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

Abstract

In order to construct an integrable system on the moduli space Hom ( π 1 ( S ) , G ) / G of a punctured sphere S , we establish a morphism between two interesting quasi-Poisson G -manifolds, G n and a subspace g ̃ n of the loop algebra of g . 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.