Cohomology and Representation Theory

摘要对于国家科学基金会的提案DMS-0400548 Nakano，主要研究人员将探索涉及代数群、李代数和有限群的表示理论和上同调的问题。一个重要的研究对象将是支撑簇，它为李代数的表示理论、上同调理论和结构理论(涉及共轭类)提供了一座桥梁。这些变体的计算将被考虑，以及将理论推广到量子群和仿射李代数。研究人员还将研究有限Chvalley群和Frobenius核的上同调群/环的计算，特别是对于小于Coxeter数的素数。预计其中几个项目可能需要使用大量的计算机计算。代数结构如群、环和李代数是自然而然产生的，对这些对象的基本理解已经用于涉及生物、物理和化学的应用中。这些代数对象具有复杂的内部对称性。有关表示和上同调理论的信息使人们能够组织和提取可在这些不同应用中使用的重要信息。首席调查员一直在积极宣传这些方法的工作知识。他目前正在组织几个关于表示/上同调理论的会议，并是佐治亚大学Vigre(研究和教育垂直整合)代数小组的联合组织者，该小组通过积极的教职员工和学生参与来促进学习。

Abstract for NSF proposal DMS-0400548 NakanoThe principal investigator will explore problems involving the representation theory and cohomology of algebraic groups, Lie algebras and finite groups. An important object of study will be support varieties which provide a bridge linking the representation theory, cohomology theory and the structure theory (involving conjugacy classes) of Lie algebras. Computations of such varieties will be considered as well as generalizations of the theory to quantum groups and affine Lie algebras. The investigator will also look at computations of cohomology groups/rings of finite Chevalley groupsand Frobenius kernels especially for primes less than the Coxeter number. It is anticipated that the use of extensive computer calculations might be necessary for several of these projects. Algebraic structures such are groups, rings and Lie algebras arise naturally, and the basic understanding of these objects have been used in applications involving biology, physics and chemistry. These algebraic objects have complicated internal symmetries. Information about the representation and cohomology theories allows one to organize and extract vital information that can be used in these various applications. The principal investigator has been actively promoting the working knowledge of these methods. He is currently organizing several conferences in representation/cohomology theory and is a co-organizer of the VIGRE (Vertical Integration of Research and Education) Algebra Group at the University of Georgia which promotes learning through active faculty and student participation.