Editing Lie–Kolchin theorem (section)

{{short description|Theorem in the representation theory of linear algebraic groups}}
In [[mathematics]], the '''Lie–Kolchin theorem''' is a theorem in the [[representation theory]] of [[linear algebraic group]]s; '''[[Lie's theorem]]''' is the analog for [[linear Lie algebra]]s.

It states that if ''G'' is a [[connected space|connected]] and [[solvable group|solvable]] linear algebraic group defined over an [[algebraically closed]] [[field (mathematics)|field]] and

:<math>\rho\colon G \to GL(V)</math>

a [[group representation|representation]] on a nonzero [[dimension (vector space)|finite-dimensional]] [[vector space]] ''V'', then there is a 1-dimensional [[linear subspace]] ''L'' of ''V'' such that

: <math>\rho(G)(L) = L.</math>

That is, ρ(''G'') has an invariant line ''L'', on which ''G'' therefore acts through a 1-dimensional representation. This is equivalent to the statement that ''V'' contains a nonzero vector ''v'' that is a common (simultaneous) [[eigenvector]] for all <math>\rho(g), \,\, g \in G</math>.

It follows directly that every [[irreducible representation|irreducible]] finite-dimensional representation of a connected and solvable linear algebraic group ''G'' has dimension 1. In fact, this is another way to state the Lie–Kolchin theorem.
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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 1-dimensional invariant subspace.-->

The result  for Lie algebras was proved by {{harvs|txt|authorlink=Sophus Lie|first=Sophus |last=Lie|year=1876}} and for algebraic groups was proved by {{harvs|txt|authorlink=Ellis Kolchin|first=Ellis|last= Kolchin|year=1948|loc=p.19}}.

The [[Borel fixed point theorem]] generalizes the Lie–Kolchin theorem.