# Lie algebroids

## 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 $M$ be a manifold^{} and let $\U0001d51b(M)$ denote the set of vector fields on $M$. Then, a
*Lie algebroid* over $M$ consists of a *vector bundle ^{} $E\mathrm{\u27f6}M$,
equipped with a Lie bracket $\mathrm{[}\mathrm{,}\mathrm{]}$ on the space of sections $\gamma \mathit{}\mathrm{(}E\mathrm{)}$,
and a bundle map^{} $\mathrm{{\rm Y}}\mathrm{:}E\mathrm{\u27f6}T\mathit{}M$*, usually called the

*anchor*. Furthermore, there is an induced map $\mathrm{{\rm Y}}:\gamma (E)\u27f6\U0001d51b(M)$, 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 :

$$[\alpha ,f\beta ]=f[\alpha ,\beta ]+(\mathrm{{\rm Y}}(\alpha ))\beta .$$ | (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 $\U0001d5a6$:

$$r,s:$$ | (0.2) |