Normal forms for Poisson maps and symplectic groupoids around Poisson transversals

TL;DR

This paper establishes normal form theorems for Poisson maps and symplectic groupoids around Poisson transversals, providing a unified framework for understanding their local structure and integrability.

Contribution

It extends previous normal form results to Poisson maps and symplectic groupoids, revealing conditions for local linearization and integrability around Poisson transversals.

Findings

Normal forms exist for Poisson maps around transversals.

Neighborhoods of Poisson transversals are integrable iff the transversals are.

Normal form theorems for symplectic groupoids around transversals.

Abstract

Poisson transversals are those submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In a previous note we proved a normal form theorem around such submanifolds. In this communication, we promote that result to a normal form theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous normal forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the normal form data of the transversals. Our second main result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a normal form theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all its structure maps in normal form. We conclude the paper by illustrating our results with examples arising from Lie algebras.