VB-algebroid morphisms and representations up to homotopy

Abstract

We show in this paper that the correspondence between $2$-term representations up to homotopy and $\mathcal{V}\mathcal{B}$-algebroids, established by Gracia-Saz and Mehta, holds also at the level of morphisms. This correspondence is hence an equivalence of categories. As an application, we study foliations and distributions on a Lie algebroid, that are compatible both with the linear structure and the Lie algebroid structure. In particular, we show how infinitesimal ideal systems in a Lie algebroid $A$ are related with subrepresentations of the adjoint representation of $A$.

References in corpus (2)

• Linear and multiplicative 2-forms
• Multiplicative Dirac structures on Lie groups

Cited by in corpus (9)

• Lie theory of multiplicative tensors
• Morita equivalences of vector bundles
• The Geometrisation of $\mathbb{N}$-manifolds of degree 2
• Double Lie algebroids and representations up to homotopy
• Differential forms with values in VB-groupoids
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Lie 2-algebroids and matched pairs of 2-representations - a geometric approach
• VB-structures and generalizations
• Categorification of VB-Lie algebroids and VB-Courant algebroids