Contents
Preface xii i
Chapter 1. Definition s an d Example s 1
1.1 Algebra s an d coalgebra s 1
1.2 Dual s o f algebra s an d coalgebra s 2
1.3 Bialgebra s 3
1.4 Convolutio n an d summatio n notatio n 6
1.5 Antipode s an d Hop f algebra s 7
1.6 Module s an d comodule s 10
1.7 Invariant s an d coinvariant s 13
1.8 Tenso r product s o f //-module s an d i/-comodule s 14
1.9 Hop f module s 14
Chapter 2 . Integral s an d Semisimplicit y 17
2.1 Integral s 17
2.2 Maschke' s Theore m 2 0
2.3 Commutativ e semisimpl e Hop f algebra s an d restricte d
enveloping algebra s 2 2
2.4 Cosemisimplicit y an d integral s o n H 2 5
2.5 Kaplansky' s conjectur e an d th e orde r o f th e antipod e 2 7
Chapter 3 . Freenes s ove r Subalgebra s 2 8
3.1 Th e Nichols-Zoelle r Theore m 2 8
3.2 Applications : Hop f algebra s o f prim e dimensio n an d semisimpl e
subHopf algebras 3
3.3 A norma l basi s fo r H ove r K 3 2
3.4 Th e adjoin t action , norma l subHop f algebras, an d quotient s 3 3
3.5 Freenes s an d faithfu l flatnes s i n th e infinite-dimensiona l cas e 3 7
Chapter 4 . Action s o f Finite-Dimensiona l Hop f Algebra s an d
Smash Product s 4 0
4.1 Modul e algebras , comodul e algebras , an d smas h product s 4 0
4.2 Integralit y an d afhn e invariants : th e commutativ e cas e 4 3
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