On the codimension growth of almost nilpotent Lie algebras

TL;DR

This paper investigates the growth of polynomial identities in certain infinite-dimensional Lie algebras, establishing the existence and integrality of the PI-exponent for extensions of nilpotent algebras by finite-dimensional semisimple algebras.

Contribution

It proves that for Lie algebras extending a nilpotent algebra with a finite-dimensional semisimple algebra, the PI-exponent exists and is a positive integer.

Findings

PI-exponent exists for the class of Lie algebras studied

PI-exponent is a positive integer

Growth behavior characterized for extensions of nilpotent algebras

Abstract

We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra $L$ is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of $L$ exists and is a positive integer.