相关论文: Degenerations of Leibniz and anticommutative algeb…

The article presents the structure of the automorphism groups of two types of non-nilpotent Leibniz algebras with a dimension of 3.

The Leibniz algebras appeared as a generalization of the Lie algebras. In this work we deal with the classification of nilpotent complex Leibniz algebras of low dimensions. Namely, the classification of nilpotent complex Leibniz algebras…

Let $\boldsymbol\Lambda_3(\mathbb C)\,(=\mathbb C^{27})$ be the space of structure vectors of $3$-dimensional algebras over $\mathbb C$ considered as a $G$-module via the action of $G={\rm GL}(3,\mathbb C)$ on $\boldsymbol\Lambda_3(\mathbb…

All complex $3$-dimensional nilalgebras were described. As a corollary, all degenerations in the variety of complex $3$-dimensional nilalgebras were obtained.

In this work, we investigate anti-derivations and biderivation of Leibniz algebras. We describe general form of anti-derivations and biderivations on null-filiform and filiform Leibniz algebras. Moreover, we show how to construct Leibniz…

We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on…

Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of four dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear…

We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding…

In this paper, we introduce the concepts of representation and dual representation for averaging Leibniz algebras. We also develop a cohomology theory for these algebras. Additionally, we explore the infinitesimal and formal deformation…

We determine the complete degeneration picture inside the variety of nilpotent associative algebras of dimension 3 over an algebraically closed field of characteristic not equal to 2. Comparing with the discussion in [Ivanova N.M. and…

We give necessary conditions for the existence of degenerations between two complex Lie superalgebras of dimension $(m,n)$. As an application, we study the variety $\mathcal{LS}^{(2,2)}$ of complex Lie superalgebras of dimension $(2,2)$.…

Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of $5-$dimensional complex non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of…

We give the classification of all $n$-dimensional anticommutative complex algebras with $(n-3)$-dimensional annihilator. Namely, we describe all central extensions of all $3$-dimensional anticommutative complex algebras.

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional nilpotent Leibniz algebras. In particular, the variety of complex $5$-dimensional nilpotent Leibniz algebras has dimension $24$ it has…

We study endomorphisms and derivations of infinite dimensional cyclic Leibniz algebra.

The aim of this work is to discuss the concepts of degeneration, deformation and rigidity, and to apply them to the geometric study of the varieties of Hopf algebras. The main result is the description of the n-dimensional rigid Hopf…

In this paper we prove that every irreducible representation of a Leibniz algebra can be obtained from irreducible representations of the semisimple Lie algebra from the Levi decomposition. We also prove that - in general - for (semi)simple…

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative…

This paper is a contribution to the development of the non associative algebras theory. More precisely, this work deals with the classification of the complex 4-dimensional Leibniz algebras. Note that the classification of 4-dimensional…

In this paper we focus on algebraic aspects of contractions of Lie and Leibniz algebras. The rigidity of algebras plays an important role in the study of their varieties. The rigid algebras generate the irreducible components of this…