Cohomology and Representation Theory

摘要NSF提案DMS-0400548中野首席研究员将探讨涉及代数群，李代数和有限群的表示论和上同调的问题。支撑簇是李代数的一个重要研究对象，它是连接李代数的表示理论、上同调理论和结构理论（包括共轭类）的桥梁。计算这些品种将被视为以及推广的理论量子群和仿射李代数。调查人员还将着眼于计算上同调群/环的有限Chevalley groupsand Frobenius内核特别是素数小于Coxeter数。预计其中几个项目可能需要使用广泛的计算机计算。代数结构，如群，环和李代数自然产生，这些对象的基本理解已被用于涉及生物学，物理学和化学的应用。这些代数对象具有复杂的内部对称性。信息的代表性和上同调理论允许一个组织和提取的重要信息，可以在这些不同的应用程序中使用。主要研究者一直在积极推广这些方法的工作知识。他目前正在组织几次会议的代表性/上同调理论，是一个共同组织者的VIGRE（垂直整合的研究和教育）代数组在格鲁吉亚大学，促进学习通过积极的教师和学生的参与。

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.