Double Lie algebroids and representations up to homotopy

TL;DR

This paper establishes an equivalence between double Lie algebroids with a linear splitting and pairs of 2-term representations up to homotopy, extending the concept of matched pairs of Lie algebroids.

Contribution

It introduces a new equivalence framework connecting double Lie algebroids with pairs of 2-term representations up to homotopy, generalizing existing matched pair concepts.

Findings

Double Lie algebroids are equivalent to pairs of 2-term representations up to homotopy.

Compatibility conditions extend the notion of matched pairs of Lie algebroids.

Detailed discussion of the tangent of a Lie algebroid.

Abstract

We show that double Lie algebroids, together with a chosen linear splitting, are equivalent to pairs of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the tangent of a Lie algebroid.