algebraicgroups

Introduction to Linear Algebraic Groups

Instructor : Dr. Anupam Singh, IISER Pune

Schedule : 2 lectures + 1 example session

Tuesday, Wednesday and Thursday : 11:30-12:30

Course Content : Affine varieties, Linear 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.

References :

    • Linear Algebraic Groups and Finite Groups of Lie Type : Malle and Testerman

    • Linear Algebraic Groups : Humphreys

    • Conjugacy Classes in Algebraic Groups : Steinberg LNM366

    • An Introduction to Algebraic Geometry and Algebraic Groups : Geck

Prerequisite : Affine varities, Morphisms, Projective varities, Dimension, Irreducible varities etc.

Notes and Examples :

Assumptions : In this course we work over algebraically closed field with affine algebraic groups until mentioned otherwise.

5 August 2015 - Affine varities, Morphisms, Affine algebraic groups, Morphism of algebraic groups.

6 August 2015 - Examples of Algebraic Groups

11 August 2015 - Irreducible variety, Connected component of an algebraic group.

12 August 2015 - How would one prove certain algebraic groups are connected? Algebraic group action on affine varities.

13 August 2015 - Examples of connected algebraic groups

18 August 2015 - Affine algebraic groups are linear!

19 August 2015 - Jordan decomposition in finite dimension

20 August 2015 - Examples - Jordan decomposition, character group

25 August 2015 - Locally finite transformations, Jordan-Chevally decomposition for elements of algebraic groups

26 August 2015 - Unipotent groups and upper triangulation, Structure of Commutative algebraic groups

27 August 2015 - Examples that semisimple elements need not be subgroup, irreducible representations of SL(2,C)

1 September 2015 - Diagonalisable groups and Characters

2 September 2015 - Characterization of diagonalisable groups using character group and group algebra of the character group.

3 September 2015 - Examples of maximal tori in classical groups

8 September 2015 - Rigidity of Tori and finiteness of Weyl group

9 September 2015 - Solvable Groups and Lie-Kolchin theorem

10 September 2015 - Examples (Action of GL(V) on flags)

15 September 2015 - Borel and Parabolic subgroups of GL and Levi decomposition

16 September 2015 - Root datum of GL_n

22 September 2015 - Root datum of SL_2 and PGL_2

23 September 2015 - Projective variety, Flag Variety, variety structure on quotient G/H

24 September 2015 - Root datum of Sp_{2l} and GSp_{2l}

6 October 2015 - Lie algebra of an algebraic group, Ad representation

7 October 2015 - Root datum (abstract definition), Root datum vs. root system

8 October 2015 - Computing co-roots

13 October 2015 - Invited lecture by Dr. Vivek Mallick on Abelian Varietes

14 October 2015 - Orthogonal groups, Maxmial tori in orthogonal groups (char not 2)

15 October 2015 - Root datum of O_{2l} (type D_l)

20 October 2015 - Root datum of O_{2l+1} (type B_l)

03 November 2015 - Borel Fixed point Theorem, Conjugacy of Borel subgroups

04 November 2015 - Structure of connected solvable groups, conjugacy of maximal tori

05 November 2015 - Example - Parabolic subgroups of GL_n and symplectic groups -

relation with simple root system

10 November 2015 - Parabolic subgroups of symplectic group

12 November 2015 - Parabolic subgroups of orthogonal group

17 November 2015 - Computing Weyl groups