0.1 Topic on Lie algebroids
This is a topic entry on Lie algebroids that focuses on their quantum applications and extensions of current algebraic theories.
Lie algebroids generalize Lie algebras, and in certain quantum systems they represent extended quantum (algebroid) symmetries. One can think of a Lie algebroid as generalizing the idea of a tangent bundle where the tangent space at a point is effectively the equivalence class of curves meeting at that point (thus
suggesting a groupoid approach), as well as serving as a site on which to study infinitesimal geometry (see, for example, ref. [Mackenzie2005]). The formal definition of a Lie algebroid is presented next.
Definition 0.1 Let be a manifold and let denote the set of vector fields on . Then, a Lie algebroid over consists of a vector bundle , equipped with a Lie bracket on the space of sections , and a bundle map , usually called the anchor. Furthermore, there is an induced map , which is required to be a map of Lie algebras, such that given sections and a differentiable function , the following Leibniz rule is satisfied :
Now suppose we have a Lie groupoid :