Lindsey Bosko-Dunbar, Jonathan Dunbar, J. T. Hird, Kristen Stagg
In this work we study Leibniz algebras whose second-maximal subalgebras are ideals. We provide a classification based on solvability, nilpotency, and the size of the derived algebra. We give specific descriptions of those Leibniz algebras whose derived algebra has codimension zero or one. This includes several algebras which are only possible over certain fields.
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We begin to study the structure of Leibniz algebras having maximal cyclic subalgebras
Conjugacy of Cartan Subalgebras in Solvable Leibniz Algebras and Real Leibniz Algebras
我们将李代数的共轭性结果推广到它们的Leibniz代数推广。李的证明依赖于反交换性。因此，有必要在莱布尼茨的情况下找到其他途径。其中一些结果涉及Cartan子代数。我们的结果可以推广到Cartan子代数上的其他结果。我们在这里展示了一个例子，其他的将在未来的工作中展示。
Lattices of subalgebras of Leibniz algebras
我描述了格的子代数的单生成元莱布尼茨代数。利用这一点，我表明，除了一个特殊的情况下，格同构之间的莱布尼茨代数L，L'映射的莱布尼茨核L的L'。
More on the Frattini Subalgebra of a Leibniz Algebra
研究了由一个元素生成的Leibniz代数的Frattini子代数。我们还研究了其真子代数都是初等的莱布尼茨代数。
A subalgebra S of a Leibniz algebra L is called self-idealizing in L if it coincides with its idealizer IL(S). In this paper we study the structure of Leibniz algebras, whose subalgebras are either ideals or self-idealizing.
Nilpotent Leibniz algebras with isomorphic maximal subalgebras are considered. The algebras are classified for coclass zero, one, and two. The results are field dependent.
In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Leibniz algebras. We study Leibniz algebras containing abelian subalgebras of codimension 1, solvable and supersolvable Leibniz algebras for codimensions 1 and 2, nilpotent Leibniz algebras in case of codimension 2 and we also analyze the case of k-abelian p-filiform Leibniz algebras. Throughout the paper, we also give examples to clarify some results and the need for restrictions on the underlying field.
Engel subalgebras of Leibniz algebras
有限维Leibniz代数的Engel子代数具有与李代数相似的性质。利用这些结果，证明了极大子代数都是右理想的左Leibniz代数是幂零的.证明了一个本原Leibniz代数在其极小理想上分裂，且其极小理想的所有补都是共轭的.一个子代数被证明是一个Cartan子代数当且仅当它是极小的恩格尔，提供了域有足够多的元素。Cartan子代数具有类似于内方差的性质。
Some radicals, Frattini and Cartan Subalgebras of Leibniz n-algebras
在目前的工作中，我们引入的概念，如$k$ -可解性，$s$ -和$K_1$ -幂零性和相应的自由基。我们证明了这些根在Leibniz $n$ -代数的导子下是不变的。研究了Leibniz $n$ -代数的Frattini和Cartan子代数.特别地，我们构造的例子表明，一个经典的结果，李代数的Cartan子代数的共轭，这也保持在莱布尼茨代数和李$n$ -代数，是不正确的莱布尼茨$n$ -代数。
Subnormal subalgebras of Leibniz algebras
Zassenhaus证明了：如果U是有限维李代数L的次正规子代数，V是有限维不可约L-模，则V的所有U-模合成因子同构. Schenkman证明了如果U是有限维李代数L的次正规子代数，则U的幂零剩余是L的理想。这些有用的结果推广到莱布尼茨代数。
An algebra is said to be a unary Leibniz algebra if every one-generated subalgebra is a Leibniz algebra. An algebra is said to be a binary Leibniz algebra if every two-generated subalgebra is a Leibniz algebra. We give characterizations of unary and binary Leibniz algebras in terms of identities.
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.
In this paper we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The fact that a non-Lie Leibniz algebra has fewer one-dimensional subalgebras in general results in a number of lattice conditions being weaker than in the Lie case.
Conjugacy of Cartan subalgebras of complex finite dimensional Leibniz algebras
本文将有限维李代数中Cartan子代数的性质及其与正则元的关系推广到Leibniz代数的情形。证明了Leibniz代数的Cartan子代数和正则元通过一个自然同态与李代数的Cartan子代数和正则元相对应。证明了Leibniz代数的Cartan子代数的共轭性。
Cartan Subalgebras of Leibniz $n$-Algebras
本文研究了Leibniz $n$ -代数中Cartan子代数和正则元的性质。建立了给定的Leibniz $n$ -代数的Cartan子代数与正则元以及相应的因子$n$ -李代数的Cartan子代数与正则元之间的关系。
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。
Leibniz Algebras and Lie Algebras
本文研究了有限维复Leibniz代数的代数结构。特别地，我们引入了左中心对称Leibniz代数，并利用取Leibniz核的结合双线性对研究了李子代数的偏序集。
Minimal nonnilpotent Leibniz algebras
利用真子代数是幂零的性质，对所有非幂零的可解Leibniz代数进行了分类。这推广了Stitzinger和Towers在李代数中的工作。我们展示了几个例子，说明Lie和莱布尼茨结果之间的差异。
Leibniz algebras with an abelian subalgebra of codimension tw0
本文刻画了特征非2域上具有余维数二阿贝尔子代数的有限维莱布尼茨代数，证明其可解并给出结构分类。
Triangulable Leibniz Algebras
发现并推广了莱布尼兹代数的一个李定理的逆定理。结果是用来寻找的情况下，广义财产，所谓的triangulable，是2-recognizable;也就是说，如果所有的2生成子代数是triangulable，那么代数也。三角可解性加入可解性，超可解性，强可解性和幂零性作为莱布尼茨代数类的2-可识别性质。
Leibniz algebras constructed by representations of General Diamond Lie algebras
本文构造了$(2m+2)$维复一般Diamond李代数的极小忠实表示$\mathfrak{D}_m(\mathbb{C})$，它同构于特殊线性李代数$\mathfrak{sl}(m+2,\mathbb{C})$的一个子代数.我们还构造了一般Diamond李代数$\mathfrak{D}_m$的一个忠实表示，它同构于特殊辛李代数$\mathfrak{sp}(2m+2,\mathbb{R})$的一个子代数。进一步，我们用相应的$(2m+2)$维广义Diamond李代数$\mathfrak{D}_m$和由元素平方生成的理想来描述Leibniz代数，从而得到$\mathfrak{D}_m$的一个忠实表示.
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 quasi-filiform algebra and the complemented space to the nilradical has maximal dimension, are described up to isomorphism.
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.
Classifying Several Classes of Leibniz Algebras
我们将李代数中的极大子代数和理想的相关结果推广到了莱布尼兹代数。特别地，我们对极小非初等Leibniz代数和具有唯一极大理想的Leibniz代数进行了分类。在这两种情况下，都有一些类型的这些代数没有李代数的类似物。我们还给出了E-Leibniz代数的一个分类，它与对应的李代数非常相似.
Modularity conditions in Leibniz algebras
本文继续研究Leibniz代数的子代数格，发现具有上半模格的可解Leibniz代数的结构特征，并探讨了弱拟理想子代数与模对称Leibniz代数的性质。
A subspace H of a Leibniz algebra L is called a quasi-ideal if [H;K] + [K;H] \subseteq H + K for every subspace K of L. They include ideals and subalgebras of codimension one in L. Quasi-ideals of Lie algebras were classified in two remarkable papers of Amayo. The objective here is to extend those results to the larger class of Leibniz algebras, and to classify those Leibniz algebras in which every subalgebra is a quasi-ideal.
Classification of some Solvable Leibniz Algebras
Leibniz代数是李代数的某种推广。本文给出了具有一维导出子代数的$\leq 8$维非Lie可解（左）Leibniz代数的分类.我们利用双线性型矩阵的同余类的标准型得到了我们的结果。我们的方法可以很容易地扩展到这些代数的高维分类。我们还重新分类的三维非李可解（左）莱布尼茨代数。
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
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 Levi-Malcev theorem for Leibniz algebras
在特征为零的域上的有限维Leibniz代数中，给出了Levi-Malcev定理成立或不成立的条件（即两个Levi子代数通过一个内自同构共轭或不共轭）.特别地，在复数域$\mathbb{C}$中，我们考虑了Levi子代数共轭和不共轭的所有可能情况.
Solvable Leibniz algebras with naturally graded non-Lie $p$-filiform nilradicals
本文描述了具有自然分次非Lie $p$ -丝状$(n-p\geq4)$诣零根和诣零根的一维可补空间的可解Leibniz代数。此外，还研究了具有交换诣零根和可补空间诣零根的极（最小，最大）维数的可解Leibniz代数。证明了具有交换诣零根的可解Leibniz代数的刚性及其补空间的最大维数。
On split Regular Hom-Leibniz algebras
引入了可裂正则Hom-Leibniz代数类，作为可裂Leibniz代数和可裂正则Hom-李代数的自然推广.通过发展这类代数根的联络技巧，我们证明了这样一个分裂正则Hom-Leibniz代数$L$是$L = U + \sum\limits_{[j] \in Λ/\sim}I_{[j]}$的形式，其中$U$是交换子代数$H$的子空间，且$L$的理想$I_{[j]}$满足$[I_{[j]}, I_{[k]}] = 0$，如果$[j]\neq [k]$ .在一定条件下，当$L$具有极大长度时，刻画了该代数的单性.
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 $L=B+C$ and $B\cap C\subseteq B_{L}$ where $B_{L}$ 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 characterisations of solvable and supersolvable Leibniz algebras generalising previous results for Lie algebras. We note that one-dimensional weak c-ideals are c-ideals, and show that a result of Turner cl…
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 characterization of perfect Leibniz algebras
本文刻画了完美 Leibniz 代数：它们恰为作为子理想嵌入时必为理想、作为理想嵌入时必为特征理想的代数，并应用于导子代数的包含关系。
In this paper, the definitions of algebras of quotients and Martandale-like qoutients of Leibniz algebras are introduced and the interactions between the two quotients are determined. Firstly, some important properties which not only hold for a Leibniz algebras but also can been lifted to its algebras of quotients are investigated. Secondly, for any semiprime Leibniz algebra, its maximal algebra of quotients is constucted and a Passman-like characterization of the maximal algebra is described. Thirdly, the relationship between a Leibniz algebra and the associative algebra which is generated by…
Leibniz algebras with associated Lie algebras sl_2\dot{+} R (dim R=2)
从有限维李代数的理论出发，我们知道，每一个有限维李代数都可分解为半单子代数与可解根的半直和。此外，由于工作的Mal'cev的研究可解李代数减少到研究幂零的。对于有限维的Leibniz代数，上述结果的类似结果尚未得到证明。为了得到一些想法如何建立的结果，我们研究的Leibniz代数的商代数关于理想所产生的代数的平方元素（表示为$I$）是一个半直和的半单李代数和极大可解理想。本文描述了一类复Leibniz代数，其中理想$I$的商代数同构于代数$sl_2$与二维可解理想$R$的半直和.
In the paper the class of all solvable extensions of a filiform Leibniz algebra in the infinite-dimensional case is classified. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of residually solvable Leibniz algebra. It is proven that the second cohomology group of the extension is trivial.
Maximal Subalgebras for Modular Graded Lie Superalgebras of Odd Cartan Type
本文的目的是确定特征为$p>3$的代数闭域上四个有限维奇Cartan型分次李超代数无穷级数的所有极大分次子代数.所有的极大分次子代数都由三种类型（\MyRoman{1}），（\MyRoman{2}）和（\MyRoman{3}）组成.（\MyRoman{3}）型极大分次子代数分为可约极大分次子代数和不可约极大分次子代数。本文对（\MyRoman{1}）、（\MyRoman{2}）型极大分次子代数和可约极大分次子代数进行了分类，将不可约极大分次子代数的分类归结为经典李超代数$\mathfrak{p}(n)$的不可约极大子代数的分类.
Derivation Algebras of Three-Dimensional Leibniz Algebras: A Complete Description
本文完整描述了任意域上三维非李左莱布尼茨代数的导子代数，利用十六类精细化分类，通过直接计算确定导子，并特别处理特征2情形，最终以表格形式给出所有类型的导子代数。
Solvable Leibniz algebras with triangular nilradicals
本文将三角诣零根可解李代数的描述推广到Leibniz代数。证明了Leibniz代数元素上的左、右算子的矩阵具有上三角形形式。证明了具有给定三角零根的极大可能维数的可解Leibniz代数是李代数。此外，还对具有低维三角形幂零根的可解Leibniz代数进行了分类。
On central characteristic ideals and quasi-Noetherian Leibniz algebras
本文定义了 Leibniz 代数的中心特征理想并给出相关必要条件，同时引入拟诺特 Leibniz 代数类，探讨其封闭性并刻画诺特 Leibniz 代数。
We give a counterexample to the conjugacy of maximal abelian diagonalizable subalgebras in extended affine Lie algebras.
The Kac Jordan superalgebra: automorphisms and maximal subalgebras
给出了Kac Jordan超代数的自同构自同构群，并利用自同构群对极大子代数进行了分类.
Some remarks on semisimple Leibniz algebras
由Levi定理可知，特征为零的域上的有限维李代数可分解为可解根与半单子代数的半直和。此外，半单部分是单理想的直和。在\cite{Bar}中，Levi定理被推广到了Leibniz代数的情形.本文研究了半单Leibniz代数，证明了半单Leibniz代数的分裂定理是不成立的。此外，我们还研究了几类特殊的半单Leibniz代数，并给出了它们分解为单理想的直和的条件。
Maximal subalgebras of Jordan superalgebras
研究了特征为0的代数闭域上有限维单特殊Jordan超代数的极大子代数。这是前一篇论文的延续由相同的作者关于极大子代数的简单结合超代数，这是工具在这里。
Maximal Subalgebras for Lie Superalgebras of Cartan Type
研究了素特征域上四类Cartan型李超代数的极大分次子代数。给出了所有极大可约分次子代数的完整刻画，并给出了它们的同构类和维数公式，将极大不可约分次子代数的分类归结为经典李超代数$\frak{gl}(m,n)$，$\frak{sl}(m,n)$和$\frak{osp}(m,n)$的极大不可约子代数的分类.
Some remarks about the second Leibniz cohomology group for Lie algebras
我们用一种非常初等的方法比较了李代数的第二伴随和平凡莱布尼茨上同调空间与通常的上同调空间。例子给出了耦合上循环。导出了莱布尼兹变形的一些性质。我们还考虑了一类Koszul 3-形式为零的李代数，并证明了它包含有限维半单李代数的Borel子代数或其诣零根的所有直子。最后，给出了维数为$\leqslant 7$的不可分解幂零李代数的Kac-Moody型.
We show that any local derivation on the solvable Leibniz algebras with model or abelian nilradicals, whose the dimension of complementary space is maximal is a derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have 1-dimension complementary space, admit local derivations which are not derivations. Moreover, similar problem concerning 1-local derivations of such algebras are investigated and an example of solvable Leibniz algebra given such that any 2-local derivation on it is a derivation, but which admit local derivations which are not derivations.
Maximal Commutative Subalgebras of Steinberg Algebras
本文在素Steinberg代数中构造了极大交换子代数的大类，推广了Leavitt路代数的已知结果。
In this paper we prove that any local automorphism on the solvable Leibniz algebras with null-filiform and naturally graded non-Lie filiform nilradicals, whose dimension of complementary space is maximal is an automorphism. Furthermore, the same problem concerning 2-local automorphisms of such algebras is investigated and we obtain the analogously results for 2-local automorphisms.
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 $\mathfrak{L}$…
Itô's theorem and metabelian Leibniz algebras
证明了著名的伊藤定理在Leibniz代数的层次上仍然成立：如果$\mathfrak{g}$是Leibniz代数，使得对于两个交换子代数$A$和$B$，$\mathfrak{g} = A + B$是亚交换的，即$[ \, [\mathfrak{g}, \, \mathfrak{g}], \, [ \mathfrak{g}, \, \mathfrak{g} ] \, ] = 0$ .证明了亚交换Leibniz/李代数的一个结构型定理。本文对所有具有$1$维导出代数的亚阿贝尔Leibniz代数进行了描述和分类，并将它们的自同构群明确地确定为与任意向量空间$P$相关的群$P^* \ltimes \bigl(k^* \times {\rm Aut}_{k} (P) \bigl)$的半直积的子群.
In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension $1$ and supersolvable Zinbiel algebras in which such subalgebras have codimension $2$, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for $α$ and $β$ for the low dimensional Zinbiel algebras over the complex field that have been classified.
On the length of a class of maximal commutative subalgebras
本文研究全矩阵代数中极大交换子代数的长度，定义并计算了一类新的极大交换子代数 $\mathcal{B}_{k,m,l}$，并通过实例说明其结论并非已有结果的简单推广。
Computing Split Maximal Toral Subalgebras of Lie algebras over Fields of Small Characteristic
一个代数群的李代数的重要子代数是它的toral子代数，或者等价地（在特征为0的域上）它的Cartan子代数。其中非常重要的是那些分裂的：它们在李代数上的作用在定义域上完全分裂。虽然存在计算分裂极大toral子代数的算法并且已经实现[Ryb 07，CM 09]，但是当李代数定义在特征为2或3的域上时，这些算法失败。给出了一个启发式算法，给定特征为2或3的有限域上的约化李代数L，求L的分裂极大toral子代数。连同早期的工作[CR 09]这些算法是非常有用的识别约化李代数在这样的领域。
Maximal graded subalgebras of Witt and Special superalgebras
由于本文的主要结果和方法与李代数情形下的结果和方法是相似的，因此作者撤回了这篇文章。
The aim of this paper is to provide an answer to the $\mathbb{C}[\partial]$-split extending structures problem for Leibniz conformal algebras, which asks that how to describe all Leibniz conformal algebra structures on $E=R\oplus Q$ up to an isomorphism such that $R$ is a Leibniz conformal subalgebra. For this purpose, an unified product of Leibniz conformal algebras is introduced. Using this tool, two cohomological type objects are constructed to classify all such extending structures up to an isomorphism. Then this general theory is applied to the special case when $R$ is a free $\mathbb{C}[…
We introduce and describe the class of split regular $Hom$-Leibniz color $3$-algebras as the natural extension of the class of split Lie algebras, split Leibniz algebras, split Lie $3$-algebras, split Lie triple systems, split Leibniz $3$-algebras, and some other algebras. More precisely, we show that any of such split regular $Hom$-Leibniz color $3$-algebras $T$ is of the form ${T}={\mathcal U} +\sum\limits_{j}I_{j}$, with $\mathcal U$ a subspace of the $0$-root space ${T}_0$, and $I_{j}$ an ideal of $T$ satisfying {for} $j\neq k:$ \[[{ T},I_j,I_k]+[I_j,{ T},I_k]+[I_j,I_k,T]=0.\] Moreover, if…
Maximal Subalgebras of Finite-Dimensional Algebras
研究了域上任意有限维代数的极大子代数，得到了这类代数的完全分类和描述结果。这是通过首先获得一个完整的分类在半单的情况下，然后提升到非半单代数。结果在代数闭域的情况下是最尖锐的，并采取特殊形式的代数表示的箭图的关系。我们还涉及表示论性质的代数及其极大和其他子代数，并提供了一系列的嵌入之间的箭图，关联代数和其他结构，涉及不可分解的表示代数和一些子代数通过诱导/限制函子。文献中的一些结果也作为特例重新得到，并给出了其他应用。
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