Coordinates for non-integrable Lie algebroids

TL;DR

This paper develops a method to construct local coordinates for the Weinstein groupoid of non-integrable Lie algebroids, enabling the attachment of a $C^*$-algebra to each Lie algebroid, thus advancing the understanding of their structure.

Contribution

It introduces an algebraic reformulation of bi-submersions and proves their existence for the Weinstein groupoid, providing new tools for analyzing non-integrable Lie algebroids.

Findings

Constructed local coordinates for the Weinstein groupoid.

Reformulated bi-submersion notion algebraically.

Established the existence of bi-submersions for the Weinstein groupoid.

Abstract

We construct local coordinates for the Weinstein groupoid of a non-integrable Lie algebroid. To this end, we reformulate the notion of bi-submersion in a completely algebraic way and prove the existence of bi-submersions as such for the Weinstein groupoid. This implies that a $C^*$-algebra can be attached to every Lie algebroid.