MATH 5143 (Spring 2022)

Overview:

Welcome to the homepage for MATH 5143: Introduction to Lie Algebras!
Send questions to eric.marberg@gmail.com or our Discord server.
For office hours, send me an email to arrange an appointment.
For information about prerequisites, grades, and course content, consult the syllabus.

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 in this semester, especially about Lie groups.

Schedule:

Lecture 1 : basic definitions, solvable and nilpotent Lie algebras (1.1–3.1)
Lecture 2 : Engel's and Lie's theorems, Jordan decomposition (3.1–4.2)
Lecture 3 : Cartan criterion, Killing form, semisimplicity (4.3–5.3)
Lecture 4 : modules, Casimir element, Weyl's theorem, abstract Jordan decomposition (6.1–6.4)
Lecture 5 : special linear representations, root space decompositions (7.1–8.2)
Lecture 6 : properties of root spaces, abstract root systems (8.3–9.4)
Lecture 7 : properties of simple roots and the Weyl group (10.1–10.4)
Lecture 8 : classification of irreducible root sytems, isomorphism and conjugacy theorems (11.1–16.5)
Lecture 9 : universal enveloping algebras (17.1–17.5)
Lecture 10 : Serre's theorem, weight spaces, standard cyclic modules (18.1–20.2)
Lecture 11 : finite-dimensional modules, multiplicity formulas, formal characters (20.3–22.5)
Lecture 12 : Harish-Chandra's theorem, more formal characters, Kostant's formula (23.1–24.2)
Lecture 13 : Weyl character formula, Chevalley bases (24.3–25.5)

Assignments: