In his study of Dirac structures, a notion which includes both Poisson structures and closed2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. Thisbracket does not satisfy the Jacobi identity except on certain subspaces. In this paper wesystematize the properties of this bracket in the definition of a Courant algebroid. Thisstructure on a vector bundle E → M, consists of an antisymmetric bracket on the sections ofE whose "Jacobi anomaly' has an explicit expression in terms of a bundle map E → TM anda field of symmetric bilinear forms on E. When M is a point, the definition reduces to that ofa Lie algebra carrying an invariant nondegenerate symmetric bilinear form.For any Lie bialgebroid (A,A*) over M (a notion defined by Mackenzie and Xu), there is anatural Courant algebroid structure on A ⊕ A* which is the Drinfel' d double of a Lie bialgebrawhen M is a point. Conversely, if A and A* are complementary isotropic subbundles of aCourant algebroid E, closed under the bracket (such a bundle, with dimension half that of E, iscalled a Dirac structure), there is a natural Lie bialgebroid structure on (A, A*) whose double is isomorphic to E. The theory of Manin triples is thereby extended from Lie algebras to Liealgebroids.Our work gives a new approach to bihamiltonian structures and a new way of combining twoPoisson structures to obtain a third one. We also take some tentative steps toward generalizingDrinfel' d's theory of Poisson homogeneous spaces from groups to groupoids.

Poisson homogeneous spaces for Poisson groupoids are classfied in terms of Dirac structuresfor the corresponding Lie bialgebroids. Applications include Drinfel’d’s classification in the caseof Poisson groups and a description of leaf spaces of foliations as homogeneous spaces of pairgroupoids.

The purpose of this paper is to establish a connection between various subjects such asdynamical r-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies onthe theory of Dirac structures developed in [17] [18]. In particular, we give a new method ofclassifying dynamical r-matrices of simple Lie algebras g, and prove that dynamical r-matricesare in one-one correspondence with certain Lagrangian subalgebras of g ⊕ g.

We study the local structure of Lie bialgebroids at regular points. In particular, we classifyall transitive Lie bialgebroids. In special cases, they are connected to classical dynamical rmatricesand matched pairs induced by Poisson group actio

We prove that, for any transitive Lie bialgebroid (A, A∗), the differential associated to the Lie algebroid structureon A∗ has the form d∗ = [Λ,•]A + Ω, where Λ is a section of ∧2A and Ω is a Lie algebroid 1-cocycle for theadjoint representation of A. Globally, for any transitive Poisson groupoid (Γ,Π), the Poisson structure has theform Π =←−Λ −−→Λ +ΠF, where ΠF is a bivector field on Γ associated to a Lie groupoid 1-cocycle. 2005 Published by Elsevier B.V.