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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. The result for Lie algebras was proved by Sophus Lie and for algebraic groups was proved by Ellis Kolchin . The Borel fixed point theorem generalizes the Lie–Kolchin theorem. (en)
- Le théorème de Lie-Kolchin est un résultat de trigonalisabilité des sous-groupes connexes et résolubles du groupe des matrices inversibles GLn(K), où K est un corps algébriquement clos de caractéristique quelconque. Démontré en 1948, il tient son nom de son auteur, Ellis Kolchin, et de son analogie avec le théorème de Lie sur les algèbres de Lie résolubles (en caractéristique nulle), démontré en 1876 par Sophus Lie. (fr)
- Теорема Ли — Колчина — это теорема теории представлений линейных алгебраических групп. Теорема Ли является аналогом для линейных алгебр Ли. (ru)
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- 4407 (xsd:nonNegativeInteger)
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- Ellis Kolchin (en)
- Sophus Lie (en)
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- Ellis (en)
- Sophus (en)
- V.V. (en)
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- Lie (en)
- Kolchin (en)
- Gorbatsevich (en)
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- 1876 (xsd:integer)
- 1948 (xsd:integer)
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- Le théorème de Lie-Kolchin est un résultat de trigonalisabilité des sous-groupes connexes et résolubles du groupe des matrices inversibles GLn(K), où K est un corps algébriquement clos de caractéristique quelconque. Démontré en 1948, il tient son nom de son auteur, Ellis Kolchin, et de son analogie avec le théorème de Lie sur les algèbres de Lie résolubles (en caractéristique nulle), démontré en 1876 par Sophus Lie. (fr)
- Теорема Ли — Колчина — это теорема теории представлений линейных алгебраических групп. Теорема Ли является аналогом для линейных алгебр Ли. (ru)
- 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 The result for Lie algebras was proved by Sophus Lie and for algebraic groups was proved by Ellis Kolchin . The Borel fixed point theorem generalizes the Lie–Kolchin theorem. (en)
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- Théorème de Lie-Kolchin (fr)
- Lie–Kolchin theorem (en)
- Теорема Ли — Колчина (ru)
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