New Unifying Structures in Lie Theory and the Cubic Dirac Operator

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

A whole array of questions in different mathematical areas seem to be converge to questions having to do with the set of alcoves in a fixed Weyl chamber of a semisimple Lie group. The areas include Kac-Moody theory, homology of loop groups, quantum cohomology, the Verlinde algebra, MacDonald identities, Schubert calculus, ideals in the Borel subalgebra, symmetric space theory and the Cartan-Weyl representation theory of compact Lie groups. In effect our proposal is to sort out what is going on to unify what seems to us to be unifiable in the subjects listed above. Group theory, and especially Lie group theory, lies at the heart of mathematics and the application of mathematics to problems in the real world. Included in the latter are applications to both classical and quantum mechanics, control theory, string theory, chemistry and crystallography. Lie group theory is extremely intricate and the extent to which it is applicable depends highly on a knowledge of its intricacies. The proposed project expects to discover highly exciting new structures in the subject and would unify many existing structures. The effect that would be to greatly increase our understanding and our ability to use these powerful structures. This award is jointly funded by the programs in Geometric Analysis and Algebra, Number Theory, & Combinatorics.