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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a class of finitary Lie algebras characterized through derivations


Authors: Matej Brešar and Antonio Fernández López
Journal: Proc. Amer. Math. Soc. 138 (2010), 4161-4166
MSC (2010): Primary 17B40, 17B65; Secondary 16W10
Published electronically: August 10, 2010
MathSciNet review: 2680042
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Abstract: Let $ L$ be an infinite-dimensional simple Lie algebra over a field of characteristic 0. Then there exist a derivation $ d$ on $ L$ and $ n\ge 2$ such that $ d^n$ is a nonzero finite rank map if and only if $ L$ is finitary and contains a nonzero finite-dimensional abelian inner ideal. This is a partial statement of our main theorem. As auxiliary results needed for the proof we establish some properties of derivations in general nonassociative algebras.


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Additional Information

Matej Brešar
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadravska ulica 19, SI-1000 Ljubljana, Slovenia – and – Faculty of Natural Sciences and Mathematics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia
Email: matej.bresar@fmf.uni-lj.si

Antonio Fernández López
Affiliation: Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Malaga, Spain
Email: emalfer@uma.es

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10644-5
PII: S 0002-9939(2010)10644-5
Keywords: Derivation, finite rank, nonassociative algebra, simple Lie algebra, finitary Lie algebra, inner ideal
Received by editor(s): October 27, 2009
Published electronically: August 10, 2010
Additional Notes: The first author was supported by the Slovenian Research Agency (Program No. P1-0288).
The second author was supported by the MEC and Fondos FEDER, MTM2007-61978
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2010 American Mathematical Society