Hopf algebras and their actions on rings : [expanded version of ten lectures given at the CBMS Conference on Hopf Algebras and thier Actions on Rings, wich took place at DePaul University in Chicago, August 10-14, 1992

The last ten years have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics and now have connections to many areas of mathematics. In addition, several conjectures of Kaplansky have been solved, the most striking of which is a kind of Lagrange's theorem for Hopf algebras. Work on actions of Hopf algebras has unified earlier results on group actions, actions of Lie algebras, and graded algebras. This book brings together many of these recent developments from the viewpoint of the algebraic structure of Hopf algebras and their actions and coactions. Quantum groups are treated as an important example, rather than as an end in themselves. The two introductory chapters review definitions and basic facts; otherwise, most of the material has not previously appeared in book form. Providing an accessible introduction to Hopf algebras, this book would make an excellent graduate textbook for a course in Hopf algebras or an introduction to quantum groups. Preface ......Page 9 §1.1 Algebras and coalgebras ......Page 11 §1.2 Duals of algebras and coalgebras ......Page 12 §1.3 Bialgebras ......Page 13 §1.4 Convolution and summation notation ......Page 16 §1.5 Antipodes and Hopf algebras ......Page 17 §1.6 Modules and comodules ......Page 20 §1.7 Invariants and coinvariants ......Page 23 §1.9 Hopf modules ......Page 24 §2.1 Integrals ......Page 27 §2.2 Maschke's Theorem ......Page 30 §2.3 Commutative semisimple Hopf algebras and restricted enveloping algebras ......Page 32 §2.4 Cosemisimplicity and integrals on H ......Page 35 §2.5 Kaplansky's conjecture and the order of the antipode ......Page 37 §3.1 The Nichols-Zoeller Theorem ......Page 38 §3.2 Applications: Hopf algebras of prime dimension and semisimple subHopfalgebras ......Page 41 §3.3 A normal basis for H over К ......Page 42 §3.4 The adjoint action, normal subHopfalgebras, and quotients ......Page 43 §3.5 Freeness and faithful flatness in the infinite-dimensional case ......Page 47 §4.1 Module algebras, comodule algebras, and smash products ......Page 50 §4.3 Trace functions and affine invariants: the non-commutative case ......Page 53 §4.4 Ideals in A#H and A as an A^H-module ......Page 58 §4.5 A Morita context relating А#H and A^H ......Page 62 §5.1 Simple subcoalgebras and the coradical ......Page 66 §5.2 The coradical filtration ......Page 70 §5.3 Injective coalgebra maps ......Page 75 §5.4 The coradical filtration of pointed coalgebras ......Page 77 §5.5 Examples: U(g) and U_q(g) ......Page 83 §5.6 The structure of pointed cocommutative Hopf algebras ......Page 86 §5.7 Semisimple cocommutative connected Hopf algebras ......Page 93 §6.1 Definitions and examples ......Page 97 §6.2 A Skolem-Noether theorem for Hopf algebras ......Page 99 §6.3 Maximal inner subcoalgebras ......Page 102 §6.4 X-inner actions and extending to quotients ......Page 106 §7.1 Definitions and examples ......Page 111 §7.2 Cleft extensions and existence of crossed products ......Page 115 §7.3 Inner actions and equivalence of crossed products ......Page 122 §7.4 Generalized Maschke theorems and semiprime crossed products ......Page 126 §7.5 Twisted H-comodule algebras ......Page 131 §8.1 Definition and examples ......Page 133 §8.2 The normal basis property and cleft extensions ......Page 138 §8.3 Galois extensions for finite-dimensional H ......Page 141 §8.4 Normal bases and Hopf algebra quotients ......Page 149 §8.5 Relative Hopf modules ......Page 154 §9.1 H° ......Page 159 §9.2 SubHopfalgebras of H° and density ......Page 163 §9.3 Classical duality ......Page 169 §9.4 Duality for actions ......Page 171 §9.5 Duality for graded algebras ......Page 182 §10.1 Quasitriangular and almost cocommutative Hopf algebras ......Page 188 §10.2 Coquasitriangular and almost commutative Hopf algebras ......Page 194 §10.3 The Drinfeld double ......Page 197 §10.4 Braided monoidal categories ......Page 207 §10.5 Hopf algebras in categories; graded Hopf algebras ......Page 213 §10.6 Byproducts and Yetter-Drinfeld modules ......Page 217 §A.1 U_q(sl(2))......Page 227 §A.3 A quantum analog of the Heisenberg Lie algebra ......Page 228 §A.4 The coordinate ring of quantum 2x2 matrices ......Page 229 §A.5 Quantum matrices, quantum SL_n(k) and quantum GL_n(k) ......Page 230 §A.6 The H-matrix approach to coordinate rings of quantum matrices ......Page 231 References ......Page 233 Index ......Page 245 There have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics. This book covers the developments from the viewpoint of the algebraic structure of Hopf algebras and their actions and coactions.