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
- if you have already taken the first half, or
- if you already know the material of the first half, or
- (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.