John Baez Algebraic groups

Algebraic groups

Some thoughts on a course.

Introductory stuff

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.

Generalization

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.