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

Abstract

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