Poisson Lie groups, representation theory, combinatorics, and integrable systems

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Yakimov will investigate the geometry of varieties of Lagrangian subalgebras, equipped with Poisson structures derived from the Belavin-Drinfeld classification of quasitriangular r-matrices for simple Lie algebras. In particular cases this involves the study of the geometry of Poisson structures on double flag varieties and the wonderful group compactifications of De Concini and Procesi. This is based on a blend of of techniques from Lie theory, combinatorics, and geometry. In the opposite direction, Yakimov will investigate applications to representation and ring theory, dynamical systems, and combinatorics. These include the study of the spectra and representations at roots of unity of the quantized universal enveloping algebras of nilradicals of parabolic subalgebras of complex simple Lie algebras and the 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. The Poisson geometric constructions from the first part suggest novel approaches in ring theory via an algebraic version of conditional expectation for operator algebras. In combinatorics, the PI will work on the construction of cluster algebras from coordinate rings of torus orbits of leaves of Poisson structures on flag varieties, double flag varieties, and wonderful compactifications. Further properties of coisotropic stratifications, related to Kazhdan-Lusztig polynomials, and explicit Poisson degenerations of Richardson varieties will be studied. In the field of completely integrable systems, the PI will study Kogan-Zelevinsky integrable systems and certain generalizations of those on Schuberts cells in (double) flag varieties and relate them to classical integrable systems, e.g. Gelfand-Tsetlin systems. Similar questions for the infinite dimensional Poisson Lie group of formal pseudo-differential operators of Khesin and Zakharevich will be studied, in the light of the interplay between Calogero-Moser systems and Wilson's adelic Grassmannian. Many objects in geometry, algebra, mathematical physics, and combinatorics posses large groups of symmetries. The investigation of theses symmetries is a key technique in the study of these objects, since it leads to a reduction of the complexity of the objects. Yakimov will study such symmetries of noncommutative objects which appear in various quantized situation and their relations to problems in dynamical systems, algebra, combinatorics, and applied mathematics.