Factorization Dynamics and Coxeter--Toda Lattices

Abstract

It is shown that the factorization relation on simple Lie groups with standard Poisson Lie structure restricted to Coxeter symplectic leaves gives an integrable dynamical system. This system can be regarded as a discretization of the Toda flow. In case of SL n the integrals of the factorization dynamics are integrals of the relativistic Toda system. A substantial part of the paper is devoted to the study of symplectic leaves in simple complex Lie groups, its Borel subgroups and their doubles.

Cluster algebras III: Upper bounds and double Bruhat cells

“…Double Bruhat cells make a natural appearance in the context of quantum groups [3], total positivity [14], and integrable systems [10,11]. The connection between cluster algebras and double Bruhat cells was hinted upon in the papers [6,7], which were devoted to the study of structural properties of cluster algebras.…”

Toda Systems, Cluster Characters, and Spectral Networks

“…Following [GK13], by a cluster integrable system we mean an integrable system whose phase space is a cluster variety equipped with its canonical Poisson structure. Examples include those studied in [GK13,FM14,HKKR00,Wil13a], and generally encompass those referred to as relativistic integrable systems in the literature [Rui90]. Roughly, cluster varieties are Poisson varieties whose coordinate rings (cluster algebras) are equipped with a canonical partial basis of functions called cluster variables [FZ02].…”

“…Double Bruhat cells are the left H-orbits of the symplectic leaves of the standard Poisson-Lie structure on a complex simple Lie group [HKKR00], and are prototypical examples of cluster varieties [BFZ05]. The cluster structure on SL c,c n { Ad H is encoded by the quiver Q n shown in Figure 1.…”

Triangular Poisson structures on Lie groups and symplectic reduction

“…symplectic leaves in Poisson Lie groups is their relation to integrable systems. Poisson Lie groups provide a very general framework for constructing integrable systems [21,18,10] whose phase spaces are the leaves of those Poisson structures. Detailed geometric information for the latter is crucial in understanding the properties of such dynamical systems.…”