Amitsur's conjecture for associative algebras with a generalized Hopf action

TL;DR

This paper proves an analog of Amitsur's conjecture for associative algebras with generalized Hopf actions, establishing asymptotic behavior of codimensions for various identities and computing the Hopf PI-exponent for a specific algebra.

Contribution

It extends Amitsur's conjecture to G- and H-identities in associative algebras with Hopf actions, including explicit exponent calculations.

Findings

Proved asymptotic behavior of codimensions for G- and H-identities.

Established the Hopf PI-exponent of Sweedler's algebra as 4.

Extended Amitsur's conjecture to broader algebraic structures.

Abstract

We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of several generalizations of polynomial identities for finite dimensional associative algebras over a field of characteristic 0, including G-identities for any finite (not necessarily Abelian) group G and H-identities for a finite dimensional semisimple Hopf algebra H. In addition, we prove that the Hopf PI-exponent of Sweedler's 4-dimensional algebra with the action of its dual equals 4.