On \(\delta\)-derivations of Lie algebras
type: Item
instance of: scholarly article
title: On \(\delta\)-derivations of Lie algebras
zbMATH Open document ID: 0936.17020
DOI: 10.1007/BF02674132
author: V. T. Filippov
published in: Siberian Mathematical Journal
publication date: 2000-04-16T00:00:00Z
review text: A linear mapping \(\varphi\) of an algebra \(A\) over a ring of scalars \(\Phi\) is called a \(\delta\)-derivation of \(A\), for a certain \(\delta\in\Phi\), if the equality \((xy)^{\varphi}=\delta(x^{\varphi}y+xy^{\varphi})\) holds for all \(x,y\in A\). The 1-derivations of \(A\) are exactly the ordinary derivations of \(A\); the elements of the centroid \(\Gamma (A)\) provide examples of 1/2-derivations; the 0-derivations of \(A\) can be described as the linear mappings with zero restrictions to \(A^2\). The \(\delta\)-derivations of these types are called trivial. The author first notices that every \(\delta\)-derivation of a Lie algebra \(L\) (with \(\delta\neq 0,1\)) provides a certain \(R\)-matrix for \(L\). The main result of the article states that a prime Lie \(\Phi\)-algebra \(L\) of characteristic different from \(2,3\), with a nondegenerate symmetric invariant bilinear form, has no nontrivial \(\delta\)-derivations, except the case in which \(\delta=-1\) and \(L\) is a central order in a simple 3-dimensional Lie algebra over a field.
Mathematics Subject Classification ID: 17B40
zbMATH DE Number: 1432396
zbMATH Keywords: \(\delta \)-derivation
zbMATH Keywords: prime Lie algebra
reviewed by: Ivan Shestakov
MaRDI profile type: Publication
cites work: Q4356756
cites work: Q4363302
full work available at URL: https://doi.org/10.1007/bf02674132
OpenAlex ID: W2317713845
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description: scientific article; zbMATH DE number 1432396