Quasi-Poisson structures and integrable systems related to the moduli space of flat connections on a punctured Riemann sphere

In order to construct an integrable system on the moduli space Hom(π1(S),G)/G of a punctured sphere SS, we establish a morphism between two interesting quasi-Poisson GG-manifolds, GnGn and a subspace g̃n of the loop algebra of gg. In particular, we prove a useful result about reduction in the quasi-Poisson context and we describe the construction of a quasi-Poisson structure coming from a Lie algebra splitting.