PlanetPhysics/Lie Algebroids

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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. ). The formal definition of a Lie algebroid is presented next.

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 \htmladdnormallink{vector {http://planetphysics.us/encyclopedia/Vectors.html} 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~:

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}:

There is an associated Lie algebroid, which in the guise of a vector bundle, it is the restriction to M of the bundle of tangent vectors along the fibers of s (ie. the s--vertical vector fields). Also, the space of sections $$\gamma (\A)$$ can be identified with the space of $$s--vertical, right--invariant vector fields $$\mathfrak X^s_{inv} (\mathsf{G})$$ which can be seen to be closed under $$[~,~]$$, and the latter induces a bracket operation on $$\gamma (A)$$ thus turning $$\A$$ into a Lie algebroid. Subsequently, a Lie algebroid $$\A$$ is integrable if there exists a Lie groupoid $$\mathsf{G}$$ inducing $$\A$$~.

Unlike Lie algebras that can be integrated to corresponding Lie groups, not all Lie algebroids are `smoothly integrable' to Lie groupoids; the subset of Lie groupoids that have corresponding Lie algebroids are sometimes called `Weinstein groupoids'.

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