Borel subgroup

Remarks

Over an arbitrary field $k$, a closed subgroup $B$ of $G$ is called a Borel subgroup if $B \times_k \bar{k}$ is a Borel subgroup of $G \times_k \bar{k}$.

Example: in $G = \mathrm{GL}_n$ the group $B$ of invertible upper triangular matrices is a Borel subgroup. It is closed, cut out by $x_{ij} = 0$ for $i > j$. It is connected and solvable, since its commutator subgroup consists of unipotent upper triangular matrices. It is maximal: by the Lie–Kolchin theorem, a connected solvable subgroup $B' \supseteq B$ fixes some full flag, and the only full flag fixed by $B$ is the standard one, so $B' \subseteq B$.

Non-example: in $\mathrm{GL}_2$, the diagonal torus $T$ is closed, connected and solvable, but it fails clause 3, since $T \subsetneq B$ for the upper triangular group $B$.

All Borel subgroups of $G$ are conjugate, and $G / B$ is a projective variety.