Nilpotent Orbits in Representation Theory

DMS-0201826主要研究者：Eric Sommers [esommers@math.umass.edu]摘要：主要研究者将研究表示论中与李代数中的幂零轨道有关的几个问题。 更具体地说，研究者将研究他定义的一个新的对偶映射（最近由Achar扩展）。 他将继续研究Lusztig双射。 他将试图完成determinationof其中幂零轨道在例外李代数有normalclosure和证明结果（在所有李代数）的结构offunctions覆盖幂零轨道。表示论是一个分支现代代数，是关注与理解对称性。 一个中心思想是，复杂的代数或几何结构可以用一组矩阵（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（以一种可能的方式）由一组特定的8个2 × 2矩阵表示。 这种表象论的观点在化学和物理学中有许多应用。 研究者的工作将有助于理解李代数和李群的表示理论。

DMS-0201826Principal Investigator: Eric Sommers [esommers@math.umass.edu]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.