Borel subgroup

In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. There is an ambiguity here, which turns out not to matter much. In algebraic group theory, solvable is defined in terms of composition series of one-dimensional groups (as successive quotients), giving one meaning. In group theory, the meaning is simply abelian quotients. Looking though at connected groups G, and at their points over an algebraically closed field, one gets the same concept of Borel subgroup, this is shown a posteriori in the theory.

For the group GL_n, the subgroup of upper triangular matrices is a Borel subgroup. In general, for G connected, there is a single conjugacy class of Borel subgroups.

Subgroups between a Borel subgroup B and G (inclusive) are called parabolic subgroups. Another characterisation of a parabolic subgroup P is that G/P is a complete variety (a posteriori, a projective variety). Therefore another way to distinguish the Borel subgroups is as minimal parabolic subgroups: B is a Borel subgroup precisely when G/B is a homogeneous space for G and a projective variety, and "as large as possible".