[Paper Review] On the integrability of subalgebroids

Ieke Moerdijk, J. Mrčun|ArXiv.org|Jun 28, 2004

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper investigates the conditions under which a Lie subalgebroid of a Lie groupoid's Lie algebroid can be integrated to a subgroupoid of the groupoid. Using foliation theory, it shows that trivial holonomy of the invariant foliation associated to the subalgebroid guarantees integration via an injective immersion, and transverse completeness ensures the closure of the image is a Lie subgroupoid. The key contribution is a foliation-theoretic criterion for integrability of subalgebroids beyond the classical group case.

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.

Motivation & Objective

• To determine when a Lie subalgebroid of a Lie groupoid's algebroid can be integrated to a subgroupoid of the groupoid, rather than just an immersed subgroupoid.
• To extend classical Lie theory results—such as the closure of immersed subgroups being a subgroup—to the context of Lie groupoids, where such properties do not always hold.
• To characterize conditions under which the image of an integration map is closed, and its closure forms a Lie subgroupoid.
• To provide a foundation for the sequel paper on integrating subalgebroids to closed subgroupoids in larger groupoids.

Proposed method

• The authors associate a right-invariant foliation to each subalgebroid of a Lie algebroid via the action of the ambient Lie groupoid.
• They use the holonomy of this foliation as a key invariant: trivial holonomy ensures the existence of an injective immersion integrating the subalgebroid.
• Transverse completeness of the foliation is used to guarantee that the closure of the image of the integration map is a Lie subgroupoid.
• The construction relies on path-lifting in leaves of the foliation to define the integration map, particularly in the case of transitive groupoids.
• The paper employs the concept of a twisted product of groupoids to describe the closure of a transitive subgroupoid in terms of closures of isotropy groups.
• It applies results from foliation theory and Lie theory, including the use of Maurer-Cartan forms and flat connections in specific examples.

Experimental results

Research questions

• RQ1 Under what conditions can a Lie subalgebroid of a Lie groupoid's algebroid be integrated to a subgroupoid of the groupoid via an injective immersion?
• RQ2 When is the closure of the image of such an integration map a Lie subgroupoid of the ambient groupoid?
• RQ3 How does the holonomy of the invariant foliation associated to a subalgebroid affect its integrability to a subgroupoid?
• RQ4 What role does transverse completeness of the foliation play in ensuring the integrability of the subalgebroid to a closed subgroupoid?
• RQ5 Can the closure of a transitive subgroupoid be described explicitly in terms of isotropy group closures and groupoid actions?

Key findings

• A subalgebroid can be integrated to an injective immersion into the groupoid if and only if the associated right-invariant foliation has trivial holonomy.
• If the foliation associated to a subalgebroid is transversely complete, then the image of the integration map is a closed subgroupoid, and its closure is a Lie subgroupoid.
• The image of the integration map is closed if and only if the integration map is an embedding.
• For transitive subgroupoids, the closure can be described as a twisted product of the original subgroupoid and the closures of its isotropy groups.
• In the case of a simply connected Lie group acting via a Maurer-Cartan form, the integration of the subalgebroid yields a subgroupoid whose closure is constructed via conjugation actions on the closures of isotropy groups.
• The results provide a foliation-theoretic criterion for integrability that generalizes classical Lie group results to the groupoid setting.