nLab Borel subgroup



Group Theory




Given an algebraic group GG a subgroup BGB\subset G is said to be a Borel subgroup if it is maximal (under inclusion) among all Zariski closed connected solvable subgroups.


They appear to be minimal parabolic subgroups (that is minimal among those PP such that G/PG/P is a projective variety).

All the Borel subgroups are mutually conjugate and the intersection of any two contains a maximal torus in GG.

If the characteristic of the ground field is zero then the tangent Lie algebra of the Borel subgroup BB is “the” Borel subalgebra of the Lie algebra of GG.


The main example is G=GL(n)G = GL(n) or G=SL(n)G = SL(n) where a corresponding Borel subgroup can be taken to be the subgroup B +B^+ of the upper triangular matrices in GG and B B^-, the subgroup of the lower triangular matrices.

These two subgroups are said to be mutually opposite in the sense that their intersection B +B B^+\cap B^- is precisely the maximal torus, which is in this case the subgroup of the diagonal matrices. If G=SL(n)G = SL(n) the quotient homogeneous spaces SL(n)/BSL(n)/B is called the flag variety and for a general semisimple Lie group GG, G/BG/B is called a generalized flag variety.


Last revised on April 21, 2014 at 22:08:07. See the history of this page for a list of all contributions to it.