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. 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 module 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 .
- 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, but every Lie algebroid gives a stacky Lie groupoid.
- 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.
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.
- 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
- 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