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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–Kolchin theorem ： ウィキペディア英語版 Lie–Kolchin theorem In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if ''G'' is a connected and solvable linear algebraic group defined over an algebraically closed field and :$\backslash rho\backslash colon\; G\; \backslash to\; GL(V)$ a representation on a nonzero finite-dimensional vector space ''V'', then there is a one-dimensional linear subspace ''L'' of ''V'' such that : $\backslash rho(G)(L)\; =\; L.$ That is, ρ(''G'') has an invariant line ''L'', on which ''G'' therefore acts through a one-dimensional representation. This is equivalent to the statement that ''V'' contains a nonzero vector ''v'' that is a common (simultaneous) eigenvector for all $\backslash rho(g),\; \backslash ,\backslash ,\; g\; \backslash in\; G$. It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group ''G'' has dimension one. In fact, this is another way to state the Lie–Kolchin theorem. Lie's theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector space over an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace. The result for Lie algebras was proved by and for algebraic groups was proved by . The Borel fixed point theorem generalizes the Lie–Kolchin theorem. == Triangularization == Sometimes the theorem is also referred to as the ''Lie–Kolchin triangularization theorem'' because by induction it implies that with respect to a suitable basis of ''V'' the image $\backslash rho(G)$ has a ''triangular shape''; in other words, the image group $\backslash rho(G)$ is conjugate in GL(''n'',''K'') (where ''n'' = dim ''V'') to a subgroup of the group T of upper triangular matrices, the standard Borel subgroup of GL(''n'',''K''): the image is simultaneously triangularizable. The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group ''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–Kolchin theorem」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.