Student Seminar: Linear Algebraic Groups (2h):
Linear Algebraic Groups
LV-Nummer: ??
Time and Place: Thursdays at 10:00 in MA 548
First seminar: 16/04/2026
The goal of this seminar is to ...
Student Seminar: Linear Algebraic Groups (2h):
Linear Algebraic Groups
LV-Nummer: ??
Time and Place: Thursdays at 10:00 in MA 548
First seminar: 16/04/2026
The goal of this seminar is to introduce the basic theory of linear algebraic groups, with a focus on concrete examples and the role of tori and representations. The seminar will follow the book Humphreys Linear Algebraic Groups.
We emphasise explicit matrix groups and their structure. The central theme of the seminar is the interaction between algebraic groups and their representations, in particular via weight space decompositions arising from torus actions.
The seminar is intended to be accessible to students with a basic background in algebra (Algebra I and Algebra 2) and some familiarity with representation theory. Some prior exposure to algebraic geometry (e.g. affine varieties) will be helpful.
The seminar complements the lecture course on Group actions, Representations and Geometric Invariant Theory, held by Peter Bürgisser.
Provisional Table of Content
Algebraic Groups
Affine algebraic groups and examples (GLn,SLn)
Morphisms and group actions
Basic properties and constructions
Lie Algebras
Lie algebra of an algebraic group
Tangent space at the identity
Differential of representations
Tori and Representations
Algebraic tori and characters
Diagonalisation of torus actions
Weight space decompositions
Examples and computations
Structure Theory
Solvable and unipotent groups
Jordan decomposition
Borel Subgroups
Definition and examples
Relation to tori and representations
If there is time:
Further Structure
Weyl groups (introductory)
Unipotent radicals