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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 Cartan's criterion ： ウィキペディア英語版 Cartan's criterion In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on $\backslash mathfrak$ defined by the formula : $K(u,v)=\backslash operatorname(\backslash operatorname(u)\backslash operatorname(v)),$ where tr denotes the trace of a linear operator. The criterion was introduced by .〔Cartan, Chapitre IV, Théorème 1〕 == Cartan's criterion for solvability == Cartan's criterion for solvability states: :''A Lie subalgebra $\backslash mathfrak$ of endomorphisms of a finite-dimensional vector space over a field of characteristic zero is solvable if and only if $Tr(ab)=0$ whenever $a\backslash in\backslash mathfrak,b\backslash in().$'' The fact that $Tr(ab)=0$ in the solvable case follows immediately from Lie's theorem that solvable Lie algebras in characteristic 0 can be put in upper triangular form. Applying Cartan's criterion to the adjoint representation gives: :''A finite-dimensional Lie algebra $\backslash mathfrak$ over a field of characteristic zero is solvable if and only if $K(\backslash mathfrak,())=0$ (where K is the Killing form).'' 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「Cartan's criterion」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.