$\mathbb{Z}$-gradings of full support on the Grassmann algebra

TL;DR

This paper studies the structure of full support $bZ$-gradings on the infinite Grassmann algebra, describing polynomial identities, providing examples of PI-equivalent but non-isomorphic gradings, and exploring central gradings and their identities.

Contribution

It introduces the first examples of PI-equivalent but non-isomorphic graded algebras with infinite support and relates central $bZ$-gradings to $bZ_2$-graded identities.

Findings

Described $bZ$-graded polynomial identities for 2-induced gradings.

Provided examples of PI-equivalent but non-isomorphic gradings.

Connected central $bZ$-gradings to $bZ_2$-graded identities.

Abstract

Let $E$ be the infinite dimensional Grassmann algebra over a field $F$ of characteristic zero. In this paper we investigate the structures of $\mathbb{Z}$-gradings on $E$ of full support. Using methods of elementary number theory, we describe the $\mathbb{Z}$-graded polynomial identities for the so-called $2$-induced $\mathbb{Z}$-gradings on $E$ of full support. As a consequence of this fact we provide examples of $\mathbb{Z}$-gradings on $E$ which are PI-equivalent but not $\mathbb{Z}$-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent and not isomorphic as graded algebras. We also present the notion of central $\mathbb{Z}$-gradings on $E$ and we show that its $\mathbb{Z}$-graded polynomial identities are closely related to the $\mathbb{Z}_{2}$-graded polynomial identities of $\mathbb{Z}_{2}$-gradings on $E$.