Lie algebroid

In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones.

Description

Just as a Lie groupoid can be thought of as a "Lie group with many objects", a Lie algebroid is like a "Lie algebra with many objects".

More precisely, a Lie algebroid is a triple $(E,[\cdot ,\cdot ],\rho )$ consisting of a vector bundle $E$ over a manifold $M$ , together with a Lie bracket $[\cdot ,\cdot ]$ on its space of sections $\Gamma (E)$ and a morphism of vector bundles $\rho :E\rightarrow TM$ called the anchor. Here $TM$ is the tangent bundle of $M$ . The anchor and the bracket are to satisfy the Leibniz rule:

$[X,fY]=\rho (X)f\cdot Y+f[X,Y]$ where $X,Y\in \Gamma (E),f\in C^{\infty }(M)$ and $\rho (X)f$ is the derivative of $f$ along the vector field $\rho (X)$ . It follows that

$\rho ([X,Y])=[\rho (X),\rho (Y)]$ for all $X,Y\in \Gamma (E)$ .

Examples

• Every Lie algebra is a Lie algebroid over the one point manifold.
• The tangent bundle $TM$ of a manifold $M$ is a Lie algebroid for the Lie bracket of vector fields and the identity of $TM$ as an anchor.
• Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.
• Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.
• To every Lie groupoid is associated a Lie algebroid, generalizing how a Lie algebra is associated to a Lie group (see also below). For example, the Lie algebroid $TM$ comes from the pair groupoid whose objects are $M$ , with one isomorphism between each pair of objects. Unfortunately, going back from a Lie algebroid to a Lie groupoid is not always possible, but every Lie algebroid gives a stacky Lie groupoid.
• Given the action of a Lie algebra g on a manifold M, the set of g -invariant vector fields on M is a Lie algebroid over the space of orbits of the action.
• The Atiyah algebroid of a principal G-bundle P over a manifold M is a Lie algebroid with short exact sequence:
$0\to P\times _{G}{\mathfrak {g}}\to TP/G{\xrightarrow {\rho }}TM\to 0.$ The space of sections of the Atiyah algebroid is the Lie algebra of G-invariant vector fields on P.
• A Poisson Lie algebroid is associated to a Poisson manifold by taking E to be the cotangent bundle. The anchor map is given by the Poisson bivector. This can be seen in a Lie bialgebroid.

Lie algebroid associated to a Lie groupoid

To describe the construction let us fix some notation. G is the space of morphisms of the Lie groupoid, M the space of objects, $e:M\to G$ the units and $t:G\to M$ the target map.

$T^{t}G=\bigcup _{p\in M}T(t^{-1}(p))\subset TG$ the t-fiber tangent space. The Lie algebroid is now the vector bundle $A:=e^{*}T^{t}G$ . This inherits a bracket from G, because we can identify the M-sections into A with left-invariant vector fields on G. The anchor map then is obtained as the derivation of the source map $Ts:e^{*}T^{t}G\rightarrow TM$ . Further these sections act on the smooth functions of M by identifying these with left-invariant functions on G.

As a more explicit example consider the Lie algebroid associated to the pair groupoid $G:=M\times M$ . The target map is $t:G\to M:(p,q)\mapsto p$ and the units $e:M\to G:p\mapsto (p,p)$ . The t-fibers are $p\times M$ and therefore $T^{t}G=\bigcup _{p\in M}p\times TM\subset TM\times TM$ . So the Lie algebroid is the vector bundle $A:=e^{*}T^{t}G=\bigcup _{p\in M}T_{p}M=TM$ . The extension of sections X into A to left-invariant vector fields on G is simply ${\tilde {X}}(p,q)=0\oplus X(q)$ and the extension of a smooth function f from M to a left-invariant function on G is ${\tilde {f}}(p,q)=f(q)$ . Therefore, the bracket on A is just the Lie bracket of tangent vector fields and the anchor map is just the identity.

Of course you could do an analog construction with the source map and right-invariant vector fields/ functions. However you get an isomorphic Lie algebroid, with the explicit isomorphism $i_{*}$ , where $i:G\to G$ is the inverse map.

Example

Consider the Lie groupoid

$\mathbb {R} ^{2}\times U(1)\rightrightarrows \mathbb {R} ^{2}$ where the target map sends

$((x,y),e^{i\theta })\mapsto {\begin{bmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}$ Notice that there are two cases for the fibers of $T^{t}(\mathbb {R} ^{2}\times U(1))$ :

{\begin{aligned}t^{-1}(0)\cong &U(1)\\t^{-1}(p)\cong &\{(a,u)\in \mathbb {R} ^{2}\times U(1):ua=p\}\end{aligned}} This demonstrating that there is a stabilizer of $U(1)$ over the origin and stabilizer-free $U(1)$ -orbits everywhere else. The tangent bundle over every $t^{-1}(p)$ is then trivial, hence the pullback $e^{*}(T^{t}(\mathbb {R} ^{2}\times U(1)))$ is a trivial line bundle.