Lie Groups, Cluster Variables and Integrable Systems

TL;DR

This paper explores Poisson structures on Lie groups, constructing integrable models using cluster variables, and extends these ideas to co-extended loop groups, revealing new integrable systems and descriptions of relativistic Toda models.

Contribution

It provides an explicit construction of integrable models on Lie groups' Poisson submanifolds and generalizes to co-extended loop groups, introducing new classes of integrable systems.

Findings

Computed integrals of motion for SL(N) using cluster variables

Presented alternative descriptions of relativistic Toda systems

Proposed a new class of integrable models from co-extended loop groups

Abstract

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.