Lie-Kolchin theorem

In mathematics, the Lie-Kolchin theorem is a theorem in the representation theory of linear algebraic groups.

It states that if G is a linear algebraic group,

<math>\rho: G \rightarrow GL(V)<math>

an irreducible representation on a finite-dimensional vector space V, and B a Borel subgroup of G, then there is a one-dimensional linear subspace L of V, such that

<math>\rho(B)(L) = L<math>.

That is, ρ(B) has an invariant line L, on which B therefore acts through a one-dimensional representation.

This result is named for Sophus Lie and Ellis Kolchin (1916–1991).

Applications

To be written