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

Source abstract

We show that the Manin triple characterization of Lie bialgebras in terms of the Drinfel’d double may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using double cotangent bundles, rather than the direct sum structures (Courant algebroids) utilized for similar purposes by Liu, Weinstein and Xu. This is achieved in terms of an abstract notion of double Lie algebroid (where “double” is now used in the Ehresmann sense) which unifies many iterated constructions in differential geometry.