Integrable systems 1. (The Yang-Baxter equation)

Course description:
The purpose of these lectures is to explain the role of the classical and quantum Yang-Baxter equations in the theory of integrable systems and to give an introduction to some of the relevant mathematical structures. Topics discussed: Liouville integrability. Lax equations and zero curvature equations. Double Lie algebras, Lie bialgebras and Poisson-Lie groups. Yang-Baxter (RTT=TTR) algebras from the quantization of quadratic Poisson bracket algebras. Applications illustrated e.g. with Toda lattices, the non-linear Schroedinger equation and solvable vertex models of statistical mechanics.

Recommended Texts:

• Korepin V.E., Bogoliubov N.M., Izergin A.G.: Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press 1993.
• Chari V., Pressley A.: A Guide to Quantum Groups, Cambridge University Press 1994.