$\mathbb{Z}_{2}$-graded identities of the Grassmann algebra over a finite field (update version)

TL;DR

This paper determines a basis for the $Z_2$-graded polynomial identities of the Grassmann algebra over a finite field with characteristic greater than 2, focusing on gradings with homogeneous underlying vector spaces.

Contribution

It provides a complete description of the $Z_2$-graded identities for the Grassmann algebra over finite fields, extending previous results to new gradings.

Findings

Established a basis for $Z_2$-graded identities

Extended known identities to finite fields with characteristic p>2

Analyzed gradings with homogeneous underlying vector spaces

Abstract

Let $F$ be a finite field with the characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis for $\mathbb{Z}_{2}$-graded polynomial identities for any non-trivial $\mathbb{Z}_{2}$-grading such that its underlying vector space is homogeneous.