Hopf algebras

1. Preliminaries. 1.1. Notation and terminology conventions. 1.2. Rank of a tensor. 1.3. Topological aspects of vector space duals -- 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. M[symbol] and M[symbol]. 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 filtration 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

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