Multiplicative quiver varieties and integrable particle systems

The main goal of this thesis is to provide a systematic study of several integrable systems defined on complex Poisson manifolds associated to extended cyclic quivers. These spaces are particular examples of multiplicative quiver varieties of Crawley-Boevey and Shaw, for which Van den Bergh observed that they can be equipped with a Poisson bracket obtained by quasi-Hamiltonian reduction. In his approach, Van den Bergh introduced the notion of double brackets to translate the geometric quasi-Hamiltonian structure associated to these varieties directly at the level of the path algebra of the quivers. We pursue this line of thought and examine these double brackets in order to find families of algebraic elements on the path algebra of extended cyclic quivers that give rise to families of Poisson commuting functions on the corresponding multiplicative quiver varieties. This provides a way to obtain candidates for Liouville integrability, and this can be adapted to the case of degenerate integrability. For specific dimensions of these spaces, we can compute the number of functionally independent elements in each family, and conclude that we can form integrable systems. They can be written in terms of local coordinates, and be related to the trigonometric spin Ruijsenaars-Schneider system or generalisations of the latter system. As part of our construction, we also prove that their flows can be obtained by the projection method from explicit integrations performed before the quasi-Hamiltonian reduction. Another application of this work consists in describing the Poisson structure in terms of local coordinates. In particular, this allows us to prove a conjecture of Arutyunov and Frolov regarding the form of the Poisson bracket for the trigonometric spin Ruijsenaars-Schneider system.