Convenient Partial Poisson Manifolds

TL;DR

This paper introduces the concept of partial Poisson structures on convenient manifolds, extending Poisson geometry to infinite-dimensional settings with applications to weak symplectic foliations.

Contribution

It defines partial Poisson structures on convenient manifolds, connecting them with weak symplectic manifolds and Lie algebroids, and explores their natural examples and limits.

Findings

Defined partial Poisson structures on infinite-dimensional manifolds.

Connected partial Poisson structures with weak symplectic foliations.

Explored limits of Banach structures in this context.

Abstract

We introduce the concept of partial Poisson structure on a manifold $M$ modelled on a convenient space. This is done by specifying a (weak) subbundle $T^{\prime}M$ of $T^{\ast}M$ and an antisymmetric morphism $P:T^{\prime}M\rightarrow TM$ such that the bracket $\{f,g\}_{P}=-<df,P(dg)>$ defines a Poisson bracket on the algebra $\mathcal{A}$ of smooth functions $f$ on $M$ whose differential $df$ induces a section of $T^{\prime}M$. In particular, to each such function $f\in\mathcal{A}$ is associated a hamiltonian vector field $P(df)$. This notion takes naturally place in the framework of infinite dimensional weak symplectic manifolds and Lie algebroids. After having defined this concept, we will illustrate it by a lot of natural examples. We will also consider the particular situations of direct (resp. projective) limits of such Banach structures. Finally, we will also give some results on the existence of (weak) symplectic foliations naturally associated to some particular partial Poisson structures.