Instructor: Amit Kumar Singh

Books:

1. Linear Algebraic Groups, by A. Borel.

2. Linear Algebraic Groups, by J. E. Humphreys.

3. Lie Algebras and Algebraic Groups, by P. Tauvel and R. W. T. Yu.

Lecture notes: L-1, L-2, L-3, L-4, L-5&6 

Topics covered

Lecture 1: Reflectios

Lecture 2: Defining the root system in a finite dimensional inner product space, and proving some basis properties of the root systems.

Lecture 3: Studied some examples of root systems of rank two. Defined the base of a root system and studied their properties that require understanding Cartan matrices and Dynkin diagrams.

Assignment 1 (due March 9, 2024)

Lecture 4: Studied the Cartan matrices and Dynkin diagrams. Proved that the Dynkin diagrams of the root systems are the same if and only if their corresponding root systems are isomorphic.

Lecture 5: Studied the classification of Dynkin diagrams

Suggested reading: Section 11 of Humphreys' book on Intro. to Lie Algebras and Representation Theory.

Lecture 6: Studied the classification of Dynkin diagrams (contd.)

Suggested reading: Section 11 of Humphreys' book on Intro. to Lie Algebras and Representation Theory.

From the next lectures onward, we will focus more on algebraic groups. This is a continuation of the course (Aug-Nov 2023, Linear Algebraic Groups)

Lecture 7: Defined the Weyl groups of a connected algebraic group G relative to a torus. Studied that for a maximal torus of G, the Weyl group W(G, T) of a connected algebraic group relative to T acts simply transitively on the set of the Borel subgroups of G containing T. 

Suggested reading:  Humphreys, section 24.1, 24.2

Assignment 2 (due March 28, 2024)

Lecture 8: Let φ: G → G'  be an epimorphism of linear algebraic groups that carries maximal torus T to a maximal torus φ(T). Then φ induces surjective maps from the set of Borel subgroups of G containing T to the set of Borel subgroups of G' containing φ(T) and  W(G, T) → W(G, φ(T)). Moreover, if the kernel lies in all the Borel subgroups of G, then the induced map from the Weyl groups is an isomorphism. 

Suggested reading:  Humphreys, section 24.1, 24.2 

Lecture 9: Regular tori, singular tori and the roots of diagonalizable groups.

Suggested reading:  Humphreys, section 24.3.

Lecture 10: Action of a maximal torus of G/B.

Suggested reading:  Humphreys, section 25.1.

Lecture 11: Action of a maximal torus of G/B (contd).

Suggested reading:  Humphreys, section 25.1.

Assignment 3 (due April 18, 2024)

Lecture 12: (Existence of enough fixed points) Let P be a proper parabolic subgroup of an affine algebraic group and T any torus in G. Then T fixes at least two points of G/P. In particular, a maximal torus lies in at least two Borel subgroups. 

Suggested reading:  Humphreys, section 25.2.

Lecture 13: A characterisation of the semisimple rank one affine linear algebraic groups. 

Suggested reading:  Humphreys, section 25.3.

Lecture 14: A characterisation of the semisimple rank one affine linear algebraic groups (contd). 

Suggested reading:  Humphreys, section 25.3.

End-sem exam (May 10, 2024)

Assessment: Assignments - 40%, End Sem - 60%