Polynomial identities of the adjoint Lie algebra of $M_{1,1}$

TL;DR

This paper determines explicit polynomial identity bases for the adjoint Lie algebra of certain matrix algebras over Grassmann algebras, revealing the fundamental identities that define their algebraic structure.

Contribution

It provides the first known identity bases for the adjoint Lie algebra of $M_{1,1}$ over Grassmann algebras, including a single identity for $E^1$ and three identities for $E$.

Findings

Identifies a single key identity for $M_{1,1}(E^1)$ in characteristic not two.

Establishes a three-identity basis for $M_{1,1}(E)$.

Shows that these identities fully characterize the algebra's polynomial identities.

Abstract

We search an identity basis for the adjoint Lie algebra of the algebra $M_{1,1}(K)$ over a field, where $K$ is either the infinite generated Grassmann algebra $E$ or $E^1$, the variant of the algebra with $1$. In particular, we prove that in the case of an infinite base field of characteristic different from two the identities of $M_{1,1}(E^1)$ are exactly all the consequences of the identity $[x,y,[z,t],v]=0$. We also find an identity basis of $M_{1,1}(E)$ consisting of three identities.