Leibniz Algebras Research Articles

Complete description of invariant, associative pseudo-Euclidean metrics on left Leibniz algebras via quadratic Lie algebras

A pseudo-Euclidean non-associative algebra (g,•) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈,〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈u•v,w〉=〈u,v•w〉 for all u,v,w∈g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra.This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5.

Normal automorphisms of free metabelian Leibniz algebras

Let $\mathfrak{M}$ be a free metabelian Leibniz algebra generating set $% X=\{x_{1},...,x_{n}\}$ over the field $K$ of characteristic $0$. An automorphism $ \phi $ of $\mathfrak{M}$ is said to be normal automorphism if each ideal of $\mathfrak{M}$ is invariant under $ \phi $. In this work, it is proven that every normal automorphism of $\mathfrak{M}$ is an IA-automorphism and the group of normal automorphisms coincides with the group of inner automorphisms.

Relative Rota--Baxter operators of arbitrary weight on Leibniz algebras and post-Leibniz algebra structures

Relative Rota--Baxter operators of arbitrary weight on Leibniz algebras and post-Leibniz algebra structures

Polynomial identities and images of polynomials on null-filiform Leibniz algebras

In this paper we study identities and images of polynomials on null-filiform Leibniz algebras. If Ln is an n-dimensional null-filiform Leibniz algebra, we exhibit a finite minimal basis for Id(Ln), the polynomial identities of Ln, and we explicitly compute the images of multihomogeneous polynomials on Ln. We present necessary and sufficient conditions for the image of a multihomogeneous polynomial f to be a subspace of Ln. For the particular case of multilinear polynomials, we prove that the image is always a vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds for Ln. We also prove similar results for an analog of null-filiform Leibniz algebras in the infinite-dimensional case.

Some Lie algebras and groups associated to representations of Leibniz algebras

Given a representation [Formula: see text] of a Leibniz algebra [Formula: see text], let [Formula: see text] (respectively,[Formula: see text][Formula: see text]) be the Lie algebra (respectively, [Formula: see text]the group) of diagonal derivations (respectively, automorphisms) of the semidirect product [Formula: see text]. We show that both [Formula: see text] and [Formula: see text] have a representation on the cohomology group [Formula: see text]. In the case that [Formula: see text] arises from an abelian extension of [Formula: see text] by [Formula: see text], such representations are applied to construct exact sequences of Wells type for [Formula: see text] and [Formula: see text] respectively.

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

Endomorphisms which preserve orbits of free metabelian Leibniz algebras

Let M be the free metabelian Leibniz algebra of rank two over a field K of characteristic 0. In this article, we show that an endomorphism of M which preserves the orbit of a nontrivial certain kind of element w ∈ M ′ is an automorphism. Additionally, we give some test elements.

Generating sets of F/R' Leibniz algebras

Abstract Let F be a free Leibniz algebra generated by the set X = { x 1 , … , x n } {X=\{x_{1},\dots,x_{n}\}} over the field K of characteristic 0 and let R be an ideal of F. In this study, a necessary and sufficient condition for n elements of the Leibniz algebra F / R ′ {F/R^{\prime}} to be a generating set is given.

Description of the automorphism groups of some 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 elements $a,b,c\in L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,b\in L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of the Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of nilpotent three-dimensional Leibniz algebras.

Crossed modules and non-abelian extensions of Rota-Baxter Leibniz algebras

We introduce the notions of crossed modules of Rota-Baxter Leibniz algebras, 2-term Rota-Baxter sh Leibniz algebras, and discuss the relationship between them and the 3-th cohomology group of Rota-Baxter Leibniz algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. Based on this classification, we study the inducibility of a pair of automorphisms about a non-abelian extension of Rota-Baxter Leibniz algebras, and give the fundamental sequence of Wells in the context of Rota-Baxter Leibniz algebras.

On the algebra of derivations of some nilpotent Leibniz algebras

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

Derivations of two-step nilpotent algebras

In this paper we study the Lie algebras of derivations of two-step nilpotent algebras. We obtain a class of Lie algebras with trivial center and abelian ideal of inner derivations. Among these, the relations between the complex and the real case of the indecomposable Heisenberg Leibniz algebras are thoroughly described. Finally we show that every almost inner derivation of a complex nilpotent Leibniz algebra with one-dimensional commutator ideal, with three exceptions, is an inner derivation.

Leibniz algebras whose solvable ideal is the maximal extension of the nilradical

The paper is devoted to the so-called complete Leibniz algebras. It is known that a Lie algebra with a complete ideal is split. We will prove that this result is valid for Leibniz algebras whose complete ideal is a solvable algebra such that the codimension of nilradical is equal to the number of generators of the nilradical.

On [formula omitted]-graded vertex algebras associated with cyclic Leibniz algebras with small dimensions

The main goals for this paper are i) to study of the algebraic structure of N-graded vertex algebras VB associated with vertex A-algebroids B when B are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras VB and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex A-algebroids B associated with given cyclic non-Lie left Leibniz algebras B. Next, we use the constructed vertex A-algebroids B to create a family of indecomposable non-simple vertex algebras VB. Finally, we use the algebraic structure of the unital commutative associative algebras A that we found to study relations between certain types of the vertex algebras VB and the vertex operator algebra associated to the rank one Heisenberg algebra.

Weak c-ideals of Leibniz algebras

A subalgebra B of a Leibniz algebra L is called a weak c-ideal of L if there is a subideal C of L such that and where is the largest ideal of L contained in B. This is analogous to the concept of a weakly c-normal subgroup, which has been studied by a number of authors. We obtain some properties of weak c-ideals and use them to give some characterizations of solvable and supersolvable Leibniz algebras generalizing previous results for Lie algebras. We note that one-dimensional weak c-ideals are c-ideals, and show that a result of Turner classifying Leibniz algebras in which every one-dimensional subalgebra is a c-ideal is false for general Leibniz algebras, but holds for symmetric ones.

On the derivations of Leibniz algebras of low dimension

Let L be an algebra over a field F. Then L is called a left Leibniz algebra if its multiplication operations [×, ×] addition- ally satisfy the so-called left Leibniz identity: [[a,b],c] = [a,[b,c]] – [b,[a,c]] for all elements a, b, c Î L. In this paper, we begin the description of the algebra of derivations of Leibniz algebras having dimension 3. It is clear that the description of the algebra of derivations of all Leibniz algebras, having dimension 3, is quite large. Therefore, in this article, we will focus on the description of the nilpotent Leibniz algebra, whose nilpotency class is 3, and the nilpotent Leibniz algebra, whose center has dimension 2.

A new model of the free monogenic digroup

It is well-known that one of open problems in the theory of Leibniz algebras is to find asuitable generalization of Lie’s third theorem which associates a (local) Lie group to any Liealgebra, real or complex. It turns out, this is related to finding an appropriate analogue of a Liegroup for Leibniz algebras. Using the notion of a digroup, Kinyon obtained a partial solution ofthis problem, namely, an analogue of Lie’s third theorem for the class of so-called split Leibnizalgebras. A digroup is a nonempty set equipped with two binary associative operations, aunary operation and a nullary operation satisfying additional axioms relating these operations.Digroups generalize groups and have close relationships with the dimonoids and dialgebras,the trioids and trialgebras, and other structures. Recently, G. Zhang and Y. Chen applied themethod of Grobner–Shirshov bases for dialgebras to construct the free digroup of an arbitraryrank, in particular, they considered a monogenic case separately. In this paper, we give a simplerand more convenient digroup model of the free monogenic digroup. We construct a new classof digroups which are based on commutative groups and show how the free monogenic groupcan be obtained from the free monogenic digroup by a suitable factorization.

Second-maximal subalgebras of Leibniz algebras

In this work we study Leibniz algebras whose second-maximal subalgebras are ideals. We provide classification based on solvability, nilpotency, and properties of the Frattini subalgebra. We give several specific descriptions of Leibniz algebras with these properties and include necessary and sufficient conditions for the non-nilpotent case.

On the Automorphism Groups for Some Leibniz Algebras of Low Dimensions

On the Automorphism Groups for Some Leibniz Algebras of Low Dimensions

Maximal Solvable Leibniz Algebras with a Quasi-Filiform Nilradical

This article is part of a study on solvable Leibniz algebras with a given nilradical. In this paper, solvable Leibniz algebras, whose nilradical is naturally graded quasi-filiform algebra and the complemented space to the nilradical has maximal dimension, are described up to isomorphism.