Preface

Chapter 1． Definitions and Examples
1．1 Algebras and coalgebras
1．2 Duals of algebras and coalgebras
1．3 Bialgebras
1．4 Convolution and summation notation
1．5 Antipodes and Hopf algebras
1．6 Modules and comodules
1．7 Invariants and coinvariants
1．8 Tensor products of H-modules and H-comodules
1．9 Hopf modules

Chapter 2． Integrals and Semisimplicity
2．1 Integrals
2．2 Maschke's Theorem
2．3 Commutative semisimple Hopf algebras and restricted enveloping algebras
2．4 Cosemisimplicity and integrals on H
2．5 Kaplansky's conjecture and the order of the antipode

Chapter 3． Freeness over Subalgebras
3．1 The Nichols-Zoeller Theorem
3．2 Applications： Hopf algebras of prime dimension and semisimple sub Hopfalgebras
3．3 A normal basis for H over K
3．4 The adjoint action， normal subHopfalgebras， and quotients
3．5 Freeness and faithful flatness in the infinite-dimensional case

Chapter 4． Actions of Finite-Dimensional Hopf Algebras and Smash Products
4．1 Module algebras， comodule algebras， and smash products
4．2 Integrality and affine invariants： the commutative case
4．3 Trace functions and affine invariants： the non-commutative case
4．4 Ideals in A#H and A as an All-module
4．5 A Morita context relating A#H and AH

Chapter 5． Coradicals and Filtrations
5．1 Simple subcoalgebras and the coradical
5．2 The coradical filtration
5．3 lnjective coalgebra maps
5．4 The coradical filtration of pointed coalgebras
5．5 Examples： U（g） and Uq（g）
5．6 The structure of pointed cocommutative Hopf algebras
5．7 Semisimple cocommutative connected Hopf algebras

Chapter 6． Inner Actions
6．1 Definitions and examples
6．2 A Skolem-Noether theorem for Hopf algebras
6．3 Maximal inner subcoalgebras
6．4 X-inner actions and extending to quotients

Chapter 7． Crossed products
7．1 Definitions and examples
7．2 Cleft extensions and existence of crossed products
7．3 Inner actions and equivalence of crossed products
7．4 Generalized Maschke theorems and semiprime crossed products
7．5 Twisted H-comodule algebras

Chapter 8． Galois Extensions
8．1 Definition and examples
8．2 The normal basis property and cleft extensions
8．3 Galois extensions for finite-dimensional H
8．4 Normal bases and Hopf algebra quotients
8．5 Relative Hopf modules

Chapter 9． Duality
9．1 H°
9．2 SubHopfalgebras of H° and density
9．3 Classical duality
9．4 Duality for actions
9．5 Duality for graded algebras

Chapter 10． New Constructions from Quantum Groups
10．1 Quasitriangular and ahnost cocommutative Hopf algebras
10．2 Coquasitriangular and almost commutative Hopf algebras
10．3 The Drinfeld double
10．4 Braided monoidal categories
10．SHopf algebras in categories; graded Hopf algebras
10．6 Biproducts and Yetter-Drinfeld modules

Appendix． Some quantum groups
References
Index