Decomposition of Lie automorphisms of upper triangular matrix algebra over commutative rings
type: Item
instance of: scholarly article
title: Decomposition of Lie automorphisms of upper triangular matrix algebra over commutative rings
zbMATH Open document ID: 1109.17007
author: Xing Tao Wang
author: Hong You
published in: Linear Algebra and its Applications
publication date: 2007-02-14T00:00:00Z
review text: Let 2 be an invertible element in the commutative ring \(R\) and let \(T_{n+1}(R)\) be the algebra of \((n+1)\times (n+1)\) upper triangular matrices with entries in \(R\). In the paper under review the authors describe the Lie automorphisms of \(T_{n+1}(R)\), i.e. the \(R\)-module automorphisms \(\varphi\) of \(T_{n+1}(R)\) with the property \(\varphi([x,y])=[\varphi(x),\varphi(y)]\), \(x,y\in T_{n+1}(R)\). The main result is that for \(n\geq 2\) every Lie automorphism \(\varphi\) is a product of four automorphisms, \(\varphi=\omega_{\varepsilon}\eta_c\theta\lambda_d\). Here the construction of the graph automorphism \(\omega_{\varepsilon}\) involves an idempotent \(\varepsilon\) of \(R\); \(\eta_c\) is a central automorphism which is identical on the strictly upper triangular matrices and is of the form \(\eta_c(e_{jj})=e_{jj}+c_je\), \(c_j\in R\), on the diagonal; the inner automorphism \(\theta\) conjugates by a unitriangular matrix; the diagonal automorphism \(\lambda_d\) conjugates by an invertible diagonal matrix. In the final remarks the authors mention that the case of \(2\times 2\) upper triangular matrices is simpler and also give the complete description of the Lie automorphisms.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 13C10
zbMATH DE Number: 5126021
zbMATH Keywords: Lie automorphism
zbMATH Keywords: triangular matrix algebra
zbMATH Keywords: commutative ring
MaRDI profile type: Publication
full work available at URL: j.laa.2006.05.012
OpenAlex ID: W1991788354
Wikidata QID: Q115344911
cites work: Automorphisms of certain Lie algebras of upper triangular matrices over a commutative ring
cites work: Automorphisms of the Lie algebra of upper triangular matrices over a connected commutative ring
cites work: Automorphisms of upper triangular matrix rings
cites work: The automorphism group of certain radical matrix rings
cites work: Decomposition of Jordan automorphisms of strictly triangular matrix algebra over local rings
DOI: 10.1016/J.LAA.2006.05.012
description: scientific article; zbMATH DE number 5126021