# Lie-Kolchin theorem

From Groupprops

## Contents

## Statement

If a connected linear algebraic group over an algebraically closed field is solvable then we can choose a basis for the vector space in which the *whole* group becomes a subgroup of the group of upper triangular matrices.

More explicitly:

If is a vector space over an algebraically closed field , and is a solvable connected closed subgroup of , then one can choose a basis for in which all the element of are represented by upper triangular matrices.

## Related results

### Similar facts

- Triangulability theorem: This states that every
*element*can be conjugated to an upper triangular matrix. - Borel fixed-point theorem
- Kolchin's theorem

### Analogues for Lie algebras

## Facts used

- Borel-Morozov theorem which in turn uses the Borel fixed-point theorem: The theorem states that any Borel subgroup contains a conjugate of any connected solvable closed subgroup

## Proof

The proof follows directly from Fact (1), applied to the group .