The classification of thick representations of simple Lie groups

量子环面的导子李代数上不可约有界权模的分类

Classification of irreducible bounded weight modules over the derivation Lie algebras of quantum tori

本文给出了有理量子环面的导子李代数上不可约有界权模的一个完全分类

Classification of finite-growth contragredient Lie superalgebras

A contragredient Lie superalgebra is a superalgebra defined by a Cartan matrix. In general, a contragredient Lie superalgebra is not finite dimensional, however it has a natural Z-grading by finite dimensional components. A contragredient Lie superalgebra has finite growth if the dimensions of these graded components depend polynomially on the degree. We discuss the classification of finite-growth contragredient Lie superalgebras. (Joint work with Vera Serganova)

Classification of minimal representations of real simple Lie groups

Based on an idea in [Gan--Savin, Represent. Theory (2005)], we give a classification of minimal representations of connected simple real Lie groups not of type $A$. Actually, we prove that there exist no new minimal representations up to infinitesimal equivalence.

A quick proof of the classification of real Lie superalgebras

This article classifies the real forms of Lie Superalgebra by Vogan diagrams, developing Borel and de Seibenthal theorem of semisimple Lie algebras for Lie superalgebras. A Vogan diagram is a Dynkin diagram of triplet $(\mathfrak{g}_{C},\mathfrak{h_{\bar{0}}},\triangle^{+})$, where $\mathfrak{g}_{C}$ is a real Lie superalgebra, $\mathfrak{h_{\bar{0}}}$ cartan subalgebra, $\triangle^{+}$ positive root system. Although the classification of real forms of contragradient Lie superalgebras is already done. But our method is a quicker one to classify.

Classification of kinematical Lie algebras

We summarise the classification of kinematical Lie algebras in arbitrary dimension and indicate which of the kinematical Lie algebras admit an invariant inner product.

Classification of irreducible integrable representations of loop toroidal Lie algebras

In this paper we classify irreducible integrable representations of loop toroidal Lie algebras with finite dimensional weight spaces. In both the cases we classify modules, when a part of center acts non-trivially and trivially on modules.

Classification of balanced toral elements of exceptional Lie algebras

Let $\mathfrak{g}$ be a simple Lie algebra of exceptional type over an algebraically closed field $k$, and let $G$ be a simple linear algebraic group with Lie algebra $\mathfrak{g}$. For $\mathrm{char} \, k =p >0$, we present a complete classification of the $G$-conjugacy classes of balanced toral elements of $\mathfrak{g}$. As a result, we also obtain the classification of conjugacy classes of balanced inner torsion automorphisms of $\mathfrak{g}$ of order $p$ when $\mathrm{char} \, k =0$.

Classification of spherical algebraic subalgebras of real simple Lie algebras of rank 1

We determine all spherical algebraic subalgebras in any simple Lie algebra of real rank 1.

Classification of Simple Cuspidal Modules over a Lattice Lie Algebra of Witt type

Let $W_π$ be the lattice Lie algebra of Witt type associated with an additive inclusion $π: \mathbb{Z}^N \hookrightarrow \mathbb{C}^2$ with $N>1$. In this article, the classification of simple $\mathbb{Z}^N$-graded $W_π$-modules, whose multiplicities are uniformly bounded, is given.

Classification of level zero irreducible integrable modules for twisted full toroidal Lie algebras

In this paper, we first construct the twisted full toroidal Lie algebra by an extension of a centreless Lie torus $LT$ which is a multiloop algebra twisted by several automorphisms of finite order and equipped with a particular grading. We then provide a complete classification of all the irreducible integrable modules with finite dimensional weight spaces for this twisted full toroidal Lie algebra having a non-trivial $LT$-action and where the centre of the underlying Lie algebra acts trivially.

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 $n,n\leq7$.

Classification of integrable representations for toroidal extended affine Lie algebras

In this paper, we classify the irreducible integrable modules with finite dimensional weight spaces and non-trivial $\widetilde{\mathfrak g}_c$-action for the nullity $2$ toroidal extended affine Lie algebra $\widetilde{\mathfrak g}$, where $\widetilde{\mathfrak g}_c$ is the core of $\widetilde{\mathfrak g}$.

单李代数中主幂零对的分类及相关问题

A classification of the principal nilpotent pairs in simple Lie algebras and related problems

给出了所有经典李代数中主幂零对和特殊幂零对的分类。由于例外单李代数中的主对的分类在前面已经得到（参见金兹伯格的数学预印本AG/9903059的附录），因此我们有了主幂零对的完整分类。

有限群的有界导出范畴的厚子范畴

Thick subcategories of the bounded derived category of a finite group

给出了有限群的模表示的有界导出范畴和稳定范畴的张量理想厚子范畴的分类的一个新的证明。更一般地，参数适用于产生一个分类的厚子范畴的有界导出范畴的artinian完全相交环。这项工作的显着特点之一是，它不需要求助于无限的建设，不像以前的证明这些结果。

三维李代数的Bianchi分类

Bianchi's classification of 3-dimensional Lie algebras revisited

我们提出了Bianchi的证明分类的真正的（和复杂的）$3$ -维李代数在一个坐标自由的版本从严格表示理论的观点。附近，我们还计算自同构群，并从这个相应的轨道在代数簇$X\subseteqΛ^2V^*\otimes V$描述所有的李括号在一个固定的向量空间$V$维度$3$的轨道维度。此外，我们澄清了哪些轨道位于一个给定的轨道的闭包，从而与$G=\mathrm{Aut}(V)$的轨道空间$X/G$的拓扑。

Classification of irreducible integrable modules for extended affine Lie algebras with center acting trivially

We classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras with fnite diemnsional weight spaces where the fnite dimensional center acts trivially. We have proved that the entire central extension part acts trivially on the modules and our modules turn out to be highest weight modules.

Cohen-Macaulay模稳定范畴的厚子范畴的分类

Classifying thick subcategories of the stable category of Cohen-Macaulay modules

三角范畴的厚子范畴的分类定理在数学的许多领域中得到了广泛的应用。本文考虑Gorenstein局部环上Cohen-Macaulay模的稳定范畴的厚子范畴的分类，作为Benson，Carlson和Rickard关于有限$p$ -群的双生成表示的稳定范畴的厚子范畴分类定理的高维版本.本文的主要结果给出了局部超曲面上Cohen-Macaulay模的稳定范畴的厚子范畴的一个完全分类，它是用包含在其奇异轨迹中的环的素理想谱的专门化闭子集来表示的.

有限维权空间的环面李代数不可约可积模的分类

Classification of Irreducible integrable modules for toroidal Lie-algebras with finite dimensional weight spaces

Toroidal李代数是以Laurent多项式为张量的有限维单李代数在多个交换变量上的泛中心扩张。本文对有限维权空间上的Toroidal李代数的不可约可积模进行了分类。它们是有限维李代数的赋值模在中心作用为零时的张量积。在中心作用非平凡的情况下，它们是由Toroidal李代数的自同构扭曲的底层仿射李代数的求值表示的张量积。自同构来自行列式为1或-1的整数的一般线性群。

具有结构识别的 Lie Amaldi 分类目录

A Catalogue of the Lie Amaldi Classification with Structures Identified

本文使用 CLAUDE 生成 Lie Amaldi 分类及其结构的目录，并解释结构描述的正确性及指导原则。

映射全环面李代数上不可约Harish-Chandra模的分类

Classification of irreducible Harish-Chandra modules over map full toroidal Lie algebras

本文分类了映射全环面李代数上的不可约Harish-Chandra模，证明其必为尖点模或最高权模，且均为单点赋值模。

Classification of simple bounded weight modules of the Lie algebra of vector fields on $\C^n$

Let $W_n^+$ be the Lie algebra of the Lie algebra of vector fields on $\C^n$. In this paper, we classify all simple bounded weight $W_n^+$ modules. Any such module is isomorphic to the simple quotient of a tensor module $F(P,M)=P\otimes M$ for a simple weight module $P$ over the Weyl algebra $K_n^+$ and a finite dimensional simple $\gl_n$ module $M$.

Classification of contraction algebras and pre-Lie algebras associated to braces and trusses

We develop tools for classification of contraction algebras and apply these to solve the problem on classification up to isomorphism of 8 and 9 dimensional algebras corresponding to 3-fold flops. We prove that there is only one up to isomorphism contraction algebra of dimension 8, and two algebras of dimension 9. The formulae for the dimension of algebra, depending on the type of the potential are obtained. In the second part of the paper we show that associated graded structure to brace and truss with appropriate descending ideal filtration is pre-Lie.

Classification of orbit closures in the variety of 4-dimensional symplectic Lie algebras

The aim of this paper is to study the natural action of the real symplectic group, $\operatorname{Sp}(4, \mathbb{R})$, on the algebraic set of $4$-dimensional Lie algebras admitting symplectic structures and to give a complete classification of orbit closures. We present some applications of such classification to the study of the Ricci curvature of left-invariant almost Kähler structures on four dimensional Lie groups.

有限维单模李超代数的分类

Towards classification of simple finite dimensional modular Lie superalgebras

给出了特征不为2的代数闭域上有限维模李（超）代数的一种构造方法。在特征2中，假设该方法只给出整数阶化的单李（超）代数和一些非阶化的单李（超）代数。该猜想是支持最新的结果计算最困难的是获得的帮助下Grozman的软件包SuperLie。

Graded simple Lie algebras and graded simple representations

For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional $Q$-graded simple Lie algebras over any algebraically closed field of characteristic $0$ based on the recent classification of gradings on finite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any $Q$-graded Lie algebra (not necessari…

限制李超代数的中心扩张与4维$p$-幂零李超代数的分类

Central extensions of restricted Lie superalgebras and classification of $p$-nilpotent Lie superalgebras in dimension $4$

本文修正已有构造建立特征p≥3下限制李超代数的一二阶限制上同调群，阐释其对代数结构的刻画作用，并应用该构造完成特征p≥3代数闭域上4维及以下p-幂零限制李超代数的分类。

Classification of Realizations of Lie Algebras of Vector Fields on a Circle

Finite-dimensional subalgebras of a Lie algebra of smooth vector fields on a circle, as well as piecewise-smooth global transformations of a circle on itself, are considered. A canonical forms of realizations of two- and three-dimensional noncommutative algebras are obtained. It is shown that all other realizations of smooth vector fields are reduced to this form using global transformations. Some combinatorial formulas for the number of inequivalent realizations of these algebras are obtained.

格上单李代数的分类

Classification of Simple Lie Algebras on a Lattice

令$Λ$是秩为$n$的格。格$Λ$上的李代数是一个李代数${\cal L}=\oplus_{λ\inΛ}\,{\cal L}_λ$，使得对所有$λ$有$\dim\,{\cal L}_λ=1$。本文对格上的单分次李代数进行了分类。

On classification of Lie algebra realizations

We study realizations of Lie algebras by vector fields. A correspondence between classification of transitive local realizations and classification of subalgebras is generalized to the case of regular local realizations. A reasonable classification problem for general realizations is rigorously formulated and an algorithm for construction of such classification is presented.

共形李超代数CK6上有限不可约模的分类

Classification of finite irreducible modules over the Lie conformal superalgebra CK6

我们分类了例外线性紧李超代数E（1，6）上的所有连续退化不可约模和例外共形李超代数CK 6上的所有有限退化不可约模，其中E（1，6）是零化代数.

圆上矩阵微分算子李代数上拟有限模的分类

Classification of Quasifinite Modules over Lie Algebras of Matrix Differential Operators on the Circle

证明了$N\times N$ -矩阵微分算子的李代数的中心扩张上的不可约拟有限模是最高或最低权模，或者是中间级数模.进一步，我们给出了不可分解一致有界模的一个完全分类。

Bounded weight modules of the Lie algebra of vector fields on ${\mathbb C}^2$

We study weight modules of the Lie algebra $W_2$ of vector fields on ${\mathbb C}^2$. A classification of all simple weight modules of $W_2$ with a uniformly bounded set of weight multiplicities is provided. To achieve this classification we introduce a new family of generalized tensor $W_n$-modules. Our classification result is an important step in the classification of all simple weight $W_n$-modules with finite weight multiplicities.

有限增长广义Kac-Moody超代数的分类

Classification of finite-growth general Kac-Moody superalgebras

逆成分李超代数是由Cartan矩阵定义的超代数。一个逆成分李超代数是有限增长的，如果其分次分支的维数（在自然分次中）多项式依赖于其次数。本文对有限增长逆成分李超代数进行了分类。以前，这样的分类只知道对称化的情况。

特征$2$下的单李超代数分类

Classification of simple Lie superalgebras in characteristic $2$

本文针对特征2情形，提出两种由单李代数构造单李超代数的程序，完成了有限维单李超代数的相对分类，还引入了李代数的(2,4)-结构及李超代数的多种对应限制结构，并指出超簇范畴下的分类问题仍待解决。

Classification of finite irreducible conformal modules over $N=2$ Lie conformal superalgebras of Block type

We introduce the $N=2$ Lie conformal superalgebras ${\frak {K}}(p)$ of Block type, and classify their finite irreducible conformal modules for any nonzero parameter $p$. where $p$ is a nonzero complex number. In particular, we show that such a conformal module admits a nontrivial extension of a finite conformal module $M$ over $K_2$ if $p=-1$ and $M$ has rank $(2+2)$, where $K_2$ is an $N=2$ conformal subalgebra of ${\frak {K}}(p)$. As a byproduct, we obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal superalgebras ${\frak k}(n)$ for $n\ge1$…

Classification of simple strong Harish-Chandra modules over the Lie superalgebra of vector fields on $\C^{m|n}$

In this paper, we classify simple strong Harish-Chandra modules over the Lie superalgebra $W_{m,n}$ of vector fields on $\C^{m|n}$. Any such module is the unique simple submodule of some tensor module $F(P,V)$ for a simple weight module $P$ over the Weyl superalgebra $\mathcal K_{m,n}$ and a simple weight module $V$ over the general linear superalgebra $\gl_{m,n}$.

Classification of irreducible $(\mathfrak{g},\mathfrak{k})$-modules associated to the ideals of minimal nilpotent orbits for simple Lie groups of type $A$

We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible $(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.

谐振子上Bianchi分类三维李代数的动力学变形

Dynamical deformations of three-dimensional Lie algebras in Bianchi classification over the harmonic oscillator

利用谐振子的运算Lax表示构造了Bianchi分类下三维实李代数的动力学变形.证明了谐振子的能量守恒与动力变形代数的雅可比恒等式有关。基于这一观察，证明了谐振子上Bianchi分类中的三维实李代数的动力学变形是李代数。

实丽超代数不可约实模的分类

Classification of irreducible real modules of real Lie superalgebras

本文通过研究奇性与共轭泛函的轨道来分类实丽超代数的不可约有限维模，揭示了实简单李代数模分类的新视角及与Kostant强正交根 cascade 的联系。

Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $\mathcal{W}(a,b)$

In this paper, under some natural condition, a complete classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $\mathcal{W}(a,b)$ is presented. Moreover, applying this result, we obtain a class of compatible left-symmetric algebraic structures on the coefficient algebra of $\mathcal{W}(a,b)$.

通过弯曲 BGG 对应对 Koszul 复形上的厚子范畴进行分类

Classifying thick subcategories over a Koszul complex via the curved BGG correspondence

本文利用弯曲 BGG 对应及完美弯曲 dg 模的厚张量子模分类，刻画了正则环中任意元素列表的 Koszul 复形上有界 dg 模导出范畴的厚子范畴，并恢复 Stevenson、Hopkins–Neeman 等结果。

Lie对C^N中向量场的有限维代数的分类

Lie's classification of finite dimensional algebras of Vector Fields in C^N

本文概述了Lie关于两变量和三变量向量场有限维子代数的经典结果的简要证明，并给出了任意N维C^N中最大秩向量场代数的结果。

基于projective geometry几何的复单李代数分类

Classification of complex simple Lie algebras via projective geometry geometry

利用齐次簇的射影几何给出了复单李代数分类的一个新证明。我们的证明收益建设齐性品种使用理想的割线和切品种的齐性品种已经建成。我们的算法不涉及根系。我们的证明使用根系的属性，但不是他们的分类。

扭Schrödinger-Virasoro李代数上具有非线性权空间的不可约权模的分类

Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over the Twisted Schrödinger-Virasoro Lie algebra

证明了Schrödinger-Virasoro李代数上具有无限维权空间的不可约权模的支集与权格一致，并且这种模的所有非平凡权空间都是无限维的.作为副积，得到了有限维权空间上的Schrödinger-Virasoro李代数上的每个单权模都是Harish-Chandra模.

Classification of finite-dimensional Lie superalgebras whose even part is a three-dimensional simple Lie algebra over a field of characteristic not two or three

Let $k$ be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ over $k$, where $\mathfrak{g}_0$ is a three-dimensional simple Lie algebra. If $\mathcal{Z}(\mathfrak{g})$ denotes the centre of $\mathfrak{g}$, the result is the following: either $\lbrace \mathfrak{g}_1,\mathfrak{g}_1 \rbrace=\lbrace 0 \rbrace$ or $\mathfrak{g}_1=(\mathfrak{g}_0 \oplus k)\oplus \mathcal{Z}(\mathfrak{g})$ or $\mathfrak{g}\cong \mathfrak{osp}_k(1|2)\oplus \mathcal{Z}(\mathfrak{g})$.

仿射李代数$A_1 ^{(1)}$的不可约Wakimoto模的分类

A classification of irreducible Wakimoto modules for the affine Lie algebra $A_1 ^{(1)}$

通过使用arXiv：math/0602181中开发的方法，我们研究了$\widehat{sl_2}$的某些Wakimoto模在临界水平上的不可约性。我们对所有的$χ\in {\Bbb C}((z))$进行分类，使得相应的Wakimoto模$W_χ$是不可约的。结果表明，Schur多项式的零点在分类结果中起着重要的作用。

Block型李代数上中间级数的Z^2-分次模的分类

Classification of Z^2-graded modules of the intermediate series over a Lie algebra of Block type

设L是特征为零的域F上的Block型李代数，定义为基{ L_{i,j}| i，j \in Z}和关系[ L_{i,j}，L_{k,l} ]=（（j+1）k-（1 +1）i）L_{i+k,j+l}。则L是Z^2-分次的。本文对中间级数的Z^2-分次L-模进行了分类。

有限秩单微分李与若尔当（超）余代数的分类

Classification of simple differential Lie and Jordan (super)coalgebras of finite rank

本文分类了有限秩单微分李与若尔当（超）余代数，并显式描述了 n=2,3,4 超共形李代数及例外李共形超代数 CK$_6$ 的 OPE 所对应的李超余代数。

小特征VI的单李代数完成分类

Simple Lie algebras of small characteristic VI. Completion of the classification

设L是F特征p>3的代数闭域上的有限维单李代数。证明了如果L的导子代数代数中L的p-包络包含最大维数的非标准环面，则p=5且L同构于其中一个Melikian代数.与我们以前的结果一起，这意味着F上的任何有限维单李代数都是经典型、Cartan型或Melikian型的。

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of…

在Lean中形式化低维可解李代数的分类定理

Formalizing a classification theorem for low-dimensional solvable Lie algebras in Lean

该研究在Lean中形式化了任意域上维数至多三的可解李代数的分类定理，涉及底层向量空间计算，并扩展了mathlib的线性代数与李理论库。

与Virasoro代数相关的李代数上Harish-Chandra模的分类

Classification of Harish-Chandra modules over some Lie algebras related to the Virasoro algebra

本文给出了一个统一的方法，对与Virasoro代数相关的李代数上的所有Harish-Chandra模进行了彻底的分类.我们首先对$s=0,\frac12$的李代数$W(\varrho)[s]$上的这类模进行分类。利用这个结果和方法，我们也可以对一些与Virasoro代数有关的李代数和超共形代数，包括目前公开的几类Schrödinger-Virasoro李代数，进行这样的工作.

仿射李超代数的可积模

Integrable modules for affine Lie superalgebras

研究了非扭仿射李超代数中具有有限维权空间的不可约非零水平模。给出了超代数A（m，n）^和C（n）^的一个完全分类.在其他情况下的分类问题是减少到分类的尖形模有限维尖形李超代数所描述的Dimitrov，Mathieu和Penkov。证明了任何不可约弱可积（Kac和Wakimoto意义下）模都是最高权模.

超级李群的么正表示及其在超级粒子分类和多重态结构中的应用

Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles

超李群范畴（SLG）等价于超Harish-Chandra对范畴（SHCP）。利用这个等价性，我们定义了超李群的酉表示范畴。我们给出了经典的诱导构造和Mackey不可导性定理在此情形下的推广。我们使用我们的结果分类的不可约酉表示的半直积的超平移群的经典李群，特别是在任意维的超庞加莱群。最后，我们比较了我们的结果与物理文献中的超多重态的结构和分类。