词条 | Lie algebroid |
释义 |
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. DescriptionJust 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 over a manifold , together with a Lie bracket on its module of sections and a morphism of vector bundles called the anchor. Here is the tangent bundle of . The anchor and the bracket are to satisfy the Leibniz rule: where and is the derivative of along the vector field . It follows that for all . Examples
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 groupoidTo 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 obtained 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. ExampleConsider the Lie groupoid where the target map sends Notice that there are two cases for the fibers of : This demonstrating that there is a stabilizer of over the origin and stabilizer-free -orbits everywhere else. The tangent bundle over every is then trivial, hence the pullback is a trivial line bundle. See also
References1. ^Marius Crainic, Rui L. Fernandes. [https://arxiv.org/abs/math/0105033 Integrability of Lie brackets], Ann. of Math. (2), Vol. 157 (2003), no. 2, 575--620 2. ^Hsian-Hua Tseng and Chenchang Zhu, Integrating Lie algebroids via stacks, Compositio Mathematica, Volume 142 (2006), Issue 01, pp 251-270, available as [https://arxiv.org/abs/math/0405003 arXiv:math/0405003] 3. ^Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, available as [https://arxiv.org/abs/math/0701024 arXiv:math/0701024] External links
7 : Lie algebras|Differential geometry|Differential topology|Differential operators|Generalizations of the derivative|Geometry processing|Vector bundles |
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