Lie algebroids

0.1 Topic on Lie algebroids

This is a topic entry on Lie algebroids that focuses on their quantum applications and extensionsPlanetmathPlanetmath of current algebraic theories.

Lie algebroids generalize Lie algebras, and in certain quantum systems they represent extended quantum (algebroid) symmetriesPlanetmathPlanetmath. One can think of a Lie algebroid as generalizing the idea of a tangent bundle where the tangent spaceMathworldPlanetmath at a point is effectively the equivalence classMathworldPlanetmathPlanetmath of curves meeting at that point (thus suggesting a groupoidPlanetmathPlanetmathPlanetmathPlanetmath approach), as well as serving as a site on which to study infinitesimalMathworldPlanetmathPlanetmath geometry (see, for example, ref. [Mackenzie2005]). The formal definition of a Lie algebroid is presented next.

Definition 0.1 Let M be a manifoldMathworldPlanetmath and let 𝔛(M) denote the set of vector fields on M. Then, a Lie algebroid over M consists of a vector bundleMathworldPlanetmath EM, equipped with a Lie bracket [,] on the space of sections γ(E), and a bundle mapMathworldPlanetmath Υ:ETM, usually called the anchor. Furthermore, there is an induced map Υ:γ(E)𝔛(M), which is required to be a map of Lie algebras, such that given sectionsPlanetmathPlanetmathPlanetmath α,βγ(E) and a differentiable function f, the following Leibniz rulePlanetmathPlanetmath is satisfied :

[α,fβ]=f[α,β]+(Υ(α))β. (0.1)
Example 0.1.

A typical example of a Lie algebroid is obtained when M is a Poisson manifold and E=T*M, that is E is the cotangent bundle of M.

Now suppose we have a Lie groupoid 𝖦:

r,s: (0.2)