Representability and Specht problem for G-graded algebras

TL;DR

This paper proves that G-graded identities of associative PI algebras over characteristic zero fields can be represented by finite dimensional algebras after field extension, and establishes the finite generation of their T-ideals.

Contribution

It demonstrates G-graded PI equivalence with finite dimensional algebras and solves the G-graded Specht problem for associative PI algebras.

Findings

Existence of a finite dimensional G-graded algebra representing the identities.

Finiteness of the G-graded T-ideal for associative PI algebras.

G-graded PI equivalence for affine 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).