MT4283/MT6423 Linear Algebraic Groups

(January - April 23, IISER Pune)

Instructor : Dr Anupam Singh

Audience : 4th year BS-MS, MSc, iPhD, PhD students at IISER Pune

Schedule : Tuesday, Thursday @ 11AM

Evaluation : mid-sem - 50%, end-sem - 50%

Prerequisite : Basic Algebra courses (Groups, Linear Algebra, Affine Variety etc)

Goal of the course : The course is aimed at graduate students interested in Lie theory or subjects where they might need the basics of algebraic group theory such as Number Theory. The course will introduce them to some aspects of this wonderful mathematics.

Proposed Content : Affine varieties, algebraic groups, examples of classical groups, connectedness, Lie algebra of algebraic groups, group actions, Jordan decomposition, Tori, solvable groups, Lie Kolchin theorem, Borel subgroups, Semisimple and reductive groups, Classification theory of Chevalley.

Text books and reference material :

A. Linear algebraic groups : Humphreys

B. Linear algebraic groups and finite groups of lie type : Malle and Testerman

C. An introduction to algebraic geometry and algebraic groups : Geck

Related courses in past:  Classical group 2007    Algebraic Groups 2015    Lie Algebra 2017

Are you bored? Try these exercises!  From Linear Algebra to Algebraic Groups

Weekly topics covered :

3 January 23 - Overview, Definition, examples

5 January 23 - examples of classical groups

10 January 23 - Irreducible components, connected component of an algebraic group

12 January 23 - Connectedness of algebraic group

17 January 23 - Group action, closeness of normalizer and centralizer, orbits

19 January 23 - Linearity of affine algebraic groups

24 January 23 - Jordan-Chevalley decomposition

26 January 23 - Holiday

31 January 23 - Root system (by Sutirtha)

2 February 23 - Root system (by Sutirtha)

7 February 23 -  No class 

9 February 23 - Classification of root system via Dynkin diagrams (by Aniketh)

14 February 23 - Commutative algebraic groups

16 February 23 - Character group

23 February 23  Mid-sem exam

25 February 23 - Extra class -- classification of diagonalisable groups using character group

28 February 23 - no class

2 March 23 -  some exercises (by Ayon)

7 March 23 - Holiday

9 March 23 - Rigidity of Tori

14 March 23 - Nilpotent and Solvable groups

16 March 23 - Lie-Kolchin Theorem 

21 March 23 - Structure of connected solvable and nilpotent groups

23 March 23  (class held on 22 March due to Maths Symposium) - Radical and unipotent radicals, definition and examples of semisimple and reductive groups

28 March 23- Borel fixed point theorem

30 March 23- Quotient of algebraic groups

31 March 23- (extra class) example of GL_n

4 April 23 - Holiday - extra class- Conjugacy of Borel subgroups and maximal tori

6 April 23 - Borel and parabolics in symplectic group

11 April 23 - Root system, root datum, example of rank 1 groups

13 April 23 - Root datum of GL_n

End Sem exam - 20 April 2023