The Fock-Rosly Poisson structure and quasitriangular r-matrices
The moduli space of flat G-connections over a Riemann surface Σ is well known to admit a natural Poisson structure. If one looks at principal G-bundles trivialized over finitely many points v1,...,vn lying in the boundary of Σ, Fock and Rosly have constructed a Poisson structure on the corresponding moduli space of flat connections which depends on the choice of an r-matrix for each point vj. We show that this Fock-Rosli Poisson structure is defined by a quasitriangular r-matrix, and is an example of a so-called mixed product Poisson structure defined by actions of pairs of dual Lie bialgebras.