Hyperbolic Spin Ruijsenaars–Schneider Model from Poisson Reduction

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 $\mathrm {GL}_{\ell }(\mathbb C)$, 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.