Nilpotent Leibniz Algebras Research Articles

Isotopisms of nilpotent Leibniz algebras and Lie racks

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J 1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks.

On classification of $7$-dimensional odd-nilpotent Leibniz algebras

In this paper we extend the method of canonical form for congruence of bilinear forms to give the classification of some subclasses of $7-$dimensional nilpotent Leibniz algebras. Odd-nilpotent Leibniz algebras are defined as that its even dimensional ideals in lower central series are all zero and the classification of $7-$dimensional complex odd-nilpotent Leibniz algebras with one dimensional Leib ideal is obtained by applying the aforementioned method.

Low-dimensional nilpotent Leibniz algebras and their automorphism groups

Low-dimensional nilpotent Leibniz algebras and their automorphism groups

On 7-Dimensional Nilpotent Leibniz Algebras With 1-Dimensional Leib Ideal

Leibniz algebras are nonanticommutative versions of Lie algebras. Lie algebras have many applications in many scientific areas as well as mathematical areas. Scientists from different disciplines have used specific examples of Lie algebras according to their needs. However, we mathematicians are more interested in generality than in obtaining a few examples. The classification problem for Leibniz algebras has an intrinsically wild nature as in Lie algebras. In this article, the approach of congruence classes of bilinear forms is extended to classify certain subclasses of seven-dimensional nilpotent Leibniz algebras over complex numbers. Certain cases of seven-dimensional complex nilpotent Leibniz algebras of those with one-dimensional Leib ideal and derived algebra of codimension two are classified.

On the cohomology of solvable Leibniz algebras

This paper is a sequel to a previous paper of the authors in which the cohomology of semi-simple Leibniz algebras was computed by using spectral sequences. In the present paper we generalize the vanishing theorems of Dixmier and Barnes for nilpotent and (super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the cohomology of the one-dimensional Lie algebra with values in an arbitrary Leibniz bimodule and show that it is periodic with period two. As a consequence, we establish the Leibniz analogue of a non-vanishing theorem of Dixmier for nilpotent Leibniz algebras. In addition, we prove a Fitting lemma for Leibniz bimodules

On the algebra of derivations of some nilpotent Leibniz algebras

We describe the algebra of derivations of some nilpotent Leibniz algebra, having dimensionality 3.

On the algebra of derivations of some low-dimensional Leibniz algebras

Let L be an algebra over a field F with the binary operations + and [,]. Then L is called a left Leibniz algebra if it satisfies the left Leibniz identity [[a,b],c]=[a,[b,c]]−[b,[a,c]] for all a,b,c∈L. We study the algebras of derivations of nilpotent Leibniz algebras of low dimensions.

Central extensions of 2-filiform Leibniz algebras

In this paper we describe the central extensions of some nilpotent Leibniz algebras and one-dimensional central extensions (up to isomorphism) of all naturally graded 2-filiform Leibniz algebras over the field of complex numbers. It is well-know that 2-filiform Leibniz algebras which are not naturally graded have not yet studied. There are such kind of algebras in the algebras obtained along this article. Also in this work we describe some subclasses of 6-dimensional nilpotent Leibniz algebras with condition quotient algebra by two-dimensional center of the algebra is isomorphic to the four-dimensional naturally graded 2-filiform nonsplit Leibniz algebra.

Two-step nilpotent Leibniz algebras

In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. Among these, we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical (2n+1)-dimensional Heisenberg Lie algebra h2n+1. Then we use the Leibniz algebras - Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. Finally we show that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the Lie algebra of that Lie group.

Pre-derivations and description of non-strongly nilpotent filiform Leibniz algebras

Abstract In this paper we give the description of some non-strongly nilpotent Leibniz algebras. We pay our attention to the subclass of nilpotent Leibniz algebras, which is called filiform. Note that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three disjoint families. We describe the pre-derivations of filiform Leibniz algebras for the first and second families and determine those algebras in the first two classes of filiform Leibniz algebras that are non-strongly nilpotent.

Characterizing nilpotent Leibniz algebras by their multiplier

In this paper, we characterize all finite dimensional nilpotent Leibniz algebras L for which (dim⁡(L))2−dim⁡(HL2(L))≤12.

Almost Inner Derivations of Some Nilpotent Leibniz Algebras

We investigate almost inner derivations of some finite-dimensional nilpotent Leibniz algebras. We show the existence of almost inner derivations of Leibniz filiform non-Lie algebras differing from inner derivations, we also show that the almost inner derivations of some filiform Leibniz algebras containing filiform Lie algebras do not coincide with inner derivations

Classification of some subclasses of 6−dimensional nilpotent Leibniz algebras

This article is a contribution to the improvement of classification theory in Leibniz algebras. We extend the method of congruence classes of matrices of bilinear forms that was used to classify complex nilpotent Leibniz algebras with one dimensional derived algebra. In this work we focus on applying this method to the classification of 6-dimensional complex nilpotent Leibniz algebras with two dimensional derived algebra.

On Degenerations and Invariants of Low-Dimensional Complex Nilpotent Leibniz Algebras

Given two algebras and , if lies in the Zariski closure of the orbit , we say that is a degeneration of . We denote this by . Degenerations (or contractions) were widely applied to a range of physical and mathematical point of view. The most well-known example oriented to the application on degenerations is limiting process from quantum mechanics to classical mechanics under that corresponds to the contraction of the Heisenberg algebras to the abelian ones of the same dimension. Research on degenerations of Lie, Leibniz and other classes of algebras are very active. Throughout the paper we are dealing with mathematical background with abstract algebraic structures. The present paper is devoted to the degenerations of low-dimensional nilpotent Leibniz algebras over the field of complex numbers. Particularly, we focus on the classification of three-dimensional nilpotent Leibniz algebras. List of invariance arguments are provided and its dimensions are calculated in order to find the possible degenerations between each pair of algebras. We show that for each possible degenerations, there exists construction of parameterized basis on parameter We proof the non-degeneration case for mentioned classes of algebras by providing some reasons to reject the degenerations. As a result, we give complete list of degenerations and non-degenerations of low-dimensional complex nilpotent Leibniz algebras. In future research, from this result we can find its rigidity and irreducible components.

Some basic properties of solvable Leibniz algebras

In this paper, we commence the systematic study of the structure of solvable Leibniz algebras and we give some basic properties on solvable non-Lie Leibniz algebras. Our main aim is to show a close relationship between solvable Leibniz algebras and nilpotent Leibniz algebras. Furthermore, we give some examples on solvable non-Lie Leibniz algebras over a field of characteristic zero.

Local and 2-Local Derivations and Automorphisms on Simple Leibniz Algebras

This paper is devoted to study local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We show that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We prove that nilpotent Leibniz algebras as a rule admit local derivations and 2-local derivations which are not derivations. Further, we consider automorphisms of simple Leibniz algebras. It is established that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and that nilpotent Leibniz algebras admit 2-local automorphisms which are not automorphisms. A similar problem concerning local automorphism on simple Leibniz algebras is reduced to the case of simple Lie algebras.

Classification of non-strongly nilpotent filiform Leibniz algebras of dimension 11 and 12:

Classification of non-strongly nilpotent filiform Leibniz algebras of dimension 11 and 12:

The second relative homology of Leibniz algebras

Let (g,n) be a pair of Leibniz algebras, where n is an ideal of g. In this article, we develop the concepts of the second relative homology and the relative stem cover of the pair (g,n). By constructing a version of the Hopf's formula for HL2(g,n), we prove the existence of relative stem covers for (g,n). We also give some inequalities and a certain upper bound for the dimension of HL2(g,n). In addition, we classify all pairs of finite dimensional nilpotent Leibniz algebras that have one or two steps distance to this upper bound.

Rota-Baxter Leibniz Algebras and Their Constructions

In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight 0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.

Characterizing nilpotent Leibniz algebras by a new bound on their second homologies

Let L be a finite dimensional nilpotent Leibniz algebra such that dim⁡(L)=n and dim⁡(L2)=m≠0. In this paper, we prove dim⁡(HL2(L))≤(n+m−2)(n−m)−m+2, where HL2(L) is the second Leibniz homology of L. As a consequence, for a non-abelian nilpotent Leibniz algebra L, we find that s(L)=(n−1)2+1−dim⁡(HL2(L))≥0. Furthermore, we determine all finite dimensional nilpotent Leibniz algebras with s(L) less than or equal to three.