Lie Algebroid - Examples

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, 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.