Infinite Dimensional Lie Algebras, Quantum Groups, and their Applications

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Among the specific problems, which this proposal addresses, is the description of finite dimensional representations of quantized affine Lie algebras in terms of the geometry of the symplectic leaves of the corresponding Poisson Lie group. When completed, this project should substantially clarify the structure of most known integrable systems. Another problem is the investigation of Weyl-type dualities for quantum affine algebras in the limit when the number of factors in the tensor product goes to infinity. Reshetikhin wants to investigate new infinite dimensional algebras that will appear in this limit and how they are related to the thermodynamics of the corresponding integrable quantum field theory. Some of the other problems he is planning to investigate are the asymptotic expansion of Witten-Reshetikhin-Turaev invariants for large levels, the deformation quantization of generic integrable systems (not necessarly regular, for which the Lagrangian fibration is a fiber bundle). He also intends to investigate characteristic classical and quantum systems and classical and quantum integrable systems related to Kac-Moody Lie algebras and Lie groups. Quantum groups appeared as an algebraic object (Hopf algebras) describing the symmetries of a wide class of quantum integrable systems. In the last decade there has been a fascinating development in structural theory of quantum groups and their representation theory. This development was stimulated by (and stimulated) various applications to integrable systems topology and geometry. Their representation theory is far more sophisticated than the representation theory of Lie groups. Quantum groups and their representation theory were instrumental in several path-breaking results: the invariants of links in 3 manifolds, integrable systems, crystal bases, and just recently, some of the ideas from the borderline between quantum groups and non-commutative geometry were used in string theory. One may say that the conceptual goal of this direction is to understand most sophisticated symmetries which may appear (and some of them appear) as symmetries of interactions of elementary particles (or strings, if they are really there instead of particles).