Identities and isomorphisms of graded simple algebras.
type: Item
instance of: scholarly article
title: Identities and isomorphisms of graded simple algebras.
zbMATH Open document ID: 1194.16016
author: Plamen Koshlukov
author: Mikhail Zaicev
published in: Linear Algebra and its Applications
publication date: 2010-05-11T00:00:00Z
review text: The authors study graded polynomial identities of finite dimensional \(G\)-graded simple algebras, where \(G\) is an Abelian group, the base field \(F\) is algebraically closed and such that the orders of all finite subgroups of \(G\) are invertible in \(F\). The main result is that two finite dimensional \(G\)-graded simple algebras are isomorphic as graded algebras if and only if they satisfy the same \(G\)-graded polynomial identities. The proof gives also the explicit form of the algebras and the graded polynomial identities which distinguish them. The algebras are of the form \(R=A\otimes_FB\), where \(A=M_n(F)\) is the \(n\times n\) matrix algebra with an elementary grading and \(B\) is a graded division algebra (i.e., the non-zero homogeneous elements of \(B\) are invertible). The identities used to distinguish the algebras are modifications of standard identities. The algebra \(B\) is a twisted group algebra \(F^\sigma[H]\) for a finite subgroup \(H\) of \(G\). Hence the classification the finite dimensional \(G\)-graded simple algebras is up to equivalence of the 2-cocycles \(\sigma\colon H\times H\to F^*\) defining the twisted group structure of \(B\).
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16S35
zbMATH DE Number: 5706972
zbMATH Keywords: graded simple algebras
zbMATH Keywords: gradings on associative algebras
zbMATH Keywords: graded division algebras
zbMATH Keywords: graded isomorphisms
zbMATH Keywords: graded identities
zbMATH Keywords: polynomial identities
MaRDI profile type: Publication
full work available at URL: j.laa.2010.01.010
OpenAlex ID: W1967760143
cites work: Graded identities of matrix algebras and the universal graded algebra
cites work: Group gradings on associative algebras
cites work: Group Gradings on Matrix Algebras
cites work: Q4474887
cites work: Finite-dimensional simple graded algebras
cites work: Distinguishing simple jordan algebras by means of polynomial identities
cites work: Polynomial identities for graded tensor products of algebras.
cites work: Q5714236
cites work: Q3663493
cites work: Polynomial identities and nonidentities of split Jordan pairs
cites work: Q4396508
DOI: 10.1016/J.LAA.2010.01.010
description: scientific article; zbMATH DE number 5706972