Classification of abelian complex structures on 6-dimensional Lie algebras
A. Andrada, M. L. Barberis, I. G. Dotti
本文分类了可赋予阿贝尔复结构的6维李代数，并在相差全纯同构的意义下对每个代数上这类结构的空间进行了参数化。
我们分类了可赋予阿贝尔复结构的6维李代数，并在每个此类代数上，将这类结构的空间在相差全纯同构的意义下进行参数化。
We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.
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Abelian complex structures on solvable Lie algebras
利用某些仿射李代数$\frak a \frak f \frak f (A)$得到了允许交换复结构的实李代数的一个特征，其中$A$是一个交换代数.这些仿射李代数是$\frak a \frak f \frak f (\Bbb C)$的自然推广，相应的李群是复仿射流形。证明了所有具有交换复结构的4维李代数都是这类仿射李代数的中心扩张。
Affine Structures on abelian Lie Groups
小维交换李代数上的完全仿射结构是众所周知的。在本文中，我们感兴趣的非完全情况。特别是，我们将所有这些结构分为2维和3维。
On abelian complex structures on Nilpotent Lie Algebras
本文提出幂零李代数上阿贝尔复结构等价分类的新方法，应用该方法得到4、6维已知分类，给出8维1-阿贝尔复结构分类及一般情况部分结果。
Abelian Hermitian geometry
利用交换结合代数研究了具有左不变交换复结构的李群的结构。此外，如果李群具有一个左不变的厄米特结构，则证明这样的厄米特结构是Kähler当且仅当李群是实双曲平面的几个副本与一个欧几里得因子的直积。此外，我们还证明了如果具有交换复结构的李群上的左不变埃尔米特度量具有平坦的第一标准联络，则该李群是交换的。
We give a counterexample to the conjugacy of maximal abelian diagonalizable subalgebras in extended affine Lie algebras.
Abelian ideals of maximal dimension for solvable Lie algebras
比较了有限维李代数的交换子代数的最大维数和交换理想的最大维数。我们发现，这些尺寸一致的可解李代数上的代数闭域的特征零。我们计算这个不变量的所有复杂的幂零李代数的维数n小于8。进一步研究了存在余维数为2的交换子代数的情形。在幂零李代数的情形下，我们明确地构造了余维为2的交换理想。
Non-abelian extensions and Wells exact sequences of Lie-Yamaguti algebras
本文研究Lie-Yamaguti代数的非阿贝尔扩张，用非阿贝尔上同调群和Maurer-Cartan元素进行分类与刻画，并给出自同构对可扩张性的等价条件及Wells基本序列。
In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).
Cohomologies, non-abelian extensions and Wells sequences of lambda-weighted Rota-Baxter Lie coalgebras
本文研究λ-权Rota-Baxter李余代数的上同调与非阿贝尔扩张，完成分类、推导Wells基本序列并讨论阿贝尔扩张情形下的对应结果。
Abelian gradings in Lie algebras
给定一个由群$G$分次的李代数$L$，若$L$不含正交分次理想，且$G$由$L$的支集生成，则$G$是一个阿贝尔群.
Non-abelian extensions of relative Rota-Baxter Lie algebras and Wells type exact sequences
本文通过引入非阿贝尔二阶上同调群分类相对Rota-Baxter李代数的非阿贝尔扩张，并研究自同构对与导子对的可诱导性问题，推导出Wells型正合序列。
We study some properties of the non-abelian tensor product of Hom-Lie algebras concerning the preservation of products and quotients, solvability and nilpotency, and describe compatibility with the universal central extensions of perfect Hom-Lie algebras.
We introduce the non-abelian tensor product of Lie superalgebras, study some of its properties including nilpotency, solvability and Engel, and we use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Miller's theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.
In this paper, first we show that under the assumption of the center of h being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra g by a regular Hom-Lie algebra h are in one-to-one correspondence with Hom-Lie algebra morphisms from g to Out(h). Then for a general Hom-Lie algebra morphism from g to Out(h), we construct a cohomology class as the obstruction of existence of a non-abelian extension that induce the given Hom-Lie algebra morphism.
Non-abelian extensions of Lie triple systems and Wells exact sequences
本文研究李三系的非阿贝尔扩张与自同构对可诱导性，通过非阿贝尔上同调群分类扩张、用Maurer-Cartan元刻画扩张，推导了李三系的Wells正合序列，并对应给出阿贝尔扩张情形的结果。
On the abelianization of the special derivation Lie algebras of free Lie algebras
本文通过Morita迹证明自由李代数特殊导数李代数的交换化包含无限多个线性无关元素，并展示其中存在被Morita迹杀死的非平凡元素。
In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of $α$-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we introduce crossed modules of Hom-Lie algebras showing their equivalence with cat$^1$-Hom-Lie algebras, and we introduce $α$-crossed modules to have a better understanding of the third cohomology group.
Non-abelian tensor product of Hom-Lie algebras is constructed and studied. This tensor product is used to describe universal ($α$-)central extensions of Hom-Lie algebras and to establish a relation between cyclic and Milnor cyclic homologies of Hom-associative algebras satisfying certain additional condition.
In this paper, we determine upper bound for the non-abelian tensor product of finite dimensional Lie superalgebra. More precisely, if $L$ is a non-abelian nilpotent Lie superalgebra of dimension $(k \mid l)$ and its derived subalgebra has dimension $(r \mid s)$, then $ \dim (L\otimes L) \leq (k+l-(r+s))(k+l-1)+2$. We discuss the conditions when the equality holds for $r=1, s=0$ explicitly.
Non-abelian cohomology of Nijenhuis Lie algebras and the inducibility of automorphisms and derivations
该研究引入Nijenhuis李代数的非阿贝尔上同调群，给出其非阿贝尔扩张同构类的参数化与阿贝尔扩张分类，并刻画了自同构、导子可诱导性的上同调阻碍。
Non-abelian extensions and automorphisms of post-Lie algebras
本文研究了后李代数的交叉模与cat$^1$-后李代数的等价性，构建了非阿贝尔上同调以分类扩展，并通过Wells精确序列解决自同构诱导性问题。
On k-abelian, p-filiform Lie algebras
对具有非交换导子代数附加性质的（n-5）-丝状李代数进行了分类。我们证明了这一性质与导子的李代数的结构密切相关;明确地证明了如果一个（n-5）-filiform代数是特征幂零的，那么它一定是2-交换的。我们还给出了k-交换李代数的应用，可解的rigis代数的建设，以及理论的nil代数的抛物子代数的例外简单模型E_{6}的例子。
$2$-term averaging $L_\infty$-algebras and non-abelian extensions of averaging Lie algebras
本文引入$2$-项平均化 $L_\infty$-代数，研究平均化李代数的非阿贝尔扩张，定义第二非阿贝尔上同调群分类等价类，并借助Wells映射刻画自同构可诱导的障碍及短正合序列。
On abelian subalgebras and ideals of maximal dimension in supersolvable lie algebras
本文的主要目的是比较有限维超可解李代数的交换子代数和最大维数理想。研究了可解李代数的极大交换子代数，并研究了含有余维为2的交换子代数的可解李代数。最后，证明了具有余维为3的交换子代数的幂零李代数包含同维的交换理想，只要其基础域的特征不为2。在整个论文中，我们还给出了几个例子来澄清一些结果。
For $c\geq 2$, the free centre-by-(nilpotent-of-class-c-1)-by abelian Lie ring on a set $X$ is the quotient $L/[(L')^c,L]$ where $L$ is the free Lie ring on $X$, and $(L')^c$ denotes the $c$th term of the lower central series of the derived ideal $L'=L^2$ of $L$. In this paper we give a complete description of the torsion subgroup of its additive group in the case where $|X|=2$ and $c$ is a prime number.
In this paper, we first give the notation of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie algebra which controls deformations of a compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible pre-Lie algebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie algebras and show that equivalent i…
The main object of study of this paper is the notion of 3-Lie superalgebras with superderivations. We consider a representation $(Φ,\mathcal{P})$ of a $3$-Lie superalgebra $\mathcal{Q}$ on $\mathcal{P}$ and construct first-order cohomologies by using superderivations of $\mathcal{P},\mathcal{Q}$ which induces a Lie superalgebra $\mathcal{T}_Φ$ and its representation $Ψ$. Then we consider abelian extensions of $3$-Lie superalgebras of the form $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\rightarrow \mathcal{Q}\rightarrow 0$ with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$ and construct an…
In this paper we study universal central extensions and non-abelian tensor product of hom-Lie-Rinehart algebras. We discuss about universal $α$- central extensions, and, lifting of automorphisms and $α$-derivations to central extensions for hom-Lie-Rinehart algebras. This is in turn provide such lifting of automorphisms and $α^k$-derivations to the central extensions for hom-Lie algebras.
In this paper, we study non-abelian extensions of 3-Lie algebras through Maurer-Cartan elements. We show that there is a one-to-one correspondence between isomorphism classes of non-abelian extensions of 3-Lie algebras and equivalence classes of Maurer-Cartan elements in a DGLA. The structure of the Leibniz algebra on the space of fundamental objects is also analyzed.
In this paper, we study non-abelian extensions of strict Lie 2-algebras via the cohomology theory. A non-abelian extension of a strict Lie 2-algebra $\g$ by $\frkh$ gives rise to a strict homomorphism from $\g$ to $\SOut(\frkh)$. Conversely, we prove that the obstruction of existence of non-abelian extensions of strict Lie 2-algebras associated to a strict Lie 2-algebra homomorphism from $\g$ to $\SOut(\frkh)$ is given by an element in the third cohomology group. We further prove that the isomorphism classes of non-abelian extensions of strict Lie 2-algebras are classified by the second cohomo…
On the tensor square of non-abelian nilpotent finite dimensional Lie algebras
对于每一个具有导子群为$p^m$阶的$p$ -群$G$，Rocco在\cite{roc}中证明了$G$的张量平方阶至多为$p^{n(n-m)}$ .作者最近在\cite{ni}中改进了这个上界。本文的目的是得到有限维非交换幂零李代数的一个类似结果。更精确地说，对于任何给定的$n$维非交换幂零李代数$L$，其导子代数的维数为$m$，我们有$\mathrm{dim} (L\otimes L)\leq (n-m)(n-1)+2$。对于$m=1$，当等式成立时，给出了$L$的显式结构.
In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product $3$-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.
Deformations and abelian extensions on anti-pre-Lie algebras
本文研究反预李代数的表示与上同调，利用Nijenhuis算子构造平凡线性形变，证明形式形变无穷小量由第二上同调群控制，并证明阿贝尔扩张由$H^2(A;V)$分类。
Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor
对于特征为零的域$\k$上的有限维李代数$\g$，定义$μ$ -函数（或$μ_{nil}$ -函数）为$V$的最小维数，使得$\g$在$V$上有一个忠实表示（或一个忠实的零表示）.设$\h_m$是维数为$2m + 1$的Heisenberg李代数，$\mathfrak{a}_n$是维数为$n$的交换李代数.本文的目的是计算所有$m,n \in \mathbb{N}$的$μ(\h_m \oplus \mathfrak{a}_n)$和$μ_{nil}(\h_m \oplus \mathfrak{a}_n)$。
Universal central extensions of Lie-Rinehart algebras
本文研究了Lie-Rinehart代数的泛中心扩张，给出了它的一个刻划，并研究了自同构和导子到中心扩张的提升。在Ellis构造李代数的非交换张量积的基础上，给出了Lie-Rinehart代数的非交换张量积的定义.我们把这个非阿贝尔张量积与泛中心扩张联系起来。
On 3-Lie algebras with abelian ideals and subalgebras
本文研究了代数闭域上m维3-李代数$L$的交换子代数的最大维数$α(L)$和交换理想的最大维数$β(L)$.我们表明，这些尺寸不一致，如果字段的特征为零，即使是幂零3-李代数。然后我们证明了具有$β(L) = m-2$的3-李代数是2-步可解的（参见第2节中的定义）。进一步，我们给出了这些具有一维或二维导代数的3-李代数的精确描述。此外，我们还利用$α(L)=\dim L-2$给出了3-李代数的一个分类。我们还得到了具有$α(L)=\dim L-1$的3-李代数及其一维导代数的分类。
Non-abelian cohomology of Lie $H$-pseudoalgebras and inducibility of automorphisms
本文研究李$H$-伪代数的非阿贝尔上同调与自同构可诱导性，用非阿贝尔上同调群分类非阿贝尔扩张等价类，证明该群可实现为微分分次李代数的Deligne广群，并刻画自同构可诱导性的阻碍为Wells映射的像。
Solvable Lie algebras having at least one Abelian descending central ideal are studied. It is shown that all such Lie algebras can be built up from canonically defined ideals. The nature of such ideals is elucidated and their construction is provided in detail. An approach to study and to classify these Lie algebras is given through the theory of extensions via appropriate cocycles and representations on which a group action is naturally defined. Also, necessary and sufficient conditions for the existence of invariant metrics on the studied extensions are given. It is shown that any solvable q…
This work is devoted to the classification of solvable Leibniz algebras with an abelian nilradical. We consider $k-1$ dimensional extension of $k$-dimensional abelian algebras and classify all $2k-1$-dimensional solvable Leibniz algebras with an abelian nilradical of dimension $k$.
We extend the classification of solvable Lie algebras with abelian nilradicals to classify solvable Leibniz algebras which are one dimensional extensions of an abelian nilradicals.
Non-abelian Extensions of Lie algebras with derivations
本文研究了利用不同方法的李代数与导数的非交换扩展，通过非交换上同调、Deligne群体、严格李2-代数的同伦范畴及（g,D）核概念来表征其理论，并探讨了非交换扩展中导数的存在性问题。
Split abelian chief factors and first degree cohomology for Lie algebras
本文研究了有限维李代数的分裂交换主因子的重数与一次上同调的关系。特别地，我们得到了模可解李代数的一次上同调为零或分裂交换主因子重数的一个刻画。这些结果的类似结果在有限群的模表示论中是众所周知的。证明这些结果的一个重要工具是Seligman关于有限维非可解李代数素特征的一阶上同调的非消灭定理的改进。作为应用，我们得到了限制李代数表示论中与有限维限制李代数的平凡不可约模的主块和投射覆盖有关的几个结果.特别地，我们得到了可解限制李代数的一个特征，即平凡不可约模的投射覆盖的第二Loewy层。
We study a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the group ring of finitely generated abelian groups. The Batalin-Vilkovisky algebra structure for finite abelian groups comes from the fact that the group ring of finite groups is a symmetric algebra, and the Batalin-Vilkovisky algebra structure for free abelian groups of finite rank comes from the fact that its group ring is a Calabi-Yau algebra.
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.
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.
The number of Frattini chief factors or of chief factors which are complemented by a maximal subalgebra of a finite-dimensional Lie algebra $L$ is the same in every chief series for $L$, by \cite[Theorem 2.3]{[11]}. However, this is not the case for the number of chief factors which are simply complemented in $L$. In this paper we determine the possible variation in that number.
Non-abelian extensions and Wells exact sequences of Bol algebras
本文研究Bol代数的非交换扩展及其Wells精确序列，探讨自同构的可扩展性条件。
We classify, up to equivalences, all abelian groups gradings on null-filiform and one-parametric filiform Leibniz algebras. Any grading on a null-filiform Leibniz algebra is toral but there are non-toral gradings on one-parametric filiform Leibniz algebras.
In our paper, we give a necessary and sufficient conditions for the kernels of the linear maps of finite Abelian group algebras to be Mathieu-Zhao spaces of $K[G]$ if $G$ is a finite Abelian group and $K$ is a split field for $G$. Hence we classify all Mathieu-Zhao spaces of the finite Abelian group algebras if $K$ is a split field for $G$.
Universal enveloping of a graded Lie algebra
本文构造了非阿贝尔群阶化下的李代数阶化泛包络代数，证明其存在唯一性，推导了Witt定理与Ado定理的阶化版本，并探讨了李阶化与阿贝尔阶化的等价性问题。
Indecomposable p-algebras and Galois subfields in generic abelian crossed products
设F是一个Hensel值域，char（F）= p，D是一个半分歧的，“非强退化”的p-代数.我们证明了D的所有伽罗瓦子域都是惯性的。以此为工具，我们研究了一般交换交叉积p-代数，证明了由非退化矩阵定义的非循环一般交换交叉积p-代数是不可分解的p-代数.为了构造这些指数为p且指数为大的不可分解p-代数的例子，我们研究了定义交换交叉积的矩阵的退化性与Severi-Brauer簇CH ^2中的挠之间的关系。
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.
In this paper, we introduce a non-abelian exterior product of Hom-Leibniz algebras and investigate its relative to the Hopf's formula. We also construct an eight-term exact sequence in the homology of Hom-Leibniz algebras. Finally, we relate the notion of capability of a Hom-Leibniz algebra to its exterior product.
In this paper, first we classify non-abelian extensions of Leibniz algebras by the second non-abelian cohomology. Then, we construct Leibniz 2-algebras using derivations of Leibniz algebras, and show that under a condition on the center, a non-abelian extension of Leibniz algebras can be described by a Leibniz 2-algebra morphism. At last, we give a description of non-abelian extensions in terms of Maurer-Cartan elements in a differential graded Lie algebra.
Split strongly abelian p-chief factors and first degree restricted cohomology
本文研究了有限维限制李代数的可裂强交换p-主因子的重数与一次限制上同调的关系。作为应用，我们得到了可解限制李代数的一个刻画，即可解限制李代数的可裂强交换p-主因子的重数。此外，我们还得到了限制李代数的表示论中与有限维限制李代数的平凡不可约模的主块和投射覆盖有关的一些结果。特别地，我们得到了有限维可解限制李代数关于平凡不可约模的投射覆盖的第二Loewy层的一个刻画。这些结果的类似物在有限群的模表示论中是众所周知的。
Deformations, cohomologies and abelian extensions of compatible $3$-Lie algebras
本文提出相容$3$-李代数概念，建立其形变与上同调理论，证明无穷小形变与第二上同调群一一对应，引入Nijenhuis算子产生平凡形变，并利用第二上同调群分类阿贝尔扩张。
In this article we show that distributive law holds for non-abelian tensor product of Lie superalgebras under certain direct sums. There by we obtain a rule for non-abelian exterior square of a Lie superalgebra. We define capable Lie superalgebra and then give some characterization. Specifically we prove that epicenter of a Lie superalgebra is equal to exterior center. Finally we classify all capable Lie superalgebras whose derived subalgebra dimension is at most one. As an application to those results we have shown that there exists at least one non-abelian nilpotent capable Lie superalgebra …
Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms
本文研究了 Rota-Baxter 李代数的非阿贝尔扩张，定义了对其进行分类的非阿贝尔上同调，揭示了自同构可诱导性的障碍，并导出了 Wells 正合序列。
Non-abelian extensions of topological Lie algebras
本文推广和改进了关于特征为零的拓扑域上的李代数扩张到拓扑李代数的几个结果。特别地，我们描述了李代数的扩张的等价类的集合，通过李代数一个具有平移群的仿射空间的不交并H ^2（\g，\z（\n））_{[S]}，其中[S]表示连续外部作用S \：\g \to \der \n的等价类。我们还讨论了拓扑交叉模，并通过证明任何连续的外部作用都产生一个交叉模，其在H^3（\g，\z（\n））_S中的阻塞类是相应交叉模的示性类，解释了它们与李代数扩张的关系。交叉模和扩张之间的对应关系进一步导致了\g的\n -扩张的描述，它是\g通过\n / \z（\n）的扩张的李代数的某些\z（\n）-扩张。我们讨论了几种类型的例子，描述应用到李代数的向量场的主丛，并在两个附录中，我们描述了一套自同构和衍生物的拓扑李代数扩展。
We show that any abelian variety that is not affine has a nontrivial strongly abelian subvariety. In later papers in this sequence we apply this result to the study of minimal abelian varieties.
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