Hopf algebras / David E. Radford.

Format:

Book

Author/Creator:

Radford, David E.

Series:

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

Series on knots and everything ; v. 49

Language:

English

Subjects (All):

Hopf algebras.

Physical Description:

1 online resource

Edition:

1st ed.

Place of Publication:

Hackensack, N.J. : 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:

Intro

Contents

Preface

1. Preliminaries

1.1 Notation and terminology conventions

1.2 Rank of a tensor

1.3 Topological aspects of vector space duals

Chapter notes

2. Coalgebras

2.1 Algebras and coalgebras, basic definitions

2.2 Comatrix identities, the fundamental theorem of coalgebras

2.3 The dual algebra

2.4 The wedge product

2.5 The dual coalgebra

2.6 Double duals

2.7 The cofree coalgebra on a vector space

3. Representations of coalgebras

3.1 Rational modules of the dual algebra

3.2 Comodules

3.3 Mr and Mr

3.4 The coradical of a coalgebra

3.5 Injective comodules

3.6 Coalgebras which are submodules of their dual algebras

3.7 Indecomposable coalgebras

4. The coradical .ltration and related structures

4.1 Filtrations of coalgebras

4.2 The wedge product and the coradical filtration

4.3 Idempotents and the coradical filtration

4.4 Graded algebras and coalgebras

4.5 The cofree pointed irreducible coalgebra on a vector space

4.6 The radical of the dual algebra

4.7 Free pointed coalgebras associated to coalgebras

4.8 Linked simple subcoalgebras

5. Bialgebras

5.1 Basic definitions and results

5.2 The dual bialgebra

5.3 The free bialgebra on a coalgebra and related constructions

5.4 The universal enveloping algebra

5.5 The cofree bialgebra on an algebra

5.6 Filtrations and gradings of bialgebras

5.7 Representations of bialgebras

6. The convolution algebra

6.1 Definition and basic properties

6.2 Invertible elements in the convolution algebra

7. Hopf algebras

7.1 Definition of Hopf algebra, the antipode

7.2 Q-binomial symbols

7.3 Two families of examples

7.4 The dual Hopf algebra

7.5 The free Hopf algebra on a coalgebra.

7.6 When a bialgebra is a Hopf algebra

7.7 Two-cocycles, pairings, and skew pairings of bialgebras

7.8 Twists of bialgebras

7.9 Filtrations and gradings on Hopf algebras

7.10 The cofree pointed irreducible Hopf algebra on an algebra

7.11 The shuffle algebra

8. Hopf modules and co-Hopf modules

8.1 Definition of Hopf module and examples

8.2 The structure of Hopf modules

8.3 Co-Hopf modules

8.4 A basic co-Hopf module and its dual

9. Hopf algebras as modules over Hopf subalgebras

9.1 Filtrations whose base term is a Hopf subalgebra

9.2 Relative Hopf modules

9.3 When Hopf algebras free over their Hopf subalgebras

9.4 An example of a Hopf algebra which is not free over some Hopf subalgebra

10. Integrals

10.1 Definition of integrals for a bialgebra and its dual algebra

10.2 Existence and uniqueness of integrals for a Hopf algebra

10.3 Integrals and semisimplicity

10.4 Integrals and the trace function

10.5 Integrals and the antipode

10.6 Generalized integrals and grouplike elements

10.7 Integrals, the center, and cocommutative elements of the dual

10.8 Integrals and co-semisimplicity

10.9 Existence and uniqueness results for integrals of the dual algebra of a Hopf algebra

11. Actions by bialgebras and Hopf algebras

11.1 Monoidal categories

11.2 Module actions and module algebras, coalgebras

11.3 Comodule actions and comodule algebras, coalgebras

11.4 Duality between the smash product and smash coproduct

11.5 Prebraiding, braiding structures on a monoidal category

11.6 Yetter-Drinfel'd modules and biproducts

11.7 Abstract characterization of biproducts

12. Quasitriangular bialgebras and Hopf algebras

12.1 The quantum Yang-Baxter and braid equations, Yang- Baxter algebras.

12.2 Almost cocommutative Hopf algebras, quasitriangular bialgebras and Hopf algebras

12.3 Grouplike and ribbon elements

12.4 Factorizable Hopf algebras

13. The Drinfel'd double of a finite-dimensional Hopf algebra

13.1 The double and its category of representations

13.2 Basic properties of the double

13.3 Characterizations of the double as a quasitriangular Hopf algebra

13.4 The dual of the double

13.5 The double of a quasitriangular Hopf algebra

13.6 The double of a factorizable Hopf algebra

13.7 Quasi-ribbon and ribbon elements of the double

13.8 Generalized doubles and their duals

14. Coquasitriangular bialgebras and Hopf algebras

14.1 Coquasitriangular and Yang-Baxter coalgebras

14.2 Coquasitriangular bialgebras and Hopf algebras

14.3 The square of the antipode of a coquasitriangular Hopf algebra

14.4 The free coquasitriangular bialgebra on a coquasitriangular coalgebra

15. Pointed Hopf algebras

15.1 Crossed products

15.2 Pointed Hopf algebras as crossed products

15.3 Cocommutative pointed Hopf algebras

the characteristic 0 case

15.4 Minimal-pointed Hopf algebras

15.5 Pointed Hopf algebras, biproducts, and Nichols algebras

15.6 Quantized enveloping algebras and their generalizations

15.7 Ore extensions and pointed Hopf algebras

16. Finite-dimensional Hopf algebras in characteristic 0

16.1 Characterizations of semisimple Hopf algebras

16.2 Isomorphism types of Hopf algebras of the same dimension

16.3 Some very basic classification results

Bibliography

Index.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786613646330

9781280669408

1280669403

9789814338660

9814338664

OCLC:

794328359