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arXiv · 2110.06377

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

Alan Guimarães·Claudemir Fidelis·Plamen Koshlukov
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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].

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Alan Guimarães, Claudemir Fidelis, Plamen Koshlukov. 2021-10-12. A note on $\mathbb{Z}$-gradings on the Grassmann algebra and Elementary Number Theory. https://arxiv.org/abs/2110.06377

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