Normal forms for principal Poisson Hamiltonian spaces

TL;DR

This paper establishes a normal form theorem for principal Hamiltonian actions on Poisson manifolds near the zero locus of the moment map, generalizing classical symplectic results and enabling linearization of quotient Poisson manifolds.

Contribution

It introduces a new normal form theorem for Poisson Hamiltonian spaces, extending classical symplectic minimal coupling to Poisson geometry and showing quotient linearizability.

Findings

Normal form theorem for Poisson Hamiltonian actions near zero moment map locus

Extension of minimal coupling construction to Poisson geometry

Linearization of quotient Poisson manifolds

Abstract

We prove a normal form theorem for principal Hamiltonian actions on Poisson manifolds around the zero locus of the moment map. The local model is the generalization to Poisson geometry of the classical minimal coupling construction from symplectic geometry of Sternberg and Weinstein. Further, we show that the result implies that the quotient Poisson manifold is linearizable, and we show how to extend the normal form to other values of the moment map.