×

On integrability of infinitesimal actions. (English) Zbl 1013.58010

The authors prove the integrability for a substantial family of Lie algebroids associated to infinitesimal actions of one algebroid on another as described in P. J. Higgins and K. C. H. Mackenzie [J. Algebra 129, 194-230 (1990; Zbl 0696.22007)]. The main results are stated as follows: Let \(\mathfrak g\) and \(\mathfrak h\) be integrable Lie algebroids and let \(\mathfrak g\ltimes\mathfrak h\) denote a semi-direct product with respect to an infinitesimal action of \(\mathfrak g\) on \(\mathfrak h\) (Theorem 5.1). If \({\mathfrak h}\) is a foliation, then \(\mathfrak g\ltimes\mathfrak h\) is integrable for any infinitesimal action of \(\mathfrak g\) on \(\mathfrak h\) (Corollary 5.4). If \(\mathfrak g\) and \(\mathfrak h\) are Lie algebroids of source-simply connected Lie groupoids \(G\rightrightarrows G_0\) and \(H\rightrightarrows H_0\) respectively, and if \(\mathfrak g\) acts on \(\mathfrak h\) along a proper map \(q: H_0\to G_0\), then there exists an action of \(G\) on \(H\) and \(\mathfrak g\ltimes\mathfrak h\) is isomorphic to the Lie algebroid of the semi-direct product groupoid \(G\ltimes H\) with respect to the groupoid action (Theorem 5.7). If \(\mathfrak h\) is a Lie algebroid of a Hausdorff Lie groupoid \(H\) and if the source map \(\alpha: H_1\to H_0\) is proper, then the semi-direct product with respect to any infinitesimal action of \(\mathfrak g\) on \(\mathfrak h\) is integrable.
In the proofs the authors use foliations and connections on principal Lie groupoid bundles. They also give a concise review of the Lie theory for groupoids including main examples and a uniform treatment of some basic results, such as the source-simply connected cover.

MSC:

58H05 Pseudogroups and differentiable groupoids
22A22 Topological groupoids (including differentiable and Lie groupoids)

Citations:

Zbl 0696.22007
PDFBibTeX XMLCite
Full Text: DOI arXiv Link Link