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A NOTE ON DERIVATIONS OF LIE ALGEBRAS

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  21 July 2011

M. SHAHRYARI
M. SHAHRYARI*
Affiliation:
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran (email: [email protected])
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Abstract

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In this note, we will prove that a finite-dimensional Lie algebra L over a field of characteristic zero, admitting an abelian algebra of derivations D≤Der(L), with the property for some n>1, is necessarily solvable. As a result, we show that if L has a derivation d:L→L such that Ln⊆d(L), for some n>1, then L is solvable.

Keywords

Lie algebrasderivationssolvable Lie algebrascompact Lie groups

MSC classification

Secondary: 17B40: Automorphisms, derivations, other operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 3 , December 2011 , pp. 444 - 446
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Benkart, G., Kostrikin, A. I. and Kuznestov, M. I., ‘Finite dimensional Lie algebras with a non-singular derivation’, J. Algebra 171 (1995), 894–916.Google Scholar
[2]Jacobson, N., ‘A note on automorphisms and derivations of Lie algebras’, Proc. Amer. Math. Soc. 6 (1955), 281–283.Google Scholar
[3]Ladisch, F., ‘Groups with anti-central elements’, Comm. Algebra 36 (2008), 2883–2894.Google Scholar