Integrability of Poisson-Lie group actions

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian action on the symplectic groupoid $\mathrm{\Sigma }(M)$. We give applications of these results to the integration of Poisson quotients $M\mathrm{/}G$, Lu-Weinstein quotients ${\mu }^{-1}(e)\mathrm{/}G$ and Poisson homogeneous spaces $G\mathrm{/}H$.