Representation theory: 797RT


The class meets on MoWeFr 10:10AM - 11:00AM
LGT 1334
My office: 1235H LGRT
Office Hours: Mo, We 11:00AM-12:00PM or by appointment
Course webpage: http://https://oblomkov-math.github.io/LieAlg.html

Overview

We are planning to cover basics of representation theory of Lie algebras. The main focus is the conctrete examples of simple Lie algebras of lower rank as well as classical simple Lie algebras.

Main Textbook

Representation theory: a first course, by W. Fulton, J. Harris
I also plan to hand some notes in class.



Other books and further reading on Representation theory and Lie Algebras

Introduction to Lie Algebras and Representation Theory, by J. Hamphreys.
Representations of $SL_2(F_q)$, by Cedric Bonnafe
Quiver Representations and Quiver Varieties, by Alexander Krillov Jr.
$SL_2(R)$, by Serge Lang.
Introduction to Soergel bimodules, by Ben Elias, Shotaro Makisumi, Ulrich Thiel, Geordie Williamson.

Syllabus


Homework

Homework will be assigned bi-weekly. We will have group solving sessions in class for the problem sets. A convenient time for homework meeting will be discussed in class.

Grades

Grade is assigned based on homework: 60%, Final presentation: 30% and class discussion participation 10%.

Presentation topics

1.Representation theory of SL(2,F_q), Drinfeld curve. Source: Bonnafe, Representation theory of SL_2(F_q)

2. BMW algebras. Source: Survey by Geordie Williamson available at http://people.mpim-bonn.mpg.de/geordie/BMW.pdf

3. Quantum groups: U_q(sl_2) and R-matrices. Source: Cassel, Quantum groups

4. Compact Lie groups and Lie algebras exponential map. Source: Hall, Lie groups, Lie algebras and representations: an elementary intrduction

5. 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, Course of arithmetics.

7. Inductive approach to representatio theory of S_n. Source: Okounkov, Vershik https://arxiv.org/abs/math/0503040

8. Homology of Lie algebras, central extensions. Source: Weibel, An introduction to homological algebra.

9. Temperley-Lieb algebras, graphical calculus and Jones-Wentzel projectors. Source: Khovanov's thesis, https://www.math.columbia.edu/~khovanov/research/thesis.pdf