640:549 Lie Groups (Spring, 2008) -- Syllabus
Text: Roe Goodman and Nolan Wallach, Representations and Invariants
of the Classical Groups
(2nd edition--download selected chapters from course web page)
| Date | Lecture | Reading | Topics |
|---|---|---|---|
| 1/23 | 1 | 1.1-1.2 | Classical groups and classical Lie algebras |
| 1/28 | 2 | 1.3 | Topological groups, Matrix power series |
| 1/30 | 3 | 1.3 | Exponential map, Lie algebra of closed subgroup of GL(n) |
| 2/04 | 4 | 1.3 | Lie algebras of classical groups Exponential coordinates on closed subgroups of GL(n) |
| 2/06 | 5 | D.1-D.2 | Manifolds and Lie groups |
| 2/11 | 6 | 1.3 | Lie group structure on closed subgroups of GL(n) Differentials of homomorphisms |
| 2/13 | 7 | 1.3, D.1 | Lie algebras and vector fields |
| 2/18 | 8 | 1.4 | Linear algebraic groups |
| 2/20 | 9 | 1.4 | Lie algebra of an algebraic group Algebraic groups as Lie groups |
| 2/25 | 10 | 1.5 | Rational representations |
| 2/27 | 11 | 1.5 1.6 |
Locally regular representations; adjoint representation
Nilpotent and unipotent matrices |
| 3/03 | 12 | 1.6 | Jordan decomposition |
| 3/05 | 13 | 1.7 | Real forms of complex algebraic groups |
| 3/10 | 14 | 1.7 | Real forms of the classical groups |
| 3/12 | 15 | 2.1 | Semisimple elements in classical groups; maximal torus |
| Spring Break | |||
| 3/24 | 16 | 2.2 | Unipotent generation of classical groups; connected groups |
| 3/26 | 17 | 2.3 | Regular representations of sl(2) |
| 3/31 | 18 | 2.3 | Regular representations of SL(2)
Complete reducibility of SL(2) |
| 4/02 | 19 | 2.4 | Adjoint representation of a classical group |
| 4/07 | 20 | 2.4 | Root spaces and sl(2) subalgebras |
| 4/09 | 21 | 2.4 | Structure of classical Lie algebras Irreduciblility of adjoint representation |
| 4/14 | 22 | 3.1 | Weyl group; Root reflections |
| 4/16 | 23 | 3.1 | Weight lattice; Dominant weights |
| 4/21 | 24 | C.2 3.2 |
Universal enveloping algebras; P-B-W Theorem
Theorem of the highest weight |
| 4/23 | 25 | 3.2 | Theorem of the highest weight |
| 4/28 | 26 | 3.3 | Reductive groups; Casimir operator |
| 4/30 | 27 | 3.3 | Algebraic proof of Lie algebra complete reducibility |
| 5/05 | 28 | 3.3 | Complete Reducibility; Irreducible Group Representations;
Fundamental Representations |
Back to home page of 640:549.
Roe Goodman / goodman "at" math "dot" rutgers "dot" edu / Revised May 6, 2008