November 5, 2011 7:30 World Scientific Book - 9in x 6in A-Hopf

Contents

Preface vii

1. Preliminaries 1 1.1 Notation and terminology conventions ...... 1 1.2 Rank of a tensor ...... 2 1.3 Topological aspects of vector space duals ...... 7

2. Coalgebras 19 2.1 Algebras and coalgebras, basic definitions ...... 19 2.2 Comatrix identities, the fundamental theorem of coalgebras 35 2.3 The dual algebra ...... 40 2.4 The wedge product ...... 52 2.5 The dual coalgebra ...... 58 2.6 Double duals ...... 66 2.7 The cofree coalgebra on a vector space ...... 70

3. Representations of coalgebras 77 3.1 Rational modules of the dual algebra ...... 77 3.2 Comodules ...... 85 r 3.3 Mr and M ...... 98 3.4 The coradical of a coalgebra ...... 98 3.5 Injective comodules ...... 103 3.6 Coalgebras which are submodules of their dual algebras . 114 3.7 Indecomposable coalgebras ...... 118

4. The coradical filtration and related structures 123

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4.1 Filtrations of coalgebras ...... 124 4.2 The wedge product and the coradical filtration ...... 129 4.3 Idempotents and the coradical filtration ...... 132 4.4 Graded algebras and coalgebras ...... 136 4.5 The cofree pointed irreducible coalgebra on a vector space 146 4.6 The radical of the dual algebra ...... 151 4.7 Free pointed coalgebras associated to coalgebras . . . . . 153 4.8 Linked simple subcoalgebras ...... 158

5. Bialgebras 165 5.1 Basic definitions and results ...... 165 5.2 The dual bialgebra ...... 177 5.3 The free bialgebra on a coalgebra and related constructions 180 5.4 The universal enveloping algebra ...... 191 5.5 The cofree bialgebra on an algebra ...... 193 5.6 Filtrations and gradings of bialgebras ...... 197 5.7 Representations of bialgebras ...... 200

6. The convolution algebra 203 6.1 Definition and basic properties ...... 203 6.2 Invertible elements in the convolution algebra ...... 206

7. Hopf algebras 211 7.1 Definition of Hopf algebra, the antipode ...... 211 7.2 Q-binomial symbols ...... 217 7.3 Two families of examples ...... 220 7.4 The dual Hopf algebra ...... 225 7.5 The free Hopf algebra on a coalgebra ...... 227 7.6 When a bialgebra is a Hopf algebra ...... 231 7.7 Two-cocycles, pairings, and skew pairings of bialgebras . . 237 7.8 Twists of bialgebras ...... 243 7.9 Filtrations and gradings on Hopf algebras ...... 246 7.10 The cofree pointed irreducible Hopf algebra on an algebra 249 7.11 The shuffle algebra ...... 250

8. Hopf modules and co-Hopf modules 259 8.1 Definition of Hopf module and examples ...... 259 8.2 The structure of Hopf modules ...... 263 November 5, 2011 7:30 World Scientific Book - 9in x 6in A-Hopf

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8.3 Co-Hopf modules ...... 268 8.4 A basic co-Hopf module and its dual ...... 270

9. Hopf algebras as modules over Hopf subalgebras 273 9.1 Filtrations whose base term is a Hopf subalgebra . . . . . 273 9.2 Relative Hopf modules ...... 276 9.3 When Hopf algebras free over their Hopf subalgebras . . . 279 9.4 An example of a Hopf algebra which is not free over some Hopf subalgebra ...... 282

10. Integrals 289 10.1 Definition of integrals for a bialgebra and its dual algebra 289 10.2 Existence and uniqueness of integrals for a Hopf algebra . 293 10.3 Integrals and semisimplicity ...... 298 10.4 Integrals and the trace function ...... 302 10.5 Integrals and the antipode ...... 305 10.6 Generalized integrals and grouplike elements ...... 313 10.7 Integrals, the center, and cocommutative elements of the dual ...... 318 10.8 Integrals and co-semisimplicity ...... 324 10.9 Existence and uniqueness results for integrals of the dual algebra of a Hopf algebra ...... 329

11. Actions by bialgebras and Hopf algebras 343 11.1 Monoidal categories ...... 345 11.2 Module actions and module algebras, coalgebras . . . . . 349 11.3 Comodule actions and comodule algebras, coalgebras . . . 355 11.4 Duality between the smash product and smash coproduct 360 11.5 Prebraiding, braiding structures on a monoidal category . 363 11.6 Yetter–Drinfel’d modules and biproducts ...... 367 11.7 Abstract characterization of biproducts ...... 380

12. Quasitriangular bialgebras and Hopf algebras 387 12.1 The quantum Yang–Baxter and braid equations, Yang- Baxter algebras ...... 387 12.2 Almost cocommutative Hopf algebras, quasitriangular bialgebras and Hopf algebras ...... 391 12.3 Grouplike and ribbon elements ...... 400 November 5, 2011 7:30 World Scientific Book - 9in x 6in A-Hopf

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12.4 Factorizable Hopf algebras ...... 405

13. The Drinfel’d double of a finite-dimensional Hopf algebra 413 13.1 The double and its category of representations ...... 413 13.2 Basic properties of the double ...... 420 13.3 Characterizations of the double as a quasitriangular Hopf algebra ...... 423 13.4 The dual of the double ...... 427 13.5 The double of a quasitriangular Hopf algebra ...... 435 13.6 The double of a factorizable Hopf algebra ...... 439 13.7 Quasi-ribbon and ribbon elements of the double ...... 440 13.8 Generalized doubles and their duals ...... 443

14. Coquasitriangular bialgebras and Hopf algebras 447 14.1 Coquasitriangular and Yang–Baxter coalgebras ...... 447 14.2 Coquasitriangular bialgebras and Hopf algebras ...... 452 14.3 The square of the antipode of a coquasitriangular Hopf algebra ...... 456 14.4 The free coquasitriangular bialgebra on a coquasitriangular coalgebra ...... 459

15. Pointed Hopf algebras 467 15.1 Crossed products ...... 468 15.2 Pointed Hopf algebras as crossed products ...... 472 15.3 Cocommutative pointed Hopf algebras; the characteristic 0 case ...... 478 15.4 Minimal-pointed Hopf algebras ...... 479 15.5 Pointed Hopf algebras, biproducts, and Nichols algebras . 485 15.6 Quantized enveloping algebras and their generalizations . 498 15.7 Ore extensions and pointed Hopf algebras ...... 509

16. Finite-dimensional Hopf algebras in characteristic 0 515 16.1 Characterizations of semisimple Hopf algebras ...... 516 16.2 Isomorphism types of Hopf algebras of the same dimension 520 16.3 Some very basic classification results ...... 525

Bibliography 537 Index 551