Actions of Ore extensions and growth of polynomial $H$-identities

TL;DR

This paper investigates the growth of polynomial $H$-identities in finite dimensional associative $H$-module algebras, reducing complex cases to $H$-simple algebras and establishing the existence of a PI-exponent for certain Hopf algebras.

Contribution

It reduces the proof of Amitsur's conjecture for $H$-codimensions to $H$-simple algebras and proves the existence of a PI-exponent for algebras acted upon by Hopf algebras constructed via Ore extensions.

Findings

Reduction of $H$-codimension conjecture to $H$-simple algebras

Existence of PI-exponent for algebras with Hopf algebra actions from Ore extensions

Structural analysis of algebras simple under Ore extension actions

Abstract

We show that if $A$ is a finite dimensional associative $H$-module algebra for an arbitrary Hopf algebra $H$, then the proof of the analog of Amitsur's conjecture for $H$-codimensions of $A$ can be reduced to the case when $A$ is $H$-simple. (Here we do not require that the Jacobson radical of $A$ is an $H$-submodule.) As an application, we prove that if $A$ is a finite dimensional associative $H$-module algebra where $H$ is a Hopf algebra $H$ over a field of characteristic $0$ such that $H$ is constructed by an iterated Ore extension of a finite dimensional semisimple Hopf algebra by skew-primitive elements (e.g. $H$ is a Taft algebra), then there exists integer $\mathop{\mathrm{PIexp}}^H(A)$. In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.