Non-Existence of Complex Structures on Filiform Lie Algebras

Index of Graded Filiform and Quasi Filiform Lie Algebras

The filiform and the quasi-filiform Lie algebras form a special class of nilpotent Lie algebras. The aim of this paper is to compute the index and provide regular vectors of this two class of nilpotent Lie algebras. we consider the graded filiform Lie algebras $L_{n}$, $Q_{n}$, the $n$-dimensional filiform Lie algebras for $n$ $<8$, also the graded quasi-filiform Lie algebras and finally Lie algebras whose nilradical is $Q_{2n}$.

There are no rigid filiform Lie algebras of low dimension

We prove that there are no rigid complex filiform Lie algebras in the variety of (filiform) Lie algebras of dimension less than or equal to 11. More precisely we show that in any Euclidean neighborhood of a filiform Lie bracket (of low dimension), there is a non-isomorphic filiform Lie bracket. This follows by constructing non trivial linear deformations in a Zariski open dense set of the variety of filiform Lie algebras of dimension 9, 10 and 11. (In lower dimensions this is well known.)

Minimal linear representations of filiform Lie algebras and their application for construction of Leibniz algebras

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.

Quasi $Q_n$-filiform Lie algebras

In this paper we explicitly determine the derivation algebra, automorphism group of quasi $Q_n$-filiform Lie algebras, and applying some properties of root vector decomposition we obtain their isomorphism theorem.

Filiform Lie algebras of dimension 8 as degenerations

For each complex 8-dimensional filiform Lie algebra we find another non isomorphic Lie algebra that degenerates to it. Since this is already known for nilpotent Lie algebras of rank $\ge 1$, only the caracteristically nilpotent ones should be considered.

Degenerations to filiform Lie algebras of dimension 9

For most complex 9-dimensional filiform Lie algebra we find another non isomorphic Lie algebra that degenerates to it. Since this is already known for nilpotent Lie algebras of rank $\geq 1$, only the characteristically nilpotent ones should be considered.

Group gradings on filiform Lie algebras

We classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank. In case of rank 0, we describe conditions to obtain non trivial $\Z_k$-gradings.

On cohomology of filiform Lie superalgebras

Suppose the ground field $\mathbb{F}$ is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras.

Completable filiform Lie algebras

We determine the solvable complete Lie algebras whose nilradical is isomorphic to a filiform Lie algebra. Moreover we show that for any positive integer $n$ there exists a solvable complete Lie algebras whose second cohomology group with values in the adjoint module has dimension at least $n$.

Naturally graded 2-filiform Leibniz algebras

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.

On classification of Filiform Hom-Lie algebras

The purpose of this paper is to introduce and study nilpotent and filiform Hom-Lie algebras. Moreover, we extend Vergne and Khakimdjanov's approach to Hom-type algebras and provide a classification of filiform Hom-Lie algebras of dimension $n,n\leq7$.

Central extensions of null-filiform and naturally graded filiform non-Lie Leibniz algebras

In this paper we describe central extensions of some nilpotent Leibniz algebras. Namely, central extensions of the Leibniz algebra with maximal index of nilpotency are classified. Moreover, non-split central extensions of naturally graded filiform non-Lie Leibniz algebras are described up to isomorphism. It is shown that $k$-dimensional central extensions ($k\geq 5$) of these algebras are split.

最大长度拟丝状李代数上的转置泊松结构

Transposed Poisson structures on quasi-filiform Lie algebras of maximum length

该研究聚焦最大长度拟丝状李代数的$\frac{1}{2}$-导子，通过建立交换结合乘法，构造出以这类李代数为伴随代数的非平凡转置泊松代数，为相关代数结构研究提供新构造。

Isomorphism classes and invariants of low dimensional filiform Leibniz algebras

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…

On one dimensional Leibniz central extensions of a naturally graded filiform Lie algebra

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.

On deformations of the filiform Lie superalgebra $L_{n,m}$

In this work, we recall that every filiform Lie superalgebra is a deformation of the superalgebra $L_{n,m}$. We study the even cocycles which give this nilpotent deformations. A family of independent bilinear maps will help us to describe this cocycles. At the end an evaluation of the dimension of the space $Z_0^2(L_{n,m},L_{n,m})$ is established.

Filiform Lie algebras with low derived length

We construct, for any integer n greater than or equal to 5, a family of complex filiform Lie algebras with derived length at most 3 and dimension n. We also give examples of n-dimensional filiform Lie algebras with derived length greater than 3.

Classification of (n-5)-filiform Lie algebras

In this paper we consider the problem of classifying the $(n-5)$-filiform Lie algebras. This is the first index for which infinite parametrized families appear, as can be seen in dimension $7.$ Moreover we obtain large families of characteristic nilpotent Lie algebras with nilpotence index 5 and show that at least for dimension 10 there is a characteristic nilpotent Lie algebra with nilpotence index 4 which is the algebra of derivations of a nilpotent Lie algebra.

Solvable Leibniz algebras with naturally graded non-Lie $p$-filiform nilradicals

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 associated with representations of filiform Lie algebras

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 $L$.

On k-abelian, p-filiform Lie algebras

We classify the (n-5)-filiform Lie algebras which have the additional property of a non-abelian derived subalgebra. We show that this property is strongly related with the structure of the Lie algebra of derivations; explicitely we show that if a (n-5)-filiform algebra is characteristically nilpotent, then it must be 2-abelian. We also give applications of k-abelian Lie algebras to the construction of solvable rigis algebras, as well as to the theory of nilalgebras of parabolic subalgebras in the example of the exceptional simple model E_{6}.

On filiform Lie algebras. Geometric and algebraic studies

A finite dimensional filiform K-Lie algebra is a nilpotent Lie algebra g whose nil index is maximal, that is equal to dim g -1. We describe necessary and sufficient conditions for a filiform algebra over an algebraically closed field of characteristic 0 to admit a contact linear form (in odd dimension) or a symplectic structure (in even dimension). If we fix a Vergne's basis, the set of filiform n-dimensional Lie algebras is a closed Zariski subset of an affine space generated by the structure constants associated with this fixed basis. Then this subset is an algebraic variety and we describe …

Solvable Leibniz algebras with quasi-filiform Lie algebras of maximum length nilradicals

In this paper solvable Leibniz algebras whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to nilradical is proved.

Solvable Leibniz algebras with filiform nilradical

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.

Extensions of solvable Lie algebras with naturally graded filiform nilradical

In this work we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras $n_{n, 1}$ and $Q_{2n}.$ We find all one-dimensional central extensions of the algebra $n_{n, 1}$ and show that any extension of $Q_{2n}$ is split. After that we find one-dimensional extensions of solvable Lie algebras with nilradical $n_{n, 1}$. We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical $n_{n, 1}$ of maximal codimension. Moreover, all one-dimensional extensions of…

Solvable Leibniz algebras with naturally graded non-Lie $p$-filiform nilradicals and maximal complemented space of its nilradical

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

Classification of solvable Leibniz algebras with naturally graded filiform nilradical

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.

括号理想与filiform李代数的Hilbert多项式

Bracket ideals and Hilbert polynomial of filiform Lie algebras

本文研究了filiform李代数的括号理想和Hilbert多项式，证明该多项式能区分无法被传统不变量区分开的同构类。

Solvable Infinite Filiform Lie Algebras

An infinite filiform Lie algebra L is residually nilpotent and its graded associated with respect to the lower central series has smallest possible dimension in each degree but is still infinite. This means that gr(L) is of dimension two in degree one and of dimension one in all higher degrees. We prove that if L is solvable, then already [L,L] is abelian. The isomorphism classes in this case are given in a paper by Bratzlavsky, but the proof there is incomplete. We make the necessary additional computations and restate Bratzlavskys result when the ground field is the complex numbers.

Isomorphism and isotopism classes of filiform Lie algebras of dimension up to seven

Since the introduction of the concept of isotopism of algebras by Albert in 1942, a prolific literature on the subject has been developed for distinct types of algebras. Nevertheless, there barely exists any result on the problem of distributing Lie algebras into isotopism classes. The current paper is a first step to deal with such a problem. Specifically, we define a new series of isotopism invariants and we determine explicitly the distribution into isotopism classes of $n$-dimensional filiform Lie algebras, for $n\leq 7$. We also deal with the distribution of such algebras into isomorphism…

标准线状李代数的中心与不变量

The center and invariants of standard filiform Lie algebras

本文研究了标准线状李代数的泛包络代数中心与不变量环，确定了商域的显式生成元及不变量环生成元的紧致形式，并证明了关于其Hilbert级数的组合结果。

On Isomorphism Classes and Invariants of Low Dimensional Complex Filiform Leibniz Algebras (PART 1)

The paper aims to investigate the classification problem of low dimensional complex none Lie filiform Leibniz algebras. There are two sources to get classification of filiform Leibniz algebras. The first of them is the naturally graded none Lie filiform Leibniz algebras and the another one is the naturally graded filiform Lie algebras. Here we do consider Leibniz algebras appearing from the naturally graded none Lie filiform Leibniz algebras. This class can be splited into two subclasses. However, isomorphisms within each class there were not investigated. Before U.D.Bekbaev and I.S.Rakhimov s…

Cohomologically rigid solvable Lie superalgebras with model filiform and model nilpotent nilradical

In this paper, we find a family $SL^{n,m}$, in any arbitrary dimensions, of cohomologically rigid solvable Lie superalgebras with nilradical the model filiform Lie superalgebra $L^{n,m}$. Moreover, we exhibit a family of cohomologically rigid solvable Lie superalgebras with nilradical the model nilpotent Lie superalgebra of generic characteristic sequence. Both cases correspond to solvable Lie superalgebras of maximal dimension for a given nilradical. Contrariwise, we will show that the family of Lie superalgebras $SL^{n,m}$ can be deformed if defined over a field of odd characteristic.

具有丝状幂零根基的可解李代数上的转置 Poisson 结构

Transposed Poisson structures on solvable Lie algebras with filiform nilradical

该研究利用具有线状幂零根基的可解李代数的 1/2-导子，建立交换结合乘法，从而构造出关联给定李代数的非平凡转置 Poisson 代数。

Filiform Lie代数的仿射上同调类

Affine cohomology classes for filiform Lie algebras

本文研究了filiform Lie代数的仿射上同调类，证明了存在仿射上同调类与代数的仿射结构之间的关系，并探讨了特定代数的仿射结构不存在性。

A distinguished example of filiform deformation

We exhibit an example of a filiform (complex) Lie algebra of dimension 13 with all its ideals of codimension 1 being characteristically nilpotent, and we construct a non trivial filiform deformation of it.

Cohomology and deformations of the infinite dimensional filiform Lie algebra m_0

Denote m_0 the infinite dimensional N-graded Lie algebra defined by basis e_i, i>= 1 and relations [e_1,e_i] = e_(i+1) for all i>=2. We compute in this article the bracket structure on H1(m_0,m_0), H2(m_0,m_0) and in relation to this, we establish that there are only finitely many true deformations of m_0 in each nonpositive weight, by constructing them explicitely. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.

Quasi-filiform Leibniz algebras of maximum length

The $n$-dimensional $p$-filiform Leibniz algebras of maximum length have already been studied with $0\leq p\leq 2$. For Lie algebras whose nilindex is equal to $n-2$ there is only one characteristic sequence, $(n-2,1,1)$, while in Leibniz theory we obtain two possibilities: $(n-2,1,1)$ and $(n-2,2)$. The first case (the 2-filiform case) is already known. The present paper deals with the second case, i.e., quasi-filiform non Lie Leibniz algebras of maximum length. Therefore this work completes the study of maximum length of Leibniz algebras with nilindex $n-p$ with $0 \leq p \leq 2$.

On Description of Isomorphism Classes of filiform Leibniz algebras in dimensions 7 and 8

The paper concerns the classification problem of a subclass of complex filiform Leibniz algebras in dimensions 7 and 8. This subclass arises from naturally graded filiform Lie algebras. We give a complete list of isomorphism classes of algebras including Lie case. In parametric families cases, the corresponding orbit functions (invariants) are given. In discrete orbits case, we show representatives of the orbits.

On Classification of Finite Dimensional Complex Filiform Leibniz Algebras (Part 2)

The paper is devoted to classification problem of finite dimensional complex none Lie filiform Leibniz algebras. Actually, the observations show there are two resources to get classification of filiform Leibniz algebras. The first of them is naturally graded none Lie filiform Leibniz algebras and the another one is naturally graded filiform Lie algebras. Using the first resource we get two disjoint classes of filiform Leibniz algebras. The present paper deals with the second of the above two classes, the first class has been considered in our previous paper. The algebraic classification here m…

On classification of complex filiform Leibniz algebras

In this paper we prove that in classifying of complex filiform Leibniz algebras, for which its naturally graded algebra is non-Lie algebra, it suffices to consider some special basis transformations. Moreover, we establish a criterion whether given two such Leibniz algebras are isomorphic in terms of such transformations. The classification problem of filiform Leibniz algebras, for which its naturally graded algebras are non-Lie in an arbitrary dimension, is reduced to the investigation of the obtained conditions.

Solvable Leibniz algebra with non-Lie and non-split naturally graded filiform nilradical and its rigidity

The description of complex solvable Leibniz algebras whose nilradical is a naturally graded filiform algebra is already known. Unfortunately, a mistake was made in that description. Namely, in the case where the dimension of the solvable Leibniz algebra with nilradical $F_n^1$ is equal to $n+2$, it was asserted that there is no such algebra. However, it was possible for us to find a unique $(n+2)$-dimensional solvable Leibniz algebra with nilradical $F_n^1$. In addition, we establish the triviality of the second group of cohomology for this algebra with coefficients in itself, which implies it…

On isomorphism classes and invariants of low dimensional complex filiform Leibniz algebras (part 2)

The paper is an implementation in low dimensional cases of the classification method presented before by Rakhimov and Bekbaev. We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from the naturally graded non-Lie filiform Leibniz algebras. A hypothetic formula for the adapted number of isomorphism classes is given.

The classification of non-characteristically nilpotent filiform Leibniz algebras

In this paper we investigate the derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension is decomposed into three non-intersected families. We found sufficient conditions under which filiform Leibniz algebras of the first family are characteristically nilpotent. Moreover, for the first family we classify non-characteristically nilpotent algebras by means of Catalan numbers. In addition, for the rest two families of filiform Leibniz algebras we describe non-characteristically nilpotent algebras, i.e., those filiform Leibniz algebras which li…

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.

Cohomology and deformations of the infinite dimensional filiform Lie algebra m_2

Denote $\fm_2$ the infinite dimensional $\N$-graded Lie algebra defined by the basis $e_i$ for $i\geq 1$ and by relations $[e_1,e_i]=e_{i+1}$ for all $i\geq 2$, $[e_2,e_j]=e_{j+2}$ for all $j\geq 3$. We compute in this article the bracket structure on $H^1(\fm_2,\fm_2)$, $H^2(\fm_2,\fm_2)$ and in relation to this, we establish that there are only finitely many true deformations of $\fm_2$ in each weight by constructing them explicitely. It turns out that in weight 0 one gets as non-trivial deformation only one formal non-converging deformation.

On low-dimensional filiform Leibniz algebras and their invariants

The paper deals with the complete classification of a subclass of complex filiform Leibniz algebras in dimensions 5 and 6. This subclass arises from the naturally graded filiform Lie algebras. We give a complete list of algebras. In parametric families cases, the corresponding orbit functions (invariants) are given. In discrete orbits case, we show a representative of the orbits.

On the Betti numbers of filiform Lie algebras over fields of characteristic two

An $n$-dimensional Lie algebra $\mathfrak{g}$ over a field $\mathbb{F}$ of characteristic two is said to be of Vergne type if there is a basis $e_1,\dots,e_n$ such that $[e_1,e_i]=e_{i+1}$ for all $2\leq i \leq n-1$ and $[e_i,e_j] = c_{i,j}e_{i+j}$ for some $c_{i,j} \in \mathbb{F}$ for all $i,j \ge 2$ with $i+j \le n$. We define the algebra $\mathfrak{m}_0$ by its nontrivial bracket relations: $[e_1,e_i]=e_{i+1}, 2\leq i \leq n-1$, and the algebra $\mathfrak{m}_2$: $[e_1, e_i ]=e_{i+1}, 2 \le i \le n-1$, $[e_2, e_j ]=e_{j+2}, 3 \le j \le n-2$. We show that, in contrast to the corresponding rea…

Infinitesimal deformations of naturally graded filiform Leibniz algebras

We describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any $n$-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra $F_n^3(0)$. We establish that in the same way any $n$-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras $F_{n}^1,$ $F_{n}^2$ and $F_{n}^3(α)$. Moreover, we describe the linear integrable deformations of above-mentioned algebras with a fixed basis of $HL^2$ in the set of all $n$-dimensional Leibni…

On Classification of Finite Dimensional Complex Filiform Leibniz Algebras (PartI)

The paper is devoted to classification problem of finite dimensional complex none Lie filiform Leibniz algebras. The motivation to write this paper is an unpublished yet result of J.R.Gomez, B.A.Omirov on necessary and sufficient conditions for two finite dimensional complex filiform Leibniz algebras to be isomorphic. We suggest another approach to this problem. The approach that we use for classification is in terms of invariants. In fact, utilizing this method for any given low dimensional case all filiform Leibniz algebras can be classified. Moreover, the results can be used for geometrical…

Almost inner derivations of Lie algebras II

We continue the algebraic study of almost inner derivations of Lie algebras over a field of characteristic zero and determine these derivations for free nilpotent Lie algebras, for almost abelian Lie algebras, for Lie algebras whose solvable radical is abelian and for several classes of filiform nilpotent Lie algebras. We find a family of $n$-dimensional characteristically nilpotent filiform Lie algebras $\mathfrak{f}_n$, for all $n\ge 13$, all of whose derivations are almost inner. Finally we compare the almost inner derivations of Lie algebras considered over two different fields $K\supseteq…

n 维完全分次丝状李代数的局部和 2 - 局部自同构

Local and 2-Local automorphisms of n-dimensional totally graded filiform Lie algebras

研究最大长度的有限维完全分次复丝状李代数族的局部和 2 - 局部自同构空间，利用代数过滤等方法，通过分析多个序列和族，明确了不同结构中局部及 2 - 局部自同构的形式特点，划分出非线性与严格线性结构的边界。

Automorphisms of some solvable Leibniz algebras

In this paper, we describe the automorphisms of solvable Leibniz algebras with null-filiform nilradical. Moreover we describe the automorphisms of solvable Leibniz algebras with naturally graded non-Lie filiform nilradicals, whose the dimension of complementary space is maximal.

A class of Solvable Lie algebras

All finite-dimensional indecomposable solvable Lie algebras g, having the filiform Lie algebra Q_(2m+1) as the nilradical, are studied and classified. It turns out that the dimension of g is at most dimQ_(2m+1)+2.