Ideals and quotients in crossed products of Hopf algebras

Abstract

Let H be a Hopf algebra over a field k, acting on the k-algebra R with action twisted by a cocycle (T such that the crossed product algebra R #, H can be constructed. This paper is concerned with the relationship between the ideals of R #, Hand those of R, and with computing the extended cen- ter and symmetric quotient ring of R #, H in terms of the extended center C and symmetric quotient ring Q of R. Our best results are obtained when H is an irreducible Hopf algebra, or more generally when H is of the form K # kG, where K is irreducible and kG is the group algebra of a group G. Some examples of irreducible Hopf algebras are enveloping algebras of Lie algebras U(L) along with their restricted counterparts u(L) in characteristic