Codimensions of polynomial identities of representations of Lie algebras

TL;DR

This paper proves an analog of Amitsur's conjecture regarding the asymptotic growth of codimensions of polynomial identities in representations of Lie algebras, extending understanding of algebraic identities.

Contribution

It establishes the asymptotic behavior of codimensions of polynomial identities for Lie algebra representations, confirming a conjecture in this context.

Findings

Proved the analog of Amitsur's conjecture for Lie algebra representations.

Established the asymptotic growth rate of codimensions.

Extended the theory of polynomial identities to Lie algebra representations.

Abstract

Consider a representation $\rho\colon L \to \mathfrak{gl}(V)$ where $L$ is a Lie algebra and $V$ is a finite dimensional vector space. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities of $\rho$.