Integrability and reduction of Poisson group actions

In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group $G$ with dual $G^{\star }$ we obtain a suitably connected integrating symplectic double groupoid $\text{calS}$. As a consequence, the cotangent lift of a Poisson action on an integrable Poisson manifold $P$ can be integrated to a Poisson action of the symplectic groupoid $\text{poidd}\text{calS}{G}^{\star }$ on the symplectic groupoid for $P$. Finally, we show that the quotient Poisson manifold $P\mathrm{/}G$ is also integrable, giving an explicit construction of a symplectic groupoid for it, by a reduction procedure on an associated morphism of double Lie groupoids.