MATH 718: Introduction to Lie Theory

Overview

We cover the basics of the representation theory of Lie algebras. The main focus is concrete examples of simple Lie algebras of lower rank, as well as the classical simple Lie algebras.

Course materials

• Syllabus
• Lecture notes, weeks 1–2
• Homework 1
• Homework 2

Textbook

Representation Theory: A First Course, by W. Fulton and J. Harris. Additional notes are handed out in class.

Further reading

• Introduction to Lie Algebras and Representation Theory, by J. Humphreys.
• Representations of $SL_2(F_q)$, by Cédric Bonnafé.
• Quiver Representations and Quiver Varieties, by Alexander Kirillov Jr.
• $SL_2(\mathbb{R})$, by Serge Lang.
• Introduction to Soergel Bimodules, by B. Elias, S. Makisumi, U. Thiel, and G. Williamson.

Homework and grading

Homework is assigned bi-weekly, with group problem-solving sessions in class. A convenient time for the homework meeting is discussed in class.

The grade is based on homework (60%), a final presentation (30%), and class discussion participation (10%).

Presentation topics

1. Representation theory of $SL(2,F_q)$, the Drinfeld curve. Source: Bonnafé, Representations of $SL_2(F_q)$.
2. BMW algebras. Source: survey by Geordie Williamson.
3. Quantum groups: $U_q(sl_2)$ and $R$-matrices. Source: Kassel, Quantum Groups.
4. Compact Lie groups, Lie algebras, and the exponential map. Source: Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction.
5. The Lie algebra of type $G_2$, roots and representation theory. Source: Humphreys, Introduction to Lie Algebras and Representation Theory.
6. Lattices, $E_8$, classification of indefinite lattices. Source: Serre, A Course in Arithmetic.
7. Inductive approach to the representation theory of $S_n$. Source: Okounkov and Vershik.
8. Homology of Lie algebras, central extensions. Source: Weibel, An Introduction to Homological Algebra.
9. Temperley–Lieb algebras, graphical calculus, and Jones–Wenzl projectors. Source: Khovanov's thesis.