Generalized derivations of lie superalgebras

Let F be a field of characteristic not equal 2 and L a finite-dimensional Lie superalgebra over F. In this article, we study the derivation superalgebra Der(L), the quasiderivation superalgebra QDer(L), and the generalized derivation superalgebra GDer(L) of L, which form a tower Der(L) subset of QDer(L) subset of GDer(L) subset of pl(L), where pl(L) denotes the general linear Lie superalgebra. More precisely, we characterize completely those Lie superalgebras L for which QDer(L) = pl(L). We prove that the quasiderivations of L can be embedded as derivations in a larger Lie superalgebra L. and, furthermore, when the annihilator of L is equal to zero, we obtain a semidirect sum decomposition of Der (L).