Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions

Understand

We introduce a notion of a weak Poisson structure on a manifold $M$ modeled on a locally convex space.

• This is done by specifying a Poisson bracket on a subalgebra $\cA \subeq C^\infty(M)$ which has to satisfy a non-degeneracy condition (the differentials of elements of $\cA$ separate tangent vectors) and we postulate the existence of smooth Hamiltonian vector fields.
• Motivated by applications to Hamiltonian actions, we focus on affine Poisson spaces which include in particular the linear and affine Poisson structures on duals of locally convex Lie algebras.
• As an interesting byproduct of our approach, we can associate to an invariant symmetric bilinear form $\kappa$ on a Lie algebra $\g$ and a $\kappa$-skew-symmetric derivation $D$ a weak affine Poisson structure on $\g$ itself.