A note on derivations of simple algebras
type: Item
instance of: scholarly article
title: A note on derivations of simple algebras
zbMATH Open document ID: 0829.17002
author: Alberto Elduque
author: Fernando Montaner
published in: Journal of Algebra
publication date: 1996-01-22T00:00:00Z
review text: A derivation of a nonassociative algebra \(A\) is called inner if it is contained in the Lie multiplication algebra \({\mathcal L}(A)\) of \(A\). For several classes of algebras it is known that the simple algebras in the respective class have only inner derivations. On the other hand, there are simple algebras (of finite dimension over an algebraically closed field of zero characteristic) which have non-inner derivations, as shown by the reviewer. Also, for simple finite dimensional algebras over an algebraically closed base field of characteristic zero there exist non- inner derivations only if every left or right multiplication by an element of \(A\) has zero trace. (In that case, derivations with nonzero trace, if they exist, are not inner.) This follows from a theorem of Sagle and Walde [\textit{A. A. Sagle} and \textit{R. E. Walde}, Introduction to Lie groups and Lie algebras (Academic Press, New York 1973; Zbl 0252.22001)]. In this paper, the authors extend the results above to arbitrary base fields of characteristic zero. (Note that derivations behave nicely under base field extensions in the characteristic zero case, due to separability!) Furthermore, they discuss the relation between semisimplicity of \({\mathcal L}(A)\) and innerness of derivations, and show that the existence of a nondegenerate trace form forces all derivations to be inner.
reviewed by: S Walcher
Mathematics Subject Classification ID: 17A36
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 589965
zbMATH Keywords: reductive Lie algebra
zbMATH Keywords: Lie multiplication algebra
zbMATH Keywords: inner derivations
zbMATH Keywords: simple algebras
zbMATH Keywords: semisimplicity
MaRDI profile type: Publication
full work available at URL: jabr.1994.1136
OpenAlex ID: W1978055888
DOI: 10.1006/JABR.1994.1136
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description: scientific article; zbMATH DE number 589965