Title of lectures
Quantization of Integrable Systems
Lecturer:
Nicolai RESHETIKHIN, U.C., Berkeley
Outline:
These lectures begin with an overview of the mathematical principles of
quantization with the focus on quantization of integrable systems. Then we
introduce special class of integrable systems related to Poisson Lie groups.
After the discussion of symplectic leaves in Poisson Lie groups and their
geometry, we give several examples of such systems with Toda-type systems as
basic examples. Then the discussion moves to quantization of such systems and
to the role of representation theory of quantum groups. Roughly speaking, the
thermodynamical limit is a limit in which the number of degrees of freedom of
a system diverges to infinity. Some questions related to the structure of the
space of states of a quantum integrable system and to the structure of the
spectrum of commuting integrals in the thermodynamical limit are the focus of
the last lecture.
Lecture 1
I will start with an overview of deformation quantization approach to quantum
theory. Then quantization of integrable systems will be discussed from this
point of view.
Lecture 2
Here I will describe special class of classical integrable systems based on
Poisson Lie groups. Toda system and its "nonlinear" version related to
Coxeter symplectic leaves in simple Poisson Lie groups will be the basic
examples. Such "nonlinear" Toda systems are known as relativistic Toda systems.
Lectures 3 and 4
Here I will focus on how to quantize integrable systems related to Poisson Lie
groups using quantum groups and their representations.
Lecture 5
Infinite dimensional integrable systems and their relation to representation
theory of quantum affine algebras will be the subject of this lecture.
References:
[1] Fedosov, B.V., The index theorem for deformation quantization, in Boundary
Value Problems, Schrodinger Operators, Deformation Quantization, (in Series
Advances in Partial Differential Equations, v. 8) Academie Verlag.
[2] Kontsevich, M., Deformation quantization of Poisson manifolds I, preprint,
q-alg/9709040.
[2] Reshetikhin, N. and Yakimov, M., Quantization of Lagrangian fibrations, in
preparation.
[3] Korogodski, L. and Soibelman, Y., Algebras of Functions on Quantum Groups: Part 1,
Mathematical Surveys and Monographs, vol. 58, published by AMS
[4] Hoffmann, T., Kellendonk, J., Kutz, N., and Reshetikhin, N., Factorization
dynamics and Coxeter-Toda systems, in preparation.
[5] Reshetikhin, N., Some algebraic and analytic structures in integrable
systems, Lecture Notes in Physics ,vol. 469, Low-Dimensional Models in
Statistical Physics and Quantum Field Theory, Proceedings of Schladming School
in Physics, 1995.
[6] Reshetikhin, N., Integrable discrete systems, in: Quantum Groups and Their
Application in Physics, Proceedings of the International School of Physics
"Enrico Fermi", IOS Press, 1996.