Rigidity of infinitesimal momentum maps

TL;DR

This paper establishes rigidity results for Poisson Lie group actions on Poisson manifolds, showing that small infinitesimal momentum maps are equivalent, with implications for integrable structures and symplectic groupoids.

Contribution

It introduces a normal form theorem for SCI spaces to prove the rigidity of infinitesimal momentum maps in Poisson geometry, extending to integrable cases.

Findings

Close infinitesimal momentum maps are equivalent

Rigidity holds for lifted actions on symplectic groupoids

Normal form theorem for SCI spaces is developed

Abstract

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.