A Course On Exceptional Groups
Speaker: Maneesh Thakur, Indian Statistical Institute, Bangalore, India.
Schedule: 3, 10, 17, 24, and 31, July 2021.
Time: 1600 Indian Standard Time (One and half an hour each) (Changed from earlier announced time 11AM)
Tutorial: Wednesday at 1730 Indian Statndard Time
Platform: Online over google meet. (Please write at grouptheory21@gmail.com to get the link). We will try our best to make the recorded lectures available on YouTube.
Host: Shripad Garge, IIT Bombay, Mumbai, India
Videos and Notes
7/7/2021 Tutorial I
10/7/2021 Lecture II Video
14/7/2021 Tutorial II
17/7/2021 Lecture III Video
21/7/2021 Tutorial III
24/7/2021 Lecture IV Video
28/7/2021 Tutorial IV
After a successful Advanced Instructional School (AIS) in 2019 on linear algebraic groups, it was expected to have a follow-up soon after. However, the pandemic forced our hands and it is only now that we are able to have this follow-up of the AIS.
In the AIS, our main reference was the book on the topic by T. A. Springer. We covered its first 10 chapters going up to the classification of simple algebraic groups over an algebraically closed field, say C. The classification theorem says that a simple algebraic group over C is isogenus to one of the following groups:
SL(n+1), SO(2n+1), Sp(2n), SO(2n), G_2, F_4, E_6, E_7 and E_8.
The first four families of groups, the blue ones, are called "classical groups" and are known to us since our graduation days. However, the last five groups are not so well-known. They are the "exceptional groups". These are called exceptional because, unlike the classical ones, these are not the automorphism groups of finite dimensional associative algebras with or without involutions.
In this course, Prof. Maneesh Thakur will give descriptions of these five exceptional groups and will discuss several of their properties.
The pre-requisites for the course are listed below:
Quadratic forms with an emphasis on Pfister forms, ([1, 2]),
Clifford algebras and spin groups, ([1, 2]),
Galois cohomology, H^1(k, Aut(X)) classifies twisted forms of X over k, ([3]),
Brauer groups, ([1]),
Theory of (affine) algebraic groups, ([3]).
References
N. Jacobson "Basic Algebra", II, Second edition, W. H. Freeman and company, 1989.
T. Y. Lam, "Introduction to quadratic forms over fields", Graduate Studies in Mathematics, 67, American Mathematical Society, 2005.
T. A. Springer, "Linear algebraic groups", reprint of the 1998 second edition, Birkhauser, 2009.