MATH 5143 (Spring 2023)

Overview:

Lectures:

Textbooks:

Our main textbook is Introduction to Lie Algebras and Representation Theory by Humphreys.
Some other online textbooks can be found here and here. These include more material than we will cover, especially about Lie groups.

Schedule:

Week 1: Sections 1.1–3.1
- Lecture 1: basic definitions, classical examples
- Lecture 2: comparison with group theory
Week 2: Sections 3.1–4.1
- Lecture 3: nilpotent Lie algebras, Engel's theorem
- Lecture 4: solvable Lie algebras, Lie's theorem
Week 3: Sections 4.2–5.3
- Lecture 5: Jordan decomposition, Cartan's criterion
- Lecture 6: Killing form, semisimplicity
Week 4: Sections 6.1–6.4
- Lecture 7: representation theory basics
- Lecture 8: Casimir element, Weyl's theorem
Week 5: Sections 7.1–8.2
- Lecture 9: abstract Jordan decomposition, special linear representations
- Lecture 10: root space decompositions
Week 6: Sections 8.3–9.4
- Lectures 11 and 12: properties of root spaces
- Lecture 13: abstract root systems
Week 7: Sections 10.1–10.4
- Lecture 14: rank two root systems and basic properties
- Lecture 15: simple roots and positive roots
Week 8: Sections 11.1–16.5
- Lecture 16: reducible root systems and Dynkin diagrams
- Lecture 17: classification of irreducible root systems
Week 9: Sections 17.1–17.5
- Lecture 18: universal enveloping algebras
- Lecture 19: PBW basis and free Lie algebras
Week 10: Sections 18.1–20.2
- Lecture 20: Serre's theorem, semisimplicity criteria
- Lecture 21: standard cyclic modules
Week 11: Sections 20.3–22.5
- Lecture 22: constructing standard cyclic modules
- Lecture 23: finite-dimensional modules
Week 12: Sections 23.1–24.3
- Lecture 24: Harish-Chandra's theorem, Kostant's formula
- Lecture 24: Weyl character formula

Assignments: