基于McKay-Slodowy对应的无限维李代数研究

McKay-Slodowy对应为群与李代数之间建立了桥梁，Killing利用Weyl群中的元素置换李代数的根获得了单李代数的分类。这两个经典且重要的结果表明群与李代数的交叉研究是相辅相成、密不可分的。本项目主要以McKay-Slodowy对应为基础，对无限维李代数相关理论中的几个关键问题进行研究。具体研究内容：从本原和非本原群的角度刻画McKay-Slodowy对应；获得仿射Coxeter变换作为螺旋位移的一个标准型；研究多边仿射李代数所对应的仿射Coxeter变换作用在多边仿射李代数的根系上所形成的轨道；探讨根格中相对应元素分解为多边仿射李代数根的和的个数，其与仿射Coxeter元素的Coxeter数的关系；考虑仿射Coxeter元素作用在根系上所形成的轨道与相对应群的Poincaré级数的密切联系；研究A_{2n}^{(1)}型仿射李代数的仿射Coxeter变换的谱问题。

The McKay-Slodowy correspondence establishes a bridge between the groups and the Lie algebras. Killing obtained the classification of the simple Lie algebras by using the elements of Weyl groups to permute the roots of the Lie algebras. These two classical and important results indicate the intersection research of the groups and the Lie algebras is complementary and inseparable. This project mainly studies several key problems in the theory of infinite-dimensional Lie algebras based on the McKay-Slodowy correspondence. We will consider to characterize the McKay-Slodowy correspondence from the perspective of primitive groups and imprimitive groups; we will describe the affine Coxeter transformation as a standard form of spiral displacement; we will research the orbit of the root system of the affine Lie algebras, which acted by the affine Coxeter transformations corresponding to the the multiply laced affine Lie algebras; we will discuss the relationship between the number of the sum of decomposing the root in root lattice into the the roots of multiply laced affine Lie algebra and the Coxeter number of the affine Coxeter element; we will consider the close connection between the orbit formed by the affine Coxeter element acting on the root system and the Poincaré series of the corresponding groups; we will research the spectrum of affine Coxeter transformation which corresponding the affine Lie algebra of type A_{2n}^{(1)}.