Every symplectic manifold is a coadjoint orbit
April 30, 2007, update May 4, 2015
It is well known that, up to coverings, a connected Hausdorff symplectic manifold, homogeneous under a symplectic action of a connected Lie group, is isomorphous to a coadjoint orbit (affine or linear) of this group. Agreeing to extend the category of Lie groups to the category of diffeological groups, we shall see that this is the only model. That is, any connected Hausdorff symplectic manifold is isomorphous to some coadjoint orbit (affine or linear, hamiltonian or not) of its group of symplectomorphims. In other
 words, every symplectic manifold is a coadjoint orbit.

Submitted to the proceedings of the 14th conference of “ Frontiers of Fundamental Physics 14 : Marseilles, July 15-18 (2014)”.http://www.cpt.univ-mrs.fr/spip.php?article419&lang=frshapeimage_3_link_0