Loop groups, Clusters, Dimers and Integrable systems

We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups $\widehat{PGL}(N)$, which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups $(\widehat{W}\times \widehat{W})^\sharp$. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the constructed by Goncharov and Kenyon out of dimer models on a two-dimensional torus and classified by the Newton polygons. We construct the correspondence between the Weyl group elements and polygons, demonstrating that each particular integrable model admits infinitely many realisations on the Poisson-Lie groups. We also discuss the particular examples, including the relativistic Toda chains and the Schwartz-Ovsienko-Tabachnikov pentagram map.