640:549 Lie Groups (Fall, 2009) -- Syllabus
Text: Roe Goodman and Nolan Wallach, Symmetry, Representations, and
Invariants
(Springer Graduate Texts in Mathematics 255; ISBN 978-0-387-79851-6)
| Date | Lecture | Reading | Topics |
|---|---|---|---|
| 9/3 | 1 | 1.1-1.2 | Classical groups and classical Lie algebras |
| 9/8 (Tues) |
2 | 1.3 | Topological groups, Matrix power series |
| 9/10 | 3 | 1.3 | Exponential map, Lie algebra of closed subgroup of GL(n) |
| 9/14 | 4 | 1.3 | Lie algebras of classical groups Exponential coordinates on closed subgroups of GL(n) |
| 9/17 | 5 | D.1-D.2 | Manifolds and Lie groups |
| 9/21 | 6 | 1.3 | Lie group structure on closed subgroups of GL(n) Differentials of homomorphisms |
| 9/24 | 7 | 1.3, D.2 | Lie algebras and vector fields |
| 9/28 | 8 | 1.4 | Linear algebraic groups |
| 10/1 | 9 | 1.4 | Lie algebra of an algebraic group |
| 10/5 | 10 | 1.4 1.5 |
Algebraic groups as Lie groups Rational representations |
| 10/8 | 11 | 1.5 | Examples of representations: dual, direct sum, tensor product, adjoint representation |
| 10/12 | 12 | 1.6 | Jordan decomposition |
| 10/15 | 13 | 1.6, 1.7 | Jordan-Chevalley decomposition in algebraic groups Real forms of complex algebraic groups |
| 10/19 | 14 | 1.7 | Complex conjugations and real forms of complex classical groups |
| 10/22 | 15 | 2.1 | Semisimple elements in classical groups; maximal torus |
| 10/26 | 16 | 2.2 | Unipotent generation of classical groups; connected groups |
| 10/29 | 17 | 2.3 | Regular representations of SL(2) |
| 11/2 | 18 | 2.3 2.4 |
Complete reducibility of SL(2) |
| 11/5 | 19 | 2.4 | Adjoint representation of a classical group |
| 11/9 | 20 | 2.4 | Root spaces and root systems |
| 11/12 | 21 | 2.4 | Structure of classical Lie algebras Irreduciblility of adjoint representation |
| 11/16 | 22 | 3.1 | Weyl group; Root reflections |
| 11/19 | 23 | 3.1 | Weight lattice; Dominant weights |
| 11/23 | 24 | C.2 3.2 |
Universal enveloping algebras; P-B-W Theorem
Theorem of the highest weight |
| 11/30 | 25 | 3.2 | Theorem of the highest weight |
| 12/3 | 26 | 3.3 | Reductive groups; Casimir operator |
| 12/7 | 27 | 3.3 | Algebraic proof of complete reducibility |
| 12/10 | 28 | D.2 | Simply-connected groups; covering groups |
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Roe Goodman / goodman "at" math "dot" rutgers "dot" edu / Revised October 18, 2009