Table of 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
3.3 Mr and Mr 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
4.1 Filtrations of coalgebras 124
4.2 The wedge product and the coradical filtration 129
4.3 Idempotents and the coradical nitration 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
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-semisimphcity 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
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
