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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 algebra bundle ： ウィキペディア英語版 Lie algebra bundle In mathematics, a weak Lie algebra bundle :$\backslash xi=(\backslash xi,\; p,\; X,\; \backslash theta)\backslash ,$ is a vector bundle $\backslash xi\backslash ,$ over a base space ''X'' together with a morphism :$\backslash theta\; :\; \backslash xi\; \backslash otimes\; \backslash xi\; \backslash rightarrow\; \backslash xi$ which induces a Lie algebra structure on each fibre $\backslash xi\_x\backslash ,$. A Lie algebra bundle $\backslash xi=(\backslash xi,\; p,\; X)\backslash ,$ is a vector bundle in which each fibre is a Lie algebra and for every ''x'' in ''X'', there is an open set $U$ containing ''x'', a Lie algebra ''L'' and a homeomorphism :$\backslash phi:U\backslash times\; L\backslash to\; p^(U)\backslash ,$ such that :$\backslash phi\_x:x\backslash times\; L\; \backslash rightarrow\; p^(x)\backslash ,$ is a Lie algebra isomorphism. Any Lie algebra bundle is a weak Lie algebra bundle, but the converse need not be true in general. As an example of a weak Lie algebra bundle that is not a strong Lie algebra bundle, consider the total space $\backslash mathfrak(3)\backslash times\backslash mathbb$ over the real line $\backslash mathbb$. Let () denote the Lie bracket of $\backslash mathfrak(3)$ and deform it by the real parameter as: :$()\_x\; =\; x\backslash cdot()$ for $X,Y\backslash in\backslash mathfrak(3)$ and $x\backslash in\backslash mathbb$. Lie's third theorem states that every bundle of Lie algebras can locally be integrated to a bundle of Lie groups. However globally the total space might fail to be Hausdorff.〔A. Weinstein, A.C. da Silva: ''Geometric models for noncommutative algebras, 1999 Berkley LNM, online readable at (), in particular chapter 16.3.〕 ==References== 〔 *A.Douady et M.Lazard, Espaces fibres en algebre de Lie et en groups, Invent. math., Vol. 1, 1966, pp. 133–151 *B.S.Kiranagi, Lie Algebra bundles, Bull. Sci. Math., 2e serie, 102(1978), 57-62. *B.S.Kiranagi, Semi simple Lie algebra bundles, Bull. Math de la Sci. Math de la R.S.de Roumaine, 27 (75), 1983, 253-257. *B.S.Kiranagi and G.Prema, On complete reducibility of Module Bundles, Bull. Austral. Math Soc., 28 (1983), 401-409. *B.S.Kiranagi and G.Prema, Cohomology of Lie algebra bundles and its applications, Ind. J. Pure and Appli. Math. 16(7): 1985, 731/735. *B.S.Kiranagi and G.Prema, Lie algebra bundles defined by Jordan algebra bundles, Bull. Math. Soc.Sci.Math.Rep.Soc. Roum., Noun. Ser. 33 (81), 1989, 255-264. *B.S.Kiranagi and G.Prema, On complete reducibility of Bimodule bundles, Bull. Math. Soc. Sci.Math. Repose; Roum, Nouv.Ser. 33 (81), 1989, 249-255. *B.S.Kiranagi and G.Prema, A decomposition theorem of Lie algebra Bundles, Communications in Algebra 18 (6), 1990, 1869-1877 . *B.S.Kiranagi, G.Prema and C.Chidambara, Rigidity theorem for Lie algebra Bundles, Communications in Algebra 20 (6), 1992, pp. 1549 – 1556. *. 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 algebra bundle」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.