Lie triple derivations of triangular algebras.
type: Item
instance of: scholarly article
title: Lie triple derivations of triangular algebras.
zbMATH Open document ID: 1253.16042
author: Feng Wei
author: Zhankui Xiao
published in: Linear Algebra and its Applications
publication date: 2012-08-03T00:00:00Z
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 15A78
Mathematics Subject Classification ID: 47L35
zbMATH DE Number: 6063148
zbMATH Keywords: Lie triple derivations
zbMATH Keywords: triangular algebras
zbMATH Keywords: nest algebras
Wikidata QID: Q115344668
MaRDI profile type: Publication
full work available at URL: j.laa.2012.04.015
OpenAlex ID: W2052858174
cites work: Q4212443
cites work: Multiplicative Lie \(n\)-derivations of triangular rings.
cites work: Lie ideals and derivations of prime rings
cites work: Commuting Traces of Biadditive Mappings, Commutativity-Preserving Mappings and Lie Mappings
cites work: Jordan derivations revisited
cites work: Q3597361
cites work: Nonlinear Lie derivations on upper triangular matrices
cites work: Lie Derivations of Triangular Algebras
cites work: Q4529271
cites work: Q3014399
cites work: Lie triple derivations of TUHF algebras
cites work: Lie triple derivations on nest algebras
cites work: Lie derivations of primitive rings
cites work: The structure of Lie derivations on \(C^{\ast}\)-algebras
cites work: Lie derivations of von Neumann algebras
cites work: Lie Triple Derivations of Von Neumann Algebras
cites work: Derivations in Prime Rings
cites work: Additive Lie (\(\xi \)-Lie) derivations and generalized Lie (\(\xi \)-Lie) derivations on nest algebras
cites work: Q3014446
cites work: Lie triple derivations of nest algebras on Banach spaces
cites work: Lie triple derivation of the Lie algebra of strictly upper triangular matrix over a commutative ring
cites work: Lie triple derivations on the parabolic subalgebras of simple Lie algebras
cites work: Nonlinear Lie-type derivations on full matrix algebras.
cites work: Nonlinear Lie derivations of triangular algebras.
cites work: Lie triple derivations of nest algebras
DOI: 10.1016/J.LAA.2012.04.015
review text: Let \(R\) be a 2-torsion free commutative ring with identity element, \(A\) and \(B\) unital \(R\)-algebras, and \(M\) an \(A\)-\(B\) bi-module that is faithful on each side. These define a formal triangular algebra \(T(A,M,B)=T=A\oplus M\oplus B\) as an additive group, with \((a,m,b)\cdot(a',m',b')=(aa',am'+mb',bb')\) the multiplication in \(T\). Note that the center of \(T\), \(Z(T)=\{(a,0,b)\in T\mid a\in Z(A),\;b\in Z(B)\), and \(am=mb\) for all \(m\in M\}\).NEWLINENEWLINE A Lie triple derivation of any ring \(W\) is an additive \(D\colon W\to W\) so that for all \(x,y,z\in W\), \(D([[x,y],z])=[[D(x),y],z]+[[x,D(y)],z]+[[x,y],D(z)]\).NEWLINENEWLINE The main result of the authors shows that if \(D\) is a Lie triple derivation of \(T\), as above, if the projections of \(Z(T)\) to \(Z(A)\) and \(Z(B)\) are surjective, and if either \(Z(A)=\{a\in A\mid [[a,x],y]=0\) for all \(x,y\in A\}\) or \(Z(B)=\{b\in B\mid [[b,x],y]=0\) for all \(x,y\in B\}\) then \(D=d+f\) for \(d\in\text{Der}(T)\) and \(f\colon T\to Z(T)\), an \(R\)-module map so that \(f([[T,T],T])=0\). One application of this result applies to any Lie triple derivation of the usual upper triangular matrix algebra \(T_n(R)\). In this case, since \(R\) is commutative, the additional hypotheses in the main theorem hold automatically, and it also happens that \(f([T,T])=0\).
description: scientific article; zbMATH DE number 6063148