Math 535: Lie Theory II

Winter Term 2023
Lior Silberman

General Information

This is the second course in our Lie Theory sequence. I shall discuss the structre and representation theory of real Lie groups.


  1. Bröcker–tom Dieck, Representations of Compact Lie Groups
  2. Knapp, Lie Groups Beyond an Introduction


  1. Problem Set 1a
  2. Problem Set 1b
  3. Problem Set 3
  4. Problem Set 4
  5. Problem Set 5
  6. Problem Set 6

Lecture-by-Lecture information

Warning: the following information is tentative and subject to change at any time

Chapter Week Date Material In-class Notes
1 M 9/1 Introduction Scan  
W 11/1 Topological groups; representations Scan  
F 13/1 Basic constructions Scan  
2 M 16/1 Compact groups Scan  
W 18/1 G-finite vectors Scan  
F 20/1 The Peter--Weyl Theorem Scan  
Lie groups 3 M 23/1 Manifolds Scan  
W 25/1 Tangent and Cotangent spaces Scan  
F 27/1 Lie Groups Scan  
4 M 30/1 Lie algebras Scan  
W 1/2 The exponential map Scan  
F 3/2 Closed subgroups Scan  
5 M 6/2 The adjoint representation Scan  
Lie groups
W 8/2 Compact Lie groups; tori Scan  
F 10/2 Centralizers of tori Scan  
6 M 13/2 Maximal tori Scan  
W 15/2 SU(2); weights Scan  
F 17/2 Roots Scan  
  20/2–25/2 Winter break    
7 M 27/2 Groups of rank 1 Scan  
W 1/3 The algebraic Weyl group Scan  
F 3/3 Weyl Chambers Scan  
8 M 6/3 Root Systems Scan  
W 8/3 The dual Weyl chamber Scan  
theory of
Lie groups
F 10/3 Represetation theory of SU(2) Scan  
9 M 13/3 (continued) Scan  
W 15/3 The universal enveloping algebra Scan  
F 17/3 Highest weight thm: uniqueness Scan  
10 M 20/3 (continued) Scan  
W 22/3 Verma modules Scan  
F 24/3 Weyl itegration formula Scan  
11 M 27/3 Weyl character formula Scan  
Lie groups
W 29/3 Semisimple Lie algebras Scan  
F 31/3 Cartan involutions Scan  
12 M 3/4 Global Cartan involution Scan  
W 5/4 No lecture due to Passover    
W 12/4 Symmetric spaces Scan  
13 M 24/4 Iwasawa decomposition Scan  
F 28/4 (continued) Scan  
14 M 1/5 Smooth vectors Scan  
F 5/5 SL2() Scan  
M 8/5 The principal series    

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