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 $\mathfrak{ sl}_2(K)$.
Categories: math.RA math.RT