Enveloping algebras of restricted Lie superalgebras satisfying non-matrix polynomial identities

TL;DR

This paper characterizes restricted Lie superalgebras L over a field of characteristic p>2 based on whether their enveloping algebra u(L) satisfies certain non-matrix polynomial identities, including conditions for solvability and nilpotency.

Contribution

It provides a complete characterization of L when u(L) satisfies non-matrix polynomial identities, especially for Lie solvable, nilpotent, or super-nilpotent cases.

Findings

u(L) satisfies a non-matrix polynomial identity if and only if L meets specific structural conditions.

Characterizations of L when u(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent are established.

The results extend understanding of polynomial identities in the context of restricted Lie superalgebras.

Abstract

Let L be a restricted Lie superalgebra with its enveloping algebra u(L) over a field F of characteristic p>2. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2\times 2 matrices over F. We characterize L when u(L) satisfies a non-matrix polynomial identity. In particular, we characterize L when u(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.