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. 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:
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 functionf 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.
