On associative superalgebras of matrices.
type: Item
instance of: scholarly article
title: On associative superalgebras of matrices.
zbMATH Open document ID: 1067.16062
author: Constantin Năstăsescu
author: Sorin Dascalescu
author: Peter D Jarvis
author: A. V. Kelarev
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-10-01T00:00:00Z
review text: Let \(G\) be a group, a \(G\)-grading of a matrix ring is called good if the matrix units \(e_{ij}\) are homogeneous. All good gradings of the full matrix ring \(M_k(F)\) were classified by \textit{S. Dăscălescu, B. Ion, C. Năstăsescu}, and \textit{J. Ríos Montes} [J. Algebra 220, No. 2, 709-728 (1999; Zbl 0947.16028)]. Now the authors consider gradings by semigroups. It is well-known that there exist five isomorphism types of semigroups with two elements. The authors classify all gradings for the full matrix algebra \(M_2(F)\) by these semigroups different from \(\mathbb{Z}_2\). It is proved that all these gradings are good. If \(F\) is not algebraically closed, the above reference yields examples of \(\mathbb{Z}_2\)-gradings for \(M_2(F)\) which are not isomorphic to good ones. The authors also classify all gradings by the semigroups with two elements for the algebra of \(2\times 2\) upper triangular matrices. The existence of non-good gradings is also considered. In some cases it is assumed that the characteristic of the base field \(F\) is different from 2.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 2104652
zbMATH Keywords: rings of matrices
zbMATH Keywords: superalgebras
zbMATH Keywords: good gradings
zbMATH Keywords: semigroup gradings
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069869
OpenAlex ID: W2050781879
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DOI: 10.1216/RMJM/1181069869
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description: scientific article; zbMATH DE number 2104652