Some Lie algebras and groups associated to representations of Leibniz algebras

Abstract

Given a representation [Formula: see text] of a Leibniz algebra [Formula: see text], let [Formula: see text] (respectively,[Formula: see text][Formula: see text]) be the Lie algebra (respectively, [Formula: see text]the group) of diagonal derivations (respectively, automorphisms) of the semidirect product [Formula: see text]. We show that both [Formula: see text] and [Formula: see text] have a representation on the cohomology group [Formula: see text]. In the case that [Formula: see text] arises from an abelian extension of [Formula: see text] by [Formula: see text], such representations are applied to construct exact sequences of Wells type for [Formula: see text] and [Formula: see text] respectively.