18.755 | Spring 2024 | Graduate

Lie Groups and Lie Algebras II

Pages

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

Homework is assigned at the beginning of each week and due at the end of the week. These Lecture Notes PDFs are both problem sets and lectures.

Homework 1: Exercises in Lecture 1 Notes (PDF) and Lecture 2 Notes (PDF)

Homework 2: Exercises in Lecture 3 Notes (PDF) and Lecture 4 Notes (PDF)

Homework 3: Exercises in Lecture 5 Notes (PDF) and Lecture 6 Notes (PDF)

Homework 4: Exercises in Lecture 7 Notes (PDF) and Lecture 8 Notes (PDF)

Homework 5: Exercises in Lecture 9 Notes (PDF) and Lecture 10 Notes (PDF)

Homework 6: Exercises in Lecture 11 Notes (PDF) and Lecture 12 Notes (PDF)

Homework 7: Exercises in Lecture 13 Notes (PDF) and Lecture 14 Notes (PDF)

Homework 8: Exercises in Lecture 15 Notes (PDF) and Lecture 16 Notes (PDF)

Homework 9: Exercises in Lecture 17 Notes (PDF) and Lecture 18 Notes (PDF)

Homework 10: Exercises in Lecture 19 Notes (PDF) and Lecture 20 Notes (PDF)

Homework 11: Exercises in Lecture 21 Notes (PDF) and Lecture 22 Notes (PDF)

Homework 12: Exercises in Lecture 23 Notes (PDF) and Lecture 24 Notes (PDF)

Homework 13: Exercises in Lecture 25 Notes (PDF)

Course Meeting Times

Lectures: 2 sessions / week, 80 minutes / session

Prerequisites

Course Description

This is the second half of the standard sequence Lie Groups and Lie Algebras I & II. The first half (18.745) covers the basic theory of Lie groups and Lie algebras, the fundamental theorems of Lie theory, nilpotent and solvable Lie algebras, Engel’s theorem, Lie’s theorem, and the structure and representation theory of finite dimensional semisimple Lie algebras. The material of the first half is contained in sections 1–26 of the full lecture notes (PDF) (although not all this material was covered in the first half).

In the second half we will give a more in-depth treatment of Lie groups (relying on what was done in the fall), with a little (but not much) more geometry and analysis compared to the first half, and with a lot of emphasis on examples. Topics will include classical groups, Haar measure on locally compact groups, the representation-theoretic understanding of the hydrogen atom, representations of compact (in particular, finite) groups, the Peter-Weyl theorem with proof, maximal tori, Cartan and Iwasawa decompositions, classification of real reductive Lie groups, topology of Lie groups, proof of the third fundamental theorem of Lie theory, Levi decomposition, Ado’s theorem, Borel subgroups, and flag manifolds. This roughly corresponds to Sections 27–51 of the full lecture notes (PDF)

So you may take this second half

  1. if you have already taken the first half, or
  2. if you already know the material of the first half, or
  3. (with caution) if you are willing to catch up.

Assignments

Homework will be assigned weekly and due in one week. It contains a lot of important material.

Grading

The grade will be given solely on the basis of homework.

Course Info

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