[Paper Review] On integrability of infinitesimal actions

Ieke Moerdijk, J. Mrčun|ArXiv.org|Jun 6, 2000

Homotopy and Cohomology in Algebraic Topology11 references4 citations

TL;DR

This paper establishes integrability conditions for semi-direct product Lie algebroids arising from infinitesimal actions of one integrable Lie algebroid on another. Using foliation theory and connections on principal Lie groupoid bundles, the authors prove that such semi-direct products are integrable under general assumptions—specifically when the target algebroid is a foliation, proper over the acting algebroid, or integrable by a source-compact, source-simply connected groupoid—generalizing results by Dazord and Nistor.

ABSTRACT

We use foliations and connections on principal Lie groupoid bundles to prove various integrability results for Lie algebroids. In particular, we show, under quite general assumptions, that the semi-direct product associated to an infinitesimal action of one integrable Lie algebroid on another is integrable. This generalizes recent results of Dazord and Nistor.

Motivation & Objective

• To generalize integrability results for Lie algebroids beyond known cases, particularly for semi-direct products of infinitesimal actions.
• To establish sufficient conditions under which the semi-direct product of two integrable Lie algebroids is itself integrable.
• To unify and extend previous results on integrability of transformation algebroids and regular algebroids with flat splittings.
• To develop a framework using foliations and connections on principal Lie groupoid bundles to analyze integrability in the context of Lie algebroid actions.

Proposed method

• Utilizes foliation theory and connections on principal Lie groupoid bundles to analyze the integrability of Lie algebroids.
• Applies the concept of source-simply connected covers of Lie groupoids to construct integral groupoids from algebroids.
• Introduces a notion of connection with values in the Lie algebroid of a Lie groupoid to control path-lifting and holonomy.
• Constructs an action of a Lie groupoid $ H $ on the monodromy groupoid $ \mathrm{Mon}(M,\mathcal{F})_1 $, using path-lifting along covering projections induced by the foliation.
• Uses the quotient $ K = \mathrm{Mon}(M,\mathcal{F})/G/H $ to construct a Lie groupoid whose Lie algebroid is isomorphic to $ \mathfrak{g} \ltimes \mathfrak{h} $.
• Applies results on integrability of subalgebroids and morphisms between algebroids to establish the main theorems.

Experimental results

Research questions

• RQ1 Under what conditions is the semi-direct product $ \mathfrak{g} \ltimes \mathfrak{h} $ of two integrable Lie algebroids itself integrable?
• RQ2 Can the integrability of transformation algebroids and regular algebroids with flat splittings be unified and generalized via a common framework?
• RQ3 How do foliations and connections on principal Lie groupoid bundles facilitate the construction of integral groupoids?
• RQ4 What role do source-simply connected covers and completeness of vector fields play in the integrability of Lie algebroids?

Key findings

• The semi-direct product $ \mathfrak{g} \ltimes \mathfrak{h} $ is integrable if $ \mathfrak{h} $ is a foliation algebroid.
• The semi-direct product is integrable if $ \mathfrak{h} $ is proper over $ \mathfrak{g} $ in a suitable sense, as formalized in Corollary 5.4.
• If $ \mathfrak{h} $ is integrable by a source-compact, source-simply connected Lie groupoid, then $ \mathfrak{g} \ltimes \mathfrak{h} $ is integrable, as shown in Theorem 5.7.
• The Lie algebra of derivations $ \mathrm{Der}(\mathfrak{h}) $ on a Lie algebroid $ \mathfrak{h} $ is isomorphic to the Lie algebra of multiplicative vector fields on its source-simply connected groupoid $ H $.
• An action of a Lie groupoid $ G $ on an integrable Lie algebroid $ \mathfrak{h} $ can be integrated to an action on the integral groupoid of $ \mathfrak{h} $, under suitable conditions.
• The construction of the quotient groupoid $ K = \mathrm{Mon}(M,\mathcal{F})/G/H $ yields a Lie groupoid whose Lie algebroid is isomorphic to $ \mathfrak{g} \ltimes \mathfrak{h} $, confirming integrability via geometric path-lifting and covering theory.