Hamiltonian integrable systems and Lie algebras
Prof. Gregorio Falqui
The aim of the couse is to provide the Ph. D. students with tools in the modern theory of Integrable Systems. Especially, I plan to center the course on those aspects related with the theory of Lie Algebras. The course will discuss both integrable systems of PDEs in 1 + 1 dimensions, and (classical and quantum) spin chain-like systems.
Contents:
- Geometry of Symplectic and Poisson manifolds. Lie Algebras and Lie Poisson brackets. Bihamiltonian manifolds and their properties.
- Hamiltonian G-actions on Poisson manifolds. The Marsden-Weinstein, Marsden-Ratiu, and Dirac reduction theorems.
- Lax pairs: properties and applications. The AKS scheme.
- Loop algebras, Drinfel'd-Sokolov type integrable hierarchies of PDEs, and differential operators on the circle. The examples of the KdV and Boussinesq hierarchies.
- Spin chains: Heisenberg, Gaudin and Toda systems. Universal enveloping algebras. Classical and quantum r-matrix theory. The algebraic Bethe Ansatz.