Mathematics – Differential Geometry
Scientific paper
2006-11-26
Mathematics
Differential Geometry
Reference added. Some rewording in introduction. 34 pages. Uses Xy-pic
Scientific paper
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.
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