Spin Ruijsenaars-Schneider models from Poisson reduction

The Spin Ruijsenaars-Schneider model is a generalization of the relativistic Calogero-Sutherland model of N interacting particles in one space dimension. It is realized by adding to every particle internal degrees of freedom, called spins, while preserving the classical integrability of the model. I will explain how to obtain the Hamiltonian structure of the model by means of symplectic reduction respect to the Poisson action of a matrix group over an appropriate initial phase space. This technique is applied to the Hyperbolic version of the model, and leads to a new, previously unknown, Poisson structure for the conserved currents which describe the degenerate integrability of the system.