Transitive Courant algebroids and double symplectic groupoids

TL;DR

This paper generalizes the construction of double symplectic groupoids to a broader class of Lie bialgebroids with transitive Courant algebroids, providing new insights and classifications in the theory.

Contribution

It extends the Lu-Weinstein construction to transitive Courant algebroids and classifies exact twisted Courant algebroids over Lie groupoids.

Findings

Generalization of double symplectic groupoids construction

Classification of exact twisted Courant algebroids over Lie groupoids

Existence of a foliation by twisted Courant algebroids

Abstract

In this work we extend the Lu-Weinstein construction of double symplectic groupoids to any Lie bialgebroid such that its associated Courant algebroid is transitive and its Atiyah algebroid integrable. We illustrate this result by showing how it generalises many of the examples of double symplectic groupoids that have appeared in the literature. As preliminary steps for this construction, we give a classification of exact twisted Courant algebroids over Lie groupoids (CA-groupoids for short) and we show the existence of a foliation by twisted Courant algebroids on the base of a twisted CA-groupoid.