On transitive Lie bialgebroids and Poisson groupoids

Abstract

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 ∧ 2 A 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 → + Π F , where Π F is a bivector field on Γ associated to a Lie groupoid 1-cocycle.