რევაზ ქურდიანი

For any Lie algebra g, the bracket [x ⊗ y, a ⊗ b] := [x, [a, b]] ⊗ y + x ⊗ [y, [a, b]] defines a Leibniz algebra structure on the vector space g ⊗ g. We let g⊗g be the maximal Lie algebra quotient of g ⊗ g. We prove that this particular Lie algebra is an abelian extension of the Lie algebra version of the nonabelian tensor product g g of Brown and Loday [1] constructed by Ellis [2], [3]. We compute this abelian extension and Leibniz homology of g ⊗ g in the case, when g is a finite dimensional semi-simple Lie algebra over a field of characteristic zero.

In this article, the n-th Hochschild cohomology group is described by (n-2) extensions. For n=2,3 the theorem coincides with known classical results. In the case n=1, we obtain a description of the differentiations group by means of extensions, and for n > 4, this theorem gives a new description of the cohomology groups

The nth Hochschild cohomology group is described by (n-2)-extensions (Theorem 1). When n = 2, 3, the theorem reduces to the well-known classical results; for n = 1, we get a description of the group of derivations by extensions; and for n ≥ 4, this result was recently obtained by Baues and Pirashvili for Shukla cohomology. However, their proof is not explicit. We provide a different and explicit proof in the case of Hochschild cohomology. One can consider this theorem as an alternative definition of cohomology theory. So, one has some kind of hint to define cohomology theory for various algebraic structures.

The aim of this work is to clarify the relationship between homology theory of commutative monoids constructed 'a la Quillen and technology of Gamma-modules.

The present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra or the universal central extension of the Lie algebra. 

We study the capability property of Leibniz algebras via the non-abelian

exterior product.