ABSTRACT.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.

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.

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1. Introduction. D. G. Higman, in [6], introduced and studied the notions of induced and produced modules of rings. These concepts are generalizations of the classical construction of induced representations of finite groups. In the present paper, we study these notions in the context of modules over Lie algebras (or equivalent^ unitary modules over the universal enveloping algebras of these Lie algebras).Induced §2 is devoted to elementary properties of induced and produced modules. We also show how produced Lie algebra modules and induced Lie group modules are related to each other. In §3 we introduce our main tool, a product structure between members of produced Lie algebra modules. This structure arises from the hyperalgebra structure possessed by universal enveloping algebras (see [10]) and reflects the natural product structure between members of induced Lie group modules. Using our product structure we prove in §4 a generalization of the Realization Theorem of Guillemin and Sternberg. The remaining two sections are devoted to proving Lie algebra analogues of theorems of G W. Mackey ([7] and [8]) concerning systems of imprimitivity and concerning irreducibility criteria for induced representations of group extensions.The present author wishes to express his thanks to Professors H. Zassenhaus, R. Steinberg, R. Arens, and S. Sternberg for conversations over a period of years bearing on the subject of this paper.