Monday, March 20, 2006
2:30 pm, Science 3-028

Hervé Sabourin

Université de Poitiers, France

Transverse Poisson Structures To Adjoint Orbits in Semi-simple Lie Algebras

Abstract: The transverse Poisson structure was introduced by A. Weinstein stating in his famous "splitting theorem" that every Poisson manifold M is locally, at each point m, the product of the symplectic leaf through m and a Poisson manifold of rank 0 at m. When M is the dual of a complex Lie algebra g, equipped with its canonical Lie-Poisson structure, we know that the symplectic leaf through x in the dual of g is the coadjoint orbit G.x of the corresponding adjoint Lie group G. Following Weinstein, we can define a transverse Poisson structure for each coadjoint orbit. Our goal is to describe more explicitly this structure, especially when the Lie algebra g is semi-simple.