Almost inner derivations of Lie superalgebras

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 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 C are inner. We also give examples of naturally occurring non-inner almost inner derivations of some quasireductive Lie superalgebras related to the Sato-Kimura classification of prehomogeneous vector spaces.