$2$-Graded Identities for the Tensor Square of the Grassmann Algebra

TL;DR

This paper investigates the graded identities of the tensor square of the Grassmann algebra, showing that in many cases these identities coincide with the ordinary identities, and providing explicit bases for certain gradings.

Contribution

It characterizes graded identities for the tensor square of the Grassmann algebra under various gradings, including non-canonical quotient gradings, and supplies explicit bases.

Findings

In many cases, graded identities match ordinary identities.

Explicit bases for graded identities are provided for quotient gradings.

The structure of graded identities depends on the type of grading applied.

Abstract

We consider the algebra $E\otimes E$ over an infinite field equipped with a $\mathbb{Z}_2$-grading where the canonical basis is homogeneous and prove that in various cases the graded identites are just the ordinary ones. If the grading is a non-canonical grading obtained as a quotient grading of the natural $\mathbb{Z}_2\times\mathbb{Z}_2$-grading we exhibit a basis for the graded identities.