Integrability of Poisson-Lie group actions

TL;DR

This paper establishes a correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative Hamiltonian actions on symplectic groupoids, providing explicit descriptions and applications to various Poisson quotients.

Contribution

It introduces a new explicit framework for lifting Poisson-Lie actions to symplectic groupoids and explores their applications to Poisson quotients and homogeneous spaces.

Findings

Established a 1:1 correspondence between actions on manifolds and groupoids.

Provided explicit descriptions of lifted Hamiltonian actions.

Applied results to integrate Poisson quotients and homogeneous spaces.

Abstract

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian action on the symplectic groupoid $\Sigma(M)$. We give applications of these results to the integration of Poisson quotients $M/G$, Lu-Weinstein quotients $\mu^{-1}(e)/G$ and Poisson homogeneous spaces $G/H$.