Exact sequences in the cohomology of a Lie superalgebra extension

Abstract

Let be an abelian extension of the Lie superalgebra . In this article we consider the problems of extending endomorphisms of and lifting endomorphisms of to certain endomorphisms of . We connect these problems to the cohomology of with coefficients in through construction of two exact sequences which are our main results. The first exact sequence is obtained using the Hochschild–Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras.