Mathematical Sciences: Group Theory and Differential Geometry

该项目涉及与经典李群和例外李群的表示和几何量化相关的各种主题的研究。研究者建议寻找新的方法来表示一个不可约的表示的特征作为一个共伴轨道的傅里叶变换。奇异表示，标志流形的量子上同环，以及在其外代数中出现的半单李代数的表示是要追求的主题之一。李群通常以对称群的形式出现。例如，所谓的酉李群描述了基本粒子的对称性。另一个例子是碳60分子，俗称巴基球。这里的全对称群是一个例外的李群。只有有限的例外群，而经典群的数量是无限的，关于如何将它们作为经典群的子群来实现，已经做了很多工作。

9625941 Kostant This project deals with research in various topics related to representations and geometric quantization of classical as well as exceptional Lie groups. The investigator proposes to find new approaches to representing the character of an irreducible representation as the Fourier transform of a coadjoint orbit. Singular representations, the quantum cohomology ring of the flag manifold, and representations of a semisimple Lie algebra appearing in its exterior algebra are among the topics to be pursued. Lie groups often arise as symmetry groups. For example, so called unitary Lie groups describe symmetries in elementary particles. Another example is provided by the Carbon 60 molecule, popularly known as the buckyball. Here the full symmetry group turns out to be an exceptional Lie group. There are only finitely many exceptional groups - whereas the number of classical groups are infinite - and much work has been done on how to realize them as subgroups of classical groups.