Reduction of Poisson manifolds with Hamiltonian Lie algebroids

TL;DR

This paper introduces a reduction theorem for Poisson manifolds using Hamiltonian Lie algebroids, defining compatible momentum sections that lead to quotient spaces which are also Poisson manifolds.

Contribution

It establishes a new reduction framework for Poisson manifolds via Hamiltonian Lie algebroids and introduces the concept of compatible momentum sections.

Findings

Compatible momentum sections are Lie algebra homomorphisms

Zero level set quotients are Poisson manifolds

Reduction extends classical Poisson reduction methods

Abstract

Reduction theorem for Poisson manifolds with Hamiltonian Lie algebroids is presented. The notion of compatibility of a momentum section is introduced to the category of Hamiltonian Lie algebroids over Poisson manifolds. It is shown that a compatible momentum section is a Lie algebra homomorphism, and then the quotient space of the zero level set of a compatible momentum section proves to be a Poisson manifold.