Lie maps on triangular algebras without assuming unity
type: Item
instance of: scholarly article
title: Lie maps on triangular algebras without assuming unity
zbMATH Open document ID: 1484.16048
author: Roonak Behfar
author: Hoger Ghahramani
published in: Mediterranean Journal of Mathematics
publication date: 2021-09-20T00:00:00Z
review text: In this paper all algebras will be considered over a commutative unital ring. The authors characterize Lie centralizer, Lie derivations and generalized Lie \(2\)-derivations at zero products on triangular algebras without assuming unity. The Lie centralizers, Lie derivations and generalized Lie \(2\)-derivations are described for triangular algebras. Several examples are presented which illustrate limitations on extending some of the theory developed. The obtained results are used for upper triangular matrix algebras (not necessarily unital) and nest algebras. Let \(A\) be an algebra. A linear mapping \(\Psi : A \to A\) is called a \textit{centralizer} if \(\Psi(ab) = \Psi(a)b = a\Psi(b)\) for any \(a, b \in A\). Lie product on \(A\) is defined as \([a, b] = ab-ba\) for any \(a, b \in A\). We say that the linear mapping \(\Phi : A \to A\) is a \textit{Lie centralizer} if \(\Phi([a, b]) = [\Phi(a), b]\) for any \(a,b\in A\). Every centralizer is a Lie centralizer but the converse is not necessarily true. One can define a \textit{Lie derivation} in a natural way, considering associative algebra \(A\) as a Lie algebra with bracket operation. Let \(\Gamma: A\to A\) be a linear mapping and \(\Delta\) be a Lie derivation on \(A\). Then \(\Gamma\) is a \textit{generalized Lie \(2\)-derivation associated with the Lie derivation \(\Delta\)} if \(\Gamma([a, b]) = [\Gamma(a), b] + [a, \Delta(b)]\) for any \(a, b \in A\). Every Lie derivation \(\Delta\) is a generalized Lie \(2\)-derivation associated with \(\Delta\). This notion is inspired by the notion of module derivation comparing with a ring derivation. In fact, the paper aimed to study algebras, related with modulo constructions (so called \textit{triangular algebras} according to J. Lewin). Suppose \(A, B\) are algebras, \(M\)~ an \(A,B\)-bimodule, then \textit{triangular algebra} associated is \[ \mathrm{Tri}(A,M,B)= \left\{ \left( \begin{array}{cc} a & m \\ 0 & b \\ \end{array} \right) \| a\in A, b\in B, m\in M \right…
reviewed by: A. Ya Belov
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 15A78
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 47B49
zbMATH DE Number: 7397613
zbMATH Keywords: triangular algebra
zbMATH Keywords: Lie centralizer
zbMATH Keywords: Lie derivation
zbMATH Keywords: generalized Lie 2-derivation
MaRDI profile type: Publication
full work available at URL: s00009-021-01836-z
OpenAlex ID: W3197985961
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DOI: 10.1007/S00009-021-01836-Z
description: scientific article; zbMATH DE number 7397613