Cohomology and Representation Theory

上同调和表示论

• 批准号: 0400548
• 负责人: Daniel Nakano
• 金额: $ 11.84万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-10-01 至 2008-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400548&HistoricalAwards=false
• 关键词: Cohomology; Representation; Theory

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.

NSF提案DMS-0400548摘要 Nakano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。一个重要的研究对象将是支持簇，它为连接李代数的表示论、上同调理论和结构理论（涉及共轭类）提供了桥梁。将考虑此类变体的计算以及理论对量子群和仿射李代数的推广。研究人员还将研究有限 Chevalley 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。群、环和李代数等代数结构自然出现，对这些对象的基本理解已应用于涉及生物学、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允许人们组织和提取可在这些不同应用中使用的重要信息。首席研究员一直在积极推广这些方法的工作知识。他目前正在组织多个表示/上同调理论会议，并且是佐治亚大学 VIGRE（研究与教育垂直整合）代数小组的共同组织者，该小组通过积极的教师和学生参与来促进学习。