problems and Propositions in Analysis J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields F. Van Oystaeyen, Ring Theory B. Kadem, Binary Tim e Series J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems R. L Sternberg e t al., Nonlinear Partial Differential Equations in Engineering and Applied Science B. R. McDonald, Ring Theory and Algebra III J. S. Golan, Structure Sheaves Over a Noncommutative Ring T. V. Narayana et al., Combinatorics, Representation Theory and Statistical Methods in Groups T. A. Burton, Modeling and Differential Equations in Biology K. H. Kim and F. W. Roush, Introduction to Mathem atical Consensus Theory

J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces O. A. Nielson, Direct Integral Theory J. E. Smith et al., Ordered Groups J. Cronin, Mathem atics of Cell Electrophysiology J. W. Brewer, Power Series Over Commutative Rings P. K. Kamthan and M. Gupta, Sequence Spaces and Series T. G. McLaughlin, Regressive Sets and the Theory of Isols T. L Herdman et al., Integral and Functional Differential Equations R. Draper, Commutative Algebra W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Repre­ sentations of Simple Lie Algebras 70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems 71. J. Van Geel, Places and Valuations in Noncommutative Ring Theory 72. C. Faith, Injective Modules and Injective Quotient Rings 73. A Fiacco, M athem atical Programming with Data Perturbations I 74. P. Schultz e t al., Algebraic Structures and Applications 75. L Bican et al., Rings, Modules, and Preradicals 76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry 77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces 78. C.-C. Yang, Factorization Theory of Meromorphic Functions 79. O. Taussky, Ternary Quadratic Forms and Norms 80. S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications 81. K. B. Hannsgen et al., Volterra and Functional Differential Equations 82. N. L. Johnson et al., Finite Geometries 83. G. I. Zapata, Functional Analysis, Holomorphy, and Approximation Theory 84. S. Greco and G. Valla, Commutative Algebra 85. A. V. Fiacco, Mathem atical Programming with Data Perturbations II 86. J.-B. Hiriart-Urruty et al., Optimization 87. A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups 88. M. Harada, Factor Categories with Applications to Direct Decomposition of Modules 89. V. I. Istr&tescu, Strict Convexity and Complex Strict Convexity 90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations 91. H. L. Manocha and J. B. Srivastava, Algebra and Its Applications 92. D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems 93. J. W. Longley, Least Squares Computations Using Orthogonalization Methods 94. L P. de Alcantara, Mathematical Logic and Formal Systems 95. C. E. Aull, Rings of Continuous Functions 96. R. Chuaqui, Analysis, Geometry, and Probability 97. L Fuchs and L. Salce, Modules Over Valuation Domains 98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics 99. W. B. Powell and C. Tsinakis, Ordered Algebraic Structures 100. G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications 101. R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications 102. J. H. Lightboume III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations 103. C. A Baker and L. M. Batten, Finite Geometries 104. J. W. Brewer et al., Linear Systems Over Commutative Rings 105. C. McCrory and T. Shifrin, Geometry and Topology 106. D. W. Kueke et al., Mathematical Logic and Theoretical Computer Science 107. B.-L. Lin and S. Simons, Nonlinear and Convex Analysis 108. S. J. Lee, Operator Methods for Optimal Control Problems 109. V. Lakshmikantham, Nonlinear Analysis and Applications 110. S. F. McCormick, Multigrid Methods 111. M .C . Tangora, Computers in Algebra 112. D. V. Chudnovsky and G. V. Chudnovsky, Search Theory 113. D. V. Chudnovsky and R. D. Jenks, Computer Algebra 114. M. C. Tangora, Computers in Geometry and Topology 115. P. Nelson et al., Transport Theory, Invariant Imbedding, and Integral Equations 116. P. Cldment et al., Semigroup Theory and Applications 117. J. Vinuesa, Orthogonal Polynomials and Their Applications 118. C. M. Dafermos et al., Differential Equations 119. E. O. Roxin, Modem Optimal Control 120. J. C. Diaz, Mathem atics for Large Scale Computing 60. 6ft. 62. 63. 64. 65. 66. 67. 68. 69.

121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183.

P. S. M ilojevi# Nonlinear Functional Analysis C. Sadosky, Analysis and Partial Differential Equations R. M. Shortt, General Topology and Applications R. Wong, Asymptotic and Computational Analysis D. V. Chudnovsky and R. D. Jenks, Computers In Mathematics W. D. W allis et al., Combinatorial Designs and Applications S. Elaydi, Differential Equations G. Chen et al., Distributed Param eter Control Systems W. N. Everitt, Inequalities H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differ­ ential Equations O. Arino et al., Mathem atical Population Dynamics S. Coen, Geom etry and Complex Variables J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andima et al., General Topology and Applications P Clement et al., Semigroup Theory and Evolution Equations K. Jarosz, Function Spaces J. M. Bayod et al., p-adic Functional Analysis G. A. Anastassbu , Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et al., Methods in Module Theory G. L M ullen and P. J.-S. Shiue , Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. Joshi and A. V. Balakrishnan, Mathem atical Theory of Control G. Komatsu and Y. Sakane, Complex Geom etry I. J. Bakelman , Geom etric Analysis and Nonlinear Partial Differential Equations T. M abuchi and S. Mukai, Einstein Metrics and Yang-M ills Connections L. Fuchs and R. Gdbel, Abelian Groups A. D. Pollington and W. M oran, Number Theory with an Emphasis on the Markoff Spectrum G. Done e t al., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedt et al., Functional Analysis K. G. Fischer et al., Computational Algebra K. D. Elworthy et al., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et al., Commutative Ring Theory S. C. C ooperand W. J. Thron, Continued Fractions and Orthogonal Functions P. CIGment and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in M athem atical Research W. O. Bray et al., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Uu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zoldsio, Boundary Control and Variation M. K ftz e k e ta l., Finite Elem ent Methods G. Da Prato and L. Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabel et al., Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twister Theory H. Cook et al., Continua D. F. Anderson and D. E. Dobbs, Zero-Dim ensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geom etry E. Casas, Control of Partial Differential Equations and Applications N. Kalton et al., Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al., Differential Equations and Control Theory P. M arcellini et al. Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Agliand, Logic and Algebra X. H. Cao e ta l., Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models

184. 185. 186. 187. 188.

J. E. Andersen et al., Geometry and Physics P.-J. Cahen et al., Commutative Ring Theory J. A. Goldstein et a/., Stochastic Processes and Functional Analysis A. Sorbi, Complexity, Logic, and Recursion Theory G. Da Prato and J.-P. Zotesio, Partial Differential Equation Methods in Control and Shape

Analysis 189. D. D. Anderson, Factorization in Integral Domains 190. N. L. Johnson, Mostly Finite Geometries 191. D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm-Liouville Problems 192. W. H. Schikhofet al., p-adic Functional Analysis 193. S. Sertdz, Algebraic Geometry 194. G. Caristi and E. M itidieri, Reaction Diffusion Systems 195. A. V. Fiacco, Mathematical Programming with Data Perturbations 196. M. K tiie k e t al., Finite Elem ent Methods: Superconvergence, Post-Processing, and A Posteriori Estimates 197. S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups 198. V. Drensky et al., Methods in Ring Theory 199. W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions 200. P. E. Newstead, Algebraic Geometry 201. D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology 202. Z. Chen et al., Advances in Computational Mathematics 203. X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs 204. C. Y. Yddirim and S. A. Stepanov, Number Theory and Its Applications 205. D. E. Dobbs et al., Advances in Commutative Ring Theory 206. F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry 207. J. Kakol et al., p-adic Functional Analysis 208. M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory 209. S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups 210. F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras 211. R. Costa et al., Nonassociative Algebra and Its Applications

Additional Volumes in Preparation

Hopf algebras and quantum groups proceedings of the Brussels conference

edited by Stefaan Caenepeel Free University of Brussels Brussels, Belgium

Freddy Van Oystaeyen

University of Antwerp/UIA Antwerp, Belgium

Boca Raton London New York

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Preface

The colloquium “Hopf Algebras and Quantum Groups” was held at the Free University of Brussels (VUB). The main lectures were delivered by N. Andruskiewitsch, S. Dascalescu, C. Kassel, L. Lebruyn, A. Masuoka, G. Militaru, S. Montgomery, H.-J. Schneider, D. §tefan, A. Van Daele, S. Woronowicz, and Y. Zhang. There were 67 participants from 19 different countries, and 46 lectures were given. The meeting was supported financially by the Flemish Research Foundation (FWO Vlaanderen), the University of Brussels (VUB), and the bilateral project of the Flemish and Romanian governments, “Hopf Algebras and (co)-Galois Theory.” Warm thanks go to Gigel Militaru and Zhu Shenglin, for helping with the local organization, and to Rieke Caenepeel, for solving numerous unexpected practical problems during the conference. Some of the manuscripts were prepared using Lyubashenko’s “t-angles” style, while some others made use of Paul Taylor’s “diagrams.” In our struggle with the various types of Tex files supplied to us by the authors, we benefitted greatly from Philippe Cara’s Tex experience, and from Bernhard Drabant’s familiarity with the t-angles package. Stefaan Caenepeel Freddy Van Oystaeyen

iii

C ontents

iii VII ix

Preface Contributors Conference Participants

1.

Lifting of Nichols Algebras of Type A2and Pointed Hopf Algebras of Order p* Nicolas Andruskiewitsch and Hans-Jiirgen Schneider

1

2.

Survey of Cross Product Bialgebras Yuri Bespalov and Bernhard Drabant

15

3.

A Morita-Takeuchi Context for Graded Coalgebras Crina Boboc

35

4.

Coalgebra-Galois Extensions from the Extension Theory Point of View Tomasz Brzezinski

47

5.

Separable Functors for the Category of Doi-Hopf Modules II Stefaan Caenepeel, Bogdan Ion, Gigel Militaru, and Shenglin Zhu

6.

Cyclic Cohomology of Coalgebras, Coderivations and De Rham Cohomology Marco A. Farinati and Andrea Solotar

105

Schur-Weyl Categories and Non-Quasiclassical Weyl Type Formula Dimitri Gurevich and Zakaria Mriss

131

7.

8.

A Generalized Power Map for Hopf Algebras Yevgenia Kashina v

69

159

Contents

VI

9.

Associated Varieties for Classical Lie Superalgebras Ian M. Musson

111

10.

Algebraic Versions of a Finite-Dimensional Quantum Groupoid Dmitri Nikshych and Leonid Vainerman

189

11.

Quasi-Hopf Algebras and the Centre of a Tensor Category Florin Panaite and Freddy Van Oystaeyen

221

12.

An Easy Proof for the Uniqueness of Integrals §erban Raianu

237

13.

The Coquasitriangular Hopf Algebra Associated to a Rigid Yang-Baxter Coalgebra Mitsuhiro Takeuchi

241

14.

On Regularity of the Algebra of Covariants for Actions of Pointed 261 Hopf Algebras on Regular Commutative Algebras Andrzej Tyc

15.

A Survey on Multiplier Hopf Algebras Alfons Van Daele and Yinhuo Zhang

269

Contributors

Nicolas Andruskiewitsch Yuri Bespalov Ukraine Crina Boboc

Universidad de Cordoba, Cordoba, Argentina

Bogolyubov Institute for Theoretical Physics, Kiev, University of Bucharest, Bucharest, Romania

Tomasz Brzezinski University of York, York, United Kingdom, and University of L6dz, L6dz, Poland Stefaan Caenepeel Free University of Brussels, Brussels, Belgium Bernhard Drabant Kingdom

University of Cambridge, Cambridge, United

Marco A. Farinati Argentina

Universidad de Buenos Aires, Buenos Aires,

Dimitri Gurevich Bogdan Ion

Universite de Valenciennes, Valenciennes, France

Princeton University, Princeton, New Jersey

Yevgenia Kashina California

University of Southern California, Los Angeles,

Gigel M ilitaru

University of Bucharest, Bucharest, Romania

Zakaria Mriss

University de Valenciennes, Valenciennes, France

Ian M. Musson Wisconsin

University of Wisconsin-Milwaukee, Milwaukee,

Dmitri Nikshych California

University of California at Los Angeles, Los Angeles,

Florin Panaite Institute of Mathematics of the Romanian Academy, Bucharest, Romania §erban Raianu

University of Bucharest, Bucharest, Romania

vii

Contributors

viii

Hans-Jiirgen Schneider Andrea Solotar

Universidad de Buenos Aires, Buenos Aires, Argentina

Mitsuhiro Takeuchi Andrzej Tyc

University of Tsukuba, Tsukuba, Ibaraki, Japan

N. Copernicus University, Torun, Poland

Leonid Vainerman Alfons Van Daele

Universite Pierre et Marie Curie, Paris, France K.U. Leuven, Heverlee, Belgium

Freddy Van Oystaeyen Yinhuo Zhang Shenglin Zhu

Universitat Miinchen, Miinchen, Germany

University of Antwerp, UIA, Antwerp, Belgium

University of Antwerp, UIA, Antwerp, Belgium Fudan University, Shanghai, China

Conference Participants Jaw ad Abuhlail, Heinrich-Heine University, Diisseldorf abuhlai @math.uni-duesseldorf.de Philippe Akueson, University de Valenciennes, Valenciennes, France akueson @univ-valenciennes.fr Nicolas Andruskiewitsch, Universidad Nacional de Cdrdoba, Cdrdoba, Argentina andrus @famaf.unc.edu.ar Anthony Bak, Universitat Bielefeld, Bielefeld, Germany [email protected] Teodor Banica, University Paris VI, Paris, France [email protected] Volodymyr Bavula, University of Edinburgh, Edinburgh, Scotland [email protected] M argaret Beattie, Mount Allison University, Sackville, New Brunswick, Canada mbeattie @mta.ca C rina Boboc, University of Bucharest, Bucharest, Romania [email protected] Alexei Bondal, Steklov Mathematical Institute, Moscow, Russia [email protected] Stefaan Caenepeel, Free University of Brussels, VUB, Brussels, Belgium scaenepe @vub. ac.be Juan Cuadra, University of Almerfa, Almerfa, Spain [email protected] Sorin Dascalescu, University of Bucharest, Bucharest, Romania sdascal @al.math.unibuc.ro Lydia Delvaux, Limburgs Universitair Centrum, LUC, Hasselt, Belgium ldelvaux @luc. ac.be Bernhard D rabant, University of Cambridge, Cambridge, England [email protected] M arco Farinati, Universidad de Buenos Aires, Buenos Aires, Argentina mfarinat @dm.uba. ar Tony Giaquinto, Texas A&M University, College Station, Texas, USA tony g @math.tamu.edu Ramon Gonzalez Rodriguez, University of Vigo, Vigo, Spain rgon @dma.uvigo.es Ralf Guenther, Universitat Miinchen, Miinchen, Germany [email protected] Dimitri Gurevich, Universit de Valenciennes, Valenciennes, France Dimitri.Gourevitch @univ-valenciennes.fr ix

X

Conference Participants

Bogdan Ion, Princeton University, Princeton, New Jersey, USA bogdan @math.princeton.edu George Janelidze. University of Tblisi, Tblisi, Georgia Yevgenia Kashina, University of Southern California, Los Angeles, California, USA yevgenia @scf .use.edu Christian Kassel, Universite Louis Pasteur, Strasbourg [email protected] Johan Kustermans, Odense Universitet,Odense, Denmark [email protected] Pascal Lambrechts, Universite d’Artois, Lens, France lambrech @gat.univ-lille 1.fr Lieven Lebruyn, Universitaiie Instelling Antwerpen, UIA, Wilrijk, Belgium lebruyn @uia.ua. ac.be Abdanacer Makhlouf, University de Haute Alsace, Mulhouse, France [email protected] Akira Masuoka, University of Tsukuba, Tsukuba, Japan akira @math, tsukuba. ac.jp Volodvmyr Mazorchuk, Kyiv Taras Shevchenko University, Kyiv, Ukraine [email protected] Claudia Menini, University di Ferrara, Ferrara, Italy men @ifeuniv.unife.it Gigel Militaru, University of Bucharest, Bucharest, Romania [email protected] Susan Montgomery, University of Southern California, Los Angeles, California, USA smontgom @mtha.usc.edu Eric Mueller, Universitat Miinchen, Miinchen, Germany [email protected] Ian Musson, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, USA [email protected] Constantin Nastasescu, University of Bucharest, Bucharest, Romania [email protected] Erna Nauwelaerts, Limburgs Universitair Centrum, LUC, Hasselt, Belgium [email protected] Chi-Keung Ng, University of Oxford, Oxford, England, UK [email protected] Dmitri Nikshych. UCLA, Los Angeles, California, USA nikshych @math.ucla.edu Christian Ohn, University de Reims, Reims, France [email protected]

Conference Participants

xi

Florin Panaite, Institute of Mathematics of the Romanian Academy, Bucharest, Romania, [email protected] Anna Paolucci, University of Leeds, Leeds, England, U.K. [email protected] §erban Raianu, University of Bucharest, Bucharest, Romania sraianu @al.math.unibuc.ro Rudolph Rentschler, Universite Pierre et Marie Curie. Paris, France rent @moka.ccr.jussieu.fr Zhong-Jin Ruan, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA [email protected] Boris Scharfschwerdt, Universitat Miinchen, Miinchen, Germany [email protected] Peter Schauenburg, Universitat Miinchen, Miinchen, Germany [email protected] Hans-Jurgen Schneider Universitat Miinchen, Miinchen, Germany [email protected] Boris Sirola. University of Zagreb, Zagreb, Croatia [email protected] Yorck Sommerhauser, Universitat Miinchen, Miinchen, Germany Sommerh @rz .mathematik.uni-muenchen.de Dragos §tefan, University of Bucharest, Bucharest, Romania [email protected] Gjenna Stippel. Universitaire Instelling Antwerpen, UIA, Wilrijk, Belgium stippel @uia.ua.ac.be Bem d Strueber, Universitat Miinchen, Miinchen, Germany [email protected] Earl Taft, Rutgers University. New Brunswick, New Jersey, USA [email protected] Andrzej Tyc, N.Copemicus University in Torun, Torun, Poland [email protected] M artine Van Gastel, Limburgs Universitair Centrum, LUC, Hasselt, Belgium m vgastel @luc.ac.be Alfons Van Daele, Katholieke Universiteit Leuven, Heverlee, Belgium [email protected] Lucien Van hamme. Free University of Brussels, VUB, Brussels, Belgium [email protected] Freddy Van Oystaeyen, Universitaire Instelling Antwerpen, UIA, Wilrijk. Belgium voy st @uia. ua.ac .be Leonid Vaynerman, Max-Planck Institut fur Mathematik, Bonn, Germany [email protected] Alain Verschoren, Universiteit Antwerpen, RUCA, Antwerp, Belgium [email protected]

xii

Conference Participants

E n rico V itale, Universite Catholique de Louvain, UCL, Louvain-la-Neuve, Belgium, vitale@ agel.ucl.ac.be P aul W auters, Limburgs Universitair Centrum, LUC, Hasselt, Belgium pwauters@ luc.ac.be S a ra W estreich, Bar-Ilan University, Ramat-Gan, Israel swestric@ ashur.cc.biu.ac.il S tanislav W oronow icz, University of Warsaw, Warsaw, Poland slworono@ fuw.edu.pl Xiao Jie, Beijing Normal University, Beijing, China xiaojie@ bnu.edu.cn Y inhuo Z han g , M ax-Planck Institut fur M athematik, Bonn, Germany yhzhang@ m pim -bonn.m pg.de Shenglin Z h u , Fudan University, Shanghai, China shlzhu@ ms.fudan.edu.cn

Lifting of Nichols Algebras of Type A 2 and Pointed Hopf Algebras of Order p 4 Nicolas Andraskiewitsch Universidad de Cordoba Cordoba, Argentina

Hans-Jurgen Schneider Universitat Munchen Miinchen, Germany

Abstract We compute all finite-dimensional Hopf algebras whose coradical is the group algebra of an abelian group T and such that gr(A) = 1. Let V be a Yetter-Drinfel’d module over k[T] with a basis x \ .......*e such that

Lifting of Nichols Algebras and Pointed HopfAlgebras

3

for all h e T and all i. Let A be a pointed Hopf algebra with gr(A) = ©(V)#k[T]. We choose elements a, in the first term of the coradical filtration of A such that the image of jr,#l in gr(A) is the canonical image of at in A \/A q. Then A is generated as an algebra by T and the elements a,. We consider another such Yetter-Drinfel’d module W over k[F] of Cartan type {atJ) of size 0 x 0 with elements h} € T, char­ acters X], and corresponding basis elements w}. Let B be a lifting of $8(W )#k[r] with corresponding elements bj. Lemma 1.2 Assume for all i j, g, g} or %, ^ %]■ (In particular this holds if for all i ^ j, the order of q, does not divide 2 —atJ). Let Q>: A -+ B be a Hopf algebra isomorphism. Then 0 = 0. q*'1 = q ^ for all i. j and there are a group isomorphism : T —►T, a permutation a and non-zero scalars a, such that

Proof 4> induces a Hopf algebra isomorphism gr(A) —►gr(S). By [3, Proposition 6.3] the assertion holds on the graded level since by our assumption the YetterDrinfel'd module V is a direct sum of pairwise non-isomorphic one-dimensional submodules lac,. Thus for all i, d>(a,) —a ,b0^ is an element in the group algebra which must be zero since the group is acting on it via the non-trivial character tla(i)tD Conversely, data .|r = (a,) = a ,ba^ if all the relations between the a, are respected. This gives a condition on the scalars a, which has to be worked out in each case. In the case of quantum linear spaces, that is if al} = 0 for all i ^ j, this is very easy. By [2, Section 5], a lifting A of a quantum linear space is given by scalars p, € {0.1} for a ll;, X,, 6 k for all i < j such that p, = 0 if g^’ = 1 or yff' ± 1, and X,j = 0 if g,gj = 1 or %,%j ± l. The relations we have to check are

If B is another such lifting with d a t a t h e n A is isomorphic to B if and only if there are a and a, as in Lemma 1.2 satisfying for all i and i < j,

2

The braided Hopf algebra R is a quantum linear space

Let p be an odd prime number. In order to classify pointed Hopf algebras of order p 4 we describe all liftings of quantum linear lines over a group T of order

4

Andruskiewitsch and Schneider

p 3 and of quantum lines or planes over T of order p 2. The following list contains all such Hopf algebras, and the remarks at the end of the last section show that Hopf algebras in two different cases of this list are non-isomorphic. We leave the details of the proof to the reader. 2.1 Index p Here T is a group of order p3. Since R is a quantum line, there exist g € T central and a character % e T such that %(g) has order p. Note that T is necessarily abelian: indeed, the center of a non-abelian group of order p3 is contained in its commutator, so x(g) = 1 f°r any g e T central, % € T. There are three abelian groups of order p 3 which we write as («i.......ur), 1 < r < 3. Then A can be presented by generators u„ 1 < / < r, a, and relations u,u, = UjUi for all 1 < i,j< r ; up = 1 if r = 1; up = 1. up = 1 if r = 2; «f = 1 for all i i f r = 3. ap = p (l —gp). with p either 0 or 1; hah~x = %(h)a. h € T .

(1)

If p = 1, then gp ^ 1 and %p = 1. The value of p is invariant under isomorphism. The Hopf algebra structure of A is determined by

A(h) — h'&h.

A ( a ) = a 3 l + |2 a ,

heT.

Up to isomorphism there are various subcases according to the order of g. s

I1T = ( h i ), we have three cases where g = u\ andx(«i) = £, 2; a root of unity of order p t_rl, 0 < s < 2. In each case, if q is another root of unity of the same order p *~1, then the corresponding Hopf algebras are isomorphic if and only if there is an integer i with %= q ' and / = 1 mod p3~s (hence %= q in case 5 = 0.1). If T = (i/i. ui ) , we take g = up and x (« i) = X(M2) = 1- £ a root of unity of order p 2 (we get an isomorphic Hopf algebra only if we replace ^ by i = 1 mod p)\ or g = ui and %{u\ ) = p, X(M2) = %a primitive p-th root of unity and p = 1 or an a priori given root of unity of order p 2; or g = u\ and x(»i) = X(“2) = 1. %a p-th root of unity. In the last two cases the Hopf algebras are pairwise non­ isomorphic. If T = (111. 111. 113), we take g = u\ and x(«i) = £ a primitive p-th root of unity, X(»2) = 1-X(M3) = 1- These Hopf algebras are pairwise non-isomorphic. 2.2 Index p2, R a quantum line There exist g e T and a character x € T such that %(g) has order p 2. Necessarily, T is isomorphic to Z /(p 2). Then there exists a primitive p 2-th root of unity £ e k such that A can be presented by generators g, a and relations 8P2 = Iap2 = 0;

gag~l - Z,a.

(2)

Lifting of Nichols Algebras and Pointed HopfAlgebras

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The Hopf algebra structure of A is determined by

M g )= S $ g -

A(a) = a& 1 + g S a .

That is, A is isomorphic to a Taft algebra of order p 4. For different roots of u n ity the Hopf algebras are non-isomorphic.

2.3 Index p2, R a quantum plane and T ~ Z /(p ) x Z /(p ) Then there exist g u g 2 € r and characters X1X2 e T such that Xi(gi) and %2 (gi) have order p, X\(g 2)%2(g\) = 1. A can be presented by generators «i, u2, a\, a2 and relations i/f = 1. / = 1.2. u \u 2 = u2u \\ « f = 0 . i = 1.2; ha,h~l = h e r . i = 1,2; w aia 2- X2 (gi)fl2 and X2- It is easy to see that either ord(g,) = p and ord(xO = p 2, i = 1.2 (first subcase) or ord(g,) = p 2 and ord(x,) = p, / = 1.2 (second, third and fourth subcases according to the different choices of p\ and P 2>First subcase. Let £ be a primitive p 2-th root of unity and integers b.c\.c 2 inte­ gers prime to p such that p\(bc\ -I-c2). A is generated by h, x\, x 2 with relations hP2 = 1 hx\ = fycih. hx2 = %bx 2h: xf = 0. Xj = 0; X2X1 —^pbClX]X2 = X ^1 —hp(ci-c 2)^ . where X is 0 or 1 (and it could be 1 only if b = —1 mod p 2 and c \+ c 2 ^ . 0 mod p ).

(4)

Andruskiewitsch and Schneider

6

carries a Hopf algebra structure given by

The only ways to get isomorphic Hopf algebras (except congruence mod p for ci.c ‘2 and mod p 2 for b) are replacing c \.c i .b .% by 5c 1 . 5 c2 .fe.Tj where 5 is an integer prime to p and rj* = or by 5 c2 . 5 c 1 .rf.rj where 5 again is prime to p, d is an integer with ferf = 1 mod p 2 and rjf = %b.

Second subcase. Let q be a primitive p -th root of unity and a. d \. d2 integers prime to p such that p|(arfi -f rf2 ). A is generated by h ,x \,x t with relations

(5) where X, is 0 or 1 (and it could be 1 only if rfi + rf2 = 0 mod p and 1 + a ^ 0 mod p). carries a Hopf algebra structure given by

The only way to get isomorphic Hopf algebras (except congruence mod p for d\.di and mod p 2 for a) is to replace d 1.d 2.aby dja. d\a.b where ab= 1 mod p 2.

Third subcase. Let q be a primitive p-th root of unity and a.d\.d 2 integers prime to p such that p\(ad\ + rf2 ). A is generated by h, x \, X2 with relations

( 6)

where X is 0 or 1 (and it could be 1 only if rfi +rf 2 = 0 mod p and 1 + a ^ 0 mod p). carries a Hopf algebra structure given by

Here the only way to get isomorphic Hopf algebras is to replace d\.d 2 by mod p congruent elements and a by a mod p 2 congruent element. In the first three subcases, X is invariant under isomorphism.

Fourth subcase. It is only here that we get an infinite family of non-isomorphic Hopf algebras. Let q be a primitive p-th root of unity and a. d \. rf2 integers prime

Lifting of Nichols Algebras and Pointed Hopf Algebras

7

to p such that p\( 1 (which is not assumed to be prime in this section) and a primitive p-th root of unity q in k such that (8 )

Thus the braiding b,7 = (g,) A < i.j < 2, is of type A2 . If C is a coalgebra, we denote by G(C) the set of group-like elements of C. If g, h 6 G(C), then we let Pg h{C) — {* € C | A(x) = g S x + x 3 h} be the space of all (g./i)-primitive elements. If A is a Hopf algebra and T = G(A) is abelian, then T acts on each Pg.h(A) by conjugation, and if X is a character of T, then we define Pg i,(A)x = { a £ Pg.h{A) | u a ir 1 = %(u)a. for all u € T}. Lemma 3.1 Let A be a Hopf algebra with coradical k[T] and a\M 2 G A such that

Andruskiewitsch and Schneider

8

Then

Proof. (1) follows easily from the assumptions. (2) and (3) follow by direct computation using also (8). (4) follows from the ^-binomial formula: let u\ = a, Z 1.1/2 = gt 0 a,. Then

Hence by the ^-binomial formula,

since q is a p-th root of unity.

In the same way we see 1/3H2 = qu2uy using the relation ajc = Xi {gi)ca2Finally

Hence by the ^-binomial formula,

This proves (5) since

To compute all liftings of type A 2 we first list the following properties of our elements g\.g 2-%\-%2 satisfying (8 ). Lemma 3.2 (1 ) g\g 2 7^ g, andg\g\ ± 1 < / < 2. (2) Ifp > 3. then %jX2 ± E-XiXi ¥=e. (3) IfT is cyclic of order 3 and X1X2 = £ or% 1X2 = E- tfien S\g 2 = 1 = gigl-

Lifting of Nichols Algebras and Pointed HopfAlgebras

9

Proof. (1) Assume g\g 2 = g 2- Then gj = 1, and we get the contradiction Xi(gi) 2 = 1. If g\g 2 — gu then gig 2 == 1> and we see from ( 8 ) that %2(gl) = 1, and Xi (g2) = 1. By ( 8) this implies q = I which is impossible. In the same way we see that glg 2 7^ gI(2) Assume X1X2 = £• Then X1X2 = X\ l . XT1(gi)Xi (gi) = 1 and Xi-1 (g2)lb(gi) = 1 by (8 ). Hence q = Xi(#2 ) = X2 (^i) which implies by ( 8 ) the contradiction q- = 1. In the same way we prove X1X2 ^ E(3) We consider the case X1X2 = £• Since T = T is of order 3, we get %i = X2Then Xi(^1^2) = Xi(si)X2(S2)Xi(g2) = 1 by (8). Therefore g\g2 = 1. Hence 51 = g 2 , a n d g i ^ = 1 . □ We also need Lemma 3.3 Let T be any group, g .h e T . a e k a n d x e k[T]. Then the following are equivalent: (1) A(x) — gh S r + r S 1 4 - a ( l -g)h% (1 —h). (2) x — a ( l —h) + X,(l —gh) for some k e k.

Proof. Both statements mean that x - a ( l - h) is (gh. l)-primitive in the group algebra k[T]. D Lemma 3.4 Let x.y. z be elements in an algebra, a. P scalars in k and n a natural number. Assume yx = ary + z. (1 ) Asssume zx = Pxz. Then yr" = a nxny + (X" ^ 1 a 'p "- 1 - ')^"-1 '- and ± P and a" = P". then (otry + ?)r " _1 = a ”r"v. (2) Assume yz = Pr\;. Then f x = a "xy" + ( X ^ 1a'P "_ 1_,)zy"_1. and if a ± p and a" = p". then y" - 1 (axy + z) = a "xy".

Proof. The first formula in (1) is easily proved by induction and the second for­ mula follows from the first since X"=To a'P "- 1 - ' = 0- (2) is dual to ( 1 ). □ Definition 3.5 Define

as a Hopf algebra in the category of left Yetter-Drinfel’d modules over k[T] with primitive elements x\ .X2 and action and coaction o fT given by

Let U := l/-#k[r] be the Radford biproduct. Identifying y#u = yu , y € U k[T] we have in U

.11 €

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Andruskiewitsch and Schneider

In quantum group theory our U is usually called U-°. Note that U~ is a welldefined braided Hopf algebra since by Lemma 3.1 the Sene relations are primitive in the free algebra generated by the primitive elements X\.X2- It follows easily from the diamond lemma that U+ has the PBW-basis /. j. I > 0. Then V := k*i + kx 2 C U+ is a Yetter-Drinfel’d module over k[T] of type Ai. Theorem 3.6 Let p\.p 2 G {0.1} and X G k be scalars such that (9) ( 10)

is a quotient Hopf algebra ofU of dimension ord(T)p3, ©(V) = U / (xf[.zp-x^), and Proof. 1) We first show that A is a quotient Hopf algebra of U . By Lemma 3.1 (4), x f - u, is (gf . l)-primitive, since u, is (gf. l)-primitive by construction. Let a := (q —l)pXiQ>2 )£^"ii- Note that p\piG = a since p is odd and 7^ 0. By Lemma 3.1 (5) and Lemma 3.3 (1),

= e if

and Hence

Thus (.xf —u\.zp —it-Xj —( 0 form a k-basis of A. This can be seen using the diamond lemma. As in the proof of [2, Proposition 5.2] we use the ordering h\ < ... A the unit of a (weak) algebra A in C. A : C —►C C is meant for the (weak) comultiplication and e : C —►I for the counit of a (weak) coalgebra C in C. w : A ®M -*■ M is the (weak) left

Cross Product Bialgebras

j7

action of an algebra A on a module M. and V /: N —►C 2 >N denotes the (weak) left coaction of a coalgebra C on a comodule N. Right actions are denoted by pr and right coactions by vr. Graphical calculus for (strict) braided monoidal categories will be used subse­ quently. Morphisms will be written as graphical symbols with incoming upper and outgoing lower strings, representing the domaine and the codomaine of the morphisms respectively. Tensor products are represented by horizontal concate­ nation of the graphics in the corresponding order. We present our own conven­ tions [1, 2, 4] in Figure 1. We often omit the assignment of a specific object to the ends of the particular strings.

2

Cross Product Bialgebras

In the first part of Section 2 we recall the definition of cocycle cross product alge­ bras. They have been considered also in [7] and are univeisal constructions [5]. Cycle cross product coalgebras will be defined afterwards by certain categorical dualization of the notion of cocycle cross product algebra. Cross product bialge­ bras are simultaneously cocycle cross product algebras and cycle cross product coalgebras with compatible bialgebra structure. Later on we will have to consider special cases called (strong) type-a cross product bialgebras which are universal constructions, too. Definition 2.1 Let (B \.m \.t\\) be an algebra andBz be an object in C. Suppose there are morphisms t |2 : I —►#2. 9 2 . 1 : B2 ® Bi -> B\ ® Bz, and d : Bz ® Bi > B\®Bz obeying the relation 92.1 = (mi id#,) o (idg, ® 6) o (■A and P : B2 —►A such that 1.

Ct is algebra morphism.

2. P o t\2 = T| A such that the iden­ tities y o (ids, ®TI2 ) = a and y o (rii ® \du2) = P hold. Rem ark 2.3 Crossed product algebras A#0H [10. 19] are special examples of cocycle cross product algebras through 92.1 := (pi ® id//) o (A// ® id^) and d := (a®m//)o(id//(g>4'//.//®id//)o(A//® A//) where A is an algebra, p i A -* A is a left H -measure on A, and a : / / ® / / - » A i s a (convolution invertible) cocycle. Cycle cross product coalgebras are somehow dual constructions to cocycle cross product algebras. The corresponding results are obtained by this kind of dualization. Definition 2.4 Let (C2 . A2 X 2 ) be a coalgebra and C\ be an object in C. Suppose there exist morphismsz\ : B\ —>I, cpj .2 : C\ 0 C2 —>C2 ® C\, p : C\ ®C 2 -* C\ ® C\ such that the relation (pi.2 = (£1 cpi.2 ) o (p ® idc2) o (idc, ® A2 ) holds. Then C = Cj ® C2 is called cycle cross product coalgebra if it is a coalgebra through

We denote C by C\ ^ MC2 . Cycle cross product coalgebras are universal constructions as well. Proposition 2.5 Let C\ MC2 be a cycle cross product coalgebra and C be a coalgebra. Suppose that there exist morphisms a and h such that

1. a : C —>C\ ande[oa — tc. 2. b : C -» C2 is coalgebra morphism. 3. £)0 Ac = (b®a)oAc. 4. (3o (a ® b) o Ac = {a ® a) o Ac. Then there exists a unique coalgebra morphism c :C identities a = (idc, ® £2 ) 0 c and b = (£1 ® idcO o c.

C\ (p,p,txiC2 obeying the

Rem ark 2.6 In [5] cocycle cross product algebras and cycle cross product coal­ gebras are called cross product algebras and cross product coalgebras respective­ ly. Rem ark 2.7 (7C-Symmetry) Definition 2.1 is not dual to Definition 2.4. How­ ever, both definitions differ from each other by a combination of duality (fol­ lowed by the usual exchanges of multiplication and comultiplication, unit and counit, cocycle and cycle, etc.) and application of the opposite tensor prod­ uct (followed by exchange of indices ” 1 2 ", and later also by exchange of ”left/right (coaction) «-* right/lefl (action)’’). This kind of symmetry between

Cross Product Bialgebras

19

cocycle cross product algebra and cycle cross product coalgebra will be called henceforth 7i-symmetry. Graphically 7t-symmetry is the rotation of the graphic (in the graphical calculus) by the angle n in the drawing plane followed by the corresponding exchanges of indices and morphism types. We will often apply this sort of transformation to obtain straightforwardly rc-symmetric results. Next we define cross product bialgebras. They are invariant under Tt-symmetry. Definition 2.8 A bialgebra B is called cross product bialgebra if its underlying algebra is a cocycle cross product algebra B\ rx]^ ( B2, and its coalgebra is a

cycle cross product coalgebra B\ ^ c x i B2- The cross product bialgebra B will be denoted by By cp/^tx^}Bn- A cross product bialgebra is called normalized if El or^ = idj (and then equivalently £2 ot|2 = idj). Cross product bialgebras are universal in the following sense. Theorem 2.9 Let B be a bialgebra in C. Then the subsequent equivalent condi­ tions hold.

1. B is bialgebra isomorphic to a normalized cross product bialgebra

2. There are idempotents f l i . II 2 6 End(B) such that

3. There are objects By and B2 and morphisms By A B By and B n ^ B ^ Z?2 where ii is algebra morphism, P2 is coalgebra morphism and p 7o i; = i&Bjfor j £ {1,2} such that ms o(iy® 12) :By®B 2 ~> B and (p i ®P 2 )°A b : B —►By ® B2 are mutually inverse isomorphisms. In Definition 2.10 below we will specialize to type-a cross product bialgebras for which (co-)modular co-cyclic structures appear naturally. Type-a cross product bialgebras are again universal. They are objects yet too general to yield an equiv­ alent (co-)modular co-cyclic characterization of cross product bialgebras. Further below we therefore introduce in Definition 2.13 strong type-a cross product bial­ gebras which are the basic objects for our discussions in Section 3. In Definition 2.8 of cross product bialgebras there emerge morphisms mj : By —y By, T|i I I —y B y . £j I By —¥ I, A 2 ' Bn —V Bn & Bn • ^2 • I —^ Bn, and

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Bespalov and Drabant

E2 -B 2 -* I- In the next Definition 2.10 additional morphisms A i: B\ -+ B \ 2 >B\, m 2 : B 2 ®B i — > B2, ju/: B2 ®B\ B\, i i r \ Bi® B\ —>• 5 2 . V / : 5 j —>• # 2 ® 5 i , vr : 5 2 —> 5 2 ® 5 i, ct : 5 2 5 |, and p : 5 2 —>5] ® 5j are required. In the graphical calculus we present all m, A, T|, e, /j , and v by the graphics given in Figure 1 respectively. The cocycle and cycle morphisms c and p will be assigned the graphics a = Y : 5 2 ® 5 2 -> B\ and p i Bi —y B\ ® fi|. Definition 2*10 A nonnalized cross product bialgebra B\ s called strong

(3) (4)

(5)

are fullfilled. According to Theorem 2.12 a bialgebra B which is isomorphic to a type-a cross product bialgebra has certain injections and projections ij. \i, Pi, and pa which are determined by the structure of the given type-a cross product bialgebra. For strong type-a cross product bialgebras the structure morphisms additionally ful­ fill the identities (3) - (5) which give rise to the following identities for the idempotents II j := i; op7, j € {1.2}.

and the corresponding Jt-symmetric counterparts. ( 6)

In (6) we used O / := (me ids) o ( / & H'b.b ) ° (Ag ® ids) for / : B -> B, and 8(,% Jjc) of Theorem 2.12.

3

Hopf Data

In the previous section we studied cross product bialgebras from a universal point of view. Now we present a (co-)-modular co-cyclic construction method for strong type-a cross product bialgebras in terms of so-called strong Hopf da­ ta. A Hopf datum consists of two objects with certain interrelated (co-)modular co-cyclic identities. We will show that the more special strong Hopf data allow

Cross Product Bialgebras

23

reconstruction of strong type-a cross product bialgebras. In particular all known cross product bialgebras [20. 15, 11, 18, 16, 4] are recovered by the Hopf data construction scheme. Hopf data consist of two objects B\ and B2, and morphisms

The precise definition of Hopf data will be given in the sequel. We use the graphi­ cal presentation of Figure 1 for these morphisms. Similarly as in the previous sec­ tion we represent a and p by a = y : B2 ® B2 —>B] and p = *.B2 — ->■B\ ® B\. Like in (1) we define certain morphisms (pi.2 . d. and p by

y

or graphically

(7) We will sometimes use the graphical abbreviations

Definition 3.1 The tuple

is called Hopf datum if 1. (B \ . m i, r| 1) is an algebra and 81 : B\ —» I is an algebra morphism. 2.

(# 2 -^ 2 : 8 2 ) is a coalgebra and T|2 : I —^ Bo is a coalgebra morphism.

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Bespalov and Drabant

3. (B\ . V / ) is left Bi-comodale. 4. (Z?2 ,)Ur)

right B i -module.

5. The identities

are satisfied. 6.

The subsequent compatibility relations hold.

Weak associativity for m2 ,

Weak coassociativity for Aj,

Weak associativity for pi,

Weak coassociativity for vr>

Module-algebra compatibility,

Comodule-coalgebra compatibility\

Cross Product Bialgebras

25

Cocycle and cycle compatibilities,

Algebra-coalgebra compatibility,

Module-coalgebra compatibility,

Comodule-algebra compatibility,

Module-comodule compatibility',

Bespalov and Drabant

26

Cycle-cocycle compatibility,

Projection compatibility. The next proposition shows that every type-a cross product bialgebra induces a Hopf datum in a canonical way. Proposition 3.2 Let B\ cp,p2cx^2^ 2 be a type-a cross product bialgebra with cor­ responding structure morphisms mi, r|i, Aj ,£j, m2, 1)2, A2, £2* Ph Pn V/. vr, p, andc. Then ((B}.m\.y\\.A\.e\).(B 2 ,m 2-T]2 -&2 -£2)*PhPf-VhVr‘p-G) is a Hopf datum. Conversely, it is not true in general that a Hopf datum canonically yields a type-a cross product bialgebra. However, in Theorem 3.4 we show that so-called strong Hopf data yield strong type-a cross product bialgebras. Definition 3.3 A Hopf datum t) is called strong if in addition the following iden­ tities hold. (8)

(9)

( 10)

Theorem 3.4 Lett) — ((B i.m i.r|i.A i.e i).(B2 .m 2 .ri 2 . A2 .£2 );/i/.jUr.V/.vr.p .a ) be a strong Hopf datum. Then B = B\ &B2 with cpi.2, 92 1, 6, and p defined ac­ cording to (7) is a strong type-a cross product bialgebra B\ ^ i B2 which we

denote by B\ Cxi ZK Strong Hopf data and strong type-a cross product bialgebras are in one-to-one correspondence.

Cross Product Bialgebras

27

The following Theorem 3.5 manifests our theory of cross product bialgebras in such a way that it provides equivalent descriptions of strong type-a cross product bialgebras either universally or in (co-)modular co-cyclic terms. We use notations and conventions of Theorem 2.12. Theorem 3.5 Let A be a bialgebra in C. Then the following statements are equivalent.

1. There is a strong Hopf datum f) such that the corresponding strong type-a cross product bialgebra B\ co B2 is bialgebra isomorphic to A. 2. There are idempotents I l i .r b € End(A) such that the conditions of Theo­ rem 2 . 12 .0 .2 and the "strongprojection" relations (6) hold. 3. There are objects B\ and Bj and morphisms B\ B B\ and B2 B B2 such that the conditions of Theorem 2.12.0.3 and the "strong projec­ tion" relations (6) hold with 11/ := i, op,. We will analyse subsequently various special cases of strong type-a cross product bialgebras. These examples include all cross product bialgebras in [20, 15. 11, 18, 16, 4]. Furthermore, new cross product bialgebra constructions emerge. In [5] a thorough analysis of all special types of strong type-a (or cocycle) cross product bialgebras can be found. Example 3.6 A type-a cross product bialgebra B\