CONTENTS i x Chapter 9 . Dualit y 14 9 9.1 14 9 9.2 SubHopfalgebra s o f an d densit y 15 3 9.3 Classica l dualit y 15 9 9.4 Dualit y fo r action s 16 1 9.5 Dualit y fo r grade d algebra s 17 2 Chapter 10 . Ne w Construction s fro m Quantu m Group s 17 8 10.1 Quasitriangula r an d almos t cocommutativ e Hop f algebra s 17 8 10.2 Coquasitriangula r an d almos t commutativ e Hop f algebra s 18 4 10.3 Th e Drinfel d doubl e 18 7 10.4 Braide d monoida l categorie s 19 7 10.5 Hop f algebra s i n categories grade d Hop f algebra s 20 3 10.6 Biproduct s an d Yetter-Drinfel d module s 20 7 Appendix. Som e quantu m group s 21 7 References 22 3 Index 235
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