# Hyperbolic spin Ruijsenaars-Schneider model from Poisson reduction

## TL;DR

This paper constructs a Hamiltonian framework for the hyperbolic spin Ruijsenaars-Schneider model using Poisson reduction, revealing its superintegrability and Poisson-Lie symmetry through classical r-matrix formalism.

## Contribution

It introduces a novel Hamiltonian structure for the model via Poisson reduction on a combined phase space involving the Heisenberg double and a deformed symplectic manifold.

## Key findings

- Model exhibits Poisson-Lie symmetry of ${\rm GL}_{\ell}({\mathbb C})$
- Demonstrates superintegrability of the system
- Aligns with recent quasi-Hamiltonian reduction results

## Abstract

We derive a Hamiltonian structure for the $N$-particle hyperbolic spin Ruijsenaars-Schneider model by means of Poisson reduction of a suitable initial phase space. This phase space is realised as the direct product of the Heisenberg double of a factorisable Lie group with another symplectic manifold that is a certain deformation of the standard canonical relations for $N\ell$ conjugate pairs of dynamical variables. We show that the model enjoys the Poisson-Lie symmetry of the spin group ${\rm GL}_{\ell}({\mathbb C})$ which explains its superintegrability. Our results are obtained in the formalism of the classical $r$-matrix and they are compatible with the recent findings on the different Hamiltonian structure of the model established in the framework of the quasi-Hamiltonian reduction applied to a quasi-Poisson manifold.