Cohomology and Representation Theory: Reductive Algebraic Groups and Related Structures

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

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 applications such as those mentioned above. This project is in the area of 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 from different 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.