Derivations of Locally Simple Lie Algebras

Karl-Hermann Neeb

Abstract: Let $\g$ be a locally finite Lie algebra over a field of characteristic zero which is a direct limit of finite-dimensional simple ones. In this short note it is shown that each invariant symmetric bilinear form on $\g$ is invariant under all derivations and that each such form defines a natural embedding der$ \g \to \g^*$. The latter embedding is used to determine der$ \g$ explicitly for all locally finite split simple Lie algebras.

Keywords: Locally finite Lie algebra, simple Lie algebra, derivation, direct limit

Classification (MSC2000): 17B65, 17B20, 17B56

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