640:550 Lie Algebras (Fall, 2004) -- Syllabus
Note: The numbering of the sections listed in the syllabus follows the revised (2nd edition) version of the Goodman-Wallach book (available through the course main page). The section Hu are in Humphrey's book Introduction to Lie Algebras and Representation Theory.
| Date | Lecture | Reading | Topics |
|---|---|---|---|
| 9/2 | 1 | 1.1.1 - 1.1.3 | Intoduction to the complex classical groups |
| 9/9 | 2 | 1.1.3, 1.2.1, 1.2.2 | Low-dimensional examples; Linear algebraic groups |
| 9/13 | 3 | 1.2.2, 1.3.1 | Regular functions on an algebraic group; Vector fields on GL(n) |
| 9/16 | 4 | 1.3.1 | Left-invariant vector fields on GL(n) |
| 9/20 | 5 | 1.3.2 | Lie algebra of a linear algebraic group |
| 9/23 | 6 | 1.3.3 | Lie algebras of the classical groups |
| 9/27 | 7 | 1.4.1; 1.4.2 | Representations of algebraic groups; Differential of a representation |
| 9/30 | 8 | 1.4.2 | Properties of the differential; Examples |
| 10/4 | 9 | 1.4.3; 1.5.1, 1.5.2 | Adjoint representation; Unipotent and semisimple one-parameter groups |
| 10/7 | 10 | 1.5.3 | Jordan-Chevalley decomposition of a linear algebraic group |
| 10/11 | 11 | 2.1.1 | Maximal tori in classical groups |
| 10/14 | 12 | 2.1.2, 2.1.3 | Unipotent generation of classical groups; connected groups |
| 10/18 | 13 | 2.2.1, 2.2.2 | Irreducible representations of sl(2) and SL(2) |
| 10/21 | 14 | 2.3.1 | Root space decomposition of classical Lie algebra |
| 10/25 | 15 | 2.3.2 | Commutation relations of root spaces |
| 10/28 | 16 | 2.3.3; 2.3.4 | Structure of classical root systems; Irreducibility of adjoint representation |
| 11/1 | 17 | 2.4.1; 2.4.2 | Reductive groups; Casimir operator |
| 11/4 | 18 | 2.4.3 | Complete reducibility of representations of classical groups |
| 11/8 | 19 | 2.5.1; 2.5.2 | Weyl group of classical group; Root reflections |
| 11/11 | 20 | 2.5.3 | Root Reflections; Weights of a representation |
| 11/15 | 21 | 2.5.4 | Dominant weights and fundamental weights |
| 11/18 | 22 | 5.1.1 | Extreme vectors and highest weights |
| 11/22 | 23 | 5.2.1-2 | Fundamental representations; Cartan product |
| 11/29 | 24 | 2.6.1-2; Hu: 3.3; 5.1-4 | Semisimple Lie algebras; Killing form; Jordan decomposition; Engel's Theorem |
| 12/2 | 25 | 2.6.2; Hu: 8.1-3 | Cartan subalgebra and Root space decomposition |
| 12/6 | 26 | 2.6.3; Hu: 8.4-5; 9.1-3 | Geometry of Root systems |
| 12/9 | 27 | 2.6.3; Hu: 10.1-4 | Simple roots and Weyl group |
| 12/13 | 28 | Hu: 11.1-5; 12.1 | Classification of root systems; Examples |
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Roe Goodman / goodman "at" math "dot" rutgers "dot" edu / Revised December 13, 2004