Transitive double Lie algebroids via core diagrams

TL;DR

This paper establishes a precise correspondence between transitive double Lie algebroids and their core diagrams, demonstrating that such structures are fully determined by their core data and linking them to transitive double Lie groupoids.

Contribution

It proves the equivalence between transitive double Lie algebroids and transitive core diagrams, and connects their integrability to transitive double Lie groupoids.

Findings

Transitive double Lie algebroids are determined by their core diagrams.

A construction of the comma double Lie algebroid related to core-anchor morphisms.

Transitive double Lie algebroids with integrable sides and core are integrable to double Lie groupoids.

Abstract

The core diagram of a double Lie algebroid consists in the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram. The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram. Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid. This research was a joint project with the sadly deceased second author. This paper is dedicated to his memory.