| Date |
Topics |
Sections |
Event |
| Week 1 |
| 1-09 |
Lie groups and Lie algebras--how they are related |
Mat Ch4.8, IT pages 13-23 |
Reading: Ziller pages 1--3, 11--15 and my two lectures |
| 1-11 |
Construction of a local Lie group from a Lie algebra (the structure coefficients) |
Week1 Notes Week1 Supplement Notes |
Reading: Warner, Warner pages 83--89 for many examples |
| Week 2 |
| 1-16 |
Homomorphisms, subgroups and subalgebras |
Ziller 1.2, 1.3 |
Reading: Ziller pages 4--10, 14--23. See Donaldson's notes for a slightly different approach of integrating a Lie algebra |
| 1-18 |
Covering groups |
IT, 1.2-1.4, and Week2 Notes |
Reading: Ziller pages 30--35 |
| Week 3 |
| 1-23 |
Representation of compact Lie groups |
Hsi Pages 1--12, and Week3 Notes-part A |
Reading: Ziller pages 44 (for the existence of a bi-invariant n-form), 90-92; Week3 Supplement Notes |
| 1-25 |
Representation of SU(2) |
Hsi Pages 13--19, and Week3 Notes-part B |
Reading: Ziller pages 93-94. Chevalley, 171-194 |
| Week 4 |
| 1-30 |
A Peter-Weyl Theorem |
Warner: pages 254--257, and Week4 Notes-part A |
Readings: Hum 1-10, Ziller: 45-47, Week4 Supplement Notes |
| 2-01 |
Basics of Lie algebras |
Ziller pages 36--40, and Week4 Notes-part B |
Readings: Hum 11-18 |
| Week 5 |
| 2-6 |
Cartan's theorem/Criterions on the semi-simplicity |
Ziller pages 40--42, Hum pages 19-22and Week5 Notes-part A |
Readings: Complexification notes |
| 2-8 |
Structure of Semi-simple Lie algebras |
Ziller pages 42--45, Hum, 5.2, 5.3 and Week5 Notes-part B |
|
| Week 6 |
| 2-13 |
Compact Lie algebras and Weyl's theorem |
Ziller pages 44--47 and Week6 Notes-part A and
Updated |
Readings: Hsiang pages 78--83; See also Ise-Takeuchi, page 61 for another analytic proof of Weyl's Theorem |
| 2-15 |
Maximal torus and the Weyl group |
Ziller pages 48--50, Ise-Takeuchi, pages 52--55 |
Readings: Hsiang pages 38--47 for slightly different arguments |
| Week 7 |
| 2-20 |
The proof of Cartan-Weyl-Hopf theorem |
Ise-Takeuchi, pages 52--57 and Week7 Notes-part A |
Reading: Computations of curvature |
| 2-22 |
The action of the Weyl group |
Ziller pages 50 and 53--54, Hsi pages 46--47, and Week7 Notes-part B |
Reading: Kronecker and Hopf Lemmas |
| Week 8 |
| 2-27 |
Root system and its properties-I |
Ziller pages 55--56, Ise-Takeuchi pages 84--85 and 89--91, and Week8 Notes-part A |
Readings: Hsi pages 48--53, 63--66 |
| 2-29 |
Root systems and its properties-II |
Ziller pages 58--62, Ise-Takeuchi pages 92--96, and Week 8 Notes-part B |
Readings: Serre, Ch 3, Roots-Examples ; Compare Ise-Takeuchi 86--89 with Humphreys 2--3 |
| Week 9 |
| 3-5 |
Cartan integers/matrices and the Dynkin diagram |
Ziller pages 62--65, and Week9 Notes |
Readings: Serre Ch 5 |
| 3-7 |
Classification of root systems |
Ziller pages 66--69, Ziller pages 62--65, Serre Ch 5 |
Readings:Humphreys Section 11.4, pages 57--63, Hsi, pages 97--102 |
| Week 10 |
| 3-12 |
The maximal root, the highest weight and Weyl's character formula-I |
Hsi 4.3, and Week10 Notes |
Readings: Serre, pages 64--65 and Ise-Takeuchi, pages 93--109 on the relation between the compact Lie algebra and the semi-simple real Lie algebras |
| 3-14 |
Weyl's character formula-II |
Hsi 4.4 |
Knapp's Representation Theory of Semi-Simple Groups, Ch 4.7, 4.10 |
