Double Lie algebroids and the double of a Lie bialgebroid

TL;DR

This paper introduces a general concept of double Lie algebroids, demonstrating their relation to double Lie groupoids, Lie bialgebroids, and matched pair structures, thereby unifying various structures in differential geometry.

Contribution

It defines the notion of an abstract double Lie algebroid and establishes its connections to double Lie groupoids, Lie bialgebroids, and matched pairs, extending the theoretical framework.

Findings

Double Lie algebroid of a double Lie groupoid fits the new definition.

Double cotangent of a Lie algebroid and its dual forms a double Lie algebroid iff they form a Lie bialgebroid.

Vacant double Lie algebroids correspond to matched pair structures.

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.