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 $\mathfrak{X}(M)$ 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)