Lie-Kolchin Theorem

The Lie-Kolchin Theorem is a result in the representation theory of linear algebraic groups, named for Sophus Lie and Ellis Kolchin. It states that a connected and solvable linear algebraic group acting on a nonzero finite-dimensional vector space over an algebraically closed field has a common eigenvector for all its elements, so the group can be represented, in a suitable basis, entirely by upper triangular matrices. The theorem is the algebraic-group analogue of Lie's theorem for linear Lie algebras, which gives the corresponding statement for solvable Lie algebras of matrices.