MANIN TRIPLES FOR LIE BIALGEBROIDS

In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisf y the Jacobi identity except on certain subspaces. In this paper we sy stematize the properties of this bracket in the definition of a Couran t algebroid. This structure on a vector bundle E --> M, consists of an antisymmetric bracket on the sections of E whose ''Jacobi anomaly'' h as an explicit expression in terms of a bundle map E --> TM and a fiel d of symmetric bilinear forms on E. When M is a point, the definition reduces to that of a Lie algebra carrying an invariant nondegenerate s ymmetric bilinear form. For any Lie bialgebroid (A, A) over M (a noti on defined by Mackenzie and Xu), there is a natural Courant algebroid structure on A + A which is the Drinfel'd double of a Lie bialgebra w hen M is a point. Conversely, if A and A are complementary isotropic subbundles of a Courant algebroid E, closed under the bracket (such a bundle, with dimension half that of E, is called a Dirac structure), t here 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 Lie algebroids. Our work gives a new approach to biha miltonian structures and a new way of combining two Poisson structures to obtain a third one. We also take some tentative steps toward gener alizing Drinfel'd's theory of Poisson homogeneous spaces from groups t o groupoids.