A note on the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded identities of $E \otimes E$ over a finite field

TL;DR

This paper characterizes the $Z_2 imes Z_2$-graded identities of a tensor product of Grassmann algebras over finite fields and explores its graded Gelfand-Kirillov dimension.

Contribution

It provides a description of the graded identities of $E_{k^*} ensor E$ and analyzes its graded GK-dimension, extending understanding of graded polynomial identities over finite fields.

Findings

Explicit $Z_2 imes Z_2$-graded identities for $E_{k^*} ensor E$

Determination of the graded GK-dimension of $E_{k^*} ensor E$

Insights into graded identities over finite fields

Abstract

Let $F$ be a finite field of $char F = p$ and size $|F| = q$. Let $E$ be the unitary infinity dimensional Grassmann algebra. In this short note, we describe the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded identities of $E_{k^{*}}\otimes E$, where $E_{k^{*}}$ is the Grassmann algebra with a specific $\mathbb{Z}_{2}$-grading. In the end, we discuss about the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded GK-dimension of $E_{k^*}\otimes E$ in $m$ variables.