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).

37 pages

References in corpus (3)

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Cited by in corpus (23)

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• Graded Polynomial Identities for Matrices with the Transpose Involution over an Infinite Field
• On group gradings on PI-algebras
• A model for the relatively free graded algebra of block triangular matrices with entries from a graded algebra
• Finite basis problem for identities with involution
• On the codimension growth of G-graded algebras
• Identities and central polynomials of real graded division algebras
• Identities of finitely generated graded algebras with involution
• Irreducible Graded Bimodules over Algebras and a Pierce Decomposition of the Jacobson Radical
• Semisimple Algebras and PI-Invariants of Finite Dimensional Algebras
• The Grassmann algebra in arbitrary characteristic and generalized sign
• Multialternating graded polynomials and growth of polynomial identities
• Finitely generated algebras with involution and their identities
• Representability of affine algebras over an arbitrary field
• Graded monomial identities and almost non-degenerate gradings on matrices
• Representability of relatively free affine algebras over a Noetherian ring
• On Shirshov bases of graded algebras
• On the growth of graded polynomial identities of sl_n
• Specht property for the $2$-graded identities of $B_m$