Graded polynomial identities of matrices
type: Item
instance of: scholarly article
title: Graded polynomial identities of matrices
zbMATH Open document ID: 1019.16011
DOI: 10.1016/S0024-3795(02)00356-7
author: Vesselin Drensky
author: Yuri Bahturin
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: Gradings by an Abelian group \(G\) on the \(p\times p\) matrix algebra \(M_p(K)\) over an algebraically closed field \(K\) were classified by \textit{Yu. A. Bahturin, S. K. Sehgal} and \textit{M. V. Zaicev} [J. Algebra 241, No. 2, 677-698 (2001; Zbl 0988.16033)]. The so called fine gradings and elementary gradings are the building blocks of this classification. The present paper studies \(G\)-graded identities of \(M_p(K)\), where \(K\) is a field of characteristic zero, and \(G\) is an arbitrary group. The authors find relations between the \(G\)-graded identities of \(M_p(K)\), and the \(G\times H\)-graded identities of the tensor product of \(M_p(K)\) and \(M_q(K)\), where the latter is endowed with a fine \(H\)-grading. This allows to reduce the study of \(G\)-graded identities (\(G\) Abelian) of matrices to the case of elementary gradings. Generalizing a result of \textit{S. Yu. Vasilovsky} [Proc. Am. Math. Soc. 127, No. 12, 3517-3524 (1999; Zbl 0935.16012)], the authors give a basis of the \(G\)-graded identities of \(M_p(K)\) with an elementary grading such that the identity component coincides with the diagonal of \(M_p(K)\).
reviewed by: Matyas Domokos
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 1864766
zbMATH Keywords: full matrix algebras
zbMATH Keywords: graded algebras
zbMATH Keywords: polynomial identities
zbMATH Keywords: graded identities
zbMATH Keywords: fine gradings
zbMATH Keywords: elementary gradings
zbMATH Keywords: bases of identities
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1864766