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
\rho\colonG\toGL(V)
a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that
\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
\rho(g),g\inG
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 and for algebraic groups was proved by .
The Borel fixed point theorem generalizes the Lie–Kolchin theorem.
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 of V the image
\rho(G)
\rho(G)
The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G.
If the field K is not algebraically closed, the theorem can fail. The standard unit circle, viewed as the set of complex numbers
\{x+iy\inC\midx2+y2=1\}
\rho(z)
z=x+iy
\begin{pmatrix}x&y\ -y&x\end{pmatrix}.