Asymptotics of H-identities for associative algebras with an H-invariant radical

TL;DR

This paper establishes the existence of the Hopf PI-exponent for finite dimensional associative algebras with a generalized Hopf action, under minimal invariance assumptions, extending classical conjectures to broader algebraic contexts.

Contribution

It proves the Hopf PI-exponent exists under weak conditions and confirms Amitsur's conjecture analogs for G-codimensions and differential codimensions.

Findings

Existence of Hopf PI-exponent for algebras with generalized Hopf action.

Validation of Amitsur's conjecture analogs for G-codimensions.

Extension of codimension growth results to broader algebraic actions.

Abstract

We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras $A$ with a generalized Hopf action of an associative algebra $H$ with $1$ over an algebraically closed field of characteristic $0$ assuming only the invariance of the Jacobson radical $J(A)$ under the $H$-action and the existence of the decomposition of $A/J(A)$ into the sum of $H$-simple algebras. As a consequence, we show that the analog of Amitsur's conjecture holds for $G$-codimensions of finite dimensional associative algebras over a field of characteristic $0$ with an action of an arbitrary group $G$ by automorphisms and anti-automorphisms and for differential codimensions of finite dimensional associative algebras with an action of an arbitrary Lie algebra by derivations.