Abelian extensions of leibniz algebras

Abstract

In this work1 we continue the study of Leibniz algebras concen-trating on their abelian extensions. We introduce the forward/backward in-duced extensions to endow the set Ext(g, N) of (classes of) abeliar. extensions of a Leibniz algebra g (by a g-module N) with a vector space structure. As an application of the above we obtain a simple proof of the product-preserving property of the second Leibniz cohomology group functor HL2(g, -). Our main new result is that to each short exact sequence of Leibniz algebras [n ↦ g → q] there corresponds a five-term natural exact sequence of vector space-valued functors defined on the category of q-modules. In the last section we use this sequence together with another one introduced in [1] to prove a"Universal Coefficient Theorem.