## Introduction to Lie Theory

### A short course to be taken after a long course of abstract algebra

The main content of the class consists of ten weeks of lectures which are posted to YouTube.

This web page contains some supplementary material, namely, PDF summaries of the YouTube lectures,
PDF files containing the homework problems, and the LaTeX templates for those problems in case you prefer that instead.

- WEEK ONE
- PDF summaries of the lectures for the week: 1-1 (the definition of a Lie algebra then review of some affine algebraic geometry)
1-2 (the definition of algebraic groups and Hopf algebra structure on their coordinate algebras)
- Homework problems: PDF TEX

- WEEK TWO
- PDF summaries of the lectures for the week: 2-1 (algebraic groups are linear)
2-2 (connected components and some other basic facts about algebraic groups) 2-3 (review of tangent space for affine varieties)
- Homework problems: PDF TEX

- WEEK THREE
- PDF summaries of the lectures for the week: 3-1 (Lie algebra structure on tangent space of an algebraic group)
3-2 (examples of calculating Lie algebras and differentials, then Int, Ad and ad) 3-3 (bad things in positive characteristic and good things in characteristic zero)
- Homework problems: PDF TEX

- WEEK FOUR
- PDF summaries of the lectures for the week: 4-1 (the philosophy of Lie theory, then quick definitions of nilpotent, solvable and semisimple Lie algebras)
4-2 (representations of Lie algebras and universal enveloping algebras) 4-3 (longer discussion of PBW theorem then shorter discussion of how the universal enveloping algebra of a connected algebraic group is related to the dual of its coordinate algebra)
- Homework problems: PDF TEX

- WEEK FIVE
- PDF summaries of the lectures for the week: 5-1 (finite-dimensional representations of sl_2)
5-2 (a bit more sl_2, then discussion of Ado's, Engel's and Lie's theorems) 5-3 (proofs of Engel and Lie then we start chapter 2)
- Homework problems: PDF TEX

- WEEK SIX
- PDF summaries of the lectures for the week: 6-1 (non-degeneracy of the Killing form)
6-2 (proof of Cartan's criterion, definition of the Casimir element) 6-3 (Weyl's theorem on complete reducibility and its application to prove the Jordan decomposition in a semisimple Lie algebra)
- Homework problems: PDF TEX

- WEEK SEVEN
- PDF summaries of the lectures for the week: 7-1 (maximal toral subalgebras and Cartan decomposition)
7-2 (root systems from semisimple Lie algebras) 7-3 (axiomatic definition of root system and their construction from the Cartan decomposition)
- Homework problems: PDF TEX

- WEEK EIGHT
- PDF summaries of the lectures for the week: 8-1 (root systems, bases and chambers)
8-2 (the Weyl group) 8-3 (classification of root system)
- Homework problems: PDF TEX

- WEEK NINE
- PDF summaries of the lectures for the week: 9-1 (back to the classification of semisimple Lie algebras)
9-2 (Serre presentation and its application to proving Existence, Conjugacy and Isomorphism Theorems)
9-3 (proof of Serre's Theorem')

- WEEK TEN
- PDF summaries of the lectures for the week: 10-1 (a glimpse of representation theory)
10-2 (more about exponentiation) 10-3 (semisimple algebraic groups from semisimple Lie algebras)

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