Home

About The Workshop

Abstract of Mini Courses

Notes

Speakers

Schedule

Participants

Application

Information

How to reach ISI

Abstracts of Mini Courses

A.  Basics of Lie Algebras (10 Lectures).
      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).
     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; 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).
    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:

  • [Humphreys]: J.E. Humphreys, Lie algebras and representation theory, Springer Verlag GTM 9.
  • [Springer]: T.A. Springer, Linear algebraic groups, Birkhauser, PM 9, 2nd edition.

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

    [ Indian Statistical Institute ] [ Stat-Math Unit ]