Strongly non-degenerate Lie algebras

Perera, Francesc; Molina, Mercedes

doi:10.1090/S0002-9939-08-09558-0

Strongly non-degenerate Lie algebras
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by Francesc Perera and Mercedes Siles Molina

Proc. Amer. Math. Soc. 136 (2008), 4115-4124

DOI: https://doi.org/10.1090/S0002-9939-08-09558-0

Published electronically: July 23, 2008

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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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