Lie Groups, Cluster Variables and Integrable Systems

Understand

We discuss the Poisson structures on Lie groups and propose an explicit construction of the integrable models on their appropriate Poisson submanifolds.

• The integrals of motion for the SL(N)-series are computed in cluster variables via the Lax map.
• This construction, when generalised to the co-extended loop groups, gives rise not only to several alternative descriptions of relativistic Toda systems, but allows to formulate in general terms some new class of integrable models.