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(*) = [&ULambda;, (.)](A) + &UOmega;, where &ULambda; is a section of &ULambda;(2)A and &UOmega; is a Lie algebroid 1-cocycle for the adjoint representation of A. Globally, for any transitive Poisson groupoid (&UGamma;, &UPi;), the Poisson structure has the form &UPi; = <(&ULambda;)over left arrow> - <(&ULambda;)over right arrow> &UPi;(F), where &UPi;(F) is a bivector field on &UGamma; associated to a Lie groupoid 1-cocycle. © 2005 Published by Elsevier B.V.Mathematics, AppliedMathematicsSCI(E)2ARTICLE3253-2742