P.I. algebras with Hopf algebra actions

Abstract

If A is a p.i. algebra in characteristic zero with action from a finite-dimensional semisimple Hopf algebra H, then A has a nilpotent H-ideal N such that A/N will be H-verbally semiprime. Every H-verbally semiprime algebra is H-p.i. equivalent to a direct sum of H-verbally prime algebras. In the case of a finite group action or a grading by an abelian group, we show that the sum can be taken to be finite. In the case of an action by a finite cyclic group G, we classify all G-p.i. algebras, up to equivalence. This paper generalizes the work of A. R. Kerner (1985, Math. USSR Izv. 25). © 1999 Academic Press.