Reductive group

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.