Equivalences of (co)module algebra structures over Hopf algebras

Abstract

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) an equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra $A$, there exists a unique universal Hopf algebra $H$ together with an $H$-(co)module structure on $A$ such that any other equivalent (co)module algebra structure on $A$ factors through the action of $H$. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf–Galois extensions, actions of algebraic groups, and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. We apply support equivalence in the study of the asymptotic behavior of codimensions of $H$-identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example, we prove this analogue for all unital $H$-module structures on the algebra $F[x]/(x^2)$ of dual numbers.