# PlanetPhysics/Lie Algebroids

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### Topic on Lie algebroidsEdit

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. ^{[1]}). The formal definition of a Lie algebroid is presented next.

Let be a manifold and let denote the set of vector fields on . Then, a
*Lie algebroid* over consists of a *\htmladdnormallink{vector* {http://planetphysics.us/encyclopedia/Vectors.html} bundle **Failed to parse (unknown function "\lra"): {\displaystyle E \lra M}**
,
equipped with a Lie bracket on the space of sections ,
and a bundle map **Failed to parse (unknown function "\lra"): {\displaystyle \Upsilon : E \lra TM}**
}, usually called the *anchor* .
Furthermore, there is an induced map **Failed to parse (unknown function "\lra"): {\displaystyle \Upsilon : \gamma (E) \lra \mathfrak X(M)}**
,
which is required to be a map of Lie algebras, such that given sections </math>\a, \beta \in
\gamma(E)f**Failed to parse (unknown function "\a"): {\displaystyle , the following Leibniz rule is satisfied~: <center><math> [ \a, f \beta] = f [\a, \beta] + (\Upsilon (\a)) \beta~. }**

A typical example of a Lie algebroid is obtained when is a Poisson manifold and , that is is the cotangent bundle of .

Now suppose we have a Lie groupoid :
**Failed to parse (unknown function "\xymatrix"): {\displaystyle r,s~:~ \xymatrix{ \mathsf{G} \ar@<1ex>[r]^r \ar[r]_s & \mathsf{G}^{(0)}}=M~. }**
There is an associated Lie algebroid **Failed to parse (unknown function "\A"): {\displaystyle \A = \A( \mathsf{G})}**
, which in the
guise of a vector bundle, it is the restriction to of the
bundle of tangent vectors along the fibers of (ie. the
--vertical vector fields). Also, the space of sections </math>\gamma
(\A)s which
can be seen to be closed under , and the latter induces a
bracket operation on thus turning **Failed to parse (unknown function "\A"): {\displaystyle \A}**
into a Lie
algebroid. Subsequently, a Lie algebroid **Failed to parse (unknown function "\A"): {\displaystyle \A}**
is integrable if
there exists a Lie groupoid inducing **Failed to parse (unknown function "\A"): {\displaystyle \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.

## All SourcesEdit

^{[1]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}K. C. H. Mackenzie: \emph{General Theory of Lie Groupoids and Lie Algebroids}, London Math. Soc. Lecture Notes Series,**213**, Cambridge University Press: Cambridge,UK (2005).