The Centralizer of Invariant Functions and Division Properties of the Moment Map

TL;DR

This paper explores the relationship between collective functions and G-invariant functions in the Poisson algebra of a symplectic manifold with a proper moment map, revealing four new theoretical results.

Contribution

It introduces four novel results connecting the centralizer of invariant functions and division properties of the moment map in symplectic geometry.

Findings

Established new properties of the centralizer of invariant functions.

Analyzed division properties related to the moment map.

Enhanced understanding of the structure of collective functions.

Abstract

Let Phi : M --> g^* be a proper moment map associated to an action of a compact connected Lie group, G, on a connected symplectic manifold, (M,\omega). A collective function is a pullback via \Phi of a smooth function on g^*. In this paper we present four new results about the relationship between the collective functions and the G-invariant functions in the Poisson algebra of smooth functions on M.