In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Lie algebroids serve the same role in the theory of Lie groupoid

Lie groupoid

In mathematics, a Lie groupoid is a groupoid where the set Ob of objects and the set Mor of morphisms are both manifolds, the source and target operationss,t : Mor \to Ob...

s that Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

s 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 consisting of a vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

  over a manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 , together with a Lie bracket

Lie bracket

Lie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...

  on its module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 of sections and a morphism of vector bundles called the anchor. Here is the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

 of . The anchor and the bracket are to satisfy the Leibniz rule:


where and is the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of along the vector field . It follows that


for all .

Examples

• Every Lie algebra
  Lie algebra
  In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
  
   is a Lie algebroid over the one point manifold.

• The tangent bundle of a manifold is a Lie algebroid for the Lie bracket of vector fields and the identity of 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
  Lie groupoid
  In mathematics, a Lie groupoid is a groupoid where the set Ob of objects and the set Mor of morphisms are both manifolds, the source and target operationss,t : Mor \to Ob...
  
   is associated a Lie algebroid, generalizing how a Lie algebra is associated to a Lie group
  Lie group
  In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
  
   (see also below). For example, the Lie algebroid comes from the pair groupoid whose objects are , 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
  Atiyah algebroid
  In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal G-bundle P over a manifold M, where G is a Lie group, is the Lie algebroid of the gauge groupoid of P...
  
   of a principal G-bundle
  Principal bundle
  In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
  
   P over a manifold M is a Lie algebroid with short exact sequence:

  

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, the units and the target map.

the t-fiber tangent space. The Lie algebroid is now the vector bundle . 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 optained as the derivation of the source map
. 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 . The target map is and the units . The t-fibers are and therefore . So the Lie algebroid is the vector bundle . The extension of sections X into A to left-invariant vector fields on G is simply and the extension of a smooth function f from M to a left-invariant function on G is . 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 , where is the inverse map.

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