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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lie algebroid ： ウィキペディア英語版 Lie algebroid 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 $(E,\; (),\; \backslash rho)$ consisting of a vector bundle $E$ over a manifold $M$, together with a Lie bracket $()$ on its module of sections $\backslash Gamma\; (E)$ and a morphism of vector bundles $\backslash rho:\; E\backslash rightarrow\; TM$ called the anchor. Here $TM$ is the tangent bundle of $M$. The anchor and the bracket are to satisfy the Leibniz rule: :$()=\backslash rho(X)f\backslash cdot\; Y\; +\; f()$ where $X,Y\; \backslash in\; \backslash Gamma(E),\; f\backslash in\; C^\backslash infty(M)$ and $\backslash rho(X)f$ is the derivative of $f$ along the vector field $\backslash rho(X)$. It follows that :$\backslash rho(())=()$ for all $X,Y\; \backslash in\; \backslash Gamma(E)$. == 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,〔Marius Crainic, Rui L. Fernandes. (Integrability of Lie brackets ), Ann. of Math. (2), Vol. 157 (2003), no. 2, 575--620〕 but every Lie algebroid gives a stacky Lie groupoid.〔Hsian-Hua Tseng and Chenchang Zhu, Integrating Lie algebroids via stacks, Compositio Mathematica, Volume 142 (2006), Issue 01, pp 251-270, available as (arXiv:math/0405003 )〕〔Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, available as (arXiv:math/0701024 )〕 * 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: *:$0\; \backslash to\; P\backslash times\_G\; \backslash mathfrak\; g\backslash to\; TP/G\backslash xrightarrow\; TM\; \backslash to\; 0.$ : The space of sections of the Atiyah algebroid is the Lie algebra of ''G''-invariant vector fields on ''P''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lie algebroid」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.