Multiplicative forms on Poisson groupoids

First, we prove a decomposition formula for any multiplicative differential form on a Lie groupoid \(\cal{G}\) $$\mathcal{G}$$ . Next, we prove that if \(\cal{G}\) $$\mathcal{G}$$ is a Poisson Lie groupoid, then the space \(\Omega_{\text{mult}}^{\bullet}(\cal{G})\) $${\mathrm{\Omega }}_{\text{mult}}^{\bullet }(\mathcal{G})$$ of multiplicative forms on \(\cal{G}\) $$\mathcal{G}$$ has a differential graded Lie algebra (DGLA) structure. Furthermore, when combined with Ω^•(M), which is the space of forms on the base manifold M of \(\cal{G},\Omega_{\text{mult}}^{\bullet}(\cal{G})\) $$\mathcal{G},{\mathrm{\Omega }}_{\text{mult}}^{\bullet }(\mathcal{G})$$ forms a canonical DGLA crossed module. This supplements a previously known fact that multiplicative multi-vector fields on \(\cal{G}\) $$\mathcal{G}$$ form a DGLA crossed module with the Schouten algebra Γ(∧^•A) stemming from the Lie algebroid A of \(\cal{G}\) $$\mathcal{G}$$ .