Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers

We study multiplicative quiver varieties associated to specific extensions of cyclic quivers with $m\geq 2$ vertices. Their global Poisson structure is characterized by quasi-Hamiltonian algebras related to these quivers, which were studied by Van den Bergh for an arbitrary quiver. We show that the spaces are generically isomorphic to the case $m=1$ corresponding to an extended Jordan quiver. This provides a set of local coordinates, which we use to interpret integrable systems as spin variants of the trigonometric Ruijsenaars–Schneider (RS) system. This generalizes to new spin cases recent works on classical integrable systems in the RS family.