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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-valued differential form ： ウィキペディア英語版 Lie algebra-valued differential form In differential geometry, a Lie algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections. == Wedge product == Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. This operation, denoted by $()$, is given by: for $\backslash mathfrak$-valued ''p''-form $\backslash omega$ and $\backslash mathfrak$-valued ''q''-form $\backslash eta$ :$()(v\_1,\; \backslash cdots,\; v\_)\; =\; \backslash sum\_\; \backslash operatorname(\backslash sigma)\; (\backslash cdots,\; v\_),\; \backslash eta(v\_,\; \backslash cdots,\; v\_))$ where ''v''''i'''s are tangent vectors. The notation is meant to indicate both operations involved. For example, if $\backslash omega$ and $\backslash eta$ are Lie algebra-valued one forms, then one has :$()(v\_1,v\_2)\; =\; (()\; -\; ()).$ The operation $()$ can also be defined as the bilinear operation on $\backslash Omega(M,\; \backslash mathfrak\; g)$ satisfying :$(\backslash otimes\; \backslash alpha)\; \backslash wedge\; (h\; \backslash otimes\; \backslash beta))\; =\; (h)\; \backslash otimes\; (\backslash alpha\; \backslash wedge\; \backslash beta)$ for all $g,\; h\; \backslash in\; \backslash mathfrak\; g$ and $\backslash alpha,\; \backslash beta\; \backslash in\; \backslash Omega(M,\; \backslash mathbb\; R)$. Some authors have used the notation $(\backslash eta)$ instead of $()$. The notation $(\backslash eta)$, which resembles a commutator, is justified by the fact that if the Lie algebra $\backslash mathfrak\; g$ is a matrix algebra then $()$ is nothing but the graded commutator of $\backslash omega$ and $\backslash eta$, i. e. if $\backslash omega\; \backslash in\; \backslash Omega^p(M,\; \backslash mathfrak\; g)$ and $\backslash eta\; \backslash in\; \backslash Omega^q(M,\; \backslash mathfrak\; g)$ then :$()\; =\; \backslash omega\backslash wedge\backslash eta\; -\; (-1)^\backslash eta\backslash wedge\backslash omega,$ where $\backslash omega\; \backslash wedge\; \backslash eta,\backslash \; \backslash eta\; \backslash wedge\; \backslash omega\; \backslash in\; \backslash Omega^(M,\; \backslash mathfrak\; g)$ are wedge products formed using the matrix multiplication on $\backslash mathfrak\; g$. 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-valued differential form」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.