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In [[mathematics]], Lie algebroids serve the same role in the theory of [[Lie groupoid]]s that [[Lie algebra]]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".
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More precisely, a '''Lie algebroid'''
is a triple <math>(E, [\cdot,\cdot], \rho)</math> consisting of a [[vector bundle]] <math>E</math> over a [[manifold]] <math>M</math>, together with a [[Lie_algebra#Definition_and_first_properties|Lie bracket]] <math>[\cdot,\cdot]</math> on its [[Module (mathematics)|module]] of sections <math>\Gamma (E)</math> and a morphism of vector bundles <math>\rho: E\rightarrow TM</math> called the '''anchor'''. Here <math>TM</math> is the [[tangent bundle]] of <math>M</math>. The anchor and the bracket are to satisfy the Leibniz rule:
 
:<math>[X,fY]=\rho(X)f\cdot Y + f[X,Y]</math>
 
where <math>X,Y \in \Gamma(E), f\in C^\infty(M)</math> and <math>\rho(X)f</math> is the [[derivative]] of <math>f</math> along the vector field <math>\rho(X)</math>. It follows that
 
:<math>\rho([X,Y])=[\rho(X),\rho(Y)] </math>
 
for all <math>X,Y \in \Gamma(E)</math>.
 
== Examples ==
 
* Every [[Lie algebra]] is a Lie algebroid over the one point manifold.
 
* The tangent bundle <math>TM</math> of a manifold <math>M</math> is a Lie algebroid for the [[Lie bracket of vector fields]] and the identity of <math>TM</math> as an anchor.
 
* Every integrable subbundle of the tangent bundle &mdash; that is, one whose sections are closed under the Lie bracket &mdash; 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 <math>TM</math> comes from the pair groupoid whose objects are <math>M</math>, with one isomorphism between each pair of objects. Unfortunately, going back from a Lie algebroid to a Lie groupoid is not always possible,<ref>Marius Crainic, Rui L. Fernandes. [http://arxiv.org/abs/math/0105033 Integrability of Lie brackets], Ann. of Math. (2), Vol. 157 (2003), no. 2, 575--620</ref> but every Lie algebroid gives a [[Algebraic stack|stacky]] Lie groupoid.<ref>Hsian-Hua Tseng and Chenchang Zhu, Integrating Lie algebroids via stacks, Compositio Mathematica, Volume 142 (2006), Issue 01, pp 251-270, available as [http://arxiv.org/abs/math/0405003 arXiv:math/0405003]</ref><ref>Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, available as [http://arxiv.org/abs/math/0701024 arXiv:math/0701024]</ref>
 
* 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 bundle|principal ''G''-bundle]] ''P'' over a manifold ''M'' is a Lie algebroid with [[short exact sequence]]:
*:<math> 0 \to P\times_G \mathfrak g\to TP/G\xrightarrow{\rho} TM \to 0.</math>
: 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, <math>e:M\to G</math> the units and <math>t:G\to M</math> the target map.
 
<math>T^tG=\bigcup_{p\in M}T(t^{-1}(p))\subset TG</math> the ''t''-fiber tangent space. The Lie algebroid is now the vector bundle <math>A:=e^*T^tG</math>.  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
<math>Ts:e^*T^tG \rightarrow TM </math>. 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 <math>G:=M\times M</math>. The target map is <math>t:G\to M: (p,q)\mapsto p</math> and the units <math>e:M\to G: p\mapsto (p,p)</math>. The ''t''-fibers are <math>p\times M</math> and therefore <math>T^tG=\bigcup_{p\in M}p\times TM \subset TM\times TM</math>. So the Lie algebroid is the vector bundle <math>A:=e^*T^tG=\bigcup_{p\in M} T_pM=TM</math>. The extension of sections ''X'' into ''A'' to left-invariant vector fields on ''G'' is simply <math>\tilde X(p,q)=0\oplus X(q)</math> and the extension of a smooth function ''f'' from ''M'' to a left-invariant function on ''G'' is <math>\tilde f(p,q)=f(q)</math>.  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 <math>i_*</math>, where <math>i:G\to G</math> is the inverse map.
 
==See also==
*[[R-algebroid]]
 
==References==
<references/>
 
==External links==
*Alan Weinstein, Groupoids: unifying internal and external
symmetry, ''AMS Notices'', '''43''' (1996), 744-752.  Also available as [http://arxiv.org/abs/math/9602220 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 [http://arxiv.org/abs/0804.2451v1 arXiv:0804.2451]
 
{{DEFAULTSORT:Lie Algebroid}}
[[Category:Lie algebras]]
[[Category:Differential geometry]]

Latest revision as of 01:27, 27 November 2014

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