Borel subgroup

Summary

A Borel subgroup of a linear algebraic group is a maximal connected solvable 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. Borel subgroups are the starting point of the structure theory of reductive groups.