Narrow Results

Let K be a field of characteristic zero and let ${𝔰𝔩}_2(K)$ 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 ${𝔰𝔩}_2(K)$, or equivalently, the ${\mathbb{Z}}_2$-graded identities for the pair $(M_3(K),{𝔰𝔩}_2(K))$. We work with the canonical grading on ${𝔰𝔩}_2(K)$ and the only nontrivial ${\mathbb{Z}}_2$-grading of the associative algebra $M_3(K)$ induced by that on ${𝔰𝔩}_2(K)$.