---
tag: 5JSYA1JNCC
kind: concept
name: Borel subgroup
msc: [20G07, 20G15]
sources:
  - url: https://en.wikipedia.org/wiki/Borel_subgroup
same_as:
  wikidata: Q4944913
  wikipedia: Borel subgroup
---

## Summary

A Borel subgroup of a [[linear algebraic group|GA81VVQHEP]] is a maximal connected [[solvable|0FXJ7YFMMX]] closed subgroup.
In $\mathrm{GL}_n$ the upper triangular matrices form a Borel subgroup.
Over an algebraically closed field all Borel subgroups are conjugate, and the quotient by one of them is a projective variety, the [[flag variety|NHRWCQQKTH]].
Borel subgroups are the starting point of the structure theory of [[reductive groups|2XS84PWE3S]].

## Explanations

### L1 human

Solvable groups are the easy groups, built up from commutative pieces.
A Borel subgroup is a largest easy piece of a linear algebraic group that is still connected.
For invertible matrices, the upper triangular ones are such a piece.
Every other Borel subgroup of $\mathrm{GL}_n$ is the stabilizer of some full flag of subspaces, a conjugate of the triangular group.