COCYCLE EQUIVALENT HOPF ALGEBRA ACTIONS

Abstract

Let H be a Hopf algebra over a field K and assume a K-algebra A is an H-module algebra under two actions; · and ∘. We call these actions cocycle equivalent if there is an action of H on M 2(A), h ♦ X, such that for h ∈ H, X ∈ M 2(A) and a,b ∈ A. Two actions are cocycle equivalent if and only if there are cocycles that relate the two actions. Using these, it is shown that cocycle equivalence is a equivalence relation. Finally let H be a finite dimensional, semisimple, cocommutative Hopf algebra and assume K is a splitting field of H. It is shown that the Connes spectrum of H acting on M 2(A) is the intersection of the Connes spectra of H acting on A under · and ∘. Denote the smash product of A and H under the action · by (A#H,·). Let A be H-prime, then (A#H,·) is prime if and only if (A)#H, ♦) is prime if and only if (M 2(A)# H, ♦) is prime.