Quantization, Quantum Groups, the Yang-Baxter Equation, Integrable Systems, and Special Functions

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The present proposal is concerned with various aspects of the theory of quantum groups. The goals of the proposal could be briefly formulated as follows. 1. To complete the theory of universal quantization of Lie bialgebras, begun in the PI's joint work with D.Kazhdan, and answer most of the remaining open questions of Drinfeld about quantization. In particular, to investigate the dependence of the quantization of an associator, and to show that the universal quantization of Kac-Moody algebras coincides with their usual quantization. To study quantization of Poisson homogeneous spaces. To study quantization theory in positive characteristic. To study quantization of solutions of the classical dynamical Yang-Baxter equations, and associated Poisson groupoids. 2. To continue to develop the theory of dynamical quantum groups, originated by Felder in 1994. In particular, to generalize the exchange construction (introduced in the PI's joint work with Varchenko) to the case of affine Lie algebras. To use this construction to establish the equivalence of suitable categories of representations for Yangians, quantum affine algebras, and elliptic algebras. To study the structure of dynamical quantum groups obtained by quantizing generalized Belavin-Drinfeld triples for simple Lie algebras. 3. To continue to develop the theory of generalized Macdonald functions, begun in the PI's joint work with Varchenko, Kirillov, and Styrkas. In particular, to use representation theory to generalize the recent results of Felder and Varchenko on elliptic hypergeometric functions from sl(2) to any simple Lie algebra. To develop Macdonald's theory for affine root systems (of type A) using the representation theory of quantum affine algebras. To develop a theory of twisted trace functions for quantum groups, and deduce difference equations for them which involve R-matrices obtained by quantizing Belavin-Drinfeld triples. 4. To work on the theory of finite dimensional Hopf algebras. To look for new examples of semisimple Hopf algebras. To study tensor products of representations of cotriangular Hopf algebras. 5. To continue to study set-theoretical solutions of the quantum Yang-Baxter equation. The theory of quantum groups is a relatively new area of mathematics which arose in mid 1980-s at the junction of the theory of groups and quantum theory. The theory of groups, which was created in 19-th century, is a mathematical theory which is designed to describe precisely the phenomenon of symmetry, and which is fundamental in modern particle physics. Symmetry is a basic property of nature. It also played a crucial role in all technological achievements of mankind, starting from the invention of a wheel: the ability of a wheel to roll comes from its symmetry with respect to rotations. Quantum theory was the most important achievement of theoretical physics of the 20-th century, which lies at the foundation of this century's technological revolution. It is a physical theory which allows to describe the behavior of very small objects, like atoms and electrons. In the late eighties, it was realized that in certain systems described by quantum theory, there is a new kind of symmetry which the theory of groups fails to describe. This new, very peculiar kind of symmetry, is called quantum symmetry. In order to describe this symmetry mathematically, mathematicians introduced new mathematical objects called quantum groups, which are generalizations of ordinary groups. The present proposal seeks to further develop this theory and to explore its interactions with other areas of mathematics and physics.