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.

arXiv comment30 pages

References in corpus (4)

• Courant algebroids, derived brackets and even symplectic supermanifolds
• Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation
• Twilled Lie-Rinehart algebras and differential Batalin-Vilkovisky algebras
• Double Lie algebras and Manin triples

Cited by in corpus (6)

• Lie algebroid structures on double vector bundles and representation theory of Lie algebroids
• VB-groupoids and representation theory of Lie groupoids
• Even symplectic supermanifolds and double field theory
• Modular Classes of Lie Groupoid Representations up to Homotopy
• Poisson double structures
• The Local Structure of Lie Bialgebroids