Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids

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, that reduces, in the case of a Lie bialgebra, to the classical Drinfel'd double. Thus one may say that the Drinfel'd double of a Lie bialgebroid is an Ehresmann double, and it follows that double Lie groupoids provide global models for the cotangent double of a Lie bialgebroid. A double vector bundle is called vacant if it is constructed from vector bundles A, B on a common base M as the simultaneous pullback A × M B. Given a matched pair (A, B) of Lie algebroids over base M, we show that A × M B has a double Lie algebroid structure, and that any double Lie algebroid structure on a vacant double vector bundle A × M B arises in this way. In particular, double Lie algebras in the sense of Lu and Weinstein, Kosmann-Schwarzbach and Magri, and Majid, have the structure of a vacant double Lie algebroid. Lastly we extend the construction of Lu (Duke Math. J. 86: 261–304, 1997) which associates a matched pair of Lie algebroids to any Poisson group action, to actions of Lie bialgebroids; this yields a double Lie algebroid which in general does not correspond to a matched pair. The methods of the paper are entirely ‘classical’ rather than utilizing super techniques.