Hopf algebra actions

Abstract

Finite groups acting on rings by automorphisms, and group-graded rings are instances of Hopf algebras H acting on H-module algebras A. We study such actions. Let A H = {a ϵ A¦h · a = ε(h) a, all h ϵ H} , the ring of H-invariants, and form the smash product A # H. We study the ring extensions A H ⊂ A ⊂ A # H. We prove a Maschke-type theorem for A # H-modules. We form an associated Morita context [ A H , A, A, A # H] and use these to get connections between the various rings.