# Lie algebroid

In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones.

## Description

Just as a Lie groupoid can be thought of as a "Lie group with many objects", a Lie algebroid is like a "Lie algebra with many objects".

More precisely, a **Lie algebroid**
is a triple consisting of a vector bundle over a manifold , together with a Lie bracket on its space of sections and a morphism of vector bundles called the **anchor**. Here is the tangent bundle of . The anchor and the bracket are to satisfy the Leibniz rule:

where and is the derivative of along the vector field . It follows that

for all .

## Examples

- Every Lie algebra is a Lie algebroid over the one point manifold.
- The tangent bundle of a manifold is a Lie algebroid for the Lie bracket of vector fields and the identity of as an anchor.
- Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.
- Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.
- To every Lie groupoid is associated a Lie algebroid, generalizing how a Lie algebra is associated to a Lie group (see also below). For example, the Lie algebroid comes from the pair groupoid whose objects are , with one isomorphism between each pair of objects. Unfortunately, going back from a Lie algebroid to a Lie groupoid is not always possible,[1] but every Lie algebroid gives a stacky Lie groupoid.[2][3]
- Given the action of a Lie algebra g on a manifold M, the set of g -invariant vector fields on M is a Lie algebroid over the space of orbits of the action.
- The Atiyah algebroid of a principal
*G*-bundle*P*over a manifold*M*is a Lie algebroid with short exact sequence:

- The space of sections of the Atiyah algebroid is the Lie algebra of
*G*-invariant vector fields on*P*.

- A Poisson Lie algebroid is associated to a Poisson manifold by taking E to be the cotangent bundle. The anchor map is given by the Poisson bivector. This can be seen in a Lie bialgebroid.

## Lie algebroid associated to a Lie groupoid

To describe the construction let us fix some notation. *G* is the space of morphisms of the Lie groupoid, *M* the space of objects, the units and the target map.

the *t*-fiber tangent space. The Lie algebroid is now the vector bundle . This inherits a bracket from *G*, because we can identify the *M*-sections into *A* with left-invariant vector fields on *G*. The anchor map then is obtained as the derivation of the source map
. Further these sections act on the smooth functions of *M* by identifying these with left-invariant functions on *G*.

As a more explicit example consider the Lie algebroid associated to the pair groupoid . The target map is and the units . The *t*-fibers are and therefore . So the Lie algebroid is the vector bundle . The extension of sections *X* into *A* to left-invariant vector fields on *G* is simply and the extension of a smooth function *f* from *M* to a left-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.

Of course you could do an analog construction with the source map and right-invariant vector fields/ functions. However you get an isomorphic Lie algebroid, with the explicit isomorphism , where is the inverse map.

### Example

Consider the Lie groupoid

where the target map sends

Notice that there are two cases for the fibers of :

This demonstrating that there is a stabilizer of over the origin and stabilizer-free -orbits everywhere else. The tangent bundle over every is then trivial, hence the pullback is a trivial line bundle.

## See also

## References

- Marius Crainic, Rui L. Fernandes. Integrability of Lie brackets, Ann. of Math. (2), Vol. 157 (2003), no. 2, 575--620
- Hsian-Hua Tseng and Chenchang Zhu, Integrating Lie algebroids via stacks, Compositio Mathematica, Volume 142 (2006), Issue 01, pp 251-270, available as arXiv:math/0405003
- Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, available as arXiv:math/0701024

## External links

- Alan Weinstein, Groupoids: unifying internal and external symmetry,
*AMS Notices*,**43**(1996), 744-752. Also available as 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 - Charles-Michel Marle,
*Differential calculus on a Lie algebroid and Poisson manifolds*(2002). Also available in arXiv:0804.2451