Representability and Specht problem for G-graded algebras

Eli Aljadeff; Alexei Kanel-Belov

doi:10.1016/j.aim.2010.04.025

Representability and Specht problem for G-graded algebras

Eli Aljadeff^*, Alexei Kanel-Belov

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

56 Scopus citations

Abstract

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let id_GG(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/2Z×G-graded algebra A over K such that id_GG(W)=id_GG(A^*) where A^* is the Grassmann envelope of A. 2. [. G-graded Specht problem] The T-ideal id_GG(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).

Original language	English
Pages (from-to)	2391-2428
Number of pages	38
Journal	Advances in Mathematics
Volume	225
Issue number	5
DOIs	https://doi.org/10.1016/j.aim.2010.04.025
State	Published - Dec 2010
Externally published	Yes

Keywords

Graded algebra
Polynomial identity

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title = "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 idGG(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/2Z×G-graded algebra A over K such that idGG(W)=idGG(A*) where A* is the Grassmann envelope of A. 2. [. G-graded Specht problem] The T-ideal idGG(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 idG(W)=idG(A).",

keywords = "Graded algebra, Polynomial identity",

author = "Eli Aljadeff and Alexei Kanel-Belov",

note = "Funding Information: ✩ The first author was partially supported by the Israel Science Foundation (grant No. 1283/08) and by the E. Schaver Research Fund. The second author was partially supported by the Israel Science Foundation (grant No. 1178/06). The second author is grateful to the Russian Fund of Fundamental Research for supporting his visit to India in 2008 (grant RFBR 08-01-91300-INDa). * Corresponding author. E-mail addresses: aljadeff@tx.technion.ac.il (E. Aljadeff), beloval@math.biu.ac.il (A. Kanel-Belov).",

year = "2010",

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doi = "10.1016/j.aim.2010.04.025",

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AU - Aljadeff, Eli

AU - Kanel-Belov, Alexei

N1 - Funding Information: ✩ The first author was partially supported by the Israel Science Foundation (grant No. 1283/08) and by the E. Schaver Research Fund. The second author was partially supported by the Israel Science Foundation (grant No. 1178/06). The second author is grateful to the Russian Fund of Fundamental Research for supporting his visit to India in 2008 (grant RFBR 08-01-91300-INDa). * Corresponding author. E-mail addresses: aljadeff@tx.technion.ac.il (E. Aljadeff), beloval@math.biu.ac.il (A. Kanel-Belov).

PY - 2010/12

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N2 - Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idGG(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/2Z×G-graded algebra A over K such that idGG(W)=idGG(A*) where A* is the Grassmann envelope of A. 2. [. G-graded Specht problem] The T-ideal idGG(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 idG(W)=idG(A).

AB - Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idGG(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/2Z×G-graded algebra A over K such that idGG(W)=idGG(A*) where A* is the Grassmann envelope of A. 2. [. G-graded Specht problem] The T-ideal idGG(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 idG(W)=idG(A).

KW - Graded algebra

KW - Polynomial identity

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VL - 225

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JO - Advances in Mathematics

JF - Advances in Mathematics

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ER -

Representability and Specht problem for G-graded algebras

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