Integrable Systems and Factorization Problems

Abstract

The present lectures were prepared for the “Faro International Summer School on Factorization and Integrable Systems” in September 2000. They were intended for participants with the background in analysis and operator theory but without special knowledge of geometry and Lie groups. The text below represents a sort of compromise: it is certainly impossible not to speak about Lie algebras and Lie groups at all; however, in order to make the main ideas reasonably clear, I tried to use only matrix algebras such as gl(n) and its natural subalgebras; Lie groups used are either GL(n) and its subgroups, or loop groups consisting of matrix-valued functions on the circle (possibly admitting an extension to parts of the Riemann sphere). I hope this makes the environment sufficiently easy to live in for an analyst. The main goal is to explain how the factorization problems (typically, the matrix Riemann problem) generate the entire small world of integrable systems along with the geometry of the phase space, Hamiltonian structure, Lax representations, integrals of motion and explicit solutions. The key tool will be the classical r-matrix (an object whose other guise is the well-known Hilbert transform). I do not give technical details, unless they may be exposed in a few lines; on the other hand, all motivations are given in full scale whenever possible. I hope that this choice agrees with the spirit of the Faro School and will help to bridge the gap between different branches of Mathematical analysis.