nLab
Borel subgroup

Context

Group Theory

Geometry

Contents

Idea

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.

Properties

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.

Example

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 sbgroup 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.

References

Revised on October 30, 2013 01:06:01 by Urs Schreiber (82.169.114.243)