Poisson Lie groups, integrable systems, and representation theory

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The primary goal of this project is to investigate the geometry of several large classes of Poisson structures on Lie groups and their homogeneous spaces. They include the Belavin-Drinfeld Poisson structures on complex simple Lie groups and their Poisson homogeneous spaces, and Poisson structures on Kac-Moody groups and real simple groups. The project is based on employing novel techniques from Lie theory and ring theory which in particular relate the above geometric problems to studies of the primitive spectra of universal enveloping algebras of Lie algebras that are in general neither solvable, nor semisimple. The principal investigator will work on applications of these problems to the study of integrable systems (whose phase spaces are symplectic leaves of the above Poisson structures) and representations of Hopf algebras, e.g. algebras of regular functions on non-standard quantum groups constructed by Etingof-Kazhdan and Etingof-Schedler-Schiffmann by explicit quantizations of Belavin-Drinfeld r-matrices. He will further address applications to combinatorics: the theory of cluster algebras of Fomin and Zelevisky, and intersections of dual Schubert cells (related to symplectic leaves of special Poisson structures on flag varieties). The second part of the project concerns applications of the infinite dimensional Poisson Lie group of formal pseudo-differential operators to problems in the theory of bispectrality of Duistermaat and Grunbaum, e.g. using the corresponding dressing action of Semenov-Tian-Shansky to construct subalgebras of infinitesimal "additional symmetries" of the KP hierarchy that preserve manifolds of bispectral wave functions. The project will further investigate relations between bispectrality and the prolate spheroidal phenomenon of Landau, Pollak, and Slepian, which first appeared in time-band limiting but consequently played an important role in random matrix theory as well. In particular this proposal targets integral operators related to the infinite dimensional class of bispectral algebras of ranks 1 and 2, all of which posses commuting differential operators, proved in previous works of the investigator. The major problems in many areas of mathematics and mathematical physics are related to the study of the symmetries (transformation groups) of the objects or models, under investigation. Recently generalized symmetries (Hopf algebras) have played an increasingly prominent role in many fields. This project investigates geometric structures related to deformations of classical symmetries (algebraic groups) to generalized symmetries (Hopf algebras) and their applications to problems in dynamical systems, algebra, combinatorics, and applied mathematics.