On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid (A, A∗), the differential associated to the Lie algebroid structure on A∗ has the form d∗=[Λ,⋅]A+Ω, where Λ is a section of ∧2A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A. Globally, for any transitive Poisson groupoid (Γ,Π), the Poisson structure has the form Π=Λ←−Λ→+ΠF, where ΠF is a bivector field on Γ associated to a Lie groupoid 1-cocycle.