Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids
| dc.creator | Mackenzie, K. C. H. | |
| dc.date | 2006-11-26 | |
| dc.date | 2006-12-22 | |
| dc.date.accessioned | 2026-07-07T07:36:49Z | |
| dc.date.available | 2026-07-07T07:36:49Z | |
| dc.description | 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. | |
| dc.description | Reference added. Some rewording in introduction. 34 pages. Uses Xy-pic | |
| dc.identifier | https://arxiv.org/abs/math/0611799 | |
| dc.identifier | http://arxiv.org/abs/math/0611799 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/120562 | |
| dc.subject | Differential Geometry | |
| dc.subject | Category Theory | |
| dc.subject | Symplectic Geometry | |
| dc.subject | Primary 53D17. Secondary 17B62, 17B66, 18D05, 22A22, 58H05 | |
| dc.title | Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids | |
| dc.type | text |