Double Lie algebroids and second-order geometry, II

TL;DR

This paper completes the construction of double Lie algebroids from double Lie groupoids, demonstrating canonical isomorphisms and the symplectic and Poisson structures arising in this context, extending classical geometric structures.

Contribution

It finalizes the construction of double Lie algebroids and reveals their canonical isomorphisms and symplectic structures, advancing the understanding of higher-order geometric structures.

Findings

The Lie algebroid of an LA-groupoid can be extended to its Lie groupoid.

For double groupoids, these extended structures are canonically isomorphic.

The cotangent of a double Lie groupoid forms a symplectic double groupoid.

Abstract

We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either LA--groupoid is canonically isomorphic to the Lie algebroid structure associated with the other; this extends many canonical isomorphisms associated with iterated tangent and cotangent structures. We also show that the cotangent of a double Lie groupoid is a symplectic double groupoid, and that the side groupoids of any symplectic double groupoid are Poisson groupoids in duality. Thus any double Lie groupoid gives rise to a dual pair of Poisson groupoids.