Connecting Representations of Algebraic Groups, Finite Groups, Lie Algebras, Quantum Groups, and Related Quivers

主要研究者计划调查详细的连接amongrepresentation理论的有限群的李类型，代数群，李代数，和箭。 本文的研究将集中于李型有限群的上同调理论与限制李代数和无穷小群概型的上同调理论的联系。目的是用代数群的高阶Frobenius核上的支集簇来描述代数群的有理模在有限域上有理点的有限群上的支集簇. 主要讲师还将研究箭图的表示与Kac-Moody李代数和量子群（及其推广）的表示之间的联系。这将涉及到利用反常层研究Hall-algebras代数的子代数，构造一般子代数的标准基，用箭图表示实现仿射李代数，并将它们与Fock空间联系起来。 数学研究的美妙之处在于将许多不同的学科领域联系在一起，将现有的理论应用于许多不同的领域。 这个项目是在数学领域被称为代表性理论。 在其核心，表征理论源于对对称性的研究，即，从亚原子粒子到行星轨道的自然物体所具有的对称性。 现在有许多不同种类的表征理论，每一种都已经发展成为处理不同类型问题的最佳方法。 这个项目的目的是发现和探索一些不同类型的表征理论之间的联系。 这样的联系不仅有利于表征理论的研究，也有利于物理学和其他使用表征理论作为标准工具的科学。

The principal investigator plans to investigate detailed connections amongrepresentation theories of finite groups of Lie types, algebraic groups, Liealgebras, and quivers. The research will concentrate on relating thecohomology theory of finite groups of Lie type and that of restricted Liealgebras and infinitesimal group schemes. The goal is to describe, for agiven rational module for an algebraic group, its support varieties overfinite groups of rational points over finite fields in terms of its supportvarieties over the higher Frobenius kernels. The principal investigatorwill also study the connections between representations of quivers andrepresentations of Kac-Moody Lie algebras and quantum groups (and theirgeneralizations). This will involve the study of subalgebras of Hallalgebras using perverse sheaves, construct canonical basis of genericsubalgebras, realize affine Lie algebras in terms of representations ofquivers, and relate them to Fock spaces. The beauty of mathematical research lies in linking many different subjectareas together to apply available theory in many different fields. This project is in the area of mathematics known as representation theory. At its core, representation theory derives from the study of symmetries, i.e., the symmetries that natural objects, from subatomic particles to planetary orbits, have. Nowadays there are many different kinds of representation theory, each having evolved as the best way to deal with different kinds of problems. The purpose of this project is to find and explore some of the links between some of the different kinds of representation theory. Such links will benefit not only the study of representation theory, but also physics and other sciences that use representation theory as a standard tool.