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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 Representation up to homotopy ： ウィキペディア英語版 Representation up to homotopy A Representation up to homotopy is a concept in differential geometry that generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. It was introduced by Abad and Crainic.〔C.A. Abad, M. Crainic: ''Representations up to homotopy of Lie algebroids'', (arXiv:0901.0319 )〕 As a motivation consider a regular Lie algebroid (''A'',''ρ'',()) (regular meaning that the anchor ''ρ'' has constant rank) where we have two natural ''A''-connections on ''g''(''A'') = ker ''ρ'' and ''ν''(''A'')= ''TM''/im ''ρ'' respectively: :$\backslash nabla\backslash colon\; \backslash Gamma(A)\backslash times\backslash Gamma(\backslash mathfrak(A))\backslash to\backslash Gamma(\backslash mathfrak(A)):\; \backslash nabla\_\backslash psi:=(),$ :$\backslash nabla\backslash colon\; \backslash Gamma(A)\backslash times\backslash Gamma(\backslash nu(A))\backslash to\backslash Gamma(\backslash nu(A)):\; \backslash nabla\_\backslash overline:=\backslash overline.$ In the deformation theory of the Lie algebroid ''A'' there is a long exact sequence〔M.Crainic, I.Moerdijk: ''Deformations of Lie brackets: cohomological aspects''. J. Eur. Math. Soc., 10:1037–1059, (2008)〕 :$\backslash dots\backslash to\; H^n(A,\backslash mathfrak(A))\backslash to\; H^n\_(A)\backslash to\; H^(A,\backslash nu(A))\backslash to\; H^(A,\backslash mathfrak(A))\backslash to\backslash dots$ This suggests that the correct cohomology for the deformations (here denoted as ''H''def) comes from the direct sum of the two modules ''g''(''A'') and ''ν''(''A'') and should be called adjoint representation. Note however that in the more general case where ''ρ'' does not have constant rank we cannot easily define the representations ''g''(''A'') and ''ν''(''A''). Instead we should consider the 2-term complex ''A''→''TM'' and a representation on it. This leads to the notion explained here. == Definition == Let (''A'',''ρ'',()) be a Lie algebroid over a smooth manifold ''M'' and let Ω(''A'') denote its Lie algebroid complex. Let further ''E'' be a ℤ-graded vector bundle over ''M'' and Ω(''A'',''E'') = Ω(''A'') ⊗ Γ(''E'') be its ℤ-graded ''A''-cochains with values in ''E''. A representation up to homotopy of ''A'' on ''E'' is a differential operator ''D'' that maps :$D\backslash colon\; \backslash Omega^\backslash bullet(A,E)\backslash to\backslash Omega^(A,E),$ fulfills the Leibniz rule :$D(\backslash alpha\backslash wedge\backslash beta)\; =\; (D\backslash alpha)\backslash wedge\backslash beta\; +\; (-1)^\backslash alpha\backslash wedge(D\backslash beta),$ and squares to zero, i.e. ''D''2 = 0. 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「Representation up to homotopy」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.