Borel subgroup - Knowlpedia

Definition

Let $G$ be a connected linear algebraic group over an algebraically closed field. A Borel subgroup $B\subseteq G$ is a maximal connected solvable closed subgroup.

For a reductive group, a Borel subgroup contains a maximal torus $T$ , and the choice $T\subseteq B$ determines a set of positive roots. Any two Borel subgroups are conjugate over an algebraically closed field.

Examples

In $GL_n$ , the invertible upper-triangular matrices form a Borel subgroup. Its image in $PGL_n$ is again a Borel subgroup. For $SL_2$ , the upper-triangular subgroup stabilizes the line spanned by the first standard basis vector.

Relation to flag varieties

The quotient $G/B$ is the complete flag variety of $G$ . Over a field that is not algebraically closed, one distinguishes Borel subgroups defined over the base field from geometric Borel subgroups after base change.

References

Armand Borel, “Groupes linéaires algébriques,” Annals of Mathematics 64 (1956), 20–82. DOI.