Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions

Abstract

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. This leads naturally to a concept of a Hamiltonian $G$-action on a weak Poisson manifold with a $\g$-valued momentum map and hence to a generalization of quasi-hamiltonian group actions.