From Symplectic to Poisson. A Study of Reduction and a Proposal Towards Implosion

TL;DR

This paper extends symplectic reduction concepts to Poisson manifolds, introduces a cross-section theorem generalizing Guillemin-Sternberg, and advances the theory of Poisson implosion with new foundational results.

Contribution

It proves a Poisson cross-section theorem generalizing Guillemin-Sternberg and clarifies the role of Poisson transversals, advancing the understanding of Poisson implosion.

Findings

Proved a cross-section theorem for Poisson manifolds.

Identified Poisson transversals as analogues of symplectic submanifolds.

Corrected a mistake in the Guillemin-Sternberg theorem proof.

Abstract

The imploded cross-section of a symplectic manifold is a stratified space allowing for an abelianization of its symplectic reduction. After recalling symplectic and Poisson reduction and reviewing the basics of symplectic implosion, we prove a cross-section theorem for Poisson manifolds, generalizing the Guillemin-Sternberg theorem for symplectic manifolds, which constitutes a first step towards Poisson implosion. On our way, we find and fix a mistake in the proof of Guillemin-Sternberg's theorem, and we identify Poisson transversals as the right analogue to symplectic submanifolds in this context.