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 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 . 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".
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).[Eckhard Meinrenken, Lie groupoids and Lie algebroids, Lecture notes, fall 2017]
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 .
-
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 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 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
-
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 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 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
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 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 form 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:
where the target map (i.e. the right action of
U(1) on
\mathbb{R}^2) is
((x,y), e^{i\theta}) \mapsto \begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}.
The
s-fibre over a point
p = (x,y) are all copies of
U(1), so that
u^*(T^s(\mathbb{R}^2\times U(1))) is the trivial vector bundle
\mathbb{R}^2 \times U(1) \to \mathbb{R}^2.
Since its anchor map \rho: \mathbb{R}^2 \times U(1) \to T\mathbb{R}^2 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(\mathbb{R}^2\times U(1)):
\begin{align}
t^{-1}(0) \cong & U(1) \\
t^{-1}(p) \cong & \{ (a,u) \in \mathbb{R}^2\times U(1) : ua = p \}
\end{align}
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
\mathrm{Lie}(G) for some Lie groupoid
G \rightrightarrows M. 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
F: A_1 \to A_2 between integrable Lie algebroids is called
integrable if it is the differential
F = d\phi_{ \mid A} for some morphism
\phi: G_1 \to G_2 between two integrations of
A_1 and
A_2. The analogue of the classical
Lie II theorem states that:
if F: \mathrm{Lie}(G_1) \to \mathrm{Lie}(G_2) is a morphism of integrable Lie algebroids, and G_1 is s-simply connected, then there exists a unique morphism of Lie groupoids \phi: G_1 \to G_2 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 Marius Crainic and Fernandes. Adopting a more general approach, one can see that every Lie algebroid integrates to a Algebraic stack Lie groupoid.
Ševera-Weinstein groupoid
Given any Lie algebroid
A, the natural candidate for an integration is given by
G(A):= P(A)/\sim, 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
-
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 always 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
A_\omega = TM \times \mathbb{R} \to M associated to a closed 2-form
\omega \in \Omega^2(M) and the
group of spherical periods associated to
\omega, i.e. the image
\Lambda:= \mathrm{Im}(\Phi) \subseteq \mathbb{R} of the following group homomorphism from the second
homotopy group of
M
\Phi: \pi_2(M) \to \mathbb{R}: \quad f \mapsto \int_{S^2} f^*\omega.
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 \Lambda \subseteq \mathbb{R} is a lattice, i.e. it is discrete. An explicit example where such condition fails is given by taking M = S^2 \times S^2 and \omega = \mathrm{pr}_1^* \sigma + \sqrt2 \mathrm{pr}_2^* \sigma \in \Omega^2(M) for \sigma \in \Omega^2(S^2) the area form. Here \Lambda turns out to be \mathbb{Z}+\sqrt2 \mathbb{Z}, which is Dense set in \mathbb{R}.
See also
Books and lecture notes
-
Alan Weinstein, Groupoids: unifying internal and external symmetry, AMS Notices, 43 (1996), 744–752. Also available at .
-
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 .
-
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.