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]
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 and is the derivative of along 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 propertiesedit
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 idealsedit
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]
Morphismsedit
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.
Examplesedit
Trivial and extreme casesedit
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 geometryedit
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.
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 algebroidsedit
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 surjectivesubmersion), 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 algebroidsedit
Totally intransitive Lie algebroidsedit
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 algebroidsedit
A Lie algebroid is called transitive if the anchor map is surjective. As a consequence:
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 algebroidsedit
A Lie algebroid is called regular if the anchor map is of constant rank. As a consequence
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.
Further related conceptsedit
Actionsedit
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.
Connectionsedit
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
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 cohomologyedit
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 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 correspondenceedit
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 functoredit
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 algebroidsedit
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]
Examplesedit
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 1edit
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 2edit
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 algebroidedit
Lie theoremsedit
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]
Weinstein groupoidedit
Given any Lie algebroid , the natural candidate for an integration is given by the Weinstein groupoid, where denotes the space of -paths and the relation of -homotopy between them. 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 examplesedit
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)
Action Lie algebroids are always integrable (but the integration is not necessarily an action Lie groupoid)[21]
Any Lie subalgebroid of an integrable Lie algebroid is integrable.[12]
A non-integrable exampleedit
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
March 14, 2024
algebroid, mathematics, vector, bundle, displaystyle, rightarrow, together, with, bracket, space, sections, displaystyle, gamma, vector, bundle, morphism, displaystyle, rightarrow, satisfying, leibniz, rule, thus, thought, many, object, generalisation, algebra. In mathematics a Lie algebroid is a vector bundle A M displaystyle A rightarrow M together with a Lie bracket on its space of sections G A displaystyle Gamma A and a vector bundle morphism r A T M displaystyle rho A rightarrow TM 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 Contents 1 Definition and basic concepts 1 1 First properties 1 2 Subalgebroids and ideals 1 3 Morphisms 2 Examples 2 1 Trivial and extreme cases 2 2 Examples from differential geometry 2 3 Constructions from other Lie algebroids 3 Important classes of Lie algebroids 3 1 Totally intransitive Lie algebroids 3 2 Transitive Lie algebroids 3 3 Regular Lie algebroids 4 Further related concepts 4 1 Actions 4 2 Connections 4 3 Representations 4 4 Lie algebroid cohomology 5 Lie groupoid Lie algebroid correspondence 5 1 Lie functor 5 2 Structures and properties induced from groupoids to algebroids 5 3 Examples 5 4 Detailed example 1 5 5 Detailed example 2 6 Integration of a Lie algebroid 6 1 Lie theorems 6 2 Weinstein groupoid 6 3 Integrable examples 6 4 A non integrable example 7 See also 8 References 9 Books and lecture notesDefinition and basic concepts editA Lie algebroid is a triple A r displaystyle A cdot cdot rho nbsp consisting of a vector bundle A displaystyle A nbsp over a manifold M displaystyle M nbsp a Lie bracket displaystyle cdot cdot nbsp on its space of sections G A displaystyle Gamma A nbsp a morphism of vector bundles r A T M displaystyle rho A rightarrow TM nbsp called the anchor where T M displaystyle TM nbsp is the tangent bundle of M displaystyle M nbsp such that the anchor and the bracket satisfy the following Leibniz rule X f Y r X f Y f X Y displaystyle X fY rho X f cdot Y f X Y nbsp where X Y G A f C M displaystyle X Y in Gamma A f in C infty M nbsp and r X f displaystyle rho X f nbsp is the derivative of f displaystyle f nbsp along the vector field r X displaystyle rho X nbsp One often writes A M displaystyle A to M nbsp when the bracket and the anchor are clear from the context some authors denote Lie algebroids by A M displaystyle A Rightarrow M nbsp suggesting a limit of a Lie groupoids when the arrows denoting source and target become infinitesimally close 2 First properties edit It follows from the definition that for every x M displaystyle x in M nbsp the kernel g x A ker r x displaystyle mathfrak g x A ker rho x nbsp is a Lie algebra called the isotropy Lie algebra at x displaystyle x nbsp the kernel g A ker r displaystyle mathfrak g A ker rho nbsp is a not necessarily locally trivial bundle of Lie algebras called the isotropy Lie algebra bundle the image I m r T M displaystyle mathrm Im rho subseteq TM nbsp is a singular distribution which is integrable i e its admits maximal immersed submanifolds O M displaystyle mathcal O subseteq M nbsp called the orbits satisfying I m r x T x O displaystyle mathrm Im rho x T x mathcal O nbsp for every x O displaystyle x in mathcal O nbsp Equivalently orbits can be explicitly described as the sets of points which are joined by A paths i e pairs a I A g I M displaystyle a I to A gamma I to M nbsp of paths in A displaystyle A nbsp and in M displaystyle M nbsp such that a t A g t displaystyle a t in A gamma t nbsp and r a t g t displaystyle rho a t gamma t nbsp the anchor map r displaystyle rho nbsp descends to a map between sections r G A X M displaystyle rho Gamma A rightarrow mathfrak X M nbsp which is a Lie algebra morphism i e r X Y r X r Y displaystyle rho X Y rho X rho Y nbsp for all X Y G A displaystyle X Y in Gamma A nbsp The property that r displaystyle rho nbsp 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 edit A Lie subalgebroid of a Lie algebroid A r displaystyle A cdot cdot rho nbsp is a vector subbundle A M displaystyle A to M nbsp of the restriction A M M displaystyle A mid M to M nbsp such that r A displaystyle rho mid A nbsp takes values in T M displaystyle TM nbsp and G A A a G A a M G A displaystyle Gamma A A alpha in Gamma A mid alpha mid M in Gamma A nbsp is a Lie subalgebra of G A displaystyle Gamma A nbsp Clearly A M displaystyle A to M nbsp admits a unique Lie algebroid structure such that G A A G A displaystyle Gamma A A to Gamma A nbsp is a Lie algebra morphism With the language introduced below the inclusion A A displaystyle A hookrightarrow A nbsp is a Lie algebroid morphism A Lie subalgebroid is called wide if M M displaystyle M M nbsp In analogy to the standard definition for Lie algebra an ideal of a Lie algebroid is wide Lie subalgebroid I A displaystyle I subseteq A nbsp such that G I G A displaystyle Gamma I subseteq Gamma A nbsp is a Lie ideal Such notion proved to be very restrictive since I displaystyle I nbsp is forced to be inside the isotropy bundle ker r displaystyle ker rho nbsp For this reason the more flexible notion of infinitesimal ideal system has been introduced 6 Morphisms edit A Lie algebroid morphism between two Lie algebroids A 1 A 1 r 1 displaystyle A 1 cdot cdot A 1 rho 1 nbsp and A 2 A 2 r 2 displaystyle A 2 cdot cdot A 2 rho 2 nbsp with the same base M displaystyle M nbsp is a vector bundle morphism ϕ A 1 A 2 displaystyle phi A 1 to A 2 nbsp which is compatible with the Lie brackets i e ϕ a b A 1 ϕ a ϕ b A 2 displaystyle phi alpha beta A 1 phi alpha phi beta A 2 nbsp for every a b G A 1 displaystyle alpha beta in Gamma A 1 nbsp and with the anchors i e r 2 ϕ r 1 displaystyle rho 2 circ phi rho 1 nbsp 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 ϕ A 1 A 2 displaystyle phi A 1 to A 2 nbsp to be a subalgebroid of the direct product A 1 A 2 displaystyle A 1 times A 2 nbsp introduced below 8 Lie algebroids together with their morphisms form a category Examples editTrivial and extreme cases edit Given any manifold M displaystyle M nbsp its tangent Lie algebroid is the tangent bundle T M M displaystyle TM to M nbsp together with the Lie bracket of vector fields and the identity of T M displaystyle TM nbsp as an anchor Given any manifold M displaystyle M nbsp the zero vector bundle M 0 M displaystyle M times 0 to M nbsp is a Lie algebroid with zero bracket and anchor Lie algebroids A displaystyle A to nbsp 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 edit Given a foliation F displaystyle mathcal F nbsp on M displaystyle M nbsp its foliation algebroid is the associated involutive subbundle F T M displaystyle mathcal F subseteq TM nbsp with brackets and anchor induced from the tangent Lie algebroid Given the action of a Lie algebra g displaystyle mathfrak g nbsp on a manifold M displaystyle M nbsp its action algebroid is the trivial vector bundle g M M displaystyle mathfrak g times M to M nbsp with anchor given by the Lie algebra action and brackets uniquely determined by the bracket of g displaystyle mathfrak g nbsp on constant sections M g displaystyle M to mathfrak g nbsp and by the Leibniz identity Given a principal G bundle P displaystyle P nbsp over a manifold M displaystyle M nbsp its Atiyah algebroid is the Lie algebroid A T P G displaystyle A TP G nbsp fitting in the following short exact sequence 0 ker r T P G r T M 0 displaystyle 0 to ker rho to TP G xrightarrow rho TM to 0 nbsp The space of sections of the Atiyah algebroid is the Lie algebra of G displaystyle G nbsp invariant vector fields on P displaystyle P nbsp its isotropy Lie algebra bundle is isomorphic to the adjoint vector bundle P G g displaystyle P times G mathfrak g nbsp and the right splittings of the sequence above are principal connections on P displaystyle P nbsp Given a vector bundle E M displaystyle E to M nbsp its general linear algebroid denoted by g l E displaystyle mathfrak gl E nbsp or D e r E displaystyle mathrm Der E nbsp is the vector bundle whose sections are derivations of E displaystyle E nbsp i e first order differential operators G E G E displaystyle Gamma E to Gamma E nbsp admitting a vector field r D X M displaystyle rho D in mathfrak X M nbsp such that D f s f D s r D f s displaystyle D f sigma fD sigma rho D f sigma nbsp for every f C M s G E displaystyle f in mathcal C infty M sigma in Gamma E nbsp The anchor is simply the assignment D r D displaystyle D mapsto rho D nbsp and the Lie bracket is given by the commutator of differential operators Given a Poisson manifold M p displaystyle M pi nbsp its cotangent algebroid is the cotangent vector bundle A T M displaystyle A T M nbsp with Lie bracket a b L p a b L p b a d p a b displaystyle alpha beta mathcal L pi sharp alpha beta mathcal L pi sharp beta alpha d pi alpha beta nbsp and anchor map p T M T M a p a displaystyle pi sharp T M to TM alpha mapsto pi alpha cdot nbsp Given a closed 2 form w W 2 M displaystyle omega in Omega 2 M nbsp the vector bundle A w T M R M displaystyle A omega TM times mathbb R to M nbsp is a Lie algebroid with anchor the projection on the first component and Lie bracket X f Y g X Y L X g L Y f w X Y displaystyle X f Y g Big X Y mathcal L X g mathcal L Y f omega X Y Big nbsp Actually the bracket above can be defined for any 2 form w displaystyle omega nbsp but A w displaystyle A omega nbsp is a Lie algebroid if and only if w displaystyle omega nbsp is closed Constructions from other Lie algebroids edit Given any Lie algebroid A M r displaystyle A to M cdot cdot rho nbsp there is a Lie algebroid T A T M r displaystyle TA to TM cdot cdot rho nbsp called its tangent algebroid obtained by considering the tangent bundle of A displaystyle A nbsp and M displaystyle M nbsp and the differential of the anchor Given any Lie algebroid A M A r A displaystyle A to M cdot cdot A rho A nbsp there is a Lie algebroid J k A M r displaystyle J k A to M cdot cdot rho nbsp called its k jet algebroid obtained by considering the k jet bundle of A M displaystyle A to M nbsp with Lie bracket uniquely defined by j k a j k b j k a b A displaystyle j k alpha j k beta j k alpha beta A nbsp and anchor r j x k a r A a x displaystyle rho j x k alpha rho A alpha x nbsp Given two Lie algebroids A 1 M 1 displaystyle A 1 to M 1 nbsp and A 2 M 2 displaystyle A 2 to M 2 nbsp their direct product is the unique Lie algebroid A 1 A 2 M 1 M 2 displaystyle A 1 times A 2 to M 1 times M 2 nbsp with anchor a 1 a 2 r 1 a 1 r 2 a 2 T M 1 T M 2 T M 1 M 2 displaystyle alpha 1 alpha 2 mapsto rho 1 alpha 1 oplus rho 2 alpha 2 in TM 1 oplus TM 2 cong T M 1 times M 2 nbsp and such that G A 1 G A 2 G A 1 A 2 a 1 a 2 p r M 1 a 1 p r M 2 a 2 displaystyle Gamma A 1 oplus Gamma A 2 to Gamma A 1 times A 2 alpha 1 oplus alpha 2 mapsto mathrm pr M 1 alpha 1 mathrm pr M 2 alpha 2 nbsp is a Lie algebra morphism Given a Lie algebroid A M A r A displaystyle A to M cdot cdot A rho A nbsp and a map f M M displaystyle f M to M nbsp whose differential is transverse to the anchor map r A T M displaystyle rho A to TM nbsp for instance it is enough for f displaystyle f nbsp to be a surjective submersion the pullback algebroid is the unique Lie algebroid f A M displaystyle f A to M nbsp with f A T M T M A M displaystyle f A TM times TM A to M nbsp the pullback vector bundle and r f A f A T M displaystyle rho f A f A to TM nbsp the projection on the first component such that f A A displaystyle f A to A nbsp is a Lie algebroid morphism Important classes of Lie algebroids editTotally intransitive Lie algebroids edit A Lie algebroid is called totally intransitive if the anchor map r A T M displaystyle rho A to TM nbsp 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 A displaystyle A nbsp is totally intransitive it must coincide with its isotropy Lie algebra bundle Transitive Lie algebroids edit A Lie algebroid is called transitive if the anchor map r A T M displaystyle rho A to TM nbsp is surjective As a consequence there is a short exact sequence0 ker r A r T M 0 displaystyle 0 to ker rho to A xrightarrow rho TM to 0 nbsp right splitting of r displaystyle rho nbsp defines a principal bundle connections on ker r displaystyle ker rho nbsp the isotropy bundle ker r displaystyle ker rho nbsp is locally trivial as bundle of Lie algebras the pullback of A displaystyle A nbsp exist for every f M M displaystyle f M to M nbsp The prototypical examples of transitive Lie algebroids are Atiyah algebroids For instance tangent algebroids T M displaystyle TM nbsp are trivially transitive indeed they are Atiyah algebroid of the principal e displaystyle e nbsp bundle M M displaystyle M to M nbsp Lie algebras g displaystyle mathfrak g nbsp are trivially transitive indeed they are Atiyah algebroid of the principal G displaystyle G nbsp bundle G displaystyle G to nbsp for G displaystyle G nbsp an integration of g displaystyle mathfrak g nbsp general linear algebroids g l E displaystyle mathfrak gl E nbsp are transitive indeed they are Atiyah algebroids of the frame bundle F r E M displaystyle Fr E to M nbsp In analogy to Atiyah algebroids an arbitrary transitive Lie algebroid is also called abstract Atiyah sequence and its isotropy algebra bundle ker r displaystyle ker rho nbsp 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 T M displaystyle T M nbsp associated to Poisson manifolds M p displaystyle M pi nbsp are transitive if and only if the Poisson structure p displaystyle pi nbsp is non degenerate Lie algebroids A w displaystyle A omega nbsp defined by closed 2 forms are transitiveThese 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 edit A Lie algebroid is called regular if the anchor map r A T M displaystyle rho A to TM nbsp is of constant rank As a consequence the image of r displaystyle rho nbsp defines a regular foliation on M displaystyle M nbsp the restriction of A displaystyle A nbsp over each leaf O M displaystyle mathcal O subseteq M nbsp 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 T M displaystyle T M nbsp associated to Poisson manifolds M p displaystyle M pi nbsp are regular if and only if the Poisson structure p displaystyle pi nbsp is regular Further related concepts editActions edit An action of a Lie algebroid A M displaystyle A to M nbsp on a manifold P along a smooth map m P M displaystyle mu P to M nbsp consists of a Lie algebra morphisma G A X P displaystyle a Gamma A to mathfrak X P nbsp such that for every p P X G A f C M displaystyle p in P X in Gamma A f in mathcal C infty M nbsp d p m a X p r m p X m p a f X f m a X displaystyle d p mu a X p rho mu p X mu p quad a f cdot X f circ mu cdot a X nbsp Of course when A g displaystyle A mathfrak g nbsp both the anchor A displaystyle A to nbsp and the map P displaystyle P to nbsp must be trivial therefore both conditions are empty and we recover the standard notion of action of a Lie algebra on a manifold Connections edit Given a Lie algebroid A M displaystyle A to M nbsp an A connection on a vector bundle E M displaystyle E to M nbsp consists of an R displaystyle mathbb R nbsp bilinear map G A G E G E a s a s displaystyle nabla Gamma A times Gamma E to Gamma E quad alpha s mapsto nabla alpha s nbsp which is C M displaystyle mathcal C infty M nbsp linear in the first factor and satisfies the following Leibniz rule a f s f a s L r a f s displaystyle nabla alpha fs f nabla alpha s mathcal L rho alpha f s nbsp for every a G A s G E f C M displaystyle alpha in Gamma A s in Gamma E f in mathcal C infty M nbsp where L r a displaystyle mathcal L rho alpha nbsp denotes the Lie derivative with respect to the vector field r a displaystyle rho alpha nbsp The curvature of an A connection displaystyle nabla nbsp is the C M displaystyle mathcal C infty M nbsp bilinear mapR G A G A H o m E E a b a b b a a b displaystyle R nabla Gamma A times Gamma A to mathrm Hom E E quad alpha beta mapsto nabla alpha nabla beta nabla beta nabla alpha nabla alpha beta nbsp and displaystyle nabla nbsp is called flat if R 0 displaystyle R nabla 0 nbsp Of course when A T M displaystyle A TM nbsp we recover the standard notion of connection on a vector bundle as well as those of curvature and flatness Representations edit A representation of a Lie algebroid A M displaystyle A to M nbsp is a vector bundle E M displaystyle E to M nbsp together with a flat A connection displaystyle nabla nbsp Equivalently a representation E displaystyle E nabla nbsp is a Lie algebroid morphism A g l E displaystyle A to mathfrak gl E nbsp The set R e p A displaystyle mathrm Rep A nbsp of isomorphism classes of representations of a Lie algebroid A M displaystyle A to M nbsp has a natural structure of semiring with direct sums and tensor products of vector bundles Examples include the following When A g displaystyle A mathfrak g nbsp an A displaystyle A nbsp connection simplifies to a linear map g g l V displaystyle mathfrak g to mathfrak gl V nbsp and the flatness condition makes it into a Lie algebra morphism therefore we recover the standard notion of representation of a Lie algebra When A g M M displaystyle A mathfrak g times M to M nbsp and V displaystyle V nbsp is a representation the Lie algebra g displaystyle mathfrak g nbsp the trivial vector bundle V M M displaystyle V times M to M nbsp is automatically a representation of A displaystyle A nbsp Representations of the tangent algebroid A T M displaystyle A TM nbsp are vector bundles endowed with flat connections Every Lie algebroid A M displaystyle A to M nbsp has a natural representation on the line bundle Q A t o p A t o p T M M displaystyle Q A wedge top A otimes wedge top T M to M nbsp i e the tensor product between the determinant line bundles of A displaystyle A nbsp and of T M displaystyle T M nbsp One can associate a cohomology class in H 1 A Q A displaystyle H 1 A Q A nbsp see below known as the modular class of the Lie algebroid 9 For the cotangent algebroid T M M displaystyle T M to M nbsp associated to a Poisson manifold M p displaystyle M pi nbsp one recovers the modular class of p displaystyle pi nbsp 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 edit Consider a Lie algebroid A M displaystyle A to M nbsp and a representation E displaystyle E nabla nbsp Denoting by W n A E G n A E displaystyle Omega n A E Gamma wedge n A otimes E nbsp the space of n displaystyle n nbsp differential forms on A displaystyle A nbsp with values in the vector bundle E displaystyle E nbsp one can define a differential d n W n A E W n 1 A E displaystyle d n Omega n A E to Omega n 1 A E nbsp with the following Koszul like formula d w a 0 a n i 1 n 1 i a i w a 0 a i a n i lt j n 1 i j 1 w a i a j a 0 a i a j a n displaystyle d omega alpha 0 ldots alpha n sum i 1 n 1 i nabla alpha i big omega alpha 0 ldots widehat alpha i ldots alpha n big sum i lt j n 1 i j 1 omega alpha i alpha j alpha 0 ldots widehat alpha i ldots widehat alpha j ldots alpha n nbsp Thanks to the flatness of displaystyle nabla nbsp W n A E d n displaystyle Omega n A E d n nbsp becomes a cochain complex and its cohomology denoted by H A E displaystyle H A E nbsp is called the Lie algebroid cohomology of A displaystyle A nbsp with coefficients in the representation E displaystyle E nabla nbsp This general definition recovers well known cohomology theories The cohomology of a Lie algebroid g displaystyle mathfrak g to nbsp coincides with the Chevalley Eilenberg cohomology of g displaystyle mathfrak g nbsp as a Lie algebra The cohomology of a tangent Lie algebroid T M M displaystyle TM to M nbsp coincides with the de Rham cohomology of M displaystyle M nbsp The cohomology of a foliation Lie algebroid F M displaystyle mathcal F to M nbsp coincides with the leafwise cohomology of the foliation F displaystyle mathcal F nbsp The cohomology of the cotangent Lie algebroid T M displaystyle T M nbsp associated to a Poisson structure p displaystyle pi nbsp coincides with the Poisson cohomology of p displaystyle pi nbsp Lie groupoid Lie algebroid correspondence editThe standard construction which associates a Lie algebra to a Lie group generalises to this setting to every Lie groupoid G M displaystyle G rightrightarrows M nbsp one can canonically associate a Lie algebroid L i e G displaystyle mathrm Lie G nbsp defined as follows the vector bundle is L i e G A u T s G displaystyle mathrm Lie G A u T s G nbsp where T s G T G displaystyle T s G subseteq TG nbsp is the vertical bundle of the source fibre s G M displaystyle s G to M nbsp and u M G displaystyle u M to G nbsp is the groupoid unit map the sections of A displaystyle A nbsp are identified with the right invariant vector fields on G displaystyle G nbsp so that G A displaystyle Gamma A nbsp inherits a Lie bracket the anchor map is the differential r d t A A T M displaystyle rho dt mid A A to TM nbsp of the target map t G M displaystyle t G to M nbsp 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 i G G displaystyle i G to G nbsp The flow of a section a G A displaystyle alpha in Gamma A nbsp is the 1 parameter bisection ϕ a ϵ B i s G displaystyle phi alpha epsilon in mathrm Bis G nbsp defined by ϕ a ϵ x ϕ a ϵ 1 x displaystyle phi alpha epsilon x phi tilde alpha epsilon 1 x nbsp where ϕ a ϵ D i f f G displaystyle phi tilde alpha epsilon in mathrm Diff G nbsp is the flow of the corresponding right invariant vector field a X G displaystyle tilde alpha in mathfrak X G nbsp This allows one to defined the analogue of the exponential map for Lie groups as exp G A B i s G exp a x ϕ a 1 x displaystyle exp Gamma A to mathrm Bis G exp alpha x phi alpha 1 x nbsp Lie functor edit The mapping G L i e G displaystyle G mapsto mathrm Lie G nbsp sending a Lie groupoid to a Lie algebroid is actually part of a categorical construction Indeed any Lie groupoid morphism ϕ G 1 G 2 displaystyle phi G 1 to G 2 nbsp can be differentiated to a morphismd ϕ L i e G 1 L i e G 1 L i e G 2 displaystyle d phi mid mathrm Lie G 1 mathrm Lie G 1 to mathrm Lie G 2 nbsp 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 edit Let G M displaystyle G rightrightarrows M nbsp be a Lie groupoid and A M r displaystyle A to M cdot cdot rho nbsp its associated Lie algebroid Then The isotropy algebras g x A displaystyle mathfrak g x A nbsp are the Lie algebras of the isotropy groups G x displaystyle G x nbsp The orbits of G displaystyle G nbsp coincides with the orbits of A displaystyle A nbsp G displaystyle G nbsp is transitive and s t G M M displaystyle s t G to M times M nbsp is a submersion if and only if A displaystyle A nbsp is transitive an action m G M P P displaystyle m G times M P to P nbsp of G displaystyle G nbsp on P M displaystyle P to M nbsp induces an action a G A X P displaystyle a Gamma A to mathfrak X P nbsp of A displaystyle A nbsp called infinitesimal action defined by a a p d 1 m p m p a m p d 1 m p p m a m p 0 displaystyle a alpha p d 1 mu p m cdot p alpha mu p d 1 mu p p m alpha mu p 0 nbsp a representation of G displaystyle G nbsp on a vector bundle E M displaystyle E to M nbsp induces a representation displaystyle nabla nbsp of A displaystyle A nbsp on E M displaystyle E to M nbsp defined by a s x d d ϵ ϵ 0 ϕ a ϵ x 1 s t ϕ a ϵ x displaystyle nabla alpha sigma x frac d d epsilon mid epsilon 0 Big phi alpha epsilon x Big 1 cdot sigma Big t phi alpha epsilon x Big nbsp Moreover there is a morphism of semirings R e p G R e p A displaystyle mathrm Rep G to mathrm Rep A nbsp which becomes an isomorphism if G displaystyle G nbsp is source simply connected there is a morphism V E k H d k G E H k A E displaystyle VE k H d k G E to H k A E nbsp called Van Est morphism from the differentiable cohomology of G displaystyle G nbsp with coefficients in some representation on E displaystyle E nbsp to the cohomology of A displaystyle A nbsp with coefficients in the induced representation on E displaystyle E nbsp Moreover if the s displaystyle s nbsp fibres of G displaystyle G nbsp are homologically n displaystyle n nbsp connected then V E k displaystyle VE k nbsp is an isomorphism for k n displaystyle k leq n nbsp and is injective for k n 1 displaystyle k n 1 nbsp 11 Examples edit The Lie algebroid of a Lie group G displaystyle G rightrightarrows nbsp is the Lie algebra g displaystyle mathfrak g to nbsp The Lie algebroid of both the pair groupoid M M M displaystyle M times M rightrightarrows M nbsp and the fundamental groupoid P 1 M M displaystyle Pi 1 M rightrightarrows M nbsp is the tangent algebroid T M M displaystyle TM to M nbsp The Lie algebroid of the unit groupoid u M M displaystyle u M rightrightarrows M nbsp is the zero algebroid M 0 M displaystyle M times 0 to M nbsp The Lie algebroid of a Lie group bundle G M displaystyle G rightrightarrows M nbsp is the Lie algebra bundle A M displaystyle A to M nbsp The Lie algebroid of an action groupoid G M M displaystyle G times M rightrightarrows M nbsp is the action algebroid g M M displaystyle mathfrak g times M to M nbsp The Lie algebroid of a gauge groupoid P P G M displaystyle P times P G rightrightarrows M nbsp is the Atiyah algebroid T P G M displaystyle TP G to M nbsp The Lie algebroid of a general linear groupoid G L E M displaystyle GL E rightrightarrows M nbsp is the general linear algebroid g l E M displaystyle mathfrak gl E to M nbsp The Lie algebroid of both the holonomy groupoid H o l F M displaystyle mathrm Hol mathcal F rightrightarrows M nbsp and the monodromy groupoid P 1 F M displaystyle Pi 1 mathcal F rightrightarrows M nbsp is the foliation algebroid F M displaystyle mathcal F to M nbsp The Lie algebroid of a tangent groupoid T G T M displaystyle TG rightrightarrows TM nbsp is the tangent algebroid T A T M displaystyle TA to TM nbsp for A L i e G displaystyle A mathrm Lie G nbsp The Lie algebroid of a jet groupoid J k G M displaystyle J k G rightrightarrows M nbsp is the jet algebroid J k A M displaystyle J k A to M nbsp for A L i e G displaystyle A mathrm Lie G nbsp Detailed example 1 edit Let us describe the Lie algebroid associated to the pair groupoid G M M displaystyle G M times M nbsp Since the source map is s G M p q q displaystyle s G to M p q mapsto q nbsp the s displaystyle s nbsp fibers are of the kind M q displaystyle M times q nbsp so that the vertical space is T s G q M T M q T M T M displaystyle T s G bigcup q in M TM times q subset TM times TM nbsp Using the unit map u M G q q q displaystyle u M to G q mapsto q q nbsp one obtain the vector bundle A u T s G q M T q M T M displaystyle A u T s G bigcup q in M T q M TM nbsp The extension of sections X G A displaystyle X in Gamma A nbsp to right invariant vector fields X X G displaystyle tilde X in mathfrak X G nbsp is simply X p q X p 0 displaystyle tilde X p q X p oplus 0 nbsp and the extension of a smooth function f displaystyle f nbsp from M displaystyle M nbsp to a right invariant function on G displaystyle G nbsp is f p q f q displaystyle tilde f p q f q nbsp Therefore the bracket on A displaystyle A nbsp is just the Lie bracket of tangent vector fields and the anchor map is just the identity Detailed example 2 edit Consider the action Lie groupoid R 2 U 1 R 2 displaystyle mathbb R 2 times U 1 rightrightarrows mathbb R 2 nbsp where the target map i e the right action of U 1 displaystyle U 1 nbsp on R 2 displaystyle mathbb R 2 nbsp is x y e i 8 cos 8 sin 8 sin 8 cos 8 x y displaystyle x y e i theta mapsto begin bmatrix cos theta amp sin theta sin theta amp cos theta end bmatrix begin bmatrix x y end bmatrix nbsp The s displaystyle s nbsp fibre over a point p x y displaystyle p x y nbsp are all copies of U 1 displaystyle U 1 nbsp so that u T s R 2 U 1 displaystyle u T s mathbb R 2 times U 1 nbsp is the trivial vector bundle R 2 U 1 R 2 displaystyle mathbb R 2 times U 1 to mathbb R 2 nbsp Since its anchor map r R 2 U 1 T R 2 displaystyle rho mathbb R 2 times U 1 to T mathbb R 2 nbsp is given by the differential of the target map there are two cases for the isotropy Lie algebras corresponding to the fibers of T t R 2 U 1 displaystyle T t mathbb R 2 times U 1 nbsp t 1 0 U 1 t 1 p a u R 2 U 1 u a p displaystyle begin aligned t 1 0 cong amp U 1 t 1 p cong amp a u in mathbb R 2 times U 1 ua p end aligned nbsp This demonstrates that the isotropy over the origin is U 1 displaystyle U 1 nbsp while everywhere else is zero Integration of a Lie algebroid editLie theorems editA Lie algebroid is called integrable if it is isomorphic to L i e G displaystyle mathrm Lie G nbsp for some Lie groupoidG M displaystyle G rightrightarrows M nbsp The analogue of the classical Lie I theorem states that 12 if A displaystyle A nbsp is an integrable Lie algebroid then there exists a unique up to isomorphism s displaystyle s nbsp simply connected Lie groupoid G displaystyle G nbsp integrating A displaystyle A nbsp Similarly a morphism F A 1 A 2 displaystyle F A 1 to A 2 nbsp between integrable Lie algebroids is called integrable if it is the differential F d ϕ A displaystyle F d phi mid A nbsp for some morphism ϕ G 1 G 2 displaystyle phi G 1 to G 2 nbsp between two integrations of A 1 displaystyle A 1 nbsp and A 2 displaystyle A 2 nbsp The analogue of the classical Lie II theorem states that 13 if F L i e G 1 L i e G 2 displaystyle F mathrm Lie G 1 to mathrm Lie G 2 nbsp is a morphism of integrable Lie algebroids and G 1 displaystyle G 1 nbsp is s displaystyle s nbsp simply connected then there exists a unique morphism of Lie groupoids ϕ G 1 G 2 displaystyle phi G 1 to G 2 nbsp integrating F displaystyle F nbsp In particular by choosing as G 2 displaystyle G 2 nbsp the general linear groupoid G L E displaystyle GL E nbsp of a vector bundle E displaystyle E nbsp it follows that any representation of an integrable Lie algebroid integrates to a representation of its s displaystyle s nbsp 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 Weinstein groupoid edit Given any Lie algebroid A displaystyle A nbsp the natural candidate for an integration is given by the Weinstein groupoid G A P A displaystyle G A P A sim nbsp where P A displaystyle P A nbsp denotes the space of A displaystyle A nbsp paths and displaystyle sim nbsp the relation of A displaystyle A nbsp homotopy between them Indeed one can show that G A displaystyle G A nbsp is an s displaystyle s nbsp simply connected topological groupoid with the multiplication induced by the concatenation of paths Moreover if A displaystyle A nbsp is integrable G A displaystyle G A nbsp admits a smooth structure such that it coincides with the unique s displaystyle s nbsp simply connected Lie groupoid integrating A displaystyle A nbsp Accordingly the only obstruction to integrability lies in the smoothness of G A displaystyle G A nbsp 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 edit 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 20 Action Lie algebroids are always integrable but the integration is not necessarily an action Lie groupoid 21 Any Lie subalgebroid of an integrable Lie algebroid is integrable 12 A non integrable example edit Consider the Lie algebroid A w T M R M displaystyle A omega TM times mathbb R to M nbsp associated to a closed 2 form w W 2 M displaystyle omega in Omega 2 M nbsp and the group of spherical periods associated to w displaystyle omega nbsp i e the image L I m F R displaystyle Lambda mathrm Im Phi subseteq mathbb R nbsp of the following group homomorphism from the second homotopy group of M displaystyle M nbsp F p 2 M R f S 2 f w displaystyle Phi pi 2 M to mathbb R quad f mapsto int S 2 f omega nbsp Since A w displaystyle A omega nbsp 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 L R displaystyle Lambda subseteq mathbb R nbsp is a lattice i e it is discrete An explicit example where such condition fails is given by taking M S 2 S 2 displaystyle M S 2 times S 2 nbsp and w p r 1 s 2 mi mathva, wikipedia, wiki, book, books, library,