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 1 0 1.7 Invariant s an d coinvariant s 1 3 1.8 Tenso r product s o f //-module s an d i/-comodule s 1 4 1.9 Hop f module s 1 4 Chapter 2 . Integral s an d Semisimplicit y 1 7 2.1 Integral s 1 7 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 1 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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