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  "body": "\n## Summary\n\nA Borel subgroup of a [[linear algebraic group|GA81VVQHEP]] is a maximal connected [[solvable|0FXJ7YFMMX]] closed subgroup.\nIn $\\mathrm{GL}_n$ the upper triangular matrices form a Borel subgroup.\nOver an algebraically closed field all Borel subgroups are conjugate, and the quotient by one of them is a projective variety, the [[flag variety|NHRWCQQKTH]].\nBorel subgroups are the starting point of the structure theory of [[reductive groups|2XS84PWE3S]].\n\n## Explanations\n\n### L1 human\n\nSolvable groups are the easy groups, built up from commutative pieces.\nA Borel subgroup is a largest easy piece of a linear algebraic group that is still connected.\nFor invertible matrices, the upper triangular ones are such a piece.\nEvery other Borel subgroup of $\\mathrm{GL}_n$ is the stabilizer of some full flag of subspaces, a conjugate of the triangular group.\n",
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