Reduction of pre-Hamiltonian actions

TL;DR

This paper establishes a reduction theorem for the tangent bundle of Poisson manifolds under pre-Hamiltonian actions of Poisson Lie groups, connecting it to classical reductions in special cases and analyzing integrability.

Contribution

It introduces a new reduction theorem for tangent bundles of Poisson manifolds with pre-Hamiltonian actions, extending classical Hamiltonian reduction results.

Findings

Reduced tangent bundle is integrable for simply connected symplectic manifolds.

Reduced symplectic groupoid corresponds to Marsden-Weinstein reduction.

Comparison with classical Marsden-Ratiu reduction in Hamiltonian case.

Abstract

We prove a reduction theorem for the tangent bundle of a Poisson manifold $(M, \pi)$ endowed with a pre-Hamiltonian action of a Poisson Lie group $(G, \pi_G)$. In the special case of a Hamiltonian action of a Lie group, we are able to compare our reduction to the classical Marsden-Ratiu reduction of $M$. If the manifold $M$ is symplectic and simply connected, the reduced tangent bundle is integrable and its integral symplectic groupoid is the Marsden-Weinstein reduction of the pair groupoid $M \times \bar{M}$.