Home 
Abstracts of Mini CoursesA. 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). R.P. Shukla. 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). Amritanshu Prasad. Basics notions; Gspaces; 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). S. Kannan. 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). B. Sury. Structure constants; A presentation for G (at least for SL(2)); Isomorphism theorem; Isogeny theorem; Existence theorem. References: Chapters 9,10 of [Springer]. References:

This page was last modified
on 25th July, 2006.