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- Lie–Kolchin_theorem abstract "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 .Because every (nonzero finite-dimensional) representation of G has a one-dimensional invariant subspace according to the Lie–Kolchin theorem, every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one, which 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 Sophus Lie (1876) and for algebraic groups was proved by Ellis Kolchin (1948, p.19).The Borel fixed point theorem generalizes the Lie–Kolchin theorem.".
- Lie–Kolchin_theorem wikiPageID "1020661".
- Lie–Kolchin_theorem wikiPageRevisionID "543852736".
- Lie–Kolchin_theorem authorlink "Ellis Kolchin".
- Lie–Kolchin_theorem authorlink "Sophus Lie".
- Lie–Kolchin_theorem first "Ellis".
- Lie–Kolchin_theorem first "Sophus".
- Lie–Kolchin_theorem first "V.V.".
- Lie–Kolchin_theorem id "l/l058710".
- Lie–Kolchin_theorem last "Gorbatsevich".
- Lie–Kolchin_theorem last "Kolchin".
- Lie–Kolchin_theorem last "Lie".
- Lie–Kolchin_theorem loc "p.19".
- Lie–Kolchin_theorem year "1876".
- Lie–Kolchin_theorem year "1948".
- Lie–Kolchin_theorem subject Category:Lie_algebras.
- Lie–Kolchin_theorem subject Category:Representation_theory_of_algebraic_groups.
- Lie–Kolchin_theorem subject Category:Theorems_in_representation_theory.
- Lie–Kolchin_theorem comment "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. ".
- Lie–Kolchin_theorem label "Lie–Kolchin theorem".
- Lie–Kolchin_theorem label "Théorème de Lie-Kolchin".
- Lie–Kolchin_theorem sameAs Lie%E2%80%93Kolchin_theorem.
- Lie–Kolchin_theorem sameAs Théorème_de_Lie-Kolchin.
- Lie–Kolchin_theorem sameAs Q3527113.
- Lie–Kolchin_theorem sameAs Q3527113.
- Lie–Kolchin_theorem wasDerivedFrom Lie–Kolchin_theorem?oldid=543852736.