Connecting Representations of Algebraic Groups, Finite Groups, Lie Algebras, Quantum Groups, and Related Quivers

Kansas State University, Manhattan KS

Investigators

Abstract

The principal investigator plans to investigate detailed connections among representation theories of finite groups of Lie types, algebraic groups, Lie algebras, and quivers. The research will concentrate on relating the cohomology theory of finite groups of Lie type and that of restricted Lie algebras and infinitesimal group schemes. The goal is to describe, for a given rational module for an algebraic group, its support varieties over finite groups of rational points over finite fields in terms of its support varieties over the higher Frobenius kernels. The principal investigator will also study the connections between representations of quivers and representations of Kac-Moody Lie algebras and quantum groups (and their generalizations). This will involve the study of subalgebras of Hall algebras using perverse sheaves, construct canonical basis of generic subalgebras, realize affine Lie algebras in terms of representations of quivers, and relate them to Fock spaces. The beauty of mathematical research lies in linking many different subject areas together to apply available theory in many different fields. This project is in the area of mathematics known as representation theory. At its core, representation theory derives from the study of symmetries, i.e., the symmetries that natural objects, from subatomic particles to planetary orbits, have. Nowadays there are many different kinds of representation theory, each having evolved as the best way to deal with different kinds of problems. The purpose of this project is to find and explore some of the links between some of the different kinds of representation theory. Such links will benefit not only the study of representation theory, but also physics and other sciences that use representation theory as a standard tool.