Borel subgroup

The following property can be evaluated for a closed subgroup of an algebraic group

Definition

A Borel subgroup of an algebraic group is a subgroup satisfying the following equivalent conditions:

1. It is maximal among connected solvable closed subgroups of the whole group.
2. It is minimal among parabolic subgroups of the whole group, i.e., it is a closed subgroup such that the quotient is a complete variety but such that there is no smaller closed subgroup for which the quotient is a complete variety.

Facts

When the base field is algebraically closed all the Borel subgroups form a single conjugacy class. In other words, upto conjugacy, we can talk of the Borel subgroup.