Chapter 1．Definitions and Examples 1
1.1 Algebras and coalgebras 1
1.2 Duals of algebras and coalgebras 2
1.3 Bialgebras 3
1.4 Convolution and summation notation 6
1.5 Antipodes and Hopf algebras 7
1.6 Modules and comodules 10
1.7 Invariants and coinvariants 13
1.8 Tensor products of H-modules and H-comodules 14
1.9 Hopf modules 14
Chapter 2．Integrals and Semisimplicity 17
2.1 Integrals 17
2.2 Maschke’s Theorem 20
2.3 Commutative semisimple Hopf algebras and restricted enveloping algebras 22
2.4 Cosemisimplicity and integrals on H 25
2.5 Kaplansky’s conjecture and the order of the antipode 27
Chapter 3．Freeness over Subalgebras 28
3.1 The Nichols-Zoeller Theorem 28
3.2 Applications:Hopf algebras of prime dimension and semisimple subHopfalgebras 31
3.3 A normal basis for H over K 32
3.4 The adjoint action,normal subHopfalgebras,and quotients 33
3.5 Freeness and faithful flatness in the infinite-dimensional case 37
Chapter 4．Actions of Finite-Dimensional Hopf Algebras and Smash Products 40
4.1 Module algebras,comodule algebras,and smash products 40
4.2 Integrality and affine invariants:the commutative case 43
4.3 Trace functions and affine invariants:the non-commutative case 45
4.4 Ideals in A#H and A as an AH module 48
4.5 A Morita context relating A#H and AH 52
Chapter 5．Coradicals and Filtrations 56
5.1 Simple subcoalgebras and the coradical 56
5.2 The coradical filtration 60
5.3 Injecti ve coalgebra maps 65
5.4 The coradical filtration of pointed coalgebras 67
5.5 Examples:U（g）and Uq（g） 73
5.6 The structure of pointed cocommutative Hopf algebras 76
5.7 Semisimple cocommutative connected Hopf algebras 83
Chapter 6．Inner Actions 87
6.1 Definitions and examples 87
6.2 A Skolem-Noether theorem for Hopf algebras 89
6.3 Maximal inner subcoalgebras 92
6.4 X-inner actions and extending to quotients 96
Chapter 7．Crossed products 101
7.1 Definitions and examples 101
7.2 Cleft extensions and existence of crossed products 105
7.3 Inner actions and equivalence of crossed products 112
7.4 Generalized Maschke theorems and semiprime crossed products 116
7.5 Twisted H-comodule algebras 121
Chapter 8．Galois Extensions 123
8.1 Definition and examples 123
8.2 The normal basis property and cleft extensions 128
8.3 Galois extensions for finite-dimensional H 131
8.4 Normal bases and Hopf algebra quotients 139
8.5 Relative Hopf modules 144
Chapter 9．Duality 149
9.1 H° 149
9.2 SubHopfalgebras of H°and density 153
9.3 Classical duality 159
9.4 Duality for actions 161
9.5 Duality for graded algebras 172
Chapter 10．New Constructions from Quantum Groups 178
10.1 Quasitriangular and almost cocommutative Hopf algebras 178
10.2 Coquasitriangular and almost commutative Hopf algebras 184
10.3 The Drinfeld double 187
10.4 Braided monoidal categories 197
10.5 Hopf algebras in categories;graded Hopf algebras 203
10.6 Biproducts and Yetter-Drinfeld modules 207
Appendix．Some quantum groups 217
References 223
Index 235