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Linear Algebraic Groups TCC 2021/2022

Welcome to the webpage for the TCC course Linear Algebraic Groups. For information on signing up etc please visit the TCC webpage.

Schedule Lectures are 10:00 - 12:00 on Thursdays for 8 weeks starting January 20th 2022. They take place online using Teams.

Notes I plan to use OneNote to handwrite my lectures. The following link should give you access to view the notebook. OneNote Notebook

Overview of the course We plan to introduce linear algebraic groups, from an algebraic point of view. So we plan to cover the classification of linear algebraic groups (over algebraically closed fields), their subgroup structure and their representation theory. Given the time constraints some things will be rather brief but the aim is to give a guide to the basic theory with plenty of examples along the way.

Pre-requisites To follow the course you need an interest and basic knowledge of group theory and parts of a first course in algebraic geometry (affine varieties, connectedness etc) although this could be learnt simultaneously. It will help if you have taken advanced algebra courses before, especially covering the basics in representation theory and/or complex Lie algebras.

References The main reference is the book Linear Algebraic Groups by Gunter Malle and Donna Testerman. There will be some further references required for the end of the course and these will be mentioned below. There are many books on linear algebraic groups that are useful, many with the title Linear Algebraic Groups, namely by Borel, Humphreys and Springer. Lots of other references are available - some will cover algebraic groups from the group scheme theoretic point of view (eg Milne's notes which are now the book Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field) which is great, but not how this course is going to introduce them.

Assessment If you need/want to take this module for credit, please send me an email. This will done through handing in of some solutions to the assignment sheets. Further details will be provided in due course.

Guide to weekly content (updated as we go along)

Week Content of lecture References/Further Reading
1 (20/1) Introduction, quick review of varieties, definition and examples of algebraic groups, connectedness, dimension, Jordan decomposition, unipotent/semisimple elements Ch. 1, 2
2 (27/1) Tori, Soluble Algebraic Groups, (briefly) quotients of algebraic groups Ch. 3,4,5
3 (3/2) Borel subgroups, Lie algebra of a LAG Ch. 6,7
4 (10/2) SL2/PGL2, root systems, coroot systems, root datum leading to the classification of semisimple LAGs Ch. 8,9
5 Representation Theory Ch. 15, 16
6 BN-pairs, parabolic subgroups (+ complete reduciblity if time) Ch. 11,12,17
7 Centralisers and Maximal subgroups Ch. 13, 14, 18, 19
8 Finite Groups of Lie Type (+ more complete reduciblity if time) Part 3: Ch 21, 22, ....