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.

Contents

DescriptionEdit

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  .

ExamplesEdit

  • 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 groupoidEdit

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.

ExampleEdit

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 alsoEdit

ReferencesEdit

  1. ^ Marius Crainic, Rui L. Fernandes. Integrability of Lie brackets, Ann. of Math. (2), Vol. 157 (2003), no. 2, 575--620
  2. ^ 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
  3. ^ Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, available as arXiv:math/0701024

External linksEdit

  • 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