On identities of infinite dimensional Lie superalgebras

TL;DR

This paper investigates the growth of polynomial identities in infinite dimensional Lie superalgebras, establishing conditions under which the PI-exponent exists and is a positive integer, especially for Grassmann envelopes of finite dimensional simple Lie algebras.

Contribution

It proves the existence and integrality of the PI-exponent for a class of infinite dimensional Lie superalgebras, extending understanding of their codimension growth.

Findings

PI-exponent exists for Grassmann envelopes of finite dimensional simple Lie algebras.

The PI-exponent is a positive integer.

Provides growth rate characterization of polynomial identities.

Abstract

We study codimension growth of infinite dimensional Lie superalgebras over an algebraically closed field of characteristic zero. We prove that if a Lie superalgebra $L$ is a Grassmann envelope of a finite dimensional simple Lie algebra then the PI-exponent of $L$ exists and it is a positive integer.