Mathematical Sciences: Algebraic Transformation Groups

小行星9401780 该项目的主要目标是将对称空间的理论推广到其他空间，其中对对称群只做一个假设：它应该是半单李群。 在这种条件下，人们能够定义平坦和小外尔群的概念。 Weyl群有三种不同的构造。 第一个是看空间的边界。 对于球面簇，有一个紧化的规范构造，该项目的一部分是证明它没有奇点。 第二个构造是通过观察对称群的某个子群的作用来完成的。 首席研究员将找到一种方法来重建所有其他数据，以描述空间的所有紧化。 最后，通过观察空间余切丛上的矩映射，可以完成一个构造。 该项目的一个主要问题是更好地理解矩图。 许多不同的代数对象可以表示为其他代数对象的变换的代数集合。 通过这些表征，可以确定它们的结构。 这个项目是关于无限维李代数的表示理论。 对这些代数的研究在整个数学和数学物理中都有应用。 ***

9401780 Knop The main goal of this project is to generalize the theory of symmetric spaces to other spaces where one makes only one assumption on the symmetry group: It should be a semisimple Lie group. Under this condition one is able to define the notion of a flat and of a little Weyl group. There exist three different constructions of the Weyl group. The first is by looking at the boundary of the space. For spherical varieties, there is a canonical construction of a compactification and one part of the project is to prove that it has no singularities. The second construction is accomplished by looking at the action of a certain subgroup of the symmetric group. The principal investigator will find a way to reconstruct all other data needed to describe all compactifications of the space. Finally, there is a construction accomplished by looking at the moment map on the cotangent bundle of the space. A major problem of the project is to obtain a better understanding of the moment map. Many different algebraic objects can be represented as algebraic sets of transformations of other algebraic objects. Through these representations their structure can be determined. This project is concerned with the representation theory of infinite dimensional Lie algebras. The study of these algebras has applications throughout mathematics and mathematical physics. ***