Notions of double for Lie algebroids

Abstract

We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated tangent bundle of an ordinary manifold, and various iterated tangent/cotangent constructions in symplectic and Poisson geometry; (2) the double of a Lie bialgebra; (3) the double of a matched pair, or twilled extension, of Lie algebras, and a corresponding construction for matched pairs of general Lie algebroids in the sense of Mokri and of J.-H. Lu. We prove that, given any abstract Lie bialgebroid, the `double cotangent' has a structure of double Lie algebroid, which reduces to the classical Drinfel'd double in the Lie bialgebra case. The key to the compatibility conditions between the structures in an abstract double Lie algebroid is an associated Lie bialgebroid in the sense of the author and Ping Xu.