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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 ρ : G → G L ( V ) {\displaystyle \rho \colon G\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 ρ ( G ) ( L ) = L .

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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 ρ : G → G L ( V ) {\displaystyle \rho \colon G\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 ρ ( G ) ( L ) = L .
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  • Lie–Kolchin theorem
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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 ρ : G → G L ( V ) {\displaystyle \rho \colon G\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 ρ ( G ) ( L ) = L . {\displaystyle \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 ρ ( g ) , g ∈ G {\displaystyle \rho (g),\,\,g\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 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.
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  • Sophus Lie
  • Ellis Kolchin
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  • Ellis
  • V.V.
  • Sophus
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  • l/l058710
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  • Lie
  • Gorbatsevich
  • Kolchin
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  • p.19
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