Lie algebroid

Lie algebroid
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.
Definition and basic concepts
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 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".
First properties
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. Such redundancy, despite being known from an algebraic point of view already before Pradine's definition, was noticed only much later.
Subalgebroids and ideals
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.
Morphisms
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. Equivalently, one can ask that the graph of to be a subalgebroid of the direct product (introduced below).
Lie algebroids together with their morphisms form a category.
Examples
Trivial and extreme cases
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
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 bracketActually, 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
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
Totally intransitive Lie algebroids
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
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
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.
Further related concepts
Actions
An action of a Lie algebroid on a manifold P along a smooth map consists of a Lie algebra morphismsuch 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
Given a Lie algebroid , an A-connection on a vector bundle consists of an -bilinear mapwhich 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 mapand 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
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. For the cotangent algebroid associated to a Poisson manifold one recovers the modular class of .
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
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
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
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
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 byMoreover, 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 .
Examples
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
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
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
Lie theorems
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: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: 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, and the first explicit example of non-integrable Lie algebroids, coming for instance from foliation theory, appeared only several years later. Despite several partial results, including a complete solution in the transitive case, the general obstructions for an arbitrary Lie algebroid to be integrable have been discovered only in 2003 by Crainic and Fernandes. Adopting a more general approach, one can see that every Lie algebroid integrates to a stacky Lie groupoid.
Weinstein groupoid
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: 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
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
Action Lie algebroids are always integrable (but the integration is not necessarily an action Lie groupoid)
Any Lie subalgebroid of an integrable Lie algebroid is integrable.
A non-integrable example
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
R-algebroid
Lie bialgebroid
References
Books and lecture notes
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.
Lie algebras
Differential geometry
Differential topology
Differential operators
Generalizations of the derivative
Geometry processing
Vector bundles