Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

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Group-graded rings, smash products, and group actions

This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, aswell as colour-Lie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied,the notion of `braided -commutative' or `braided-cocommutative' Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group aut(C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.

Algebras and Hopf Algebras in Braided Categories

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include \"inner\" group gradings of algebras. We prove that if 7r : H —► H is a Hopf algebra epimorphism which is split as a coalgebra map, then H is algebra isomorphic to A #„ H, a crossed product of H with the left Hopf kernel A of it. Given any crossed product A #CT H with H (weakly) inner on A, then A #CT if is isomorphic to a twisted product AT[H] with trivial action. Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #„ H; in particular this is true if the (weak) action of H is inner. Introduction. The purpose of this paper is to begin to lay the foundations of a general theory of actions of Hopf algebras on noncommutative algebras. The importance of such a theory derives from three special cases in which the Hopf algebra is a group algebra, an enveloping algebra of a Lie algebra, or the dual of a group algebra. These cases show that the theory encompasses the study of actions of groups as automorphisms of algebras, the study of actions of Lie algebras as derivations of algebras, and the study of group graded rings, respectively. Each of these areas of study have been quite active lately (see [16, 9, 5]). A fundamental concept in the first two cases has been the notion of inner action. One of the major purposes of this paper is to study inner actions of Hopf algebras; as a new example, we will investigate what is meant by an inner grading. Another important concept in the first two cases is that of a semidirect product (smash product): for group actions we have skew group rings and for Lie algebra actions we have differential polynomial rings. This notion is also defined for Hopf algebra actions. More generally one can consider crossed products A ffa H of an algebra A with a Hopf algebra H, where the multiplication of the copy of H in A ffa H is twisted by a cocycle o, and their study is the other major purpose of this paper. It turns out that these two concepts (inner actions and crossed products) are closely interrelated. Now inner actions and crossed products of Hopf algebras were both studied by Sweedler [24] in the context of the cohomology theory of Hopf algebras. The present paper owes a great debt to Sweedler's work, especially in §§1 and 4. However, his set-up was restricted at crucial places to the situation where H is a cocommutative Received by the editors December 12, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A24; Secondary 16A72, 16A03, 46L40. The third author was partially supported by NSF Grant DMS 8500959 and by a Guggenheim Memorial Foundation Fellowship. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

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Crossed products and inner actions of Hopf algebras

Normal basis and transitivity of crossed products for Hopf algebras

Let R be a group graded ring . The map ( , ): R × R →1 defined by: (x,y) = (xy)1 , is an inner product on R. In this paper we investigate aspects of nondegeneracy of the product, which is a generalization of the notion of strongly G —graded rings,introduced by Dade. We show that various chain conditions are satisfied by R if and only if they are satisfied by R1 , and that when R1 is simple artinian, then R is a crossed product R1 * G. We give conditions for simple R-modules to be completely reducible R1 -modules . Finally, we prove an incomparability theorem,when G is finite abelian.

Miriam Cohen

Papers

Group-graded rings, smash products, and group actions

Crossed products and inner actions of Hopf algebras

Group graded rings

Hopf Galois extensions, smash products, and Morita equivalence

Hopf algebra actions