Some thoughts on a course.
The idea of Klein geometry.
Euclidean geometry, projective geometry.
Lines, planes, etc. in projective geometry. Grassmannians. Grassmannians as projective varieties: Plücker embedding.
Flags and flag varieties.
Define algebraic sets, affine algebraic varieties, linear algebraic groups.
Give definition and examples of linear algebraic groups $G$:
$GL(n,k)$
$SL(n,k)$
$O(n,k)$
$SO(n,k)$
$Sp(n,k)$.
Define projective varieties.
A parabolic subgroup $P$ of a linear algebraic group $G$ is one such that the quotient variety $G/P$ is projective. Examples from $G = SL(n)$.
If $P$ is parabolic we call $G/P$ a (generalized) flag variety. If $P$ is a maximal parabolic we call $G/P$ a (generalized) Grassmannian.
A minimal parabolic is called a Borel subgroup. Equivalently - and this would take work to prove, I think - it’s maximal connected solvable algebraic subgroup of a linear algebraic group $G$.
A subgroup $P$ is parabolic iff it contains some Borel subgroup.
All Borel subgroups of $GG are conjugate - if$k$ is algebraically closed?
Definition of a maximal torus.
The intersection of any two Borel subgroups of $G$ contains a maximal torus of G; if this intersection is a maximal torus, such Borel subgroups are said to be opposite. Opposite Borel subgroups exist in $G$ if and only if $G$ is reductive.
Last revised on August 21, 2016 at 09:37:29. See the history of this page for a list of all contributions to it.