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 GG:

  • GL(n,k)GL(n,k)

  • SL(n,k)SL(n,k)

  • O(n,k)O(n,k)

  • SO(n,k)SO(n,k)

  • Sp(n,k)Sp(n,k).

Define projective varieties.

A parabolic subgroup PP of a linear algebraic group GG is one such that the quotient variety G/PG/P is projective. Examples from G=SL(n)G = SL(n).

If PP is parabolic we call G/PG/P a (generalized) flag variety. If PP is a maximal parabolic we call G/PG/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 GG.

A subgroup PP is parabolic iff it contains some Borel subgroup.

All Borel subgroups of GGareconjugateifGG are conjugate - if k$ is algebraically closed?

Definition of a maximal torus.

The intersection of any two Borel subgroups of GG 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 GG if and only if GG is reductive.

Last revised on August 21, 2016 at 05:37:29. See the history of this page for a list of all contributions to it.