$\mathbb{Z}_3$-graded identities of the pair $(M_3(K),gl_3(K))$

TL;DR

This paper characterizes the polynomial identities that define the $bZ_3$-graded structure of the pair consisting of 3x3 matrices and their Lie algebra over an infinite integral domain, providing explicit bases.

Contribution

It explicitly describes the basis for the $bZ_3$-graded identities of the pair $(M_3(K), gl_3(K))$, extending understanding of graded identities in matrix and Lie algebra pairs.

Findings

Explicit basis for $bZ_3$-graded identities provided

Describes $G$-graded identities for elementary gradings

Extends graded identity theory to matrix-Lie algebra pairs

Abstract

Let $M_n(K)$ be the algebra of $n \times n$ matrix over an infinite integral domain $K$. Let $gl_n(K)$ be the Lie algebra of $n \times n$ matrix with the usual Lie product over $K$. Let $G = \{g_1,\ldots,g_n\}$ be a group of order $n$. We describe the polynomials that form a basis for the $G$-graded identities of the pair $(M_n(K),gl_n(K))$ with an elementary $G$-grading induced by the $n$-tuple $(g_1,\ldots,g_n)$. In the end, we describe an explicit basis for the $\mathbb{Z}_3$-graded identities of the pair $(M_3(K),gl_3(K))$.