nLab Borel subgroup

Idea

Given an algebraic group $G$ a subgroup $B\subset G$ 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 $P$ such that $G/P$ is a projective variety).

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

If the characteristic of the ground field is zero then the tangent Lie algebra of the Borel subgroup $B$ is “the” Borel subalgebra of the Lie algebra of $G$.

Example

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

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

Related concepts

• Weyl group,

• parabolic subgroup,

• Schubert calculus

References

• A. Borel, Groupes linéaires algébriques, Ann. of Math. (2) , 64 : 1 (1956) pp. 20–82
• Armand Borel, Linear algebraic groups, Springer
• wikipedia Borel subgroup
• eom: Borel subgroup