Graded identities of some simple Lie superalgebras

TL;DR

This paper investigates the graded identities of certain simple Lie superalgebras, establishing asymptotic bounds on their codimensions and deriving upper bounds for their PI-exponents.

Contribution

It provides new asymptotic bounds on the graded identities and PI-exponents of simple Lie superalgebras of type b(t), extending understanding of their polynomial identities.

Findings

The n-th codimension is asymptotically less than (im b(t))^n.

An upper bound for the PI-exponent of each simple Lie superalgebra b(t), t  3.

Results apply to Lie superalgebras over fields of characteristic zero.

Abstract

We study $\mathbb{Z}_2$-graded identities of Lie superalgebras of the type $b(t), t\ge 2$, over a field of characteristic zero. Our main result is that the $n$-th codimension is strictly less than $(\dim b(t))^n$ asymptotically. As a consequence we obtain an upper bound for ordinary (non-graded) PI-exponent for each simple Lie superalgebra $b(t), t\ge 3$.