| Preface | p. vii |
| Preliminaries | p. 1 |
| Notation and terminology conventions | p. 1 |
| Rank of a tensor | p. 2 |
| Topological aspects of vector space duals | p. 7 |
| Coalgebras | p. 19 |
| Algebras and coalgebras, basic definitions | p. 19 |
| Comatrix identities, the fundamental theorem of coalgebras | p. 35 |
| The dual algebra | p. 40 |
| The wedge product | p. 52 |
| The dual coalgebra | p. 58 |
| Double duals | p. 66 |
| The cofree coalgebra on a vector space | p. 70 |
| Representations of coalgebras | p. 77 |
| Rational modules of the dual algebra | p. 77 |
| Comodules | p. 85 |
| Mr and Mr | p. 98 |
| The coradical of a coalgebra | p. 98 |
| Injective comodules | p. 103 |
| Coalgebras which are submodules of their dual algebras | p. 114 |
| Indecomposable coalgebras | p. 118 |
| The coradical filtration and related structures | p. 123 |
| Filtrations of coalgebras | p. 124 |
| The wedge product and the coradical filtration | p. 129 |
| Idempotents and the coradical nitration | p. 132 |
| Graded algebras and coalgebras | p. 136 |
| The cofree pointed irreducible coalgebra on a vector space | p. 146 |
| The radical of the dual algebra | p. 151 |
| Free pointed coalgebras associated to coalgebras | p. 153 |
| Linked simple subcoalgebras | p. 158 |
| Bialgebras | p. 165 |
| Basic definitions and results | p. 165 |
| The dual bialgebra | p. 177 |
| The free bialgebra on a coalgebra and related constructions | p. 180 |
| The universal enveloping algebra | p. 191 |
| The cofree bialgebra on an algebra | p. 193 |
| Filtrations and gradings of bialgebras | p. 197 |
| Representations of bialgebras | p. 200 |
| The convolution algebra | p. 203 |
| Definition and basic properties | p. 203 |
| Invertible elements in the convolution algebra | p. 206 |
| Hopf algebras | p. 211 |
| Definition of Hopf algebra, the antipode | p. 211 |
| Q-binomial symbols | p. 217 |
| Two families of examples | p. 220 |
| The dual Hopf algebra | p. 225 |
| The free Hopf algebra on a coalgebra | p. 227 |
| When a bialgebra is a Hopf algebra | p. 231 |
| Two-cocycles, pairings, and skew pairings of bialgebras | p. 237 |
| Twists of bialgebras | p. 243 |
| Filtrations and gradings on Hopf algebras | p. 246 |
| The cofree pointed irreducible Hopf algebra on an algebra | p. 249 |
| The shuffle algebra | p. 250 |
| Hopf modules and co-Hopf modules | p. 259 |
| Definition of Hopf module and examples | p. 259 |
| The structure of Hopf modules | p. 263 |
| Co-Hopf modules | p. 268 |
| A basic co-Hopf module and its dual | p. 270 |
| Hopf algebras as modules over Hopf subalgebras | p. 273 |
| Filtrations whose base term is a Hopf subalgebra | p. 273 |
| Relative Hopf modules | p. 276 |
| When Hopf algebras free over their Hopf subalgebras | p. 279 |
| An example of a Hopf algebra which is not free over some Hopf subalgebra | p. 282 |
| Integrals | p. 289 |
| Definition of integrals for a bialgebra and its dual algebra | p. 289 |
| Existence and uniqueness of integrals for a Hopf algebra | p. 293 |
| Integrals and semisimplicity | p. 298 |
| Integrals and the trace function | p. 302 |
| Integrals and the antipode | p. 305 |
| Generalized integrals and grouplike elements | p. 313 |
| Integrals, the center, and cocommutative elements of the dual | p. 318 |
| Integrals and co-semisimphcity | p. 324 |
| Existence and uniqueness results for integrals of the dual algebra of a Hopf algebra | p. 329 |
| Actions by bialgebras and Hopf algebras | p. 343 |
| Monoidal categories | p. 345 |
| Module actions and module algebras, coalgebras | p. 349 |
| Comodule actions and comodule algebras, coalgebras | p. 355 |
| Duality between the smash product and smash coproduct | p. 360 |
| Prebraiding, braiding structures on a monoidal category | p. 363 |
| Yetter-Drinfel'd modules and biproducts | p. 367 |
| Abstract characterization of biproducts | p. 380 |
| Quasitriangular bialgebras and Hopf algebras | p. 387 |
| The quantum Yang-Baxter and braid equations, Yang-Baxter algebras | p. 387 |
| Almost cocommutative Hopf algebras, quasitriangular bialgebras and Hopf algebras | p. 391 |
| Grouplike and ribbon elements | p. 400 |
| Factorizable Hopf algebras | p. 405 |
| The DrinfelÆd double of a finite-dimensional Hopf algebra | p. 413 |
| The double and its category of representations | p. 413 |
| Basic properties of the double | p. 420 |
| Characterizations of the double as a quasitriangular Hopf algebra | p. 423 |
| The dual of the double | p. 427 |
| The double of a quasitriangular Hopf algebra | p. 435 |
| The double of a factorizable Hopf algebra | p. 439 |
| Quasi-ribbon and ribbon elements of the double | p. 440 |
| Generalized doubles and their duals | p. 443 |
| Coquasitriangular bialgebras and Hopf algebras | p. 447 |
| Coquasitriangular and Yang-Baxter coalgebras | p. 447 |
| Coquasitriangular bialgebras and Hopf algebras | p. 452 |
| The square of the antipode of a coquasitriangular Hopf algebra | p. 456 |
| The free coquasitriangular bialgebra on a coquasitriangular coalgebra | p. 459 |
| Pointed Hopf algebras | p. 467 |
| Crossed products | p. 468 |
| Pointed Hopf algebras as crossed products | p. 472 |
| Cocommutative pointed Hopf algebras; the characteristic 0 case | p. 478 |
| Minimal-pointed Hopf algebras | p. 479 |
| Pointed Hopf algebras, biproducts, and Nichols algebras | p. 485 |
| Quantized enveloping algebras and their generalizations | p. 498 |
| Ore extensions and pointed Hopf algebras | p. 509 |
| Finite-dimensional Hopf algebras in characteristic 0 | p. 515 |
| Characterizations of semisimple Hopf algebras | p. 516 |
| Isomorphism types of Hopf algebras of the same dimension | p. 520 |
| Some very basic classification results | p. 525 |
| Bibliography | p. 537 |
| Index | p. 551 |
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