Poisson Reduction

TL;DR

This paper develops a theory for reducing classical systems with Poisson Lie group symmetries, utilizing Lu's momentum map to define Poisson reduced spaces, advancing the understanding of symmetry reduction in Poisson geometry.

Contribution

It introduces a novel reduction framework for Poisson systems with Lie group symmetries using Lu's momentum map, expanding the tools for Poisson geometry analysis.

Findings

Defined Poisson reduced spaces using Lu's momentum map

Extended reduction techniques to systems with Poisson Lie group symmetries

Provided local descriptions of Poisson manifolds and Lie groups

Abstract

In this paper we develope a theory of reduction for classical systems with Poisson Lie groups symmetries using the notion of momentum map introduced by Lu. The local description of Poisson manifolds and Poisson Lie groups and the properties of Lu's momentum map allow us to define a Poisson reduced space.