Cohomology, Support Varieties, and Representation THeory

University Of Oklahoma Norman Campus, Norman OK

Investigators

Abstract

This project addresses questions in the representation theory and cohomology of supergroups, Lie superalgebras, finite groups, and related algebras. The principal investigator is particularily interested in the interplay between the representation theory, geometry, and algebraic combinatorics of these objects. For example, one focus of the investigator's work will be the continued development of support varieties for complex Lie superalgebras. These geometric objects provide links between the representation theory, cohomology, and combinatorics of these algebras. Additionally, he intends to investigate questions involving the modular representations of the symmetric group and related algebraic objects. This project is in the area of mathematics known as representation theory. Algebraic structures such as groups and Lie algebras arise in nature as the symmetries of some object. Thus it is natural to study these objects in terms of how they interact with other mathematical objects. This is the point of view of representation theory. Because of the insight it provides regarding the underlying structure and symmetries of an object, it is not surprising that representation theory has proven valuable in other areas of mathematics, physics, chemistry, biology, cryptography, networking, computer graphics, and art. This proposal focuses on ``super'' versions of these algebraic objects; that is, ones which involve both symmetries and anti-symmetries. Inspired by their usefulness in mathematical physics, mathematicians have found that super algebraic objects play an important role similar to their classical counterparts. The investigator expects this trend to continue as new applications of ``super'' representation theory are discovered.