L. M. Camacho, J. R. Gómez, R. M. Navarro, B. A. Omirov
The aim of this work is to present the description up to isomorphism of Leibniz superalgebras with characteristic sequence (n | m-1, 1) and nilindex n+m.
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In this article, we introduce a deformation cohomology of Leibniz superalgebras. Also, we introduce formal deformation theory of Leibniz superalgebras. Using deformation cohomology we study the formal deformation theory of Leibniz superalgebras.
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…
This article deals with a Leibniz superalgebras $L=L_0\oplus L_1,$ whose even part is a simple Lie algebra $\mathfrak{sl}_2$. We describe all such Leibniz superalgebras when odd part is an irreducible Leibniz bi-module on $\mathfrak{sl}_2 $. We show that there exist such Leibniz superalgebras with nontrivial odd part only in case of $dim L_1=2.$
Flat pseudo-Euclidean Leibniz superalgebras
本文引入预李和预莱布尼茨超代数，定义平坦伪欧几里得左莱布尼茨超代数，证明二次情形下平坦等价于对称且2步幂零，并利用双扩张刻画所有此类超代数。
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 work the classification of filiform Leibniz superalgebras of nilindex $n+m,$ where $n$ and $m$ ($m \ne 0$) are dimensions of even and odd parts, respectively, is obtained.
In this paper, we elaborate Gröbner-Shirshov bases method for Leibniz (super)algebras. We show that there is a unique reduced Gröbner-Shirshov basis for every (graded) ideal of a free Leibniz (super)algebra. As applications, we construct linear bases of free metabelian Leibniz superalgebras and new linear bases of free metabelian Lie algebras. We present a complete characterization of extensions of a Leibniz (super)algebra by another Leibniz (super)algebra, where the former is presented by generators and relations.
We present the description up to isomorphism of Leibniz superalgebras with characteristic sequence $(n|m_1,...,m_k)$ and nilindex $n+m,$ where $m=m_1+ >...+m_k,$ $n$ and $m$ ($m\neq 0$) are dimensions of even and odd parts, respectively.
Consider a Leibniz superalgebra $\mathfrak L$ additionally graded by an arbitrary set $I$ (set grading). We show that $\mathfrak L$ decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case of ${\mathfrak L}$ being of maximal length, the simplicity of ${\mathfrak L}$ is also characterized in terms of connections.
Solvable Leibniz superalgebras whose nilradical has the characteristic sequence $(n-1, 1 \mid m)$ and nilindex $n+m$
本文完整分类了幂零根基具有特征序列 $(n-1, 1 \mid m)$ 和幂零指数 $n+m$ 的可解 Leibniz 超代数，给出了其存在可解扩张的参数条件，并分类了根基为最大幂零指数 Lie 超代数的情形。
In this paper we investigate the description of the complex Leibniz superalgebras with nilindex n+m, where n and m ($m\neq 0$) are dimensions of even and odd parts, respectively. In fact, such superalgebras with characteristic sequence equal to $(n_1, ..., n_k | m_1, ..., m_s)$ (where $n_1+... +n_k=n, m_1+ ... + m_s=m$) for $n_1\geq n-1$ and $(n_1, ..., n_k | m)$ were classified in works \cite{FilSup}--\cite{C-G-O-Kh1}. Here we prove that in the case of $(n_1, ..., n_k| m_1, ..., m_s)$, where $n_1\leq n-2$ and $m_1 \leq m-1$ the Leibniz superalgebras have nilindex less than n+m. Thus, we compl…
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 dimensions less than 3 is extended to the dimension four. {\it AMS Subject Classifications}: 16D70, 17A30, 17A60, 17B30 {\it Key words:} Leibniz algebra, associative algebra, nilpotence, nulfiliform Leibniz algebra, filiform Leibniz algebra.
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…
Nilpotent Leibniz algebras with isomorphic maximal subalgebras are considered. The algebras are classified for coclass zero, one, and two. The results are field dependent.
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 pre-derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three non-intersected families. We describe the pre-derivation of filiform Leibniz algebras for the first and second families. We found sufficient conditions under which filiform Leibniz algebras are strongly nilpotent. Moreover, for the first and second families, we give the description of characteristically nilpotent algebras which are non-strongly nilpotent.
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…
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.
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.
This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property (there is an irreducible component formed by only nilpotent algebras), but on the other hand, it has Vergne Property (there are no rigid nilpotent 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.
The aim of this work is to present the first problems that appear in the study of nilpotent Leibniz superalgebras. These superalgebras and so the problems, will be considered as a natural generalization of nilpotent Leibniz algebras and Lie superalgebras.
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 algebraic and geometric classification of nilpotent binary and mono Leibniz algebras
研究给出复5维幂零二元Leibniz代数和4维幂零一元Leibniz代数的完整代数与几何分类，并由此推出幂零指数3的复4维幂零代数分类。
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.
This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the variety of Leibniz algebras. It is shown that, up to isomorphism, there exist three Leibniz algebras of level two, one of which is solvable, and two of which are nilpotent. Moreover we describe all algebras of level two in the variety of nilpotent algebras.
Moens proved that a finite-dimensional Lie algebra over field of characteristic zero is nilpotent if and only if it has an invertible Leibniz-derivation. In this article we prove the analogous results for finite-dimensional Malcev, Jordan, (-1,1)-, quasiassociative, quasialternative, right alternative and Malcev-admissible noncommutative Jordan algebras over the field of characteristic zero. Also, we describe all Leibniz-derivations of semisimple Jordan, right alternative and Malcev algebras.
Computational Methods for Rota-type operators on 4-dimensionnal nilpotent Leibniz Algebras
本文研究相容幂零Leibniz代数，给出了维数小于四的代数及其Rota型算子的分类。
On naturally graded Lie and Leibniz superalgebras
本文将自然分次概念推广至 Lie 与非 Lie Leibniz 超代数，给出定义，并刻画最大超幂零指数情形下的自然分次。
In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras $NF^{n,m}$. One of them is a Leibniz algebra (that is $m=0$) and the second one is a pure Leibniz superalgebra (that is $m\neq 0$) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable defor…
The algebraic and geometric classification of nilpotent Leibniz algebras
本文对复5维幂零Leibniz代数进行了完整的代数与几何分类，并确定其代数簇维数为24，包含10个不可约分支及一个刚性代数。
Two-step nilpotent Leibniz algebras
本文基于Kronecker模完成两步幂零Leibniz代数分类，推广经典Heisenberg李代数得到不可分解Heisenberg Leibniz代数的复实情形，证明实幂零Leibniz代数可整体积分并给出应用，还揭示了积分Leibniz代数的李quandle与李群共轭的关系。
On split Leibniz superalgebras
本文研究了分裂莱布尼茨超代数的结构，证明其可分解为一个阿贝尔分次子代数的子空间与若干两两交换的分次理想之和，并在最大长度情形下用根连接刻画了其单性。
We describe degenerations of four-dimensional Zinbiel and four-dimensional nilpotent Leibniz algebras over C. In particular, we describe all irreducible components in the corresponding varieties.
Computational Methods for Biderivations of 4-dimensional nilpotent complex leibniz algebras
本文针对4维幂零复Leibniz代数，基于现有分类开发算法计算导子、反导子及双导子，并借助Mathematica和Maple软件给出详细示例。
The goal of this paper is to study the structure of split regular BiHom-Leiniz superalgebras, which is a natural generalization of split regular Hom-Leiniz algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Leiniz superalgebras $\mathfrak{L}$ is of the form $\mathfrak{L}=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $\mathfrak{L}$, satisfying $[I_\a, I_\b]= 0$ if $[\a]\neq [\b]$. In the case of $\mathfrak{L}$…
The aim of this paper is to study the cohomology of Hom-Leibniz superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and provide as application the computations of the derivations and second cohomology group. Moreover, we extend to graded case the Takhtajan's construction of a cohomology of $n$-ary Hom-Nambu-Lie algebras starting from cohomology of Hom-Leibniz algebras.
In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n_1,...,n_k|m)$ and nilindex n+m, where $n=n_1+...+n_k,$ n and m (m is not equal to zero) are dimensions of even and odd parts, respectively. Such superalgebras with condition n_1 > n-2 were classified in \cite{FilSup}--\cite{C-G-O-Kh}. Here we prove that in the case $n_1 < n-1$ the Leibniz superalgebras have nilindex less than $n+m.$ Thus, we get the classification of Leibniz superalgebras with characteristic sequence $(n_1, ...,n_k|m)$ and nilindex n+m.
The paper studies nilpotent $n$-Lie superalgebras. More specifically speaking, we first prove Engel's theorem for $n$-Lie superalgebras. Second, we research some properties of nilpotent $n$-Lie superalgebras, Finally, we give several sufficient conditions that an $n$-Lie superalgebra is nilpotent.
Isotopisms of nilpotent Leibniz algebras and Lie racks
该文研究两步幂零代数的同痕类，刻画了换位子理想为一维的幂零Leibniz代数的同痕分类，建立其与对应Lie racks同痕性的等价关系，并引入新的同痕不变量。
In this article we define the $c$-nilpotent multiplier of a finite dimensional Lie suepralgebra. We characterize the structure of $2$-nilpotent multiplier of finite dimensional nilpotent Lie superalgebras whose derived subalgebras have dimension at most one. Then we give an upper bound on the dimension of $2$-nilpotent multiplier of any finite dimensional nilpotent Lie superalgebra. Moreover, we discuses the $2$-capability of special as well as odd Heisenberg Lie superalgebras and abelian Lie superalgebras.
Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra
本文研究了非幂零非李李代数的一维导出子代数，证明其可分解为二维非幂零非李代数与阿贝尔代数的直和，并扩展了[11]的结果，同时研究了导出代数的李代数、自同构群及双导出代数。
In this paper we study the varieties of nilpotent Lie superalgebras of dimension $\leq 5$. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a by product we construct rigid nilpotent Lie superalgebras of arbitrary dimension.
The main result of this paper is to prove that if a (right) Leibniz algebra $L$ is \textit{right nilpotent} of degree $n$, then $L$ is \textit{strongly nilpotent} of degree less or equal to $4n^2-2n+1$.
Nilpotentizers and the Nilpotent Graphs: Structural Insights into Lie Superalgebras
本文研究了李超代数的nilpotentizer和nilpotent graph，揭示了其结构特性，并引入了nilpotency measure来量化nilpotency，同时利用范畴论建立了与nilpotent子结构的联系。
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 nilpotent complex Leibniz algebras was obtained in [1]. Therefore we will only consider non nilpotent case in this work.
In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m.$ We prove that such superalgebras with the condition $m_2\neq0$ have nilindex less than $n+m$. Therefore the complete classification of Leibniz algebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m$ is reduced to the classification of filiform Leibniz superalgebras of nilindex equal to $n+m,$ which was provided in \cite{AOKh} and \cite{GKh}.
On the dimension of the Schur multiplier of nilpotent Lie superalgebras
本研究针对有限维幂零李超代数的Schur乘子维数问题，提出了一个比已有李超代数Schur乘子维数界更精确的界，改进了相关结果。
Let $L$ be a nilpotent Lie superalgebra of dimension $(m\mid n)$ and $s(L) = \frac{1}{2}[(m + n - 1)(m + n -2)]+ n+ 1 - \dim \mathcal{M}(L)$, where $\mathcal{M}(L)$ denotes the Schur multiplier of $L$. Here $s(L)\geq 0$ and the structure of all non-abelian nilpotent Lie superalgebras with $s(L)=0$ is known((\cite{Nayak2019}). This paper is devoted to obtain all nilpotent Lie superalgebras when $s(L) \leq 2$. Further, we apply those results to list all non-abelian nilpotent Lie superalgebras $L$ with $ t(L) \leq 4$.
Capability of Nilpotent Lie Superalgebras of Small Dimension
该研究引入部分可容李超代数概念，并据此完成维数不超过五的可容幂零李超代数的完整分类。
In this paper, we introduce the concept of (super-)multiplier-rank for Lie superalgeras and classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank $\leq 2$ over an algebraically closed field of characteristic zero. In the process, we also determine the multipliers of Heisenberg superalgebras.
In this article, we discuss the category $\mathcal{SN}_2$ where the objects are finite-dimensional nilpotent Lie superalgebras of class two and the category $\mathcal{SSKE}$ where the objects are skew-supersymmetric bilinear maps. We establish relation between $\mathcal{SN}_2$ and $\mathcal{SSKE}$. As a result, we discuss the capability of nilpotent Lie superalgebras of class two.
We give algebraic and geometric classifications of $4$-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are $41$ one-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $18$ two-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $2$ three-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by $21$ additional isomorphism classes. The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the…
In this article, we define the capable pairs of Lie superalgebras. We classify all capable pairs of abelian and Heisenberg Lie superalgebras. After that we discuss on pairs of Lie superalgebras with derived subalgebra of dimension one and a non-abelian ideal. Finally, we determine the structure of the Schur multiplier of pairs of Heisenberg Lie superalgebras.
Some three-dimensional non-nilpotent Leibniz algebras: automorphism groups
本文研究了三维非幂零 Leibniz 代数，明确给出了两类该代数的自同构群的具体结构。
In this paper, we study the cup products and Betti numbers over cohomology superspaces of two-step nilpotent Lie superalgebras with coefficients in the adjoint modules over an algebraically closed field of characteristic zero. As an application, we prove that the cup product over the adjoint cohomology superspaces for Heisenberg Lie superalgebras is trivial and we also determine the adjoint Betti numbers for Heisenberg Lie superalgebras by means of Hochschild-Serre spectral sequences.
We show that the finite dimensional nilpotent complex Lie superalgebras g whose injective hulls of simple U(g)-modules are locally Artinian are precisely those whose even part g_0 is isomorphic to a nilpotent Lie algebra with an abelian ideal of codimension 1 or to a direct product of an abelian Lie algebra and a certain 5-dimensional or a certain 6-dimensional nilpotent Lie algebra.
Local (Anti-)Superderivations on Nilpotent Lie Superalgebras
研究有限维幂零李超代数上的局部（反）超导子，证明2步幂零李超代数存在纯局部（反）超导子，并给出n>2时存在性的充分判据，特别地3步幂零李超代数存在纯局部超导子。
Upper bounds on the dimension of the Schur $\mathsf{Lie}$-multiplier of $\mathsf{Lie}$-nilpotent Leibniz $n$-algebras
本文研究了李-丝状 Leibniz n-代数的 Schur 李-乘子维数上界，发现其与 Pascal 三角形性质相关，并改进了具有 d 维李-换位子的 m 维李-幂零 Leibniz n-代数的现有上界。
Many theorems and formulas of Lie algebras run quite parallel to Lie superalgebra case, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebra immediately to Lie superalgebra case, as these type of algebras have wide applications in physics and related theories. Using the concept of isoclinism, F. Saeedi and S. Sheikh-Mohseni recently studied the central derivation of nilpotent Lie algebra with nilindex 2. The purpose of the present paper is to continue and extend the investigation to obtain some similar results for Lie superalgebras, as isoclini…
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