Singular subalgebroids

TL;DR

This paper extends the concept of Lie subalgebroids to singular subalgebroids, develops their associated holonomy groupoids, and provides a functorial construction suitable for noncommutative geometry.

Contribution

It introduces singular subalgebroids, constructs their holonomy groupoids, and develops a functorial framework for morphisms, broadening Lie theory to singular cases.

Findings

Constructed holonomy groupoids for singular subalgebroids.

Extended Lie theory to include singular subalgebroids.

Provided a functorial approach for morphisms in this setting.

Abstract

We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion of Lie subalgebroid by dropping the constant rank requirement. We lay the bases of a Lie theory for singular subalgebroids: we construct the associated holonomy groupoids, adapting the procedure of Androulidakis-Skandalis for singular foliations, in a way that keeps track of the choice of Lie groupoid integrating the ambient Lie algebroid. The holonomy groupoids are topological groupoids, and are suitable for noncommutative geometry as they allow for the construction of the associated convolution algebras. Further we carry out the construction for morphisms in a functorial way.