Bialgebra

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. 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 module of sections $\mathrm{\Gamma }(E)$ and a morphism of vector bundles $\rho :E\to TM$ 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 $X,Y\in \mathrm{\Gamma }(E),f\in C^{\mathrm{\infty }}(M)$ 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 $X,Y\in \mathrm{\Gamma }(E)$.

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:

  $0\to P{\times }_G\mathfrak{g}\to TP/G\stackrel{\rho }{\to }TM\to 0.$

The space of sections of the Atiyah algebroid is the Lie algebra of G-invariant vector fields on P.

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 $Ts:e^{*}{T}^{t}G\to TM$. 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.

See also

• R-algebroid

External links

• Alan Weinstein, Groupoids: unifying internal and external

symmetry, AMS Notices, 43 (1996), 744-752. Also available as arXiv:math/9602220

• Kirill Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge U. Press, 1987.

• Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge U. Press, 2005

• Charles-Michel Marle, Differential calculus on a Lie algebroid and Poisson manifolds (2002). Also available in arXiv:0804.2451