Poisson Lie groups, integrable systems, and representation theory

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The primary goal of this project is to investigate the geometry of severallarge classes of Poisson structures on Lie groups and their homogeneousspaces. They include the Belavin-Drinfeld Poisson structures on complex simpleLie groups and their Poisson homogeneous spaces, and Poisson structures onKac-Moody groups and real simple groups. The project is based on employing noveltechniques from Lie theory and ring theory which in particular relatethe above geometric problems to studies of the primitive spectra of universalenveloping algebras of Lie algebras that are in general neither solvable,nor semisimple. The principal investigator will work on applications of theseproblems to the study of integrable systems (whose phase spaces are symplecticleaves of the above Poisson structures) and representations of Hopf algebras,e.g. algebras of regular functions on non-standard quantum groups constructedby Etingof-Kazhdan and Etingof-Schedler-Schiffmann by explicit quantizations ofBelavin-Drinfeld r-matrices. He will further address applications to combinatorics:the theory of cluster algebras of Fomin and Zelevisky, and intersections of dualSchubert cells (related to symplectic leaves of special Poisson structures onflag varieties). The second part of the project concerns applications of theinfinite dimensional Poisson Lie group of formal pseudo-differential operators toproblems in the theory of bispectrality of Duistermaat and Grunbaum, e.g. using thecorresponding dressing action of Semenov-Tian-Shansky to construct subalgebrasof infinitesimal "additional symmetries" of the KP hierarchy that preserve manifoldsof bispectral wave functions. The project will further investigate relationsbetween bispectrality and the prolate spheroidal phenomenon of Landau, Pollak, andSlepian, which first appeared in time-band limiting but consequently played animportant role in random matrix theory as well. In particular this proposal targetsintegral operators related to the infinite dimensional class of bispectral algebrasof ranks 1 and 2, all of which posses commuting differential operators, provedin previous works of the investigator.The major problems in many areas of mathematics and mathematical physics arerelated to the study of the symmetries (transformation groups) of theobjects or models, under investigation. Recently generalized symmetries (Hopfalgebras) have played an increasingly prominent role in many fields. This projectinvestigates geometric structures related to deformations of classical symmetries(algebraic groups) to generalized symmetries (Hopf algebras) and their applicationsto problems in dynamical systems, algebra, combinatorics, and applied mathematics.