Abstracts of Mini Courses
A. Habib and U. Kulkarni.
Basic definitions and examples; Solvable and Nilpotent Lie algebras; Theorems of Lie and Cartan; Killing form; Complete reducibility; Representations of sl(2); Root space decomposition.
References: Chapters 1, 2 of [Humphreys].
B. Root systems and the classification theorem (5 Lectures).
Root systems; Weyl group; Classification for irreducible root systems; Automorphisms of root systems; Isomorphism and conjugacy theorems (especially for Cartan and Borel subalgebras of a semisimple Lie algebra).
References: Chapters 3,4 of [Humphreys].
C. Representations of Lie algebras (6 Lectures).
Anupam Kumar Singh.
Abstract theory of weights; Universal enveloping algebra; PBW theorem; Classification of finite dimensional representations; Time permitting, Weyl's character and dimension formulas.
References: Sections 13, 17, 20, 21 of [Humphreys].
D. Existence theorems for Lie algebras (4 lectures).
N.S.N. Sastry and R.P.Shukla.
Serre's theorem on generators and relations; Existence of Chevalley basis; construction of Chevalley groups.
References: Sections 18, 25 of [Humphreys].
E. Basics of algebraic groups (7 lectures).
Basics notions; G-spaces; Borel's closed orbit lemma; Jordan decomposition; Abelian algebraic groups; Connected one dimensional algebraic group; Classification of tori.
References: Chapters 2,3 of [Springer].
F. Geometric aspects of algebraic groups (7 lectures).
Tangent space; smooth/simple points; The Lie algebra of an algebraic group; Quotients of algebraic groups.
References: Chapters 4,5 of [Springer].
G. Reductive algebraic groups (15 lectures).
Maneesh Thakur and K.N. Raghavan.
Parabolic and Borel subgroups; Maximal tori; Weyl group; Semisimple rank one group is SL(2) or PSL(2); Root data and root datum; Internal structure of a reductive group; Borel subgroup and positive roots; Bruhat decomposition; Classification of parabolic subgroups.
References: Chapters 6,7,8 of [Springer].
H. Existence and Uniqueness theorems (6 Lectures).
Structure constants; A presentation for G (at least for SL(2)); Isomorphism theorem; Isogeny theorem; Existence theorem.
References: Chapters 9,10 of [Springer].
This page was last modified
on 25th July, 2006.
[ Indian Statistical Institute ] [ Stat-Math Unit ]