Equivalences of (co)module algebra structures over Hopf algebras

Joost Vercruysse,Alexey Gordienko,Ana L Agore

doi:10.4171/jncg/428

Abstract

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) 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 behaviour 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.

Highlights

We introduce the notion of support equivalence formodule algebras, which generalizes in a natural way equivalence of gradings
It is easy to see that a similar universal group exists for group actions too (Remark 2.12). Generalizing these constructions, we show in Theorems 3.8 and 4.5 that for a givenmodule algebra A not necessarily finite dimensional, there exists a unique Hopf algebra H with aaction on A, which is universal among all Hopf algebras that admit a support equivalentaction on A
In Proposition 5.6, we show that the universal Hopf algebra of a group action can contain non-trivial primitive elements

Summary

Introduction

Module and comodule algebras over Hopf algebras (see the definitions in Sections 3.1 and 4.1 below) appear in many areas of mathematics and physics. Using the correspondence between H∗-actions and H-coactions for finite dimensional Hopf algebras H, we define support equivalence for module algebra structures and, in particular, for group actions. It is easy to see that a similar universal group exists for group actions too (Remark 2.12) Generalizing these constructions, we show in Theorems 3.8 and 4.5 that for a given (co)module algebra A not necessarily finite dimensional, there exists a unique Hopf algebra H with a (co)action on A, which is universal among all Hopf algebras that admit a support equivalent (co)action on A. For a connected affine algebraic group G over an algebraically closed field F of characteristic 0 and its associated Lie algebra g, we show that the U (g)- and the F G-module structures on a finite dimensional algebra with a rational action are support equivalent (see Theorem 5.2).

Preliminaries

Equivalences of group gradings and group actions

Comodule algebras

Module structures on algebras

Applications