  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.

Differential Geometry;Category Theory;Symplectic Geometry