Transitive double Lie algebroids via core diagrams

Lean Madeleine Jotz; Mackenzie Kirill C. H.

doi:10.3934/jgm.2021023

Article Contents

2021, Volume 13, Issue 3: 403-457. Doi: 10.3934/jgm.2021023

This issue Previous Article On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras Next Article From Schouten to Mackenzie: Notes on brackets

Transitive double Lie algebroids via core diagrams

Lean Madeleine Jotz^1, , and
Mackenzie Kirill C. H.²

1.
Institute for Mathematics, Julius-Maximilians-Universität Würzburg, Germany
2.
In memoriam, School of Mathematics and Statistics, The University of Sheffield, United Kingdom

^*Corresponding author: Madeleine Jotz Lean
^*Corresponding author: Madeleine Jotz Lean

Received: March 2021

Revised: July 2021

Early access: August 2021

Published: September 2021

This research was a joint project with the sadly deceased second author. This paper is dedicated to his memory

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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.

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Mathematics Subject Classification: Primary: 58H05, 53C05, 53D17; Secondary: 18N10.

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Figure 1. The core diagram of a double Lie algebroid

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Figure 2. Core diagram of Lie groupoids

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Figure 3. The transitive core diagram of a transitive double Lie algebroid

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Figure 4. A morphism of core diagrams

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Figure 6. The core diagram of align="right"

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Figure 7. Square of Lie algebroid morphisms

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Figure 8. Setting of Section 5, the transitive core diagram $ \mathcal C $

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