Representability and Specht problem for G-graded algebras

Abstract

Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of F and a finite dimensional Z/2ZxG-graded algebra A over K such that id_{G}(W)=id_{G}(A^{*}) where A^{*} is the Grassmann envelope of A. 2. {[G-graded Specht problem]} The T-ideal id_{G}(W) is finitely generated as a T-ideal. 3. {[G-graded PI-equivalence for affine algebras]} Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite dimensional algebra A over K such that id_{G}(W)=id_{G}(A).Comment: 37 page

On regular 𝐺-gradings

“…In the first part of this section we recall some general facts and terminology on G-graded PI-theory which will be used in the proofs of the main results (we refer the reader [3] for a detailed account on this topic). In the second part of this section we present some additional examples of regular gradings on finite and infinite dimensional algebras.…”

“…It is well known that applying this operation, one can extend the solution of the Specht problem and proof of "representability" from affine to nonaffine PI-algebras (see [10], [3]). Interestingly, the property satisfied by the Grassmann algebra we just mentioned follows from the fact that the Z /2Z-grading on E is regular, and indeed in Theorem 6 we show that a similar property holds for arbitrary regular graded algebras.…”

“…We recall from [3] that the T -ideal I = Id G (W ) is generated by multilinear polynomials and so it does not change when passing toF, the algebraic closure of F, in the sense that the ideal of identities of WF overF is the span (overF) of the T -ideal of identities of W over F.…”

On the graded identities of the Grassmann algebra

“…We recall that the works by Vasilovsky and by Azevedo are a generalization of the work by Di Vincenzo for 2 by 2 matrices (see [11]). As well as in the ordinary case, if A is an associative algebra graded by a finite group G, then T G (A) is finitely generated as a T G -ideal (see [1]). …”

Kemer's theory for H-module algebras with application to the PI exponent

“…In all of these frameworks analogs of these problems exist and in some of them also solved: For finite group-graded algebras satisfying an ordinary PI see [3] (it is worth mentioning that in [15] the special case of abilean finite groups is treated). For algebras with involutions satisfying an ordinary PI see [16].…”