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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 Frölicher-Nijenhuis bracket ： ウィキペディア英語版 Frölicher–Nijenhuis bracket In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of connections, notably the Ehresmann connection, as well as in the more general study of projections in the tangent bundle. It was introduced by Alfred Frölicher and Albert Nijenhuis (1956) and is related to the work of Schouten (1940). It is related to but not the same as the Nijenhuis–Richardson bracket and the Schouten–Nijenhuis bracket. ==Definition== Let Ω *(''M'') be the sheaf of exterior algebras of differential forms on a smooth manifold ''M''. This is a graded algebra in which forms are graded by degree: :$\backslash Omega^$ *(M) = \bigoplus_^\infty \Omega^k(M). A graded derivation of degree ℓ is a mapping :$D:\backslash Omega^$ *(M)\to\Omega^(M) which is linear with respect to constants and satisfies :$D(\backslash alpha\backslash wedge\backslash beta)\; =\; D(\backslash alpha)\backslash wedge\backslash beta\; +\; (-1)^\backslash alpha\backslash wedge\; D(\backslash beta).$ Thus, in particular, the interior product with a vector defines a graded derivation of degree ℓ = −1, whereas the exterior derivative is a graded derivation of degree ℓ = 1. The vector space of all derivations of degree ℓ is denoted by DerℓΩ *(''M''). The direct sum of these spaces is a graded vector space whose homogeneous components consist of all graded derivations of a given degree; it is denoted :$\backslash mathrm\backslash ,\; \backslash Omega^$ *(M) = \bigoplus_^\infty \mathrm_k\, \Omega^ *(M). This forms a graded Lie superalgebra under the anticommutator of derivations defined on homogeneous derivations ''D''1 and ''D''2 of degrees ''d''1 and ''d''2, respectively, by :$()\; =\; D\_1\backslash circ\; D\_2\; -\; (-1)^D\_2\backslash circ\; D\_1.$ Any vector-valued differential form ''K'' in Ω''k''(''M'', T''M'') with values in the tangent bundle of ''M'' defines a graded derivation of degree ''k'' − 1, denoted by ''i''''K'', and called the insertion operator. For ω ∈ Ωℓ(''M''), :$i\_K\backslash ,\backslash omega(X\_1,\backslash dots,X\_)=\backslash frac\backslash sum\_\}\backslash textrm\backslash ,\backslash sigma\; \backslash cdot$ \omega(K(X_,\dots,X_),X_,\dots,X_) The Nijenhuis–Lie derivative along ''K'' ∈ Ωk(''M'', T''M'') is defined by :$\backslash mathcal\_K\; =\; ()\; =d\backslash ,\backslash ,\; i\_K-(-1)^i\_K\backslash ,\; d$ where ''d'' is the exterior derivative and ''i''K is the insertion operator. The Frölicher–Nijenhuis bracket is defined to be the unique vector-valued differential form :$(\backslash cdot)\_\; :\; \backslash Omega^k(M,\backslash mathrmM)\; \backslash times\; \backslash Omega^\backslash ell(M,\backslash mathrmM)\; \backslash to\; \backslash Omega^(M,\backslash mathrmM)\; :\; (K,\; L)\; \backslash mapsto\; (L)\_$ such that :$\backslash mathcal\_\_K,"\; TITLE="\backslash mathcal\_K,">\backslash mathcal\_L\; ).$ Hence, :$$ (L )_=-(-1)^()_. If ''k'' = 0, so that ''K'' ∈ Ω0(''M'', T''M'') is a vector field, the usual homotopy formula for the Lie derivative is recovered :$\backslash mathcal\_K\; =\; ()\; =d\; \backslash ,\backslash ,\; i\_K+i\_K\; \backslash ,\backslash ,\; d.$ If ''k''=''ℓ''=1, so that ''K,L'' ∈ Ω1(''M'', T''M''), one has for any vector fields ''X'' and ''Y'' :$$ (L )_(X,Y) = (LY )+(KY )+(KL+LK)()-K(()+(LY ))-L(()+(KY )). If ''k''=0 and ''ℓ''=1, so that ''K=Z''∈ Ω0(''M'', T''M'') is a vector field and ''L'' ∈ Ω1(''M'', T''M''), one has for any vector field ''X'' :$$ (L )_(X) = (LX )-L(). An explicit formula for the Frölicher–Nijenhuis bracket of $\backslash phi\backslash otimes\; X$ and $\backslash psi\backslash otimes\; Y$ (for forms φ and ψ and vector fields ''X'' and ''Y'') is given by :$\backslash left.\backslash right.(\backslash otimes\; X,\backslash psi\; \backslash otimes\; Y)\_\; =\; \backslash phi\backslash wedge\backslash psi\backslash otimes\; ()\; +\; \backslash phi\backslash wedge\backslash mathcal\_X\; \backslash psi\backslash otimes\; Y\; -\; \backslash mathcal\_Y\; \backslash phi\backslash wedge\backslash psi\; \backslash otimes\; X\; +(-1)^(d\backslash phi\; \backslash wedge\; i\_X(\backslash psi)\backslash otimes\; Y\; +i\_Y(\backslash phi)\; \backslash wedge\; d\backslash psi\; \backslash otimes\; X).$ 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「Frölicher–Nijenhuis bracket」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.