18.755 | Spring 2024 | Graduate

Lie Groups and Lie Algebras II

Calendar

Lecture 1: Representations of \(GL_n\), I  

Lecture 2: Representations of \(GL_n\), II 

Problem set 1 due

Lecture 3: Representations of \(GL_n\), III

Lecture 4: Fundamental and Minuscule Weights

Problem set 2 due

Lecture 5: Fundamental Representations of Classical Lie Algebras

Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual Representations  

Problem set 3 due

Lecture 7: Differential Forms, Partitions of Unity  

Lecture 8: Integration on Manifolds  

Problem set 4 due

Lecture 9: Representations of Compact Lie Groups  

Lecture 10: Proof of the Peter-Weyl Theorem  

Problem set 5 due

Lecture 11: Representations of Compact Topological Groups  

Lecture 12: The Hydrogen Atom, I 

Problem set 6 due

Lecture 13: The Hydrogen Atom, II 

Lecture 14: Forms of Semisimple Lie Algebras over an Arbitrary Field 

Problem set 7 due

Lecture 15: Classification of Real Forms of Semisimple Lie Algebras 

Lecture 16: Real Forms of Exceptional Lie Algebras  

Problem set 8 due

Lecture 17: Classification of Connected Compact and Complex Reductive Groups  

Lecture 18: Maximal Tori in Compact Groups, Cartan Decomposition  

Problem set 9 due

Lecture 19: Topology of Lie Groups and Homogeneous Spaces, I  

Lecture 20: Topology of Lie Groups and Homogeneous Spaces, II

Problem set 10 due

Lecture 21: Topology of Lie Groups and Homogeneous Spaces, III

Lecture 22: Levi Decomposition  

Problem set 11 due

Lecture 23: The Third Fundamental Theorem of Lie Theory  

Lecture 24: Ado’s Theorem  

Problem set 12 due

Lecture 25: Borel Subgroups and the Flag Manifold of a Complex Reductive Lie Group 

Problem set 13 due

Course Info

Instructor
Departments
As Taught In
Spring 2024
Level
Learning Resource Types
Online Textbook
Lecture Notes
Problem Sets