Editing Lie–Kolchin theorem

{{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.

== 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 (linear algebra)|basis]] of ''V'' the image <math>\rho(G)</math> has a ''triangular shape''; in other words, the image group <math>\rho(G)</math> is conjugate in GL(''n'',''K'') (where ''n'' = dim&thinsp;''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 algebraic group|semisimple]] [[linear algebraic group]] ''G''.

== Counter-example ==

If the field ''K'' is not algebraically closed, the theorem can fail. The standard [[unit circle]], viewed as the set of [[complex number]]s <math> \{ x+iy \in \mathbb{C} \mid x^2+y^2=1 \} </math> of absolute value one is a 1-dimensional [[abelian group|commutative]] (and therefore solvable) [[linear algebraic group]] over the [[real number]]s which has a 2-dimensional representation into the [[special orthogonal group]] SO(2) without an invariant (real) line.  Here the image <math>\rho(z)</math> of <math>z=x+iy</math> is the [[orthogonal matrix]]

: <math>\begin{pmatrix} x & y \\ -y & x \end{pmatrix}.</math>

==References==
{{refbegin}}
*{{SpringerEOM|first=V.V.|last= Gorbatsevich|title=Lie-Kolchin theorem}}
*{{Citation | last1=Kolchin | first1=E. R. | title=Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations | jstor=1969111 | mr=0024884 | zbl=0037.18701 | year=1948 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=49 | issue=1 | pages=1–42 | doi=10.2307/1969111}}
*{{Citation | last1=Lie | first1=Sophus | author1-link=Sophus Lie | title=Theorie der Transformationsgruppen. Abhandlung II | url=https://archive.org/details/archivformathem02sarsgoog | year=1876 | journal=Archiv for Mathematik og Naturvidenskab | volume=1 | pages=152–193}}
*{{citation |author1-link=William C. Waterhouse |chapter=10. Nilpotent and Solvable Groups §10.2 The Lie-Kolchin Triangularization Theorem |first1=William C. |last1=Waterhouse |title=Introduction to Affine Group Schemes |chapter-url=https://books.google.com/books?id=SpfwBwAAQBAJ&pg=PR1 |date=2012 |orig-year=1979 |publisher=Springer |series=Graduate texts in mathematics |volume=66 |isbn=978-1-4612-6217-6 |pages=74–75}}
{{refend}}

{{DEFAULTSORT:Lie-Kolchin theorem}}
[[Category:Lie algebras]]
[[Category:Representation theory of algebraic groups]]
[[Category:Theorems in representation theory]]