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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 groupoid ： ウィキペディア英語版 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 operations :$s,t\; :\; Mor\; \backslash to\; Ob$ are submersions, and all the category operations (source and target, composition, and identity-assigning map) are smooth. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Just as every Lie group has a Lie algebra, every Lie groupoid has a Lie algebroid. ==Examples== *Any Lie group gives a Lie groupoid with one object, and conversely. So, the theory of Lie groupoids includes the theory of Lie groups. *Given any manifold $M$, there is a Lie groupoid called the pair groupoid, with $M$ as the manifold of objects, and precisely one morphism from any object to any other. In this Lie groupoid the manifold of morphisms is thus $M\; \backslash times\; M$. *Given a Lie group $G$ acting on a manifold $M$, there is a Lie groupoid called the translation groupoid with one morphism for each triple $g\; \backslash in\; G,\; x,y\; \backslash in\; M$ with $gx\; =\; y$. *Any foliation gives a Lie groupoid. *Any principal bundle $P\backslash to\; M$ with structure group ''G'' gives a groupoid, namely $P\backslash times\; P/G$ over ''M'', where ''G'' acts on the pairs componentwise. Composition is defined via compatible representatives as in the pair groupoid. 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 groupoid」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.