Reduction of pre-Hamiltonian actions

We prove a reduction theorem for the tangent bundle of a Poisson manifold (M,π) endowed with a pre-Hamiltonian action of a Poisson-Lie group (G,πG). In the special case of a Hamiltonian action of a Lie group, we are able to compare our reduction to the classical Marsden-Ratiu reduction of M. If the manifold M is symplectic and simply connected, the reduced tangent bundle is integrable and its integral symplectic groupoid is the Marsden-Weinstein reduction of the pair groupoid M×M̄.