Representability and Specht problem for G-graded algebras

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In the early 1980's A. R. Kemer [Ideals of identities of associative algebras, translated from the Russian by C. W. Kohls, Transl. Math. Monogr., 87, Amer. Math. Soc., Providence, RI, 1991; MR1108620] proved the following basic results of the theory of polynomial identities: Let $A$ be a PI-algebra over a field of characteristic zero and let ${\rm Id}(A)$ be its $T$-ideal of polynomial identities. Then: (1) if $A$ is finitely generated, there exists a finite-dimensional algebra $B$ such that ${\rm Id}(A)={\rm Id}(B)$; (2) more generally, there exists a finite-dimensional ${Z}_2$-graded algebra $B=B_0\oplus B_1$ such that ${\rm Id}(A)={\rm Id}(E(B))$, where $E=E_0\oplus E_1$ is the Grassmann algebra with its natural ${Z}_2$-grading and $E(B)=(E_0\otimes B_0)\oplus (E_1\otimes B_1)$ is the Grassmann envelope of $B$; (3) ${\rm Id}(A)$, and thus any $T$-ideal of the free algebra, is finitely generated as a $T$-ideal. In this paper the authors prove similar results for PI-algebras graded by a finite group $G$ and their ideals of graded identities. Let $W$ be a PI-algebra over a field of characteristic zero and suppose that $W$ is graded by a finite group $G$. If ${\rm Id}_G(W)$ is the ideal of $G$-graded identities of $W,$ then: (1) if $W$ is finitely generated, there exists a field extension $K$ of $F$ and a finite-dimensional $G$-graded algebra $A$ over $K$ such that ${\rm Id}_G(W)={\rm Id}_G(A)$; (2) more generally, there exists a field extension $K$ of $F$ and a finite-dimensional ${Z}_2 \times G$-graded algebra $A$ over $K$ such that ${\rm Id}_G(W)={\rm Id}_G(E(A))$, where $E(A)$ is the Grassmann envelope of $A$; (3) ${\rm Id}(W)$ is finitely generated as a $G$-graded $T$-ideal. The proofs of these results follow closely the general scheme of the original proofs of Kemer as presented by A. Ya. Kanel-Belov and L. H. Rowen [Computational aspects of polynomial identities, Res. Notes Math., 9, A K Peters, Wellesley, MA, 2005; MR2124127]. In addition to the difficulties of understanding and explaining the original proofs, here the authors face the difficulties arising from the grading. It must be pointed out that these same theorems have been recently proved independently by I. Sviridova ["Identities of PI-algebras graded by a finite abelian group'', Comm. Algebra, to appear] when $G$ is an abelian group. As a consequence of these results, the theory of group-graded polynomial identities of PI-algebras has been further developed in recent years. For instance Aljadeff, Giambruno and La Mattina have recently proved in three different papers the existence of the graded exponent of any $G$-graded PI-algebra. Reviewed by Antonio Giambruno

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