---
tag: VK7402274F
kind: definition
name: Reductive group
msc: [20G15, 20G07]
defines: 2XS84PWE3S
uses: [GA81VVQHEP, 07BSPZPSMF, 0CB0DX5BVV, 0FCHE902D7]
sources:
  - url: https://en.wikipedia.org/wiki/Reductive_group
---

## Definition

Let $k$ be an algebraically closed field, and let $G$ be a smooth connected [[linear algebraic group|GA81VVQHEP]] over $k$.
An element $g \in \mathrm{GL}_n(k)$ is *unipotent* if $g - I$ is a [[nilpotent matrix|07BSPZPSMF]].
A linear algebraic group is *unipotent* if all its points over $k$ are unipotent in some, equivalently every, faithful representation.
The *unipotent radical* $R_u(G)$ is the largest smooth connected unipotent [[normal subgroup|0CB0DX5BVV]] of $G$.
The group $G$ is *reductive* if $R_u(G)$ is trivial.

## Remarks

Over an arbitrary field $k$, a smooth connected linear algebraic group $G$ is called reductive if $G \times_k \bar{k}$ is reductive.

Example: $\mathrm{GL}_n$ is reductive.
Let $U$ be a connected normal unipotent subgroup.
By Kolchin's theorem, the vectors fixed by $U$ form a nonzero subspace of $k^n$.
Since $U$ is normal, $\mathrm{GL}_n$ preserves this subspace, so it is all of $k^n$ and $U$ is trivial.

Non-example: the group $B$ of invertible upper triangular $2 \times 2$ matrices is not reductive.
The subgroup of matrices $\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix}$ is normal in $B$, connected and unipotent.

In characteristic $0$, a smooth connected linear algebraic group is reductive if and only if all its finite-dimensional representations are semisimple.