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-ÂKolchin 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
Reductive Groups : J. S. Milne
Linear Algebraic Groups : T. A. Springer
Linear Algebraic Groups : Borel
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