Full graph and complete Lie algebra

Faithful completely reducible representations of modular Lie algebras

The Ado-Iwasawa Theorem asserts that a finite-dimensional Lie algebra $L$ over a field $F$ has a finite-dimensional faithful module $V$. There are several extensions asserting the existence of such a module with various additional properties. In particular, Jacobson has proved that if the field has characteristic $p>0$, then there exists a completely reducible such module $V$. I strengthen Jacobson's Theorem, proving that if $L$ has dimension $n$ over the field $F$ of characteristic $p>0$, then $L$ has a faithful completely reducible module $V$ with $\dim(V) \le p^{n^2-1}$.

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 $n$.

Complete LR-structures on solvable Lie algebras

An LR-structure on a Lie algebra is a bilinear product, satisfying certain commutativity relations, and which is compatible with the Lie product. LR-structures arise in the study of simply transitive affine actions on Lie groups. In particular one is interested in the question which Lie algebras admit a complete LR-structure. In this paper we show that a Lie algebra admits a complete LR-structure if and only if it admits any LR-structure.

完备与完备的李代数及其注入型和投射型性质

Complete and cocomplete Lie algebras with injective- and projective-type properties

本文研究了有限维李代数中延伸的分裂现象，定义了平凡和半平凡分裂，并证明了完备李代数的等价条件，同时指出投射型性质不存在，且通过谱条件描述几乎阿贝尔李代数的完备性。

Derivation Algebras of Centerless Perfect Lie Algebras Are Complete

It is proved that the derivation algebra of a centerless perfect Lie algebra of arbitrary dimension over any field of arbitrary characteristic is complete and that the holomorph of a centerless perfect Lie algebra is complete if and only if its outer derivation algebra is centerless.

三维可解李代数上的李 affgebra 结构

Lie affgebra structures on three-dimensional solvable Lie algebras

本文完整分类了三维可解李代数上的复李 affgebra 结构。

The algebraic and geometric classification of nilpotent binary Lie algebras

The paper is devoted to give the complete algebraic classification of nilpotent binary Lie algebras of dimension $\leq 6$ over an arbitrary base field ${\mathbb{F}}$ of characteristic not $2$ and the complete geometric classification of nilpotent binary Lie algebras of dimension $6$ over $\mathbb C.$ As an application, we have the algebraic and geometric classification of nilpotent anticommutative $\mathfrak{CD}$-algebras of dimension $\leq 6.$

Classification of multiplicative simple BiHom-Lie algebras

In this paper, we classify flnite dimensional multiplicative simple BiHom-Lie algebras by investigating the corresponding semisimple Lie algebras and we give a complete classification of the complex 3-dimensional multiplicative BiHom-Lie algebras.

The subalgebras of the rank two symplectic Lie algebra

The semisimple subalgebras of the rank $2$ symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{C})$ are well-known, and we recently classified its Levi decomposable subalgebras. In this article, we classify the solvable subalgebras of $\mathfrak{sp}(4,\mathbb{C})$, up to inner automorphism. This completes the classification of the subalgebras of $\mathfrak{sp}(4,\mathbb{C})$. More broadly speaking, in completing the classification of the subalgebras of $\mathfrak{sp}(4,\mathbb{C})$ we have completed the classification of the subalgebras of the rank $2$ semisimple Lie algebras.

Non Complete Affine Connections on Filiform Lie Groups

We give some examples of non-complete invariant affine connections on nilpotent and filiform Lie groups. This permits to describe non-nilpotent faithful representations on the model of filiform n-dimensional Lie algebras and, in particular, on the 3 dimensional Heisenberg algebra.

Centralizers in partially commutative Lie algebras (in Russian)

In this paper, the complete description of centralizers of elements in partially commutative Lie algebras is obtained. The description is given explicitly in the terms of generators.

The classification of $ω$-Lie algebras

Zusmanovich gave a fundamental result on the structure of $ω$-Lie algebras. But up to now, the classification of $ω$-Lie algebras is still open. In this paper, we give a complete classification of $ω$-Lie algebras over $\mathbb C$.

Sur les algèbres de Lie quasi-filiformes complétables [On completable quasi-filiform Lie algebras]

The aim of this work is to determine the quasi-filiform Lie algebras that are completable. We further prove that for any positive integer $m$ there exists a complete Lie algebra, the second cohomology group of which has dimension greater or equal than $m$.

可解辛李代数的完整描述

A complete description of solvable symplectic Lie algebras

通过辛双扩张过程，完整刻画了可解辛李代数，证明了任何此类代数要么是辛不可约的，要么可通过有限次线或平面的辛双扩张从辛不可约李代数构造得到，并分类了维数不超过6的辛李代数。

由Leibniz-dendriform双代数和Zinbiel-dendriform双代数诱导的无穷维pre-Lie双代数

Infinite-dimensional pre-Lie bialgebras induced from Leibniz-dendriform bialgebras and Zinbiel-dendriform bialgebras

通过张量积构造，在Leibniz-dendriform双代数与Zinbiel代数、Zinbiel-dendriform双代数与Leibniz代数上建立完备pre-Lie双代数结构，并利用YBE解构造S-方程的解。

The Pro-Lie Group Aspect of Weakly Complete Algebras and Weakly Complete Group Hopf Algebras

A weakly complete vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ is isomorphic to $\mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is illustrated by the fact that the underlying vector space of the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly complete real or complex associative algebras are studied because they are necessarily projective limits of finite dimensional algebras. The group of units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the asso…

Simple completable contractions of nilpotent Lie algebras

We study a certain class of non-maximal rank contractions of the nilpotent Lie algebra $\frak{g}_{m}$ and show that these contractions are completable Lie algebras. As a consequence a family of solvable complete Lie algebras of non-maximal rank is given in arbitrary dimension.

n元Hom-Nambu超代数与Hom-Lie超代数的完备性

On Completeness of $n$-ary Hom-Nambu and Hom-Lie Superalgebras

本文研究乘法n元Hom-Nambu超代数的完备性，建立相关判据，给出非n元Hom-Lie超代数的完备例子，分析直和、扭曲下中心性质及低维分类中的完备成员。

上界为1的一致有界单李共形代数的分类

Classification of uniformly bounded simple Lie conformal algebras with upper bound one

本文证明一致有界单李共形代数必为有限生成，并完全分类了上界为1的一致有界单李共形代数。

Lie solvable enveloping algebras of characteristic two

Lie solvable restricted enveloping algebras were characterized by Riley and Shalev except when the ground field is of characteristic 2. We resolve the characteristic 2 case here which completes the classification. As an application of our result, we obtain a characterization of ordinary Lie algebras over any field whose enveloping algebra is Lie solvable.

Structures of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras

This paper is devoted to investigating the structure theory of a class of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we completely determine the derivation algebras, the automorphism groups and the second cohomology groups of these Lie superalgebras.

On 3-dimensional complex Hom-Lie algebras

We study and classify the 3-dimensional Hom-Lie algebras over $\mathbb{C}$. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space $\mathfrak{g}$. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps $\mathfrak{g} \to \mathfrak{g}$ that turn $g$ into a Hom-Lie algebra, thus characterizing the corres…

一些ω-李代数的几何与分类

Geometry and classifications of some $ω$-Lie algebras

本文通过群作用和轨道-稳定子方法研究有限维ω-李代数的几何结构，建立同构类与稳定子的轨道空间的一一对应关系，并利用计算理想理论探讨3维ω-李代数的仿射变种几何结构，证明其为6维不可约仿射变种和完全交集。

Sympathetic Lie algebras and adjoint cohomology for Lie algebras

We study sympathetic Lie algebras, namely perfect and complete Lie algebras. They arise among other things in the study of adjoint Lie algebra cohomology. This is motivated by a conjecture of Pirashvili, which says that a non-trivial finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several results on sympathetic Lie algebras and the adjoint Lie algebra cohomology of Lie algebras in general, using the Hochschild-Serre formula. For certain semidirect products we obtain explicit results for the adjoint cohomology.

Affine Structures on abelian Lie Groups

The complete affine structures on abelian Lie algebras in small dimensions are well known. In this paper we are interested by the non complete case. In particular we classify all these structures in dimensions 2 and 3.

The Annihilation theorem for the completely reducible Lie superalgebras

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1] a family of such "singular" Verma modules for osp(1,2l) cases. In this article we give a…

Complete Description of Invariant, Associative Pseudo-Euclidean Metrics on Left Leibniz Algebras via Quadratic Lie Algebras

A pseudo-Euclidean non-associative algebra $(\mathfrak{g}, \bullet)$ is a real algebra of finite dimension that has a metric, i.e., a bilinear, symmetric, and non-degenerate form $\langle\;\rangle$. The metric is considered $\mathrm{L}$-invariant (resp. $\mathrm{R}$-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if $\langle u\bullet v,w\rangle= \langle u,v\bullet w\rangle$ for all $u, v, w \in \mathfrak{g}$. These three notions coincide when $\mathfrak{g}$ is a Lie algebra and in this case $\mathfrak{g}$ endowed with th…

Derivations of Lie Algebras of Dominant Upper Triangular Ladder Matrices

We explicitly describe the Lie algebras $M_L$ of ladder matrices in $M_n$ associate with dominant upper triangular ladders $L$, and completely characterize the derivations of these $M_L$ over a field $F$ with $char(F) \neq 2$. We also completely characterize the derivations of Lie algebras $[M_L, M_L]$ where $L$ are strongly dominant upper triangular ladders and $char(F) \neq 2, 3$.

完全罗塔-巴克斯李代数的正式积分

Formal integration of complete Rota-Baxter Lie algebras

本文提出了一种完全罗塔-巴克斯李代数的正式积分理论，通过后李马格努斯展开给出了罗塔-巴克斯算子的显式公式，并展示了如何从过滤罗塔-巴克斯群构造分层罗塔-巴克斯李环。

On complete affine structures in Lie groups

Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket; left-symmetric algebras can be defined as Lie-admissible algebras such that the multiplication by left defines a representation of the underlying Lie algebra). An affine structure (and the corresponding left symmetric algebra) is complete if $G$ is affinely equivalent to $\mathfrak g$. By the main result of this paper, a complete left symmetric algebra admits a …

Einstein Lorentzian solvable unimodular Lie groups

The goal of this paper is to show that many key results found in the study of Einstein Lorentzian nilpotent Lie algebras can still hold in the more general settings of unimodular Lie algebras and (completely) solvable Lie algebras.

完备 Leibniz 代数的双导子

Biderivations of complete Leibniz algebras

本文针对文献中完备 Leibniz 代数的两种定义，研究其双导子，给出双线性映射为双导子的充要条件，并刻画对称与反对称双导子的结构性质。

The Jacobson--Morozov theorem and complete reducibility of Lie subalgebras

In this paper we determine the precise extent to which the classical sl_2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson--Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer--Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example i…

关于有限莫利等级的p-李代数

On p-Lie algebras of finite Morley rank

本文研究有限莫利等级的p-李代数理论，特别是在可解情况下的完整特征化。

Homology of Lie algebra of supersymmetries and of super Poincare Lie algebra

We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare Lie algebra in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions $\leq 11$. For dimensions $D=10,11$ we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry algebra.