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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 Courant algebroid ： ウィキペディア英語版 Courant algebroid In a field of mathematics known as differential geometry, a Courant algebroid is a structure which, in a certain sense, blends the concepts of Lie algebroid and of quadratic Lie algebra. This notion, which plays a fundamental role in the study of Hitchin's generalized complex structures, was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997.〔Z-J. Liu, A. Weinstein, and P. Xu: ( Manin triples for Lie Bialgebroids ), Journ. of Diff.geom. 45 pp.647–574 (1997).〕 Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier in 1990〔T.J. Courant: ''Dirac Manifolds'', Transactions of the AMS, vol. 319, pp.631–661 (1990).〕 the standard prototype of Courant algebroid through his discovery of a skew symmetric bracket on $TM\backslash oplus\; T^$ *M, called Courant bracket today, which fails to satisfy the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids. ==Definition== A Courant algebroid consists of the data a vector bundle $E\backslash to\; M$ with a bracket $():\backslash Gamma\; E\; \backslash times\; \backslash Gamma\; E\; \backslash to\; \backslash Gamma\; E$, a non degenerate fiber-wise inner product $\backslash langle.,.\backslash rangle:\; E\backslash times\; E\backslash to\; M\backslash times\backslash R$, and a bundle map $\backslash rho:E\backslash to\; TM$ subject to the following axioms, :$,\; \backslash psi]\; +\; +\; +$ :$()=\; \backslash tfrac12\; D\backslash langle\; \backslash phi,\backslash phi\backslash rangle$ :$\backslash rho(\backslash phi)\backslash langle\; \backslash psi,\backslash psi\backslash rangle=\; 2\backslash langle\; (),\backslash psi\backslash rangle$ where ''φ,ψ,χ'' are sections of ''E'' and ''f'' is a smooth function on the base manifold ''M''. ''D'' is the combination $\backslash kappa^\backslash rho^T\; d$ with ''d'' the de Rham differential, $\backslash rho^T$ the dual map of $\backslash rho$, and ''κ'' the map from ''E'' to $E^$ * induced by the inner product. 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「Courant 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.