Manin pairs and moment maps

A Lie group G in a group pair (D, G), integrating the Lie algebra g in a Ma nin pair (d, g), has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, and show that they generalize the Poisson ac tions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are hamiltonian. These moment maps take values in the homogeneous space D/G. We prove an analogue of the hamiltonian reduc tion theorem for quasi-Poisson group actions, and we study the symplectic l eaves of the orbit spaces of hamiltonian quasi-Poisson spaces.