Hopf Algebras

The following sections are included:

• Preface

• Contents

The following sections are included:

• Notation and terminology conventions

• Rank of a tensor

• Topological aspects of vector space duals

The following sections are included:

• Algebras and coalgebras, basic definitions

• Comatrix identities, the fundamental theorem of coalgebras

• The dual algebra

• The wedge product

• The dual coalgebra

• Double duals

• The cofree coalgebra on a vector space

The following sections are included:

• Rational modules of the dual algebra

• Comodules

• M_r and M^r

• The coradical of a coalgebra

• Injective comodules

• Coalgebras which are submodules of their dual algebras

• Indecomposable coalgebras

The following sections are included:

• Filtrations of coalgebras

• The wedge product and the coradical filtration

• Idempotents and the coradical filtration

• Graded algebras and coalgebras

• The cofree pointed irreducible coalgebra on a vector space

• The radical of the dual algebra

• Free pointed coalgebras associated to coalgebras

• Linked simple subcoalgebras

The following sections are included:

• Basic definitions and results

• The dual bialgebra

• The free bialgebra on a coalgebra and related constructions

• The universal enveloping algebra

• The cofree bialgebra on an algebra

• Filtrations and gradings of bialgebras

• Representations of bialgebras

The following sections are included:

• Definition and basic properties

• Invertible elements in the convolution algebra

The following sections are included:

• Definition of Hopf algebra, the antipode

• Q-binomial symbols

• Two families of examples

• The dual Hopf algebra

• The free Hopf algebra on a coalgebra

• When a bialgebra is a Hopf algebra

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

• Twists of bialgebras

• Filtrations and gradings on Hopf algebras

• The cofree pointed irreducible Hopf algebra on an algebra

• The shuffle algebra

The following sections are included:

• Definition of Hopf module and examples

• The structure of Hopf modules

• Co-Hopf modules

• A basic co-Hopf module and its dual

The following sections are included:

• Filtrations whose base term is a Hopf subalgebra

• Relative Hopf modules

• When Hopf algebras free over their Hopf subalgebras

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

The following sections are included:

• Definition of integrals for a bialgebra and its dual algebra

• Existence and uniqueness of integrals for a Hopf algebra

• Integrals and semisimplicity

• Integrals and the trace function

• Integrals and the antipode

• Generalized integrals and grouplike elements

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

• Integrals and co-semisimplicity

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

The following sections are included:

• Monoidal categories

• Module actions and module algebras, coalgebras

• Comodule actions and comodule algebras, coalgebras

• Duality between the smash product and smash coproduct

• Prebraiding, braiding structures on a monoidal category

• Yetter—Drinfel’d modules and biproducts

• Abstract characterization of biproducts

The following sections are included:

• The quantum Yang—Baxter and braid equations, Yang-Baxter algebras

• Almost cocommutative Hopf algebras, quasitriangular bialgebras and Hopf algebras

• Grouplike and ribbon elements

• Factorizable Hopf algebras

The following sections are included:

• The double and its category of representations

• Basic properties of the double

• Characterizations of the double as a quasitriangular Hopf algebra

• The dual of the double

• The double of a quasitriangular Hopf algebra

• The double of a factorizable Hopf algebra

• Quasi-ribbon and ribbon elements of the double

• Generalized doubles and their duals

The following sections are included:

• Coquasitriangular and Yang—Baxter coalgebras

• Coquasitriangular bialgebras and Hopf algebras

• The square of the antipode of a coquasitriangular Hopf algebra

• The free coquasitriangular bialgebra on a coquasitriangular coalgebra

The following sections are included:

• Crossed products

• Pointed Hopf algebras as crossed products

• Cocommutative pointed Hopf algebras; the characteristic 0 case

• Minimal-pointed Hopf algebras

• Pointed Hopf algebras, biproducts, and Nichols algebras

• Quantized enveloping algebras and their generalizations

• Ore extensions and pointed Hopf algebras

The following sections are included:

• Characterizations of semisimple Hopf algebras

• Isomorphism types of Hopf algebras of the same dimension

• Some very basic classification results

The following sections are included:

• Bibliography

• Index