Double Lie algebroids and the double of a Lie bialgebroid
arXiv:math/9808081
Abstract
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (A, A*) is a Lie bialgebroid; (3) that a vacant double Lie algebroid structure is equivalent to a matched pair structure on the side Lie algebroids.
19 pages, LaTeX
Cited by in corpus (22)
- Higher vector bundles and multi-graded symplectic manifolds
- Lie algebroid structures on double vector bundles and representation theory of Lie algebroids
- Supergroupoids, double structures, and equivariant cohomology
- Vector bundles over Lie groupoids and algebroids
- VB-groupoids and representation theory of Lie groupoids
- The Atiyah class of a dg-vector bundle
- Linear duals of graded bundles and higher analogues of (Lie) algebroids
- LA-Courant algebroids and their applications
- Characteristic classes associated to Q-bundles
- Mackenzie theory and Q-manifolds
- Higher Lie algebra actions on Lie algebroids
- Q-manifolds and Mackenzie theory: an overview
- Deformations of Linear Lie Brackets
- On the integration of LA-groupoids and duality for Poisson groupoids
- Modular Classes of Lie Groupoid Representations up to Homotopy
- Poisson double structures
- Polyvector fields and polydifferential operators associated with Lie pairs
- Higher omni-Lie algebroids
- Deformations of symplectic groupoids
- Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras
- Pseudo-Dirac Structures
- Categorification of VB-Lie algebroids and VB-Courant algebroids