Almost inner derivations of Lie superalgebras

Abstract

An almost inner derivation of a Lie algebra $L$ is a derivation that coincides with an inner derivation on each one-dimensional subspace of $L$. The almost inner derivations form a subalgebra $aDer(L)$ of the Lie algebra $Der(L)$ of all derivations of $L$, containing the inner derivations $iDer(L)$ as an ideal. If $L$ is a simple finite-dimensional Lie algebra, then $aDer(L)=iDer(L)$, since all derivations of $L$ are inner. In this paper, we introduce and study almost inner derivations derivations of Lie superalgebras. Since simple Lie superalgebras may admit non-inner outer derivations, the existence of non-inner almost inner derivations becomes a nontrivial question. Nevertheless, we show that all almost inner derivations of finite-dimensional simple Lie superalgebras over $\mathbb{C}$ are inner. We also give examples of naturally occurring non-inner almost inner derivations derivations of some pseudo-reductive Lie superalgebras related to the Sato-Kimura classification of prehomogeneous vector spaces.