higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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.
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 $G$.
The main example is $G = GL(n)$ or $G = SL(n)$ where a corresponding Borel subgroup can be taken to be the sbgroup $B^+$ 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.