Yvette Kosmann-Schwarzbach (École Polytechnique)
The problem of defining a bracket on the direct sum of dual Lie algebroids forming a Lie bialgebroid was solved by Zhangju Liu, Alan Weinstein and Ping Xu in "Manin triples for Lie bialgebroids" (1997), where they defined the general concept of Courant algebroids and their Dirac sub-bundles. As a preliminary to these definitions, I shall recall the Dorfman bracket (treated as a derived bracket) and the Courant bracket (treated as the skew-symmetrization of the Dorfman bracket) on the generalized tangent bundle of a manifold. I shall determine the Dorfman bracket on the Drinfeld double of a Lie bialgebroid, and more generally, of a proto-bialgebroid (by means of the big bracket). Dirac structures on manifolds generalize both Poisson and pre-symplectic structures. Lie groupoids equipped with a multiplicative Dirac structures generalize both Poisson and pre-symplectic groupoids.