Uniqueness of the momentum map

We give a detailed discussion of existence and uniqueness of the momentum map associated to Poisson Lie actions, which was defined by Lu. We introduce a weaker notion of momentum map, called infinitesimal momentum map, which is defined on one-forms and we analyze its integrability to the Lu’s momentum map. Finally, the uniqueness of the Lu’s momentum map is studied by describing, explicitly, the tangent space to the space of momentum maps.