Borel subgroup


In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group GLn, the subgroup of invertible upper triangular matrices is a Borel subgroup.
For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups.
Borel subgroups are one of the two key ingredients in understanding the structure of simple algebraic groups, in Jacques Tits' theory of groups with a pair. Here the group B is a Borel subgroup and N is the normalizer of a maximal torus contained in B.
The notion was introduced by Armand Borel, who played a leading role in the development of the theory of algebraic groups.

Parabolic subgroups

Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups.
Parabolic subgroups P are also characterized, among algebraic subgroups, by the condition that G/P is a complete variety.
Working over algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense. Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is "as large as possible".
For a simple algebraic group G, the set of conjugacy classes of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding Dynkin diagram; the Borel subgroup corresponds to the empty set and G itself corresponding to the set of all nodes.

Example

Let. A Borel subgroup of is the set of upper triangular matrices
and the maximal proper parabolic subgroups of containing are
Also, a maximal torus in is
This is isomorphic to the algebraic torus.

Lie algebra

For the special case of a Lie algebra with a Cartan subalgebra, given an ordering of, the Borel subalgebra is the direct sum of and the weight spaces of with positive weight. A Lie subalgebra of containing a Borel subalgebra is called a parabolic Lie algebra.