Editing Lie algebroid (section)

=== Lie theorems ===
A Lie algebroid is called '''integrable''' if it is isomorphic to ''<math>\mathrm{Lie}(G)</math>'' for some Lie groupoid'' <math>G \rightrightarrows M</math>''. The analogue of the classical '''Lie I theorem''' states that:<ref name=":1">{{Cite journal|last1=Moerdijk|first1=Ieke|last2=Mrcun|first2=Janez|date=2002|title=On integrability of infinitesimal actions|url=http://muse.jhu.edu/content/crossref/journals/american_journal_of_mathematics/v124/124.3moerdijk.pdf|journal=American Journal of Mathematics|language=en|volume=124|issue=3|pages=567–593|doi=10.1353/ajm.2002.0019|issn=1080-6377|arxiv=math/0006042|s2cid=53622428}}</ref><blockquote>if ''<math>A</math>'' is an integrable Lie algebroid, then there exists a unique (up to isomorphism) ''<math>s</math>''-simply connected Lie groupoid ''<math>G</math>'' integrating ''<math>A</math>''.</blockquote>Similarly, a morphism ''<math>F: A_1 \to A_2</math>'' between integrable Lie algebroids is called '''integrable''' if it is the differential ''<math>F = d\phi_{ \mid A} </math>'' for some morphism ''<math>\phi: G_1 \to G_2</math>'' between two integrations of ''<math>A_1</math>'' and ''<math>A_2</math>''. The analogue of the classical '''Lie II theorem''' states that:<ref>{{Cite journal|last1=Mackenzie|first1=Kirill|last2=Xu|first2=Ping|date=2000-05-01|title=Integration of Lie bialgebroids|url=https://www.sciencedirect.com/science/article/pii/S004093839800069X|journal=Topology|language=en|volume=39|issue=3|pages=445–467|doi=10.1016/S0040-9383(98)00069-X|issn=0040-9383|arxiv=dg-ga/9712012|s2cid=119594174}}</ref> <blockquote>if ''<math>F: \mathrm{Lie}(G_1) \to \mathrm{Lie}(G_2)</math>'' is a morphism of integrable Lie algebroids, and ''<math>G_1</math>'' is ''<math>s</math>''-simply connected, then there exists a unique morphism of Lie groupoids ''<math>\phi: G_1 \to G_2</math>'' integrating ''<math>F</math>''.</blockquote>In particular, by choosing as ''<math>G_2</math>'' the general linear groupoid ''<math>GL(E)</math>'' of a vector bundle ''<math>E</math>'', it follows that any representation of an integrable Lie algebroid integrates to a representation of its ''<math>s</math>''-simply connected integrating Lie groupoid.

On the other hand, there is no analogue of the classical '''Lie III theorem''', i.e. going back from any Lie algebroid to a Lie groupoid is not always possible. Pradines claimed that such a statement hold,<ref>{{Cite journal|last=Pradines|first=Jean|date=1968|title=Troisieme théorème de Lie pour les groupoides différentiables|url=https://gallica.bnf.fr/ark:/12148/bpt6k480295b/f24.item.r=pradine|journal=Comptes Rendus de l'Académie des Sciences, Série A|language=fr|volume=267|pages=21–23}}</ref> and the first explicit example of non-integrable Lie algebroids, coming for instance from foliation theory, appeared only several years later.<ref>{{Cite journal|last1=Almeida|first1=Rui|last2=Molino|first2=Pierre|date=1985|title=Suites d'Atiyah et feuilletages transversalement complets|journal=Comptes Rendus de l'Académie des Sciences, Série I|language=fr|volume=300|pages=13–15}}</ref> Despite several partial results, including a complete solution in the transitive case,<ref>{{Cite book|last=Mackenzie|first=K.|url=https://www.cambridge.org/core/books/lie-groupoids-and-lie-algebroids-in-differential-geometry/6B8B1D00E8B7672ABA80A0C950FD4979|title=Lie Groupoids and Lie Algebroids in Differential Geometry|date=1987|publisher=Cambridge University Press|isbn=978-0-521-34882-9|series=London Mathematical Society Lecture Note Series|location=Cambridge|doi=10.1017/cbo9780511661839}}</ref> the general obstructions for an arbitrary Lie algebroid to be integrable have been discovered only in 2003 by [[Marius Crainic|Crainic]] and [[Rui Loja Fernandes|Fernandes]].<ref name=":2">{{cite journal|last1=Crainic|first1=Marius|last2=Fernandes|first2=Rui L.|year=2003|title=Integrability of Lie brackets|journal=Ann. of Math.|series=2|volume=157|issue=2|pages=575–620|arxiv=math/0105033|doi=10.4007/annals.2003.157.575|s2cid=6992408}}</ref> Adopting a more general approach, one can see that every Lie algebroid integrates to a [[Algebraic stack|stacky]] Lie groupoid.<ref>{{cite journal|author1=Hsian-Hua Tseng|author2=Chenchang Zhu|year=2006|title=Integrating Lie algebroids via stacks|journal=Compositio Mathematica|volume=142|issue=1|pages=251–270|arxiv=math/0405003|doi=10.1112/S0010437X05001752|s2cid=119572919}}</ref><ref>{{cite arXiv|eprint=math/0701024|author1=Chenchang Zhu|title=Lie II theorem for Lie algebroids via stacky Lie groupoids|date=2006}}</ref>