Convenient Partial Poisson Manifolds

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 $P:T^{\prime }M\to TM$ 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.

References in corpus (3)

• Queer Poisson brackets
• Banach Lie algebroids
• Banach-Lie groupoids and generalized inversion

Cited by in corpus (6)

• Covariant canonical formulations of classical field theories
• Null Hamiltonian Yang-Mills theory: Soft symmetries and memory as superselection
• A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting
• Poisson structures in the Banach setting: comparison of different approaches
• Banach Lie groupoid of partial isometries over restricted Grassmannian
• Banach Poisson-Lie group structure on $U(\mathcal{H})$