On the graded identities of the Grassmann algebra

Abstract

Abstract:We survey the results concerning the graded identities of the infinite dimensional Grassmann algebra.

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

“…The next result is an adaptation of [5,Proposition 5.2]. It will be important for our goals in this section.…”

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

“…We shall denote the above canonical basis of E by B E . The Grassmann algebra has a natural Z 2 -grading E can = E (0) ⊕ E (1) , where E (0) is the vector space spanned by 1 and all products e i1 • • • e i k with even k while E (1) is the vector space spanned by the products with odd k. It is well known that E (0) is the centre of E and E (1) is the "anticommuting" part of E. Recall that the theory developed by Kemer [14] has been shaping much of the theory of PI algebras during the last three decades. In Kemer's theory the Grassmann algebra E together with its natural grading, plays a crucial role in it.…”

“…The graded identities for the natural (or canonical) grading E can are well known, see for example [7]. More generally, in the papers [1,3,9], all homogeneous Z 2 -gradings on E and the respective graded identities were studied. (Here and in what follows Z 2 stands for the cyclic group of order 2.)…”

“…When the field K is infinite, one has to consider multihomogeneous identities, for more details of this technique we recommend [5,Section 4.2]. The methods one uses in this case are mostly based on Combinatorial Algebra [1,12,13]. A technique that has been used recently is based on Elementary Number Theory.…”

“…Without loss of generality, assume m ≥ n. In this case,u 1 • • • u t ∈ A r−nc , and u ′ 1 • • • u ′ k e j1 e j2 • • • e jm−n ∈ A r−nc .It follows that there exists q ∈ Z such that qd = r − nc, and therefore A qd ∩ Z(E) and A qd ∩ E 1 are non-trivial, thus contradicting Lemma 4.13. Therefore either A r ⊆ Z(E) or A r ⊆ E(1) .Notice that gcd(d, c) = 1 implies that c is a generator of the group Z d . It follows that the set {0, c, 2c, .…”