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 [[../CoIntersections/|algebraic]] theories.

Lie algebroids generalize [[../BilinearMap/|Lie algebras]], and in certain quantum [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] they represent extended quantum ([[../Algebroids/|algebroid]]) symmetries. One can think of a Lie algebroid as generalizing the idea of a tangent bundle where the [[../BilinearMap/|tangent space]] at a point is effectively the equivalence class of curves meeting at that point (thus suggesting a [[../QuantumOperatorAlgebra5/|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 [[../NoncommutativeGeometry4/|manifold]] and let $𝔛(M)$ denote the set of [[../NeutrinoRestMass/|vector fields]] on $M$. Then, a Lie algebroid over $M$ 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 [[../IsomorphicObjectsUnderAnIsomorphism/|sections]] $\gamma (E)$, 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)$andadifferentiable[[../Bijective/|function]]$fFailed 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 $M$ is a Poisson manifold and $E=T^{*}M$, that is $E$ is the cotangent bundle of $M$.

Now suppose we have a Lie groupoid $𝖦$: 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 $M$ of the bundle of tangent vectors along the fibers of $s$ (ie. the $s$--vertical vector fields). Also, the space of sections </math>\gamma (\A)$canbeidentifiedwiththespaceof$s$--vertical,right--invariantvectorfields<math>{𝔛}_{inv}^s(𝖦)$ which can be seen to be closed under $[,]$, and the latter induces a bracket [[../Cod/|operation]] on $\gamma (A)$ 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 $𝖦$ inducing Failed to parse (unknown function "\A"): {\displaystyle \A} ~.

Unlike Lie algebras that can be integrated to corresponding [[../BilinearMap/|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 [[../Bijective/|relation]] of the Lie algebroids to [[../HamiltonianAlgebroid3/|Hamiltonian algebroids]], also concerning recent developments in (relativistic) [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum gravity theories]].