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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 : a representation on a nonzero finite-dimensional vector space ''V'', then there is a one-dimensional linear subspace ''L'' of ''V'' such that : 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 . 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 has a ''triangular shape''; in other words, the image group 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
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