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

首席研究员计划研究李型有限群、代数群、李代数和颤振的表示理论之间的详细联系。本文将集中研究李型有限群的同调理论、限制李代数的同调理论和无穷小群方案的同调理论。目的是描述给定一个代数群的有理模，它的支持簇在有限域上的有理点群的支持簇在高Frobenius核上的支持簇。首席研究员还将研究颤振的表示与Kac-Moody李代数和量子群的表示（及其推广）之间的联系。这将涉及到使用逆束的hall代数的子代数的研究，构造泛子代数的正则基，用颤振的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称为表示理论。表征理论的核心源于对对称性的研究，即从亚原子粒子到行星轨道等自然物体所具有的对称性。现在有许多不同类型的表征理论，每一种都发展成为处理不同类型问题的最佳方式。这个项目的目的是寻找和探索一些不同类型的表征理论之间的联系。这种联系不仅有利于表征理论的研究，也有利于将表征理论作为标准工具的物理学和其他科学。

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.