Poisson-Lie analogues of spin Sutherland models

TL;DR

This paper introduces Poisson-Lie analogues of spin Sutherland models through Hamiltonian reduction on Lie groups, revealing new integrable systems with explicit solutions and connections to hyperbolic Ruijsenaars--Schneider models.

Contribution

It generalizes spin Sutherland models using Poisson-Lie group techniques, providing explicit symplectic structures and solution methods, and relates to previously known hyperbolic models.

Findings

Reduced models reproduce spin Sutherland Hamiltonians

Solutions follow from geodesics on Lie groups

Models possess many first integrals

Abstract

We present generalizations of the well-known trigonometric spin Sutherland models, which were derived by Hamiltonian reduction of `free motion' on cotangent bundles of compact simple Lie groups based on the conjugation action. Our models result by reducing the corresponding Heisenberg doubles with the aid of a Poisson-Lie analogue of the conjugation action. We describe the reduced symplectic structure and show that the `reduced main Hamiltonians' reproduce the spin Sutherland model by keeping only their leading terms. The solutions of the equations of motion emerge from geodesics on the compact Lie group via the standard projection method and possess many first integrals. Similar hyperbolic spin Ruijsenaars--Schneider type models were obtained previously by L.-C. Li using a different method, based on coboundary dynamical Poisson groupoids, but their relation with spin Sutherland models was not discussed.