### 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.