Nilpotent Orbits in Representation Theory

表示论中的幂零轨道

• 批准号: 0201826
• 负责人: Eric Sommers
• 金额: $ 9.65万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-15 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0201826&HistoricalAwards=false
• 关键词: Nilpotent; Orbits; Representation; Theory

DMS-0201826Principal Investigator: Eric Sommers [[e-mail]]Abstract:The principal investigator will study several questions in representationtheory which are related to nilpotent orbits in Lie algebras. Morespecifically, the investigator will study a new duality map that hedefined (and that has recently been extended by Achar). He will continueto study the Lusztig bijection. He will try to complete the determinationof which nilpotent orbits in the exceptional Lie algebras have normalclosure and prove results (in all Lie algebras) about the structure offunctions on covers of nilpotent orbits.Representation theory is a branch of modern algebra that is concerned withunderstanding symmetries. A central idea is that complicated algebraic orgeometric structures can be represented by a certain set of matrices(arrays of numbers), which are easier to understand than the originalstructure. For example, the symmetries of the square can be represented(in one possible way) by a certain set of eight two-by-two matrices. This view of representation theory has many applications in chemistry andphysics. The investigator's work will contribute to understanding therepresentation theory of Lie algebras and Lie groups.

主要研究人员：Eric Sommers[[e-mail]]摘要：主要研究人员将研究表示论中与李代数中幂零轨道有关的几个问题。更具体地说，研究人员将研究他定义的新的对偶映射(最近由Achar扩展)。他将继续研究Lusztig双射。他将试图完成确定例外李代数中哪些幂零轨道具有正规闭包，并证明(在所有李代数中)关于幂零轨道覆盖上的函数结构的结果。表示论是现代代数中与理解对称性有关的一个分支。一个中心思想是，复杂的代数或几何结构可以用一组特定的矩阵(数组)来表示，这比原始结构更容易理解。例如，正方形的对称性可以用一组8个2乘2的矩阵来表示(以一种可能的方式)。这种表象理论在化学和物理中有很多应用。研究人员的工作将有助于理解李代数和李群的表示理论。