Graded polynomial identities, group actions, and exponential growth of Lie algebras

TL;DR

This paper proves an analog of Amitsur's conjecture concerning the exponential growth of polynomial identities in finite-dimensional Lie algebras with group actions or gradings, confirming conjectured asymptotic behaviors.

Contribution

It establishes the asymptotic behavior of codimensions of polynomial G-identities and graded identities in finite-dimensional Lie algebras, extending Amitsur's conjecture to these contexts.

Findings

Proved the analog of Amitsur's conjecture for G-identities in Lie algebras.

Confirmed the exponential growth rate of graded codimensions for Abelian group gradings.

Extended the conjecture's validity to Lie algebras with finite group actions.

Abstract

Consider a finite dimensional Lie algebra L with an action of a finite group G over a field of characteristic 0. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial G-identities of L. As a consequence, we prove the analog of Amitsur's conjecture for graded codimensions of any finite dimensional Lie algebra graded by a finite Abelian group.