In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections \Gamma(A) 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 such that the anchor and the bracket satisfy the following Leibniz rule: where. Here \rho(X)f is the image of f via the derivation \rho(X), i.e. the Lie derivative of f along the vector field \rho(X). The notation denotes the (point-wise) product between the function \rho(X)f and the vector field Y. One often writes A \to M 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 all. The property that \rho 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 A'\to M' of the restriction such that takes values in TM' and is a Lie subalgebra of \Gamma(A). Clearly, A'\to M' 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 M' = M. 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 I is forced to be inside the isotropy bundle \ker(\rho). 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 M 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

Examples from differential geometry

Constructions from other Lie algebroids

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 A 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: The prototypical examples of transitive Lie algebroids are Atiyah algebroids. For instance: In analogy to Atiyah algebroids, an arbitrary transitive Lie algebroid is also called abstract Atiyah sequence, and its isotropy algebra bundle \ker(\rho) is also called adjoint bundle. However, it is important to stress that not every transitive Lie algebroid is an Atiyah algebroid. For instance: 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 For instance:

Further related concepts

Actions

An action of a Lie algebroid A \to M on a manifold P along a smooth map ' consists of a Lie algebra morphismsuch that, for every ',Of course, when , both the anchor A \to {} and the map P \to {} 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 A \to M, an A-connection on a vector bundle E \to M consists of an \mathbb{R}-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 \nabla is the '-bilinear mapand \nabla is called flat' if .'' Of course, when A=TM, 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 A \to M is a vector bundle E \to M together with a flat A-connection \nabla. Equivalently, a representation (E, \nabla) is a Lie algebroid morphism . The set of isomorphism classes of representations of a Lie algebroid A \to M has a natural structure of semiring, with direct sums and tensor products of vector bundles. Examples include the following: 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 A \to M and a representation (E, \nabla). Denoting by ' the space of n-differential forms on A with values in the vector bundle E, one can define a differential ' with the following Koszul-like formula:Thanks to the flatness of \nabla, ' becomes a cochain complex and its cohomology, denoted by H^*(A,E), is called the Lie algebroid cohomology' of A with coefficients in the representation (E, \nabla)''. This general definition recovers well-known cohomology theories:

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: 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\to G. 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

Examples

Detailed example 1

Let us describe the Lie algebroid associated to the pair groupoid. Since the source map is, the s-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 f from M to a right-invariant function on G is. Therefore, the bracket on A 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 U(1) on ) is The s-fibre over a point p = (x,y) are all copies of U(1), 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 U(1), 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 A is an integrable Lie algebroid, then there exists a unique (up to isomorphism) s-simply connected Lie groupoid G integrating A."Similarly, a morphism ' between integrable Lie algebroids is called integrable if it is the differential ' for some morphism between two integrations of A_1 and A_2. The analogue of the classical Lie II theorem states that: "if ' is a morphism of integrable Lie algebroids, and G_1 is s-simply connected, then there exists a unique morphism of Lie groupoids ' integrating F."In particular, by choosing as G_2 the general linear groupoid GL(E) of a vector bundle E, it follows that any representation of an integrable Lie algebroid integrates to a representation of its s-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.

Ševera-Weinstein groupoid

Given any Lie algebroid A, the natural candidate for an integration is given by , where P(A) denotes the space of A-paths and \sim the relation of A-homotopy between them. This is often called the Weinstein groupoid or Ševera-Weinstein groupoid. Indeed, one can show that G(A) is an s-simply connected topological groupoid, with the multiplication induced by the concatenation of paths. Moreover, if A is integrable, G(A) admits a smooth structure such that it coincides with the unique s-simply connected Lie groupoid integrating A. Accordingly, the only obstruction to integrability lies in the smoothness of G(A). 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

A non-integrable example

Consider the Lie algebroid associated to a closed 2-form and the group of spherical periods associated to \omega, i.e. the image of the following group homomorphism from the second homotopy group of M Since A_\omega 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 \Lambda turns out to be, which is dense in \mathbb{R}.

Books and lecture notes