Cores of Symplectic Double Groupoids via Reduction

TL;DR

This paper introduces a new symplectic reduction method to construct the core of a symplectic double groupoid, revealing how the double structure descends to the core and relating it to cotangent lifts of Lie groupoid maps.

Contribution

It provides a novel reduction-based construction of the core of symplectic double groupoids and clarifies the relationship between the double structure and the core groupoid.

Findings

Core of symplectic double groupoid as leaf space of characteristic foliations

Reduction process yields cotangent lifts of structure maps

Double groupoid structure descends to the core groupoid

Abstract

We use symplectic reduction to give a new construction of the core $C$ of a symplectic double groupoid $D$ as the common leaf space of characteristic foliations associated to various coisotropic submanifolds of $D$. In the case of the cotangent double groupoid of a Lie groupoid $G$, the canonical relations arising from this process turn out to be cotangent lifts of structure maps associated to $G$. We also show that under this reduction procedure the double groupoid structure on $D$ descends to a groupoid structure on the leaf space above, recovering the core groupoid structure on $C$ of Brown and Mackenzie.