Aïssa Wade

Conformal Dirac Structures

The Courant bracket defined originally on the sections of a vector bundle TM⊕T*M→M is extended to the direct sum of the 1-jet vector bundle and its dual. The extended bracket allows one to interpret many structures encountered in differential geometry, in terms of Dirac structures. We give here a new approach to conformal Jacobi structures.

Contact manifolds and generalized complex structures

We give simple characterizations of contact 1-forms in terms of Dirac structures. We also relate normal almost contact structures to the theory of Dirac structures.

Generalized contact structures

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odddimensional manifolds. We name the latter strong generalized contact structures. Using a Boothby-Wang construction bridging symplectic structures and contact structures, we find examples to demonstrate that, within the category of generalized contact structures, classical contact structures have non-trivial deformations. Using deformation theory of Lie bialgebroids, we construct new families of strong generalized contact structures on the threedimensional Heisenberg group and its co-compact quotients. Address: Department of Mathematics, University of California at Riverside, Riverside CA 92521, U.S.A., Email: ypoon@ucr.edu. Partially supported by UCMEXUS and NSF-0906264 Address: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A., Email: wade@math.psu.edu

Normalisation formelle de structures de Poisson

Resume Nous donnons ici, pour les structures de Poisson nulles en un point, une forme normale formelle relevant la decomposition de Levi de l’algebre de Lie, appelee linearisee, qui est associee a la partie lineaire. Ce resultat nous permet de generaliser le resultat de Weinstein, qui dit que toute structure de Poisson singuliere admettant une linearisee semi-simple, est formellement linearisable.

Integration of Dirac?Jacobi structures

We study precontact groupoids whose infinitesimal counterparts are Dirac?Jacobi structures. These geometric objects generalize contact groupoids. We also explain the relationship between precontact groupoids and homogeneous presymplectic groupoids. Finally, we present some examples of precontact groupoids.

Formes normales de structures de poisson ayant un 1 jet nul en un point

Abstract Singularities of Poisson structures where 1-jet vanishes appear in a stable manner because they are generically not destroyed by small perturbations of the Poisson structure. In this paper we study these singularities. We give first a normal form for Poisson structures with zero 1-jet but with a “generic” 2-jet at a point. We give also a “quadratisation” result in class C∞.

Nambu–Dirac Structures for Lie Algebroids

The theory of Nambu–Poisson structures on manifolds is extended to the context of Lie algebroids in a natural way based on the derived bracket associated with the Lie algebroid differential. A new way of combining Nambu–Poisson structures and triangular Lie bialgebroids is described in this work. Also, we introduce the concept of a higher order Dirac structure on a Lie algebroid. This allows to describe both Nambu–Poisson structures and Dirac structures on manifolds in the same setting.

Generalized contact bundles

Abstract In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators [8] , it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.

Bicrossed products induced by Poisson vector fields and their integrability

First we show that, associated to any Poisson vector field E on a Poisson manifold (M,π), there is a canonical Lie algebroid structure on the first jet bundle J1M which, depends only on the cohomology class of E. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and L∞-algebras.