The Fock-Rosly Poisson structure and quasitriangular \(r\)-matrices

Abstract: The moduli space of flat $G$-connections over a Riemann surface $\Sigma$ is well known to admit a natural Poisson structure. If one looks at principal $G$-bundles trivialized over finitely many points $v_1, ..., v_n$ lying in the boundary of $\Sigma$, 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 $v_j$. 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.