# On $\mathbb{Z}_{2}$-graded polynomial identities of $sl_{2}(F)$ over a finite field

## TL;DR

This paper determines a basis for the $Z_2$-graded polynomial identities of the Lie algebra $sl_2(F)$ over a finite field with characteristic greater than 3, enhancing understanding of its algebraic structure.

## Contribution

It provides the first explicit basis for the $Z_2$-graded identities of $sl_2(F)$ over finite fields, filling a gap in the algebraic theory.

## Key findings

- Explicit basis for $Z_2$-graded identities of $sl_2(F)$
- Characterization of identities over finite fields with char > 3
- Advances understanding of graded polynomial identities in Lie algebras

## 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 $F$. In this paper, we find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.