Lie 2-algebroids and matched pairs of 2-representations - a geometric approach

Abstract

Li-Bland's correspondence between linear Courant algebroids and Lie $2$-algebroids is explained and shown to be an equivalence of categories. Decomposed VB-Courant algebroids are shown to be equivalent to split Lie 2-algebroids in the same manner as decomposed VB-algebroids are equivalent to 2-term representations up to homotopy (Gracia-Saz and Mehta). Several classes of examples are discussed, yielding new examples of split Lie 2-algebroids. We prove that the bicrossproduct of a matched pair of $2$-representations is a split Lie $2$-algebroid and we explain this result geometrically, as a consequence of the equivalence of VB-Courant algebroids and Lie $2$-algebroids. This explains in particular how the two notions of double" of a matched pair of representations are geometrically related. In the same manner, we explain the geometric link between the two notions of double of a Lie bialgebroid.

This is the improved and completed second part of the work arXiv:1504.00880 (N-manifolds of degree 2 and metric double vector bundles)

References in corpus (1)

• The Geometrisation of $\mathbb{N}$-manifolds of degree 2

Cited by in corpus (4)

• Deformation spaces and normal forms around transversals
• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Stability of fixed points in Poisson geometry and higher Lie theory
• Categorification of VB-Lie algebroids and VB-Courant algebroids