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

Abstract

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Zgradings on E of full support. Using methods of elementary number theory, we describe the Z-graded polynomial identities for the so-called 2-induced Z-gradings on E of full support. As a consequence of this fact we provide examples of Z-gradings on E which are PI-equivalent but not 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 Z-gradings on E and we show that its Z-graded polynomial identities are closely related to the Z2-graded polynomial identities of Z2-gradings on E.

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

“…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].…”

“…Group gradings on the Grassmann algebra have been studied in several papers, see [2,3,10,11,13]. There are two important types of gradings on E. Definition 2.2 A G-grading E = ⊕ g∈G E g on the Grassmann algebra is said to be:…”

“…A technique that has been used recently is based on Elementary Number Theory. We recall that such an approach appeared implicitly in [4], and more explicitly in [6,11]. Although elementary as a method, the results obtained suggest that a good understanding of such a relation between Number theory and PI theory could give us some insights for the nature of the gradings and graded identities not only on E, but also in the case of other important algebras.…”

“…of the non-negative integers as support. Furthermore in the paper [11], the authors dealt with Z-gradings on E of full support; they described the Z-graded identities for these structures providing a necessary and sufficient condition for two structures of this type to be isomorphic as Z-graded algebras. The authors of the latter paper pointed out that the support of a grading on E can have interesting properties as a subset of Z.…”

“…In this paper, we shall study gradings on E by the infinite cyclic group Z. Throughout, following the line of research presented in [11], we deal with the structures of type E (v1,...,vn) (g1,...,gn) , where G = Z. In this paper we consider initially the case n = 3.…”