Borel subgroup

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

Definition

A Borel subgroup of a linear algebraic group over a field 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.

Equivalence of definitions

Further information: equivalence of definitions of Borel subgroup

Caveat

Zariski topology differs from Lie group topology

Note that both connected and closed are used for the Zariski topology on the underlying algebraic variety. In particular, a Borel subgroup need not be connected with respect to other topologies, such as the topology arising from analytic structures under the natural interpretation of the group as a Lie group when working over the field of real numbers, field of complex numbers, or field of p-adic numbers.

Definitions do not work over finite fields

The definition as presented here does not work over finite fields because Zariski topologies for varieties over finite fields are discrete. In such cases, we typically pass to an algebraically closed field containing the finite field, find a Borel subgroup there, and then look at those points in the Borel subgroup that are defined over the original finite field.

Facts

• Borel fixed-point theorem: This states under certain conditions, an action of a Borel subgroup must have a fixed point.
• Borel-Morozov theorem: 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.
• Borel subgroup is conjugate-dense in connected algebraic group
• Maximal solvable subgroup in connected algebraic group need not be Borel subgroup