VB-algebroid morphisms and representations up to homotopy

TL;DR

This paper extends the known correspondence between 2-term representations up to homotopy and VB-algebroids to include morphisms, establishing an equivalence of categories and exploring applications to Lie algebroid structures.

Contribution

It proves that the correspondence between 2-term representations up to homotopy and VB-algebroids is an equivalence of categories at the morphism level.

Findings

The correspondence holds at the level of morphisms.

Infinitesimal ideal systems relate to subrepresentations of the adjoint.

Applications to foliations and distributions on Lie algebroids.

Abstract

We show in this paper that the correspondence between $2$-term representations up to homotopy and $\mathcal{VB}$-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 $A$.