Hopf algebras / David E. Radford.

Format:

Book

Author/Creator:

Radford, David E.

Series:

K & E series on knots and everything ; v. 49.

K & E series on knots and everything

Language:

English

Subjects (All):

Hopf algebras.

Physical Description:

xxii, 559 pages : illustrations ; 24 cm.

Place of Publication:

Singapore ; Hackensack, NJ : World Scientific, [2012]

Summary:

The book provides a detailed account of basic coalgebra and Hopf algebra theory with emphasis on Hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. It is intended to be a graduate text as well as a research monograph.

Contents:

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 M_r and M^r 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.

Notes:

Includes bibliographical references and index.

ISBN:

9789814335997

9814335991

OCLC:

681485777