# Lie algebroids

## 0.1 Topic on Lie algebroids

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 $M$ be a manifold  and let $\mathfrak{X}(M)$ denote the set of vector fields on $M$. Then, a Lie algebroid over $M$ consists of a vector bundle  $E{\longrightarrow}M$, equipped with a Lie bracket $[~{},~{}]$ on the space of sections $\gamma(E)$, and a bundle map  $\Upsilon:E{\longrightarrow}TM$, usually called the anchor. Furthermore, there is an induced map $\Upsilon:\gamma(E){\longrightarrow}\mathfrak{X}(M)$, which is required to be a map of Lie algebras, such that given sections   $\alpha,\beta\in\gamma(E)$ and a differentiable function $f$, the following Leibniz rule  is satisfied :

 $[\alpha,f\beta]=f[\alpha,\beta]+(\Upsilon(\alpha))\beta~{}.$ (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 $\mathsf{G}$:

 $r,s~{}:~{}\hbox{}$ (0.2)