Convenient Partial Poisson Manifolds

Understand

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.