Abelian extensions of Leibniz algebras

In this work(1) we continue the study of Leibniz algebras concentrating on their abelian extensions. We introduce the forward/backward induced extensions to endow the set Ext(g, N) of (classes of) abelian 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 0 --> Der(q, -) --> Der(g, -) --> Hom(q)(n(ab), -) --> HL2(q, -) --> HL2(g, -) 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".