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arXiv · 1308.4617

Reductivity of the Lie algebra of a bilinear form

S. Ruhallah Ahmadi·Martin Chaktoura·Fernando Szechtman
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Abstract

Let $f:V\times V\to F$ be a totally arbitrary bilinear form defined on a finite dimensional vector space $V$ over a a field $F$, and let $L(f)$ be the subalgebra of $\gl(V)$ of all skew-adjoint endomorphisms relative to $f$. Provided $F$ is algebraically closed of characteristic not 2, we determine all $f$, up to equivalence, such that $L(f)$ is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for $L(f)$ to be simple, semisimple or isomorphic to $\sl(n)$ for some $n$.

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S. Ruhallah Ahmadi, Martin Chaktoura, Fernando Szechtman. 2013-08-21. Reductivity of the Lie algebra of a bilinear form. https://arxiv.org/abs/1308.4617

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