Poisson smooth structures on stratified symplectic spaces

TL;DR

This paper develops a framework for smooth and Poisson structures on stratified symplectic spaces, extending key symplectic properties and providing numerous examples of such structures.

Contribution

It introduces new notions of smooth, Poisson, and weakly symplectic structures on stratified spaces, refining previous concepts and enabling the extension of symplectic properties.

Findings

Existence of smooth partitions of unity on these spaces

Extension of Hamiltonian flow and homology isomorphisms

Many examples of stratified symplectic spaces with Poisson structures

Abstract

In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show that these smooth spaces possess several important properties, e.g. the existence of smooth partitions of unity. Furthermore, under mild conditions many properties of a symplectic manifold can be extended to a symplectic stratified space provided with a smooth Poisson structure, e.g. the existence and uniqueness of a Hamiltonian flow, the isomorphism between the Brylinski-Poisson homology and the de Rham homology, the existence of a Leftschetz decomposition on a symplectic stratified space. We give many examples of stratified symplectic spaces possessing a Poisson smooth structure which is also weakly symplectic.