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. [1]). 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 $E \lra M$ , equipped with a Lie bracket $[~,~]$ on the space of sections $\gamma(E)$ , and a bundle map $\Upsilon : E \lra TM$ , usually called the anchor. Furthermore, there is an induced map $\Upsilon : \gamma (E) \lra \mathfrak X(M)$ , which is required to be a map of Lie algebras, such that given sections $\a, \beta \in \gamma(E)$ and a differentiable function $f$ , the following Leibniz rule is satisfied :

\begin{equation} [ \a, f \beta] = f [\a, \beta] + (\Upsilon (\a)) \beta~. \end{equation}

Now suppose we have a Lie groupoid :

(0.1)

Note also the relation of the Lie algebroids to Hamiltonian algebroids, also concerning recent developments in (relativistic) quantum gravity theories.

Bibliography

1

K. C. H. Mackenzie: General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture Notes Series, 213, Cambridge University Press: Cambridge,UK (2005).