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Abstract of Mini Courses







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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].


  • [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.

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