Reductivity of the Lie algebra of a bilinear form

TL;DR

This paper characterizes when the Lie algebra of skew-adjoint endomorphisms relative to a bilinear form is reductive, providing a classification that includes conditions for simplicity and semisimplicity over various fields.

Contribution

It determines all bilinear forms for which the associated Lie algebra is reductive, extending the classification to arbitrary fields and identifying conditions for simplicity and semisimplicity.

Findings

Classification of forms with reductive Lie algebra $L(f)$

Necessary and sufficient conditions for $L(f)$ to be simple or semisimple

Explicit description of $L(f)$ as $ ext{sl}(n)$ in certain cases

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 $n$.