Coactions of weak multiplier Hopf algebras

In this article, we investigate the notion of a coaction for a weak multiplier Hopf algebra. We show that there is a duality for actions and coactions of weak algebraic quantum groups. Then we obtain the well known Blattner-Montgomery duality theorem for weak algebraic quantum groups in the framework of dual pairs. We form a Morita context when the weak multiplier Hopf algebra admits a two-sided faithful integral and show that the coaction is Galois if and only if the map in the Morita context is surjective. As an application, we obtain the result that a grading by any groupoid $G$ can be considered as a $\mathrm{C}G$-comodule algebra. We also construct a Morita context connecting the smash product $A\mathrm{\#}\mathrm{C}(G)$ and the coinvariant subalgebra $A^{\mathrm{C}(G)}$. Finally we construct an isomorphism between $(R\mathrm{\#}\mathrm{C}(G))\mathrm{\#}\mathrm{C}G$ and $R\otimes (\mathrm{C}(G)\mathrm{\#}\mathrm{C}G)$ when $R$ is graded by $G$.