Cohomology and Representation Theory: Reductive Algebraic Groups and Related Structures

首席研究员将研究涉及约化代数群的表示理论的问题。与约化群相关的结构包括Lie代数、Chevalley群、Weyl群和正化代数。研究中涉及的方法和结构将是代数的以及几何的。潜在的主题将是建立对象的上同调和表征之间的联系，以证明关于这些结构的新的和有趣的结果。建立这样的关系也有助于提供具体的计算。被称为“群”、“环”和“李代数”的代数对象出现在生物学、化学和物理学的许多不同的物理应用中。这些代数对象通常具有复杂的内部结构和对称性。提取关于这些结构的信息可以提供重要的信息，这些信息可以用于一系列的应用，如上面提到的那些。这个项目属于表示理论领域，它现在是数学的中心领域，因为它为研究复杂的代数结构提供了系统的方法。粗略地说，表征可以被认为是一些代数对象从不同视角的“快照”。这些快照通过显式描述的矩阵提供。通过将这些表示的信息放在一起，围绕这些复杂代数系统的许多问题就可以得到解答。

The principal investigator will investigate problems involving the representation theory of reductive algebraic groups. The structures related to reductive groups include Lie algebras, Chevalley groups, Weyl groups and centralizer algebras. The methods and constructions involved in the study will be algebraic as well as geometric. The underlying theme will be to establish connections between the cohomology and representations of the objects in order to prove new and interesting results about these structures. Establishing such relationships also lends itself to providing concrete calculations. The algebraic objects known as "groups," "rings" and "Lie algebras" arise in many different physical applications in biology, chemistry and physics. These algebraic objects in general have complex internal structures and symmetries. Extracting information about these structures can provide vital information which can be used in a range of applicationssuch as those mentioned above. This project is in the areaof representation theory, which is now a central area of mathematics because it provides a systematic method for studying complicated algebraic structures. Roughly speaking, representations can be thought of as ``snapshots'' of some algebraic object fromdifferent viewing angles. These snapshots are provided via explicitly described matrices. By putting together the information from the representations, many questions surrounding these complicated algebraic systems can be answered.