classicalgroups

This is the topic for a course which I will be giving at IMSc, Chennai this semester (August-November 2007). It will be aimed at first year graduate students.

Course Description:

The prerequisite for this course is elementary Linear Algebra, Group Theory and Ring Theory. The following topics will be covered in that order.

1. Classical Groups (bilinear forms, quadratic forms, orthogonal groups, symplectic groups and unitary groups)

Reference: a) Classical Groups and Geometric Algebras by L.C. Groves

b) Classical Groups by P. J. Cameron http://www.maths.qmul.ac.uk/~pjc/class_gps/

c) Representations and Invariants of Classical groups by Goodman and Wallach

2. Central Simple Algebras (central simple algebra, Wedderburn's theorem, Splitting fields)

Reference: a) Noncommutative algebra by Farb and Dennis

b) TIFR lecture notes on Semisimple algebras

(pdf file is here, thanks to Shripad!)

3. Algebras with Involution and Classical Groups (description of maximal tori in classical groups, classical groups defined over k etc.)

Reference: Chapter 1 of "The Book of Involutions" by KMRT

4. Finite Groups of Lie type, Introduction to Chevalley groups (Root systems, Semisimple Lie algebra, Chevalley Groups)

Reference: "Simple Groups of Lie type" by Carter.

(Warning: The last topic may not be covered in this course.)

Schedule :

The course will meet twice a week on Tuesday and Friday from 13:45 to 15:15.

I am planning to devote one of the lectures every alternate week for exercises (hope we will have enough of them!)

Notes for the Course :

I am planning to put up notes for this course here. Though I may not manage to write everything in details in time. I will try to make available the rough notes as course progresses with reference to all materials.