A note on $\mathbb{Z}$-gradings on the Grassmann algebra and Elementary Number Theory

TL;DR

This paper investigates specific $Z$-gradings on the Grassmann algebra of an infinite-dimensional vector space, linking elementary number theory with polynomial identity theory to classify structures and identities.

Contribution

It introduces a criterion for the support of certain $Z$-gradings to form subgroups and describes their graded identities, connecting number theory with PI theory.

Findings

Support of gradings can coincide with subgroups of $Z$ under certain conditions.

Provides a classification of graded identities for these gradings.

Establishes a novel link between elementary number theory and polynomial identity theory.

Abstract

Let $E$ be the Grassmann algebra of an infinite dimensional vector space $L$ over a field of characteristic zero. In this paper, we study the $\mathbb{Z}$-gradings on $E$ having the form $E=E_{(r_{1},r_{2}, r_{3})}^{(v_{1},v_{2}, v_{3})}$, in which each element of a basis of $L$ has $\mathbb{Z}$-degree $r_{1}$, $r_{2}$, or $r_{3}$. We provide a criterion for the support of this structure to coincide with a subgroup of the group $\mathbb{Z}$, and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in [11].