A-Poisson Structures on Weil Bundles

Let M be a paracompact differentiable manifold, A a Weil algebra and M A a manifold of infinitely near points on M of kind A. We define the notion of A-Poisson manifold on M A . We show that when M is a Poisson manifold, then M A is an A-Poisson manifold. We also show that if (M,Ω) is a symplectic manifold, the structure of A-Poisson manifold on M A defined by Ω A coincide with the prolongation on M A of the Poisson structure on M defined by the symplectic form Ω.