On exceptional nilpotents in semisimple Lie algebras
A. G. Elashvili, V. G. Kac, E. B. Vinberg
本文对在单有理W-代数的分类中出现的半单李代数的所有对（m，e）进行了分类，其中m是正整数，e是幂零元.
We classify all pairs (m,e), where m is a positive integer and e is a nilpotent element of a semisimple Lie algebra, which arise in the classification of simple rational W-algebras.
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On the second cohomology of nilpotent orbits in exceptional Lie algebras
本文研究非紧致非复例外李代数中幂零轨道的第二上同调，计算了大多数情形的维数，并对其余情形给出上界。
Computing with nilpotent orbits in simple Lie algebras of exceptional type
设G是一个代数闭域上的单代数群，其上有李代数g.然后对g的幂零元在G的伴随作用下的轨道进行了分类.我们描述了一个简单的算法，找到一个代表的幂零轨道。我们用它来计算这些轨道的特殊类型的李代数的代表列表。然后我们给出两个应用。第一个是关于解决Elashvili关于幂零轨道中心化子指数的猜想，其中李代数是例外型的。第二类是双扶正器的最小尺寸问题。
In the paper "The Second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee and C. Maity homotopy types of nilpotent orbits are explicitly described in the case of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is not semisimple. In this paper we extend the above description of homotopy type of nilpotent orbits to the remaining cases of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is semisimple.
Special representations of nilpotent Lie groups and the associated Poisson representations of current groups
在本文中，我们描述了一个新的模型的表示的电流群与一个半单李群的秩为1。在早期的论文70-80日（荒木，Gelfand-Graev-Vershik）提出了问题的不可约表示的当前组为$SL(2,R)$，并用于这一著名的福克空间结构，建设可以适用于任意局部紧群，是基于所谓的特殊表示的原组$G$，与非平凡1-上同调。利用幂零群的一维扩张（半直积）的特殊性质，给出了一种新的构造方法，该方法可以直接构造出群的特殊表示，然后利用作者以前的文章中的拟Poisson构造方法构造出当前群的表示.秩为1的半单群的抛物子群具有这样的半直积，它的特殊表示可以推广到整个半单群上。从而得到了所需的半单流群的不可约表示的新模型。
On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits
给出了实半单李群中伴随轨道上的Kostant-Kirillov型是精确的一个判据。我们明确地计算了每个复单李代数中所有幂零伴随轨道的第二上同调。
Let $\mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$. We show that, except for some explicit exceptional cases, every derivation of the nilpotent subalgebra $\mathfrak{n}$ that preserves its restricted root space decomposition is of the form $\text{ad}( W)$, where $W \in \mathfrak{m}\oplus .
Harish-Chandra integrals as nilpotent integrals
最近的相关函数的所谓的Itzykson-Zuber/Harish-Chandra积分计算（由作者和合作者之一）为所有典型群使用的积分公式，涉及积分在紧群的Haar措施和高斯积分在一个极大幂零李子代数的复杂化。由于后验积分公式对于经典级数具有相同的形式，因此提出了一个猜想，即这样的公式对于任意半单李群都成立。我们证明这个猜想使用一个抽象的李理论的方法。
Stepwise Square Integrable Representations: the Concept and Some Consequences
幂零李群的酉表示的Plancherel公式和矩阵系数的增长性有了新的发展。这对弱对称空间的几何学和实半单李群的抛物子群的分析，以及对（无限维）局部幂零李群的分析都有一些影响。其中许多后果仍在研究之中。在本说明中，我将调查一些这些新的方面的表示论的幂零李群和抛物子群。
Small spherical nilpotent orbits and K-types of Harish Chandra modules
设G是连通线性半单李群，其李代数g和极大紧子群K。设K_C -> Aut（p_C）是对称空间在单位陪集上的复化迷向表示.设O是p_C中的一个幂零K_C-轨道，bar（O）是它在p_C中的Zapriki闭包.本文研究了棒（O）上的正则函数环的K-型分解，当O是球形的且"小“的时.我们还证明了这种分解给出了任何伴随簇为bar（O）的不可约（g_C，K）-模中K-型的渐近方向。
The structure of the nilpotent cone, the Kazhdan-Lusztig map and algebraic group analogues of the Slodowy slices
定义了复半单李代数g中Slodowy横截片与伴随轨道的代数群类似.这些新的切片横截于具有李代数g的代数群G的共轭类。这些切片与g的外尔群W的元素s（的共轭类）相关联。对于这样的切片，我们证明了一个类似的Kostant截面定理的幂幺群的行动。
In this paper the authors introduce an analog of the nilpotent cone, ${\mathcal N}$, for a classical Lie superalgebra, ${\mathfrak g}$, that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$ with $\text{Lie }G_{\bar{0}}={\mathfrak g}_{\bar{0}}$, it is shown that there are finitely many $G_{\bar{0}}$-orbits on ${\mathcal N}$. Later the authors prove that the Duflo-Serganova commuting variety, ${\mathcal X}$, is contained in ${\mathcal N}$ for any classical …
New directions in the study of prime ideals in rational, nilpotent Iwasawa algebras
本文探讨了幂零p-可值（紧p进李）群的完成群代数（伊瓦萨代数）素理想，验证了关于其标准形式的猜想，排除了类型C的例外情况。
Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (re…
Series of nilpotent orbits
利用triality模型将特殊复李代数中的幂零轨道组织成级数，并证明了在每个级数中，对于f_4，e_6，e_7，e_8，轨道的维数分别是自然参数a= 1，2，4，8的线性函数.我们还以统一的方式获得明确的代表。我们观察到类似的中心化子的幂零元元在一个系列和分次组件在相关的分次的环境李代数。更引人注目的是，对于一个大于1，相应的Chevalley群的幂幺特征的次数，通过施普林格对应关系与这些级数相关联，由多项式给出，这些多项式具有关于a的统一表达式。
Irreducible A_1 Subgroups of Exceptional Algebraic Groups
本文分类了例外代数群的所有不可约 A_1 子群，并证明了此类子群的共轭类可由其在伴随模上的合成因子决定。
Let $G$ be a connected reductive algebraic group over an algebraically closed field $\mathbf{k}$, and let Lie$(G)$ be its associated Lie algebra. In his series of papers on unipotent elements in small characteristic, Lusztig defined a partition of the unipotent variety of $G$. This partition is very useful when working with representations of $G$. Equivalently, one can consider certain subsets of the nilpotent variety of Lie$(G)$ called pieces. This approach appears in Lusztig's article from 2011. The pieces for the exceptional groups of type $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$ in bad charac…
In this paper we classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was done by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology to obtain the cl…
Leibniz algebras are a non-anticommutative version of Lie algebras. They play an important role in different areas of mathematics and physics and have attracted much attention over the last thirty years. In this paper we investigate whether conditions such as being a Lie algebra, cyclic, simple, semisimple, solvable, supersolvable or nilpotent in such an algebra are preserved by lattice isomorphisms.
A note on the Jordan decomposition
本文证明了有限维向量空间的自同态$X$（或同构$g$）的椭圆、双曲和幂零（或幂单）的加（或乘）Jordan分支由$X$（或$g$）中的多项式给出.利用这一点，我们给出了一个简单的证明：对于线性半单李代数$\g$（或线性半单连通李群$G$的$g$）的一个元素$X$，它的三个Jordan分支又位于代数（群）中.这是以前未知的线性李群，然后$\Int(\g)$。这意味着，对于这类代数和群，通常的线性若尔当分解与抽象的若尔当分解一致。
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