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

20 pages

References in corpus (1)

• $\mathbb{Z}$-graded identities of the Lie algebras $U_1$