Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive) algebraic groups, in Jacques Tits' theory of groups with a (B, N) 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. (In general, each node of the Dynkin diagram determines a simple negative root and thus a one-dimensional 'root group' of G. A subset of the nodes thus yields a parabolic subgroup, generated by B and the corresponding negative root groups. Moreover, any parabolic subgroup is conjugate to such a parabolic subgroup.) The corresponding subgroups of the Weyl group of G are also called parabolic subgroups, see Parabolic subgroup of a reflection group.
Example
Let . A Borel subgroup of is the set of upper triangular matrices
and the maximal proper parabolic subgroups of containing are