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 group GL_n (n x n invertible matrices), 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 (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.