2006/11/26 by Kirill C. H. Mackenzie, Mackenzie, K. C. H.
Mathematics · Medicine · #17B66 #18D05 #22A22 #58H05 #Advanced Topics in Algebra #Category Theory (math.CT) #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #Primary 53D17. Secondary 17B62 #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.math/0611799
openalex publication_date 2006/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The word `double' was used by Ehresmann to mean `an object X in the category of all X'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this paper we use the duality of double vector bundles to define a notion of double Lie algebroid, and we show that this abstracts the infinitesimal structure (at second order) of a double Lie groupoid. We further show that the cotangent of either Lie algebroid in a Lie bialgebroid has a double Lie algebroid structure, and that a pair of Lie algebroid structures on dual vector bundles forms a Lie bialgebroid if and only if the structures which they canonically induce on their cotangents form a double Lie algebroid. In particular, the Drinfel'd double of a Lie bialgebra has a double Lie algebroid structure. We also show that matched pairs of Lie algebroids, as used by J.-H. Lu in the classification of Poisson group actions, are in bijective correspondence with vacant double Lie algebroids.