**Contemporary Mathematics**

Volume: 585;
2013;
334 pp;
Softcover

MSC: Primary 16; 17; 18; 19;

**Print ISBN: 978-0-8218-7564-3
Product Code: CONM/585**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

**Electronic ISBN: 978-0-8218-9498-9
Product Code: CONM/585.E**

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# Hopf Algebras and Tensor Categories

Share this page *Edited by *
*Nicolás Andruskiewitsch; Juan Cuadra; Blas Torrecillas*

This volume contains the proceedings of the Conference on Hopf
Algebras and Tensor Categories, held July 4–8, 2011, at the
University of Almería, Almería, Spain.

The articles in this volume cover a wide variety of topics related
to the theory of Hopf algebras and its connections to other areas of
mathematics. In particular, this volume contains a survey covering
aspects of the classification of fusion categories using Morita
equivalence methods, a long, self-contained, and comprehensive
introduction to Hopf algebras in the category of species, and a
summary of the status to date of the classification of Hopf algebras
of dimensions up to 100. Among other articles in this volume are a
study of normalized class sum and generalized character table for
semisimple Hopf algebras, a contribution to the classification program
of finite dimensional pointed Hopf algebras, an analysis of conjugacy
classes related to a conjecture of De Concini, Kac, and Procesi on
representations of quantum groups at roots of unity, a categorical
approach to the Drinfeld double of a braided Hopf algebra via Hopf
monads, an overview of Hom-Hopf algebras, and several discussions on
the crossed product construction in different settings.

#### Readership

Graduate students and research mathematicians interested in Hopf algebra theory, tensor categories, quantum groups, representation theory, and combinatorics and conformal field theory.

# Table of Contents

## Hopf Algebras and Tensor Categories

- Preface vii8 free
- Conference program ix10 free
- Unified products and split extensions of Hopf algebras 118 free
- Hopf monoids in the category of species 1734
- Introduction 1835
- 1. Preliminaries on compositions and partitions 2239
- 2. Species and Hopf monoids 2845
- 3. The Hadamard product 3552
- 4. Set species and set-theoretic Hopf monoids 3855
- 5. Connected Hopf monoids 4259
- 6. The free monoid 4562
- 7. The free commutative monoid 4966
- 8. The free Lie monoid 5269
- 9. Examples of Hopf monoids 5673
- 10. Higher associativity and compatibility 6784
- 11. The bimonoids of set (de)compositions 7592
- 12. Series and functional calculus 7996
- 13. The characteristic operations (Hopf powers) 97114
- 14. Classical idempotents 107124
- 15. The Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems 116133
- 16. The dimension sequence of a connected Hopf monoid 119136
- References 121138

- Classifying Hopf algebras of a given dimension 125142
- On the double crossed product of weak Hopf algebras 153170
- The doubles of a braided Hopf algebra 175192
- Induced Conjugacy Classes and Induced 𝒰_{𝜖}(𝒢)-Modules 199216
- Recovering information from character tables of Hopf algebras: normality, dimensions and quotients 213230
- Hom-quasi-bialgebras 227244
- On twisted conjugacy classes of type D in sporadic simple groups 247264
- Partial and unified crossed products are weak crossed products 261278
- The Green rings of the generalized Taft Hopf algebras 275292
- 1. Introduction 275292
- 2. Representation theory of the generalized Taft algebra 𝐻_{𝑛,𝑑} 277294
- 3. Generators and relations for the Green ring 𝑟(𝐻_{𝑛,𝑑}) 279296
- 4. Roots of the generalized Fibonacci polynomials 281298
- 5. Nilpotent elements in 𝑟(𝐻_{𝑛,𝑑}) and representation theory of 𝑅(𝐻_{𝑛,𝑑}) 283300
- 6. Comparing the Green ring and the projective class ring of 𝐻_{𝑛,𝑑} 286303
- Acknowledgments 287304
- References 287304

- Morita equivalence methods in classification of fusion categories 289306
- 1. Introduction 289306
- 2. Definitions and basic notions 290307
- 3. Module categories and categorical Morita equivalence 300317
- 4. Braided fusion categories 307324
- 5. Characterization of Morita equivalence 317334
- 6. Classification results for fusion categories of integral dimensions 319336
- References 324341