Lie algebroid

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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 Γ(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 ρ(X)f is the derivative of f along the vector field ρ(X). It follows that

ρ([X,Y]) = [ρ(X),ρ(Y)]

for all X,Y \in \Gamma(E).

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[edit] 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[1], but every Lie algebroid gives a stacky Lie groupoid[2][3].
  • 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.
  • Atiyah algebroid. Given a vector bundle V over a smooth manifold M consider its derivations, i.e. smooth \Bbb R-linear maps \psi:\Gamma(V)\to \Gamma(V) for which exists a vector field X such that they fulfill the Leibniz rule ψ(fv) = X[f]v + fψ(v) for all smooth functions f and all sections v into the vector bundle. The association \psi\to X is clearly linear and thus comes from a map of vector bundles \rho:A(V)\to TM (if you find the bundle whose sections give the derivations). The Atiyah algebroid is further characterized by fitting into the following short exact sequence: 0\to \mathrm{End}_M(V)\to A(V)\to TM\to 0 To see that the Atiyah algebroid exists for every vector bundle note that it is the Lie algebroid associated to the Lie groupoid coming from the frame bundle of the vector bundle V.

[edit] 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^tG=\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 * TtG. This inherits a bracket from G, because we can identify the M-sections into A with left-invariant vector fields on G. 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^tG=\bigcup_{p\in M}p\times TM \subset TM\times TM. So the Lie algebroid is the vector bundle A:=e^*T^tG=\bigcup_{p\in M} T_pM=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.

[edit] References

  1. ^ Marius Crainic, Rui L. Fernandes: Integrability of Lie brackets, available as arXiv:math/0105033
  2. ^ Hsian-Hua Tseng and Chenchang Zhu, Integrating Lie algebroids via stacks, available as arXiv:math/0405003
  3. ^ Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, available as arXiv:math/0701024

[edit] 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

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