Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras
type: Item
instance of: scholarly article
title: Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras
zbMATH Open document ID: 0933.15029
DOI: 10.1016/S0024-3795(98)10182-9
author: L. W. Marcoux
author: A. R. Sourour
published in: Linear Algebra and its Applications
publication date: 2000-03-21T00:00:00Z
review text: A linear map \(\varphi\) on an algebra is said to be commutativity preserving in both directions when the condition \(ab=ba\) holds if and only if \(\varphi(a)\varphi(b)=\varphi(b)\varphi(a)\). In this paper, the algebra \(\mathcal T_n\) of upper triangular \(n\times n\) matrices over a field is considered. It is shown that for \(n>2\) the commutativity preserving linear map in both directions \(\varphi: \mathcal T_n\rightarrow\mathcal T_n\) is either the sum of an algebra automorphism and a map into the center, or the sum of the negative of an algebra anti-automorphism and a map into the center. That is, \(\varphi(T)=cS^{-1}TS+f(T)I\) or \(\varphi(T)=cS^{-1}JT ^t JS+f(T)I\), where \(c\) is a non-zero scalar, \(S\in\mathcal T_n\) is invertible, \(f(T)\) is a linear functional, \(J=(\delta_{i,n+1-i})\). This result is false for \(n=2\). As a consequence, \(\varphi\) is a Lie automorphism if and only if \(\varphi(T)=S^{-1}TS+\tau(T)I\) or \(\varphi(T)=-S^{-1}JT ^t J S+\tau(T)I\), where \(\tau\) is a linear functional satisfying \(\tau(AB-BA)=0\) for every \(A,B\in\mathcal T_n\) and \(\tau(I)\neq-1\).
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A27
zbMATH DE Number: 1331336
zbMATH Keywords: commutativity preserving
zbMATH Keywords: Lie automorphism
zbMATH Keywords: triangular matrix
MaRDI profile type: Publication
Wikidata QID: Q115339982
cites work: Linear maps preserving commutativity
cites work: Linear transformations on symmetric matrices that preserve commutativity
cites work: On operators preserving commutativity
cites work: Commutativity-preserving operators on symmetric matrices
cites work: Linear maps that preserve commuting pairs of matrices
cites work: Q4529271
cites work: Automorphisms of the Lie algebra of upper triangular matrices over a connected commutative ring
cites work: Lie isomorphisms of nest algebras
full work available at URL: s0024-3795(98)10182-9
OpenAlex ID: W2084725810
description: scientific article; zbMATH DE number 1331336