Crossed products of semisimple cocommutative Hopf algebras

Abstract

We provide a short proof of an analog of Nagata’s theorem for finite-dimensional Hopf algebras. The result, proved Hopf-algebraically by Sweedler and using group schemes by Demazure and Gabriel, says that a finite-dimensional cocommutative semisimple irreducible Hopf algebra is commutative. With mild base field assumptions such a Hopf algebra is just the dual of a p p -group algebra. We give en route an easy proof of a version of Hochschild’s theorem on semisimple restricted enveloping algebras. Let R # t H R{\# _t}H denote a crossed product with an invertible cocycle t t , where H H is a semisimple cocommutative Hopf algebra H H over a perfect field. The result above is applied to show that R # t H R{\# _t}H is semiprime if and only if R R is H H -semiprime. The approach relies on results on ideals of the crossed product that are stable under the action of the dual of H H and the Fisher-Montgomery theorem for crossed products of finite groups.