In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.

We describe all degenerations of three-dimensional anticommutative algebras $\mathfrak{A}\mathfrak{c}\mathfrak{o}\mathfrak{m}_{3}$ and of three-dimensional Leibniz algebras $\mathfrak{L}\mathfrak{e}\mathfrak{i}\mathfrak{b}_{3}$ over $\mathbb{C}$. In particular, we describe all irreducible components and rigid algebras in the corresponding varieties.

This article presents results which are consistent with conjectures about Leibniz (co)homology for discrete groups, due to J. L. Loday in 2003. We prove that rack cohomology has properties very close to the properties expected for the conjectural Leibniz cohomology. In particular, we prove the existence of a graded dendriform algebra structure on rack cohomology, and we construct a graded associative algebra morphism H•(−) → HR•(−) from group cohomology to rack cohomology which is injective for ● = 1.

In this work we consider deformations of Leibniz algebras over a field of characteristic zero. The main problem in deformation theory is to describe all non-equivalent deformations of a given object. We give a method to solve this problem completely, namely work out a construction of a versal deformation for a given Leibniz algebra, which induces all non-equivalent deformations and is unique on the infinitesimal level.