We rederive the Hamiltonian structure of 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 Nℓ conjugate pairs of dynamical variables. We show that the model enjoys the Poisson-Lie symmetry of the spin group GLℓ(C) which explains its superintegrability. Our results are obtained in the formalism of the classical r-matrix and they confirm the recent findings on the Hamiltonian structure of the model established in the different framework of the quasi-Hamiltonian reduction applied to a quasi-Poisson manifold.
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 Nℓ conjugate pairs of dynamical variables. We show that the model enjoys the Poisson-Lie symmetry of the spin group GLℓ(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.