Adjoint orbits of the Jacobi group

极小伴随轨道的Katzarkov-Kontsevich-Pantev猜想

Katzarkov-Kontsevich-Pantev conjecture for minimal adjoint orbits

本文证明了极小半单伴随轨道的LG模型满足关于新Hodge理论不变量的Katzarkov-Kontsevich-Pantev猜想。

Lefschetz fibrations on adjoint orbits

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the example of sl(2,C) we compute the Fukaya--Seidel category of Lagrangian vanishing cycles.

Enhanced adjoint action and their orbits for the general linear group

We studied an enhanced adjoint action of the general linear group on a product of its Lie algebra and a vector space consisting of several copies of defining representations and its duals. We determined regular semisimple orbits (i.e., closed orbits of maximal dimension) and the structure of enhanced null cone, including its irreducible components and their dimensions.

Complex adjoint orbits in Lie theory and geometry

This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which adjoint orbits feature prominently (ex. hyperkähler geometry, Landau-Ginzburg models, and the theory of symplectic singularities). Also included are a few arguments and observations that, to the author's knowledge, have not yet appeared in the research literature.

Equations determining the orbit of the highest weight vector in the adjoint representation

We explicitly construct a set of quadratic equations defining the highest weight vector orbit for adjoint representations of Chevalley groups of types D_l, E_6, E_7, and E_8. The combinatorics of these equations is related to the combinatorics of embeddings of the root system of type A_3. We believe that the constructed equations provide a prominent framework for calculations with exceptional groups in adjoint representations, which is particularly interesting for groups of type E_8.

Slodowy slices and the complete integrability of Mishchenko-Fomenko subalgebras on regular adjoint orbits

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve upon this result, showing that each Mishchenko-Fomenko subalgebra yields a completely integrable system on all regular orbits (i.e. orbits of maximal dimension). Our approach incorporates the theory of regular $\mathfrak{sl}_2$-triples and associated Slodowy slices, as developed …

双有理等价的朗之万-金斯堡模型在余切丛和伴随轨道上

Birationally equivalent Landau-Ginzburg models on cotangent bundles and adjoint orbits

研究了最小半单伴随轨道上的李势能与庞特里亚金势能的双有理等价性，并探讨了变形家族中朗之万-金斯堡模型的变形对镜像的影响。

极小伴随轨道的内蕴镜像与奇点范畴

Intrinsic mirrors for minimal adjoint orbits and categories of singularities

该研究探讨$\frak{sl}(n)$极小半单伴随轨道的Landau-Ginzburg模型镜像，验证Gross-Siebert方法的对应性，并通过构造镜像与范畴等价证明了$\frak{sl}(2)$情形下$LG(2)$的同调镜像对称。

余伴随轨道的融合积

Fusion product of co-adjoint orbits

本文的结果是一个与Hamilton圈群作用有关的不动点公式的应用。我们得到了一个G-空间，它是Verlinde融合积的几何对偶。并给出了Verlinde公式的一个证明。不动点集上权的计算是基于仿射李代数的根系的基本性质。这篇论文写于1997年初，目前的版本有一个计算机时间戳4/4/97。

极小伴随轨道的内蕴镜像（ICM G&T 2022）

Intrinsic Mirrors for Minimal Adjoint Orbit (ICM G&T 2022)

本文作为《ZAG现代代数几何手册》第77篇文稿，总结了作者在2022年哥本哈根ICM几何与拓扑分会场发表的简短报告内容。

最小伴随轨道的高阶割线簇

Higher secant varieties of the minimal adjoint orbit

我们试图把经典单李代数的（类属）元素写成长根向量轨道C中尽可能少的元素之和。如果所讨论的李代数是sl_n或sp_n，则C由该李代数中的所有秩为1的矩阵组成，并且秩为k的元素是C的k个元素之和。另一方面，如果李代数是o_n，则C由所有秩为2且平方为零的矩阵组成，情况要复杂得多。在这种情况下，我们也明确地描述了C的k个元素的所有和的集合的闭包，也称为C的第（k-1）正割簇。

在伴随轨道中具有规定条目的矩阵枚举

Enumerating matrices with prescribed entries in an adjoint orbit

研究有限域上方阵共轭类与仿射坐标子空间的交集，给出部分定义线性映射扩展为具有给定相似不变量的自同态数的Hall标量积公式，并应用于计数具有给定Smith标准形和行列式的矩阵多项式。

有理齐性空间嵌入Nilpotent轨道的等变Legendrian分类

Classification of Equivariant Legendrian Embeddings of Rational Homogeneous Spaces into Nilpotent Orbits

本文研究了有理齐性空间到adjoint种的等变Legendrian嵌入分类，通过分析Nilpotent轨道的复接触结构，提出了一种基于稳定子群作用的齐性子变种分类方法。