In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.

Lie algebroids play a similar same role in the theory of Lie groupoids that Lie algebras play in the theory of Lie groups: reducing global problems to infinitesimal ones. Indeed, any Lie groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every Lie algebroid arises from a Lie groupoid.

Lie algebroids were introduced in 1967 by Jean Pradines.[1]

Definition and basic concepts

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A Lie algebroid is a triple   consisting of

  • a vector bundle   over a manifold  
  • a Lie bracket   on its space of sections  
  • a morphism of vector bundles  , called the anchor, where   is the tangent bundle of  

such that the anchor and the bracket satisfy the following Leibniz rule:

 

where  . Here   is the image of   via the derivation  , i.e. the Lie derivative of   along the vector field  . The notation   denotes the (point-wise) product between the function   and the vector field  .

One often writes   when the bracket and the anchor are clear from the context; some authors denote Lie algebroids by  , suggesting a "limit" of a Lie groupoids when the arrows denoting source and target become "infinitesimally close".[2]

First properties

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It follows from the definition that

  • for every  , the kernel   is a Lie algebra, called the isotropy Lie algebra at  
  • the kernel   is a (not necessarily locally trivial) bundle of Lie algebras, called the isotropy Lie algebra bundle
  • the image   is a singular distribution which is integrable, i.e. its admits maximal immersed submanifolds  , called the orbits, satisfying   for every  . Equivalently, orbits can be explicitly described as the sets of points which are joined by A-paths, i.e. pairs   of paths in   and in   such that   and  
  • the anchor map   descends to a map between sections   which is a Lie algebra morphism, i.e.
 

for all  .

The property that   induces a Lie algebra morphism was taken as an axiom in the original definition of Lie algebroid.[1] Such redundancy, despite being known from an algebraic point of view already before Pradine's definition,[3] was noticed only much later.[4][5]

Subalgebroids and ideals

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A Lie subalgebroid of a Lie algebroid   is a vector subbundle   of the restriction   such that   takes values in   and   is a Lie subalgebra of  . Clearly,   admits a unique Lie algebroid structure such that   is a Lie algebra morphism. With the language introduced below, the inclusion   is a Lie algebroid morphism.

A Lie subalgebroid is called wide if  . In analogy to the standard definition for Lie algebra, an ideal of a Lie algebroid is wide Lie subalgebroid   such that   is a Lie ideal. Such notion proved to be very restrictive, since   is forced to be inside the isotropy bundle  . For this reason, the more flexible notion of infinitesimal ideal system has been introduced.[6]

Morphisms

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A Lie algebroid morphism between two Lie algebroids   and   with the same base   is a vector bundle morphism   which is compatible with the Lie brackets, i.e.   for every  , and with the anchors, i.e.  .

A similar notion can be formulated for morphisms with different bases, but the compatibility with the Lie brackets becomes more involved.[7] Equivalently, one can ask that the graph of   to be a subalgebroid of the direct product   (introduced below).[8]

Lie algebroids together with their morphisms form a category.

Examples

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Trivial and extreme cases

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  • Given any manifold  , its tangent Lie algebroid is the tangent bundle   together with the Lie bracket of vector fields and the identity of   as an anchor.
  • Given any manifold  , the zero vector bundle   is a Lie algebroid with zero bracket and anchor.
  • Lie algebroids   over a point are the same thing as Lie algebras.
  • More generally, any bundles of Lie algebras is Lie algebroid with zero anchor and Lie bracket defined pointwise.

Examples from differential geometry

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  • Given a foliation   on  , its foliation algebroid is the associated involutive subbundle  , with brackets and anchor induced from the tangent Lie algebroid.
  • Given the action of a Lie algebra   on a manifold  , its action algebroid is the trivial vector bundle  , with anchor given by the Lie algebra action and brackets uniquely determined by the bracket of   on constant sections   and by the Leibniz identity.
  • Given a principal G-bundle   over a manifold  , its Atiyah algebroid is the Lie algebroid   fitting in the following short exact sequence:
     
The space of sections of the Atiyah algebroid is the Lie algebra of  -invariant vector fields on  , its isotropy Lie algebra bundle is isomorphic to the adjoint vector bundle  , and the right splittings of the sequence above are principal connections on  .
  • Given a vector bundle  , its general linear algebroid, denoted by   or  , is the vector bundle whose sections are derivations of  , i.e. first-order differential operators   admitting a vector field   such that   for every  . The anchor is simply the assignment   and the Lie bracket is given by the commutator of differential operators.
  • Given a Poisson manifold  , its cotangent algebroid is the cotangent vector bundle  , with Lie bracket   and anchor map  .
  • Given a closed 2-form  , the vector bundle   is a Lie algebroid with anchor the projection on the first component and Lie bracket Actually, the bracket above can be defined for any 2-form  , but   is a Lie algebroid if and only if   is closed.

Constructions from other Lie algebroids

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  • Given any Lie algebroid  , there is a Lie algebroid  , called its tangent algebroid, obtained by considering the tangent bundle of   and   and the differential of the anchor.
  • Given any Lie algebroid  , there is a Lie algebroid  , called its k-jet algebroid, obtained by considering the k-jet bundle of  , with Lie bracket uniquely defined by   and anchor  .
  • Given two Lie algebroids   and  , their direct product is the unique Lie algebroid   with anchor   and such that   is a Lie algebra morphism.
  • Given a Lie algebroid   and a map   whose differential is transverse to the anchor map   (for instance, it is enough for   to be a surjective submersion), the pullback algebroid is the unique Lie algebroid  , with   the pullback vector bundle, and   the projection on the first component, such that   is a Lie algebroid morphism.

Important classes of Lie algebroids

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Totally intransitive Lie algebroids

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A Lie algebroid is called totally intransitive if the anchor map   is zero.

Bundle of Lie algebras (hence also Lie algebras) are totally intransitive. This actually exhaust completely the list of totally intransitive Lie algebroids: indeed, if   is totally intransitive, it must coincide with its isotropy Lie algebra bundle.

Transitive Lie algebroids

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A Lie algebroid is called transitive if the anchor map   is surjective. As a consequence:

  • there is a short exact sequence 
  • right-splitting of   defines a principal bundle connections on  ;
  • the isotropy bundle   is locally trivial (as bundle of Lie algebras);
  • the pullback of   exist for every  .

The prototypical examples of transitive Lie algebroids are Atiyah algebroids. For instance:

  • tangent algebroids   are trivially transitive (indeed, they are Atiyah algebroid of the principal  -bundle  )
  • Lie algebras   are trivially transitive (indeed, they are Atiyah algebroid of the principal  -bundle  , for   an integration of  )
  • general linear algebroids   are transitive (indeed, they are Atiyah algebroids of the frame bundle  )

In analogy to Atiyah algebroids, an arbitrary transitive Lie algebroid is also called abstract Atiyah sequence, and its isotropy algebra bundle   is also called adjoint bundle. However, it is important to stress that not every transitive Lie algebroid is an Atiyah algebroid. For instance:

  • pullbacks of transitive algebroids are transitive
  • cotangent algebroids   associated to Poisson manifolds   are transitive if and only if the Poisson structure   is non-degenerate
  • Lie algebroids   defined by closed 2-forms are transitive

These examples are very relevant in the theory of integration of Lie algebroid (see below): while any Atiyah algebroid is integrable (to a gauge groupoid), not every transitive Lie algebroid is integrable.

Regular Lie algebroids

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A Lie algebroid is called regular if the anchor map   is of constant rank. As a consequence

  • the image of   defines a regular foliation on  ;
  • the restriction of   over each leaf   is a transitive Lie algebroid.

For instance:

  • any transitive Lie algebroid is regular (the anchor has maximal rank);
  • any totally intransitive Lie algebroids is regular (the anchor has zero rank);
  • foliation algebroids are always regular;
  • cotangent algebroids   associated to Poisson manifolds   are regular if and only if the Poisson structure   is regular.
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Actions

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An action of a Lie algebroid   on a manifold P along a smooth map   consists of a Lie algebra morphism such that, for every  , Of course, when  , both the anchor   and the map   must be trivial, therefore both conditions are empty, and we recover the standard notion of action of a Lie algebra on a manifold.

Connections

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Given a Lie algebroid  , an A-connection on a vector bundle   consists of an  -bilinear map which is  -linear in the first factor and satisfies the following Leibniz rule: for every  , where   denotes the Lie derivative with respect to the vector field  .

The curvature of an A-connection   is the  -bilinear map and   is called flat if  .

Of course, when  , we recover the standard notion of connection on a vector bundle, as well as those of curvature and flatness.

Representations

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A representation of a Lie algebroid   is a vector bundle   together with a flat A-connection  . Equivalently, a representation   is a Lie algebroid morphism  .

The set   of isomorphism classes of representations of a Lie algebroid   has a natural structure of semiring, with direct sums and tensor products of vector bundles.

Examples include the following:

  • When  , an  -connection simplifies to a linear map   and the flatness condition makes it into a Lie algebra morphism, therefore we recover the standard notion of representation of a Lie algebra.
  • When   and   is a representation the Lie algebra  , the trivial vector bundle   is automatically a representation of  
  • Representations of the tangent algebroid   are vector bundles endowed with flat connections
  • Every Lie algebroid   has a natural representation on the line bundle  , i.e. the tensor product between the determinant line bundles of   and of  . One can associate a cohomology class in   (see below) known as the modular class of the Lie algebroid.[9] For the cotangent algebroid   associated to a Poisson manifold   one recovers the modular class of  .[10]

Note that there an arbitrary Lie groupoid does not have a canonical representation on its Lie algebroid, playing the role of the adjoint representation of Lie groups on their Lie algebras. However, this becomes possible if one allows the more general notion of representation up to homotopy.

Lie algebroid cohomology

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Consider a Lie algebroid   and a representation  . Denoting by   the space of  -differential forms on   with values in the vector bundle  , one can define a differential   with the following Koszul-like formula: Thanks to the flatness of  ,   becomes a cochain complex and its cohomology, denoted by  , is called the Lie algebroid cohomology of   with coefficients in the representation  .

This general definition recovers well-known cohomology theories:

  • The cohomology of a Lie algebroid   coincides with the Chevalley-Eilenberg cohomology of   as a Lie algebra.
  • The cohomology of a tangent Lie algebroid   coincides with the de Rham cohomology of  .
  • The cohomology of a foliation Lie algebroid   coincides with the leafwise cohomology of the foliation  .
  • The cohomology of the cotangent Lie algebroid   associated to a Poisson structure   coincides with the Poisson cohomology of  .

Lie groupoid-Lie algebroid correspondence

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The standard construction which associates a Lie algebra to a Lie group generalises to this setting: to every Lie groupoid   one can canonically associate a Lie algebroid   defined as follows:

  • the vector bundle is  , where   is the vertical bundle of the source fibre   and   is the groupoid unit map;
  • the sections of   are identified with the right-invariant vector fields on  , so that   inherits a Lie bracket;
  • the anchor map is the differential   of the target map  .

Of course, a symmetric construction arises when swapping the role of the source and the target maps, and replacing right- with left-invariant vector fields; an isomorphism between the two resulting Lie algebroids will be given by the differential of the inverse map  .

The flow of a section   is the 1-parameter bisection  , defined by  , where   is the flow of the corresponding right-invariant vector field  . This allows one to defined the analogue of the exponential map for Lie groups as  .

Lie functor

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The mapping   sending a Lie groupoid to a Lie algebroid is actually part of a categorical construction. Indeed, any Lie groupoid morphism   can be differentiated to a morphism   between the associated Lie algebroids.

This construction defines a functor from the category of Lie groupoids and their morphisms to the category of Lie algebroids and their morphisms, called the Lie functor.

Structures and properties induced from groupoids to algebroids

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Let   be a Lie groupoid and   its associated Lie algebroid. Then

  • The isotropy algebras   are the Lie algebras of the isotropy groups  
  • The orbits of   coincides with the orbits of  
  •   is transitive and   is a submersion if and only if   is transitive
  • an action   of   on   induces an action   of   (called infinitesimal action), defined by  
  • a representation of   on a vector bundle   induces a representation   of   on  , defined by Moreover, there is a morphism of semirings  , which becomes an isomorphism if   is source-simply connected.
  • there is a morphism  , called Van Est morphism, from the differentiable cohomology of   with coefficients in some representation on   to the cohomology of   with coefficients in the induced representation on  . Moreover, if the  -fibres of   are homologically  -connected, then   is an isomorphism for  , and is injective for  .[11]

Examples

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  • The Lie algebroid of a Lie group   is the Lie algebra  
  • The Lie algebroid of both the pair groupoid   and the fundamental groupoid   is the tangent algebroid  
  • The Lie algebroid of the unit groupoid   is the zero algebroid  
  • The Lie algebroid of a Lie group bundle   is the Lie algebra bundle  
  • The Lie algebroid of an action groupoid   is the action algebroid  
  • The Lie algebroid of a gauge groupoid   is the Atiyah algebroid  
  • The Lie algebroid of a general linear groupoid   is the general linear algebroid  
  • The Lie algebroid of both the holonomy groupoid   and the monodromy groupoid   is the foliation algebroid  
  • The Lie algebroid of a tangent groupoid   is the tangent algebroid  , for  
  • The Lie algebroid of a jet groupoid   is the jet algebroid  , for  

Detailed example 1

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Let us describe the Lie algebroid associated to the pair groupoid  . Since the source map is  , the  -fibers are of the kind  , so that the vertical space is  . Using the unit map  , one obtain the vector bundle  .

The extension of sections   to right-invariant vector fields   is simply   and the extension of a smooth function   from   to a right-invariant function on   is  . Therefore, the bracket on   is just the Lie bracket of tangent vector fields and the anchor map is just the identity.

Detailed example 2

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Consider the (action) Lie groupoid

 

where the target map (i.e. the right action of   on  ) is

 

The  -fibre over a point   are all copies of  , so that   is the trivial vector bundle  .

Since its anchor map   is given by the differential of the target map, there are two cases for the isotropy Lie algebras, corresponding to the fibers of  :

 

This demonstrates that the isotropy over the origin is  , while everywhere else is zero.

Integration of a Lie algebroid

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Lie theorems

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A Lie algebroid is called integrable if it is isomorphic to   for some Lie groupoid  . The analogue of the classical Lie I theorem states that:[12]

if   is an integrable Lie algebroid, then there exists a unique (up to isomorphism)  -simply connected Lie groupoid   integrating  .

Similarly, a morphism   between integrable Lie algebroids is called integrable if it is the differential   for some morphism   between two integrations of   and  . The analogue of the classical Lie II theorem states that:[13]

if   is a morphism of integrable Lie algebroids, and   is  -simply connected, then there exists a unique morphism of Lie groupoids   integrating  .

In particular, by choosing as   the general linear groupoid   of a vector bundle  , it follows that any representation of an integrable Lie algebroid integrates to a representation of its  -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,[14] and the first explicit example of non-integrable Lie algebroids, coming for instance from foliation theory, appeared only several years later.[15] Despite several partial results, including a complete solution in the transitive case,[16] the general obstructions for an arbitrary Lie algebroid to be integrable have been discovered only in 2003 by Crainic and Fernandes.[17] Adopting a more general approach, one can see that every Lie algebroid integrates to a stacky Lie groupoid.[18][19]

Ševera-Weinstein groupoid

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Given any Lie algebroid  , the natural candidate for an integration is given by  , where   denotes the space of  -paths and   the relation of  -homotopy between them. This is often called the Weinstein groupoid or Ševera-Weinstein groupoid.[20][17]

Indeed, one can show that   is an  -simply connected topological groupoid, with the multiplication induced by the concatenation of paths. Moreover, if   is integrable,   admits a smooth structure such that it coincides with the unique  -simply connected Lie groupoid integrating  .

Accordingly, the only obstruction to integrability lies in the smoothness of  . This approach led to the introduction of objects called monodromy groups, associated to any Lie algebroid, and to the following fundamental result:[17]

A Lie algebroid is integrable if and only if its monodromy groups are uniformly discrete.

Such statement simplifies in the transitive case:

A transitive Lie algebroid is integrable if and only if its monodromy groups are discrete.

The results above show also that every Lie algebroid admits an integration to a local Lie groupoid (roughly speaking, a Lie groupoid where the multiplication is defined only in a neighbourhood around the identity elements).

Integrable examples

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  • Lie algebras are always integrable (by Lie III theorem)
  • Atiyah algebroids of a principal bundle are always integrable (to the gauge groupoid of that principal bundle)
  • Lie algebroids with injective anchor (hence foliation algebroids) are alway integrable (by Frobenius theorem)
  • Lie algebra bundle are always integrable[21]
  • Action Lie algebroids are always integrable (but the integration is not necessarily an action Lie groupoid)[22]
  • Any Lie subalgebroid of an integrable Lie algebroid is integrable.[12]

A non-integrable example

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Consider the Lie algebroid   associated to a closed 2-form   and the group of spherical periods associated to  , i.e. the image   of the following group homomorphism from the second homotopy group of  

 

Since   is transitive, it is integrable if and only if it is the Atyah algebroid of some principal bundle; a careful analysis shows that this happens if and only if the subgroup   is a lattice, i.e. it is discrete. An explicit example where such condition fails is given by taking   and   for   the area form. Here   turns out to be  , which is dense in  .

See also

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References

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  1. ^ a b Pradines, Jean (1967). "Théorie de Lie pour les groupoïdes dif́férentiables. Calcul différentiel dans la caté́gorie des groupoïdes infinitésimaux". C. R. Acad. Sci. Paris (in French). 264: 245–248.
  2. ^ Meinrenken, Eckhard (2021-05-08). "On the integration of transitive Lie algebroids". arXiv:2007.07120 [math.DG].
  3. ^ J. C., Herz (1953). "Pseudo-algèbres de Lie". C. R. Acad. Sci. Paris (in French). 236: 1935–1937.
  4. ^ Kosmann-Schwarzbach, Yvette; Magri, Franco (1990). "Poisson-Nijenhuis structures". Annales de l'Institut Henri Poincaré A. 53 (1): 35–81.
  5. ^ Grabowski, Janusz (2003-12-01). "Quasi-derivations and QD-algebroids". Reports on Mathematical Physics. 52 (3): 445–451. arXiv:math/0301234. Bibcode:2003RpMP...52..445G. doi:10.1016/S0034-4877(03)80041-1. ISSN 0034-4877. S2CID 119580956.
  6. ^ Jotz Lean, M.; Ortiz, C. (2014-10-01). "Foliated groupoids and infinitesimal ideal systems". Indagationes Mathematicae. 25 (5): 1019–1053. doi:10.1016/j.indag.2014.07.009. ISSN 0019-3577. S2CID 121209093.
  7. ^ Mackenzie, Kirill C. H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9781107325883. ISBN 978-0-521-49928-6.
  8. ^ Eckhard Meinrenken, Lie groupoids and Lie algebroids, Lecture notes, fall 2017
  9. ^ Evens, S; Lu, J-H; Weinstein, A (1999-12-01). "Transverse measures, the modular class and a cohomology pairing for Lie algebroids". The Quarterly Journal of Mathematics. 50 (200): 417–436. arXiv:dg-ga/9610008. doi:10.1093/qjmath/50.200.417. ISSN 0033-5606.
  10. ^ Weinstein, Alan (1997). "The modular automorphism group of a Poisson manifold". Journal of Geometry and Physics. 23 (3–4): 379–394. Bibcode:1997JGP....23..379W. doi:10.1016/S0393-0440(97)80011-3.
  11. ^ Crainic, Marius (2003-12-31). "Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes". Commentarii Mathematici Helvetici. 78 (4): 681–721. arXiv:math/0008064. doi:10.1007/s00014-001-0766-9. ISSN 0010-2571. S2CID 6392715.
  12. ^ a b Moerdijk, Ieke; Mrcun, Janez (2002). "On integrability of infinitesimal actions" (PDF). American Journal of Mathematics. 124 (3): 567–593. arXiv:math/0006042. doi:10.1353/ajm.2002.0019. ISSN 1080-6377. S2CID 53622428.
  13. ^ Mackenzie, Kirill; Xu, Ping (2000-05-01). "Integration of Lie bialgebroids". Topology. 39 (3): 445–467. arXiv:dg-ga/9712012. doi:10.1016/S0040-9383(98)00069-X. ISSN 0040-9383. S2CID 119594174.
  14. ^ Pradines, Jean (1968). "Troisieme théorème de Lie pour les groupoides différentiables". Comptes Rendus de l'Académie des Sciences, Série A (in French). 267: 21–23.
  15. ^ Almeida, Rui; Molino, Pierre (1985). "Suites d'Atiyah et feuilletages transversalement complets". Comptes Rendus de l'Académie des Sciences, Série I (in French). 300: 13–15.
  16. ^ Mackenzie, K. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511661839. ISBN 978-0-521-34882-9.
  17. ^ a b c Crainic, Marius; Fernandes, Rui L. (2003). "Integrability of Lie brackets". Ann. of Math. 2. 157 (2): 575–620. arXiv:math/0105033. doi:10.4007/annals.2003.157.575. S2CID 6992408.
  18. ^ Hsian-Hua Tseng; Chenchang Zhu (2006). "Integrating Lie algebroids via stacks". Compositio Mathematica. 142 (1): 251–270. arXiv:math/0405003. doi:10.1112/S0010437X05001752. S2CID 119572919.
  19. ^ Chenchang Zhu (2006). "Lie II theorem for Lie algebroids via stacky Lie groupoids". arXiv:math/0701024.
  20. ^ Ševera, Pavol (2005). "Some title containing the words "homotopy" and "symplectic", e.g. this one" (PDF). Travaux mathématiques. Proceedings of the 4th Conference on Poisson Geometry: June 7-11, 2004. 16. Luxembourg: University of Luxembourg: 121–137. ISBN 978-2-87971-253-6.
  21. ^ Douady, Adrien; Lazard, Michel (1966-06-01). "Espaces fibrés en algèbres de Lie et en groupes". Inventiones Mathematicae (in French). 1 (2): 133–151. Bibcode:1966InMat...1..133D. doi:10.1007/BF01389725. ISSN 1432-1297. S2CID 121480154.
  22. ^ Dazord, Pierre (1997-01-01). "Groupoïde d'holonomie et géométrie globale". Comptes Rendus de l'Académie des Sciences, Série I. 324 (1): 77–80. doi:10.1016/S0764-4442(97)80107-3. ISSN 0764-4442.

Books and lecture notes

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  • Alan Weinstein, Groupoids: unifying internal and external symmetry, AMS Notices, 43 (1996), 744-752. Also available at arXiv:math/9602220.
  • Kirill Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge U. Press, 1987.
  • Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge U. Press, 2005.
  • Marius Crainic, Rui Loja Fernandes, Lectures on Integrability of Lie Brackets, Geometry&Topology Monographs 17 (2011) 1–107, available at arXiv:math/0611259.
  • Eckhard Meinrenken, Lecture notes on Lie groupoids and Lie algebroids, available at http://www.math.toronto.edu/mein/teaching/MAT1341_LieGroupoids/Groupoids.pdf.
  • Ieke Moerdijk, Janez Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge U. Press, 2010.