Journal of Geometric Mechanics | 2021
Transitive double Lie algebroids via core diagrams
Abstract
The core diagram of a double Lie algebroid consists of 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.