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 $p$-group algebra. We give en route an easy proof of a version of Hochschild’s theorem on semisimple restricted enveloping algebras. Let $R{\# _t}H$ denote a crossed product with an invertible cocycle $t$, where $H$ is a semisimple cocommutative Hopf algebra $H$ over a perfect field. The result above is applied to show that $R{\# _t}H$ is semiprime if and only if $R$ is $H$-semiprime. The approach relies on results on ideals of the crossed product that are stable under the action of the dual of $H$ and the Fisher-Montgomery theorem for crossed products of finite groups.