Manin pairs and moment maps

Abstract

A Lie group G in a group pair (D, G), integrating the Lie algebra g in a Manin pair (∂, 𝔤), has a quasi-Poisson structure. We define the quasi- Poisson actions of such Lie groups G, and show that they generalize the Poisson actions 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 reduction theorem for quasi-Poisson group actions, and we study the symplectic leaves of the orbit spaces of hamiltonian quasi-Poisson spaces. © 2000 Applied Probability Trust.