Course Meeting Times
Lectures: 2 sessions / week, 80 minutes / session
Prerequisites
- 18.701 Algebra I or 18.703 Modern Algebra
- Real Analysis—either 18.100A, 18.100B, 18.100P, or 18.100Q
Course Overview
We will discuss both Lie algebras and Lie groups together in both halves. (These subjects are so intimately related that it is more natural to study them together.) But still the first semester will be mostly algebra while the second one will involve a bit more geometry and analysis. There will be a lot of emphasis on examples.
In the first half (18.745) we will essentially follow Alexander Kirillov Jr.’s An Introduction to Lie Groups and Lie Algebras (PDF), excluding the more advanced material on representations of Lie groups (such as Haar measure, Peter-Weyl theorem, etc.), but with more detailed treatment of Lie algebras. Namely, chapters 2 and 3 (without proofs of the harder theorems, such as Lie’s third theorem), 4.1 to 4.4, and chapters 5 to 8, classification of simple complex Lie algebras with proof. The course also discusses universal enveloping algebras, the Poincare-Birkhoff-Witt theorem, free Lie algebras, and the Campbell-Hausdorff formula.
In the second half (18.755) we will give a more in-depth treatment of Lie groups (relying on what was done in the first half). Topics will include representations of GL(n) and other classical groups; fundamental and minuscule representations; applications of representation theory of Lie groups to quantum mechanics of the hydrogen atom; Haar measure and integration on compact groups; representations of compact (in particular, finite) groups; the Peter-Weyl theorem with proof; maximal tori in compact Lie groups; complex reductive Lie groups; Borel and parabolic subgroups; flag variety; classification of real reductive groups and their structure (maximal tori, maximal compact subgroups, Cartan and Iwasawa decompositions); cohomology of Lie groups and Lie algebras; classification of finite-dimensional simple complex Lie algebras with proof; Levi decomposition; proofs of the third fundamental theorem of Lie theory; Ado’s theorem.
Assignments
Homework will be assigned weekly (first one after 1 week of class) and due in one week. It contains a lot of important material.
Grading
The grade will be given solely on the basis of homework.
