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Strongly non-degenerate Lie algebras

Authors: Francesc Perera and Mercedes Siles Molina
Journal: Proc. Amer. Math. Soc. 136 (2008), 4115-4124
MSC (2000): Primary 17B60; Secondary 16W25
Published electronically: July 23, 2008
MathSciNet review: 2431022
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Abstract: Let $ A$ be a semiprime $ 2$- and $ 3$-torsion free non-commutative associative algebra. We show that the Lie algebra $ \mathcal{D}\mathrm{er}(A)$ of (associative) derivations of $ A$ is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of $ A$. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra $ A$ with involution and the Lie algebra $ \mathrm{SDer}(A)$ of involution preserving derivations of $ A$.

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Francesc Perera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Mercedes Siles Molina
Affiliation: Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain

Received by editor(s): April 13, 2007
Received by editor(s) in revised form: September 26, 2007
Published electronically: July 23, 2008
Additional Notes: The first author was partially supported by the DGI MEC-FEDER through Project MTM2005-00934 and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
The second author was partially supported by the MEC and Fondos FEDER jointly through project MTM2004-06580-C02-02 and by the Junta de Andalucía PAI, projects FQM-336 and FQM-1215.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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