We introduce the notion of a generalized metric n-Leibniz algebra and show that there is a one-to-one correspondence between generalized metric n-Leibniz algebras and faithful generalized orthogonal representations of metric Lie algebras (called Lie triple data). We further show that there is also a one-to-one correspondence between generalized orthogonal derivations (resp. generalized orthogonal automorphisms) on generalized metric n-Leibniz algebras and Lie triple datas.
In this paper we introduce the concept of $n$-Lie-isoclinism on non-Lie Leibniz algebras. Among the results obtained, we provide several characterizations of $n$-Lie-isoclinic classes of Leibniz algebras. Also, we provide a characterization of $n$-Lie-stem Leibniz algebras, and prove that every $n$-Lie-isoclinic class of Leibniz algebras contains a $n$-Lie-stem Leibniz algebra.
In this paper we study the notion of isoclinism on Lie-central extensions of Leibniz algebras, this yields to introduce the concept of Lie-isoclinic Leibniz algebras. We provide several equivalent conditions under which Leibniz algebras are Lie-isoclinic. We also define the concept of Schur Lie-multiplier and analyze its connection with Lie-isoclinism.
In this article, we study the notion of central extension of Leibniz $n$-algebras relative to $n$-Lie algebras to study properties of Schur $\mathsf{Lie}$-multiplier and $\mathsf{Lie}$-covers on Leibniz $n$-algebras. We provide a characterization of $\mathsf{Lie}$-perfect Leibniz $n$-algebras by means of universal $\mathsf{Lie}$-central extensions. It is also provided some inequalities on the dimension of the Schur $\mathsf{Lie}$-multiplier of Leibniz $n$-algebras. Analogue to Wiegold [38] and Green [17] results on groups or Moneyhun [26] result on Lie algebras, we provide upper bounds for the…
We introduce 3-irreducible modules, even roots and odd roots for Leibniz algebras, produce a basis for a root space of a Leibniz algebra with a semisimple Lie factor, and classify finite dimensional simple Leibniz algebras with Lie factor $s\ell_2$.
We introduce the notion of c-nilpotent Schur Lie-multiplier of Leibniz algebras. We obtain exact sequences and formulas of the dimensions of the underlying vector spaces relating the c-nilpotent Schur Lie-multiplier of a Leibniz algebra Q and its quotient by a two-sided ideal. These tools are used to characterize Lie-nilpotency and c-Lie-stem covers of Leibniz algebras. We prove the existence of c-Lie-stem covers for finite dimensional Leibniz algebras and the non existence of c-covering on certain Lie-nilpotent Leibniz algebras with non trivial c-nilpotent Schur Lie-multiplier, and we provide…
This paper concerns the algebraic structure of finite-dimensional complex Leibniz algebras. In particular, we introduce left central and symmetric Leibniz algebras, and study the poset of Lie subalgebras using an associative bilinear pairing taking values in the Leibniz kernel.
In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ${\sf ID}_*-Lie$-derivations. A ${\sf ID}_*-Lie$-derivation of a Leibniz algebra G is a Lie-derivation of G in which …
The aim of this paper is to consider the relation between Lie-isoclinism and isomorphism of two pairs of Leibniz algebras. We show that, unlike the absolute case for finite dimensional Lie algebras, these concepts are not identical, even if the pairs of Leibniz algebras are Lie-stem. Moreover, throughout the paper we provide some conditions under which Lie-isoclinism and isomorphism of Lie-stem Leibniz algebras are equal. In order to get this equality, the concept of factor set is studied as well.
Along this paper we show that under certain conditions the method for describing of solvable Lie and Leibniz algebras with maximal codimension of nilradical is also extensible to Lie and Leibniz superalgebras, respectively. In particular, we totally determine the solvable Lie and Leibniz superalgebras with maximal codimension of model filiform and model nilpotent nilradicals. Finally, it is established that the superderivations of the obtained superalgebras are inner.
In this paper some characterizations of Hom-Leibniz superalgebras are given and some of their basic properties are found. These properties can be seen as a generalization of corresponding well-known properties of Hom-Leibniz algebras. Considering the Hom-Akivis superalgebra associated to a given Hom-Leibniz superalgebra, it is observed that the Hom-super Akivis identity leads to an additional property of Hom-Leibniz superalgebras, which in turn gives a necessary and sufficient condition for Hom-super Lie admissibility of Hom-Leibniz superalgebras. We show also that every (left) Hom-Leibniz sup…
We study some properties on $\mathsf{Lie}$-centroids related to central $\mathsf{Lie}$-derivations, generalized $\mathsf{Lie}$-derivations and almost inner $\mathsf{Lie}$-derivations. We also determine the $\mathsf{Lie}$-centroid of the tensor product of a commutative associative algebra and a Leibniz algebra.
Adjoint functors between the categories of crossed modules of dialgebras and Leibniz algebras are constructed. The well-known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules.
Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.
We extend results on finite dimensional nilpotent Lie algebras to Leibniz algebras and counterexamples to others are found. One generator algebras are used in these examples and are investigated further.
Basing ourselves on the categorical notions of central extensions and commutators in the framework of semi-abelian categories relative to a Birkhoff subcategory, we study central extensions of Leibniz algebras with respect to the Birkhoff subcategory of Lie algebras, called Lie-central extensions. We obtain a six-term exact homology sequence associated to a Lie-central extension. This sequence, together with the relative commutators, allows us to characterize several classes of Lie-central extensions, such as Lie-trivial, Lie-stem and Lie-stem cover, to introduce and characterize Lie-unicentra…
In this paper we find minimal faithful representations of several classes filiform Lie algebras by means of strictly upper-triangular matrices. We investigate Leibniz algebras whose corresponding Lie algebras are filiform Lie algebras such that the action $I \times L \to I$ gives rise to a minimal faithful representation of a filiform Lie algebra. The classification up to isomorphism of such Leibniz algebras is given for low-dimensional cases.
Given a symmetric Leibniz algebra $(\mathcal{L},.)$, the product is Lie-admissible and defines a Lie algebra bracket $[\;,\;]$ on $\mathcal{L}$. Let $G$ be the connected and simply-connected Lie group associated to $(\mathcal{L},[\;,\;])$. We endow $G$ with a Lie rack structure such that the right Leibniz algebra induced on $T_eG$ is exactly $(\mathcal{L},.)$. The obtained Lie rack is said to be associated to the symmetric Leibniz algebra $(\mathcal{L},.)$. We classify symmetric Leibniz algebras in dimension 3 and 4 and we determine all the associated Lie racks. Some of such Lie racks give ris…
In this paper solvable Leibniz algebras with naturally graded non-Lie $p$-filiform $(n-p\geq4)$ nilradical and with one-dimensional complemented space of nilradical are described. Moreover, solvable Leibniz algebras with abelian nilradical and extremal (minimal, maximal) dimensions of complemented space nilradical are studied. The rigidity of solvable Leibniz algebras with abelian nilradical and maximal dimension of its complemented space is proved.
Leibniz algebras are certain generalization of Lie algebras. In this paper we survey the important results in Leibniz algebras which are analogs of corresponding results in Lie algebras. In particular we highlight the differences between Leibniz algebras and Lie algebras.
Leibniz algebras are certain generalization of Lie algebras. It is natural to generalize concepts in Lie algebras to Leibniz algebras and investigate whether the corresponding results still hold. In this paper we introduce the notion of complete Leibniz algebras as generalization of complete Lie algebras. Then we study properties of complete Leibniz algebras and their holomorphs.
In extending results from Lie to Leibniz algebras, it is helpful to have techniques which translate results from the former to the latter without having to repeat the (perhaps modified) arguments. Such a technique is developed in this work, and some applications are given.
The Leibniz algebras appear as a generalization of the Lie algebras \cite{loday}. The classification of naturally graded $p$-filiform Lie algebras is known \cite{C-G-JM}, \cite{J.Lie.Theory}, \cite{AJM}, \cite{Ve}. In this work we deal with the classification of 2-filiform Leibniz algebras. The study of $p$-filiform Leibniz non Lie algebras is solved for $p=0$ (trivial) and $p=1$ \cite{Omirov1}. In this work we get the classification of naturally graded non Lie 2-filiform Leibniz algebras.
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 variety. We deal with Leibniz algebras which are generalizations of Lie algebras. In Lie algebras case, there are different kind of rigidities (rigidity, absolutely rigidity, geometric rigidity and e.c.t.). We explore the relations of these rigidities with Leibniz algebra rigidity. Necessary conditions for a Lie algebra to be Leibniz rigid are discussed.
W. A. Moens proved that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. In this paper we show that with the definition of Leibniz-derivation from W. A. Moens the similar result for non Lie Leibniz algebras is not true. Namely, we give an example of non nilpotent Leibniz algebra which admits an invertible Leibniz-derivation. In order to extend the results of paper W. A. Moens for Leibniz algebras we introduce a definition of Leibniz-derivation of Leibniz algebras which agrees with Leibniz-derivation of Lie algebras case. Further we prove that a Leibniz alge…
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra $n_{n,1}.$ We introduce a Fock module for the algebra $n_{n,1}$ and provide classification of Leibniz algebras $L$ whose corresponding Lie algebra $L/I$ is the algebra $n_{n,1}$ with condition that the ideal $I$ is a Fock $n_{n,1}$-module, where $I$ is the ideal generated by squares of elements from $L$.
Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques.
The present article is a part of the study of solvable Leibniz algebras with a given nilradical. In this paper solvable Leibniz algebras, whose nilradicals is naturally graded $p$-filiform non-Lie Leibniz algebra $(n-p\geq4)$ and the complemented space to nilradical has maximal dimension, are described up to isomorphism. Moreover, among obtained algebras we indicate the rigid and complete algebras
In this paper we study symmetric Leibniz and related algebras, namely symmetric dialgebras and symmetric Perm-algebras. We also calculate their Koszul duals, if not known. This will give us Lie-admissible algebras and new types of algebras, which we call commutative and associative admissible algebras. Finally we prove that all operads describing the considered algebras are Koszul.
Leibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. In this paper we define subinvariant subalgebras of Leibniz algebras and study their properties. It is shown that the signature results on subinvariance in Lie algebras have analogs for Leibniz algebras.
This paper deals with the classification of Leibniz central extensions of a naturally graded filiform Lie algebra. We choose a basis with respect to that the table of multiplication has a simple form. In low dimensional cases isomorphism classes of the central extensions are given. In parametric family orbits cases invariant functions (orbit functions) are provided.
We extend results related to maximal subalgebras and ideals from Lie to Leibniz algebras. In particular, we classify minimal non-elementary Leibniz algebras and Leibniz algebras with a unique maximal ideal. In both cases, there are types of these algebras with no Lie algebra analogue. We also give a classification of E-Leibniz algebras which is very similiar to its Lie algebra counterpart.
The first aim of this paper is to introduce and study symmetric (Bi)Hom-Leibniz algebras, which are left and right Leibniz algebras. We discuss $α^kβ^l$-generalized derivations, $α^kβ^l$ -quasi-derivations and $α^kβ^l$-quasi-centroid of (Bi)Hom-Leibniz algebras and colour BiHom-Leibniz algebras. The second aim is to define a new type of BiHom-Lie algebras satisfies the following hierarchy \begin{equation*} \displaystyle \{ \text{ BiHom-Lie type $B_1$} \}\supseteq_{β=id} \{ \text{ Hom-Lie} \}\supseteq_{α=id} \{ \text{ Lie} \}. \end{equation*} Moreover, define representations and a cohomology of…
In this paper we show that the method for describing solvable Lie algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case of Leibniz algebras. Using this method we extend the classification of solvable Lie algebras with naturally graded filiform Lie algebra to the case of Leibniz algebras. Namely, the classification of solvable Leibniz algebras whose nilradical is a naturally graded filiform Leibniz algebra is obtained.
Leibniz algebras are certain generalization of Lie algebras. In this paper we give classification of non-Lie solvable (left) Leibniz algebras of dimension $\leq 8$ with one dimensional derived subalgebra. We use the canonical forms for the congruence classes of matrices of bilinear forms to obtain our result. Our approach can easily be extended to classify these algebras of higher dimensions. We also revisit the classification of three dimensional non-Lie solvable (left) Leibniz algebras.
It is well-known that all saturated formations of finite soluble groups are locally defined and, except for the trivial formation, have many different local definitions. I show that for Lie and Leibniz algebras over a field of characteristic 0, the formations of all nilpotent algebras and of all soluble algebras are the only locally defined formations and that the latter has many local definitions. Over a field of non-zero characteristic, a saturated formation of soluble Lie algebras has at most one local definition but a locally defined saturated formation of soluble Leibniz algebras other th…
We extend conjugacy results from Lie algebras to their Leibniz algebra generalizations. The proofs in the Lie case depend on anti-commutativity. Thus it is necessary to find other paths in the Leibniz case. Some of these results involve Cartan subalgebras. Our results can be used to extend other results on Cartan subalgebras. We show an example here and others will be shown in future work.
Since Leibniz algebras were introduced by Loday as a generalization of Lie algebras, there has been a lot of interest in which results of the latter extend to the former. Cyclic algebras, those generated by one element, are a useful tool for studying these concepts. In fact, they have no Lie algebra counterpart. Their simple structure lends itself to elegant counterexamples to the extension of several important results from Lie algebras to Leibniz algebras. In this paper, we give a classification of cyclic Leibniz algebras.
We classify all nonnilpotent, solvable Leibniz algebras with the property that all proper subalgebras are nilpotent. This generalizes the work of Stitzinger and Towers in Lie algebras. We show several examples which illustrate the differences between the Lie and Leibniz results.
From the theory of finite dimensional Lie algebras it is known that every finite dimensional Lie algebra is decomposed into a semidirect sum of semisimple subalgebra and solvable radical. Moreover, due to work of Mal'cev the study of solvable Lie algebras is reduced to the study of nilpotent ones. For the finite dimensional Leibniz algebras the analogues of the mentioned results are not proved yet. In order to get some idea how to establish the results we examine the Leibniz algebra for which the quotient algebra with respect to the ideal generated by squares elements of the algebra (denoted b…
In this paper we continue the description of solvable Leibniz algebras whose nilradical is a filiform algebra. In fact, solvable Leibniz algebras whose nilradical is a naturally graded filiform Leibniz algebra are described in \cite{Campo} and \cite{SolGFil}. Here we extend the description to solvable Leibniz algebras whose nilradical is a filiform algebra. We establish that solvable Leibniz algebras with filiform Lie nilradical are Lie algebras.
In this paper we construct a minimal faithful representation of the $(2m+2)$-dimensional complex general Diamond Lie algebra, $\mathfrak{D}_m(\mathbb{C})$, which is isomorphic to a subalgebra of the special linear Lie algebra $\mathfrak{sl}(m+2,\mathbb{C})$. We also construct a faithful representation of the general Diamond Lie algebra $\mathfrak{D}_m$ which is isomorphic to a subalgebra of the special symplectic Lie algebra $\mathfrak{sp}(2m+2,\mathbb{R})$. Furthermore, we describe Leibniz algebras with corresponding $(2m+2)$-dimensional general Diamond Lie algebra $\mathfrak{D}_m$ and ideal …
In this note we point out that the definition of the universal enveloping dialgebra for a Leibniz algebra is consistent with the interpretation of a Leibniz algebra as a generalization not of a Lie algebra, but of the adjoint representation of a Lie algebra. From this point of view, the formal integration problem of Leibniz algebras is, essentially, trivial.
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 forms and some other techniques to obtain our result.
We introduce a non-abelian exterior product of two crossed modules of Leibniz algebra and investigate its relation to the low dimensional Leibniz homology. Later this non-abelian exterior product is applied to the construction of eight term exact sequence in Leibniz homology. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the second Lie and Leibniz homologies of a Lie algebra.
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.
In this study, we extend the result on classification of a subclass of filiform Leibniz algebras in low dimensions to dimensions seven and eight based on the technique used by Rakhimov and Bekbaev for classification of subclasses which arise from naturally graded non-Lie filiform Leibniz algebras. The subclass considered here arises from naturally graded filiform Lie algebras. This subclass contains the class of filiform Lie algebras and consequently, by classifying this subclass, we again re-examine the classification of filiform Lie algebras. Our resulting list of filifom Lie algebras is com…
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 bilinear forms to classify the case $\dim(A^2)=3$ and $\dim(Leib(A))=1$ which can be applied to higher dimensions. The remaining cases are classified via direct method.
A classification exists for Lie algebras whose nilradical is the triangular Lie algebra $T(n)$. We extend this result to a classification of all solvable Leibniz algebras with nilradical $T(n)$. As an example we show the complete classification of all Leibniz algebras whose nilradical is $T(4)$.
