On graded polynomial identities of $s{l}_2(F)$ over a finite field

Abstract

Let $F$ be a finite field of $charF>3$ and $s{l}_2(F)$ 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 $s{l}_2(F)$; Find a basis for the ${\mathbb{Z}}_3$-graded identities of $s{l}_2(F)$ when $F$ contains a primitive 3rd root of one; Find a basis for the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded identities of $s{l}_2(F)$.

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