Some Lie algebras and groups associated to representations of Leibniz algebras

Given a representation (M; l, r) of a Leibniz algebra g, let D(g, M) (respectively, G(g, M)) be the Lie algebra (respectively, the group) of diagonal derivations (respectively, automorphisms) of the semidirect product g x M. We show that both D(g, M) and G(g, M) have a representation on the cohomology group HL2(g, M). In the case that (M; l, r) arises from an abelian extension of g by M, such representations are applied to construct exact sequences of Wells type for D(g, M) and G(g, M), respectively.