Poisson bivectors on infinite dimensional manifolds

TL;DR

This paper investigates the structure of Poisson bivectors on infinite-dimensional manifolds, demonstrating a factorization property of the dual map of a Poisson bracket within a specific geometric framework.

Contribution

It introduces a new factorization result for the dual map of Poisson brackets on infinite-dimensional manifolds modeled on convenient spaces.

Findings

Dual map of Poisson bracket factors as a smooth section

Applicable to manifolds with bornological approximation property

Advances understanding of Poisson structures in infinite dimensions

Abstract

We show that, on a smoothly paracompact convenient manifold $M$ modeled on a convenient space with the bornological approximation property, the dual map of a Poisson bracket factors as a smooth section of the vector bundle $L_{skew}^2(T^*M,\mathbb R)$.