On graded polynomial identities of $sl_{2}(F)$ over a finite field

Abstract

Let $F$ be a finite field of $char F > 3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. This paper aims for the following goals: Find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$; Find a basis for the $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$ when $F$ contains a primitive 3rd root of one; Find a basis for the $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.