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
subHopfalgebras
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
