On the integrability of subalgebroids

TL;DR

This paper investigates conditions under which subalgebroids of a Lie algebroid can be integrated into subgroupoids of the original Lie groupoid, focusing on invariant foliations and closure properties.

Contribution

It provides new criteria for the integrability of subalgebroids and the conditions for their closure to form subgroupoids, advancing understanding of Lie groupoid and algebroid relationships.

Findings

Identifies conditions for integrability of subalgebroids via invariant foliations.

Establishes criteria for the closure of subgroupoids to remain subgroupoids.

Provides insights into the structure of integrable subalgebroids.

Abstract

Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the case of Lie groups, not every subalgebroid of g can be integrated by a subgroupoid of G. In this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible. We also consider the problem of integrability by closed subgroupoids, and we give conditions under which the closure of a subgroupoid is again a subgroupoid.