Lie Algebroids

Lie algebroids are vector bundles with a bunch more data.

 called the anchor map.
and a bracket on sections.

There is also a Leibniz identity with the bracket and anchor

Some examples:
1. Take M to be a point, E to be a regular Lie algebra lying on top. The anchor map is stupid in this case, and the bracket is just the bracket of the Lie algebra.
2. E is the tangent bundle itself with the anchor map being the identity. The bracket is the Lie bracket of vector fields.
3. The cotangent bundle of a Poisson manifold. The anchor map is given as
  and the bracket is   

From a Lie algebroid you get the supermanifold E[1] by viewing the fiber coordinates in degree 1. (Remember a previous post where we caution that this is not functorial so there is actually a subtlety between vector bundles and supermanifolds even though we can go from one to another.)



is the sheaf of smooth functions.

You get a derivation on this space. Call it . It squares to 0 making this space a Q manifold.

We also get an action of   when we view  for the appropriate G groupoid. Infinetesimal generators of this are v which reads off the degree and .

If this supermanifold happens to have a symplectic structure invariant under our vector fields above, we can ask what is the degree of this symplectic form.

This is where we come to a theorem:
Symplectic Q manifolds of degree 1 are in bijection with Poisson manifolds.

The next step is natural. What about degree 2 symplectic Q manifolds? The answer needs Courant algebroids.

This comes from a talk given at GRASP on 1/27/12.