de Mathématiques de Luminy
On moment maps associated to a twisted Heisenberg double
We review the concept of the (anomalous) Poisson-Lie symmetry in a way that emphasises the notion of Poisson-Lie Hamiltonian. The language that we develop turns out to be very useful for several applications: we prove that the left and the right actions of a group G on its twisted Heisenberg double (D, κ) realize the (anomalous) Poisson-Lie symmetries and we explain in a very transparent way the concept of the Poisson-Lie subsymmetry and that of Poisson-Lie symplectic reduction. Under some additional conditions, we construct also a non-anomalous moment map corresponding to a sort of quasiadjoint action of G on (D, κ). The absence of the anomaly of this ”quasi-adjoint” moment map permits to perform the gauging of deformed WZW model.