Graded Identities for the Adjoont Representation of $s{l}_2$

Abstract

Let $K$ be a field of characteristic zero and let ${\mathfrak{sl}}_2(K)$ be the 3-dimensional simple Lie algebra over $K$. In this paper we describe a finite basis for the ${\mathbb{Z}}_2$-graded identities of the adjoint representation of ${\mathfrak{sl}}_2(K)$, or equivalently, the ${\mathbb{Z}}_2$-graded identities for the pair $(M_3(K),{\mathfrak{sl}}_2(K))$. We work with the canonical grading on ${\mathfrak{sl}}_2(K)$ and the only nontrivial ${\mathbb{Z}}_2$-grading of the associative algebra $M_3(K)$ induced by that on ${\mathfrak{sl}}_2(K)$.

20 pages, no figures