On \(F\)-algebras of algebraic matrices over a subfield \(F\) of the center of a division ring
type: Item
instance of: scholarly article
title: On \(F\)-algebras of algebraic matrices over a subfield \(F\) of the center of a division ring
zbMATH Open document ID: 1108.15016
author: Bamdad R. Yahaghi
published in: Linear Algebra and its Applications
publication date: 2006-12-07T00:00:00Z
review text: The author considers the structure of irreducible subalgebras of \(n\)-by-\(n\) matrices over a fixed division ring \(D\). Suppose \(F\subseteq D\) is a division subring, contained in the center of \(D\). The main result classifies irreducible \(F\)-subalgebras in \(M_n(D)\) that consist of \(F\)-algebraic matrices. It is proven that, modulo similarity, such an algebra equals \(M_{n/r}({\mathcal D}_r)\) for some integer \(r\mid n\), and some irreducible division subring \({\mathcal D}_r\subset M_r(D)\). Application to irreducibility and to simultaneous triangularizability of subalgebras, consisting of \(F\)-algebraic matrices, are also given. In case of commutative \(D\), the main result, for instance, implies that (i) irreducible \(F\)-subalgebra of \(M_n(D)\) equals \(M_n(F)\) modulo similarity, provided the spectrum of any its element is contained in \(F\), and (ii) that an irreducible semigroup \({\mathcal S}\subset M_n(D)\) lies in \(M_n(F)\) modulo similarity, provided that \(\{0\}\not=\) Tr\(\,{\mathcal S}\subseteq F\), and \(F\) enjoys certain algebraic property. Reviewer's remark: A more general problem than (i) asks whether the irreducible \textit{semigroup} lies in \(M_n(F)\), modulo similarity, provided the spectrum of any its element is contained in \(F\). This was already answered positively for some special fields \(F\). Besides the references listed in the article, \(p\)-adic fields were treated in a paper by \textit{J. Bernik} [Pac. J. Math. 208, No. 2, 201--207 (2003; Zbl 1057.16013)]. The reviewer was informed that J. Bernik recently managed to give a positive answer for arbitrary fields \(F\).
reviewed by: Bojan Kuzma
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 5077998
zbMATH Keywords: irreducible subalgebra of matrices
zbMATH Keywords: triangularizability
zbMATH Keywords: trace
zbMATH Keywords: spectra
zbMATH Keywords: inner eigenvalues
zbMATH Keywords: semigroup of matrices
zbMATH Keywords: \(F\)-algebra
MaRDI profile type: Publication
full work available at URL: j.laa.2006.03.014
OpenAlex ID: W2075409444
cites work: Q3963168
cites work: Q4214899
cites work: Permutability of characters on algebras
cites work: Structure of algebras
cites work: Q4044675
cites work: Q3998992
cites work: Q5614003
cites work: Burnside's Theorem on Algebras of Matrices
cites work: Q4138841
cites work: Triangularizability of algebras over division rings
cites work: On semigroups of matrices with traces in a subfield
cites work: A finiteness lemma, brauer's theorem and other iIrreducibility results
cites work: Burnside's theorem for matrix rings over division rings
cites work: On one-sided ideals of rings of linear transformations
cites work: A Trace Condition Equivalent to Simultaneous Triangularizability
cites work: Simultaneous triangularization
cites work: Block triangularization of algebras of matrices
cites work: On irreducible semigroups of matrices with traces in a subfield
cites work: Near Triangularizability Implies Triangularizability
cites work: Q2714335
cites work: Theorems of Burnside and Wedderburn revisited
DOI: 10.1016/J.LAA.2006.03.014
description: scientific article; zbMATH DE number 5077998