Amitsur's conjecture for polynomial H-identities of H-module Lie algebras

TL;DR

This paper proves an analog of Amitsur's conjecture for polynomial H-identities in finite dimensional H-module Lie algebras over characteristic zero fields, extending to graded and G-codimensions under certain conditions.

Contribution

It establishes the asymptotic behavior of codimensions of polynomial H-identities for H-module Lie algebras, including cases with semisimple Hopf algebras, graded, and group actions.

Findings

Amitsur's conjecture holds for finite dimensional semisimple H.

The conjecture extends to graded codimensions of graded Lie algebras.

The conjecture applies to G-codimensions with rational group actions.

Abstract

Consider a finite dimensional H-module Lie algebra L over a field of characteristic 0 where H is a Hopf algebra. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial H-identities of L under some assumptions on H. In particular, the conjecture holds when H is finite dimensional semisimple. As a consequence, we obtain the analog of Amitsur's conjecture for graded codimensions of any finite dimensional Lie algebra graded by an arbitrary group and for G-codimensions of any finite dimensional Lie algebra with a rational action of a reductive affine algebraic group G by automorphisms and anti-automorphisms.