Course Syllabus

Overview

This is the first of the sequence (18.745/737) on Lie algebras/Algebraic groups. The class aims to cover the following: Structure of finite-dimensional Lie algebras. Theorems of Engel and Lie. Cartan subalgebras and regular elements. Trace form and Cartan's criterion. Chevalley's conjugacy theorem. Classification and construction of semisimple Lie algebras. Weyl group. Universal enveloping algebra and the Casimir operator. Weyl's complete reducibility theorem, Levi and Maltsev theorems. Verma modules. Classification of irreducible finite-dimensional representations of semisimple Lie algebras. Weyl's character and dimension formulas. 

Pre-requisites

18.701, 18.702, 18.100B

References

Main text: James E. Humphreys: Introduction to Lie algebras and representation theory, Springer GTM

Other references:

•Lecture note by Pavel Etingof: Lie groups and Lie algebras https://arxiv.org/pdf/2201.09397.pdf

•Groupes et Algebres de Lie, N. Bourbaki Chapter 4,5,6 

• Alexander Kirillov Jr. : An introduction to Lie groups and Lie algebras, Cambridge University Press, 2008.

Lectures and Schedule

We will march through Humphrey's at a rate 1 or 2 chapters per class. We might reorder the sections occasionally. The weekly lecture plans will be posted on the google spreadsheet.

Reading 

Each chapter in the textbook often contains more material than a lecture and it won't be covered thoroughly. You should read the textbook supplementing the lectures.

Homework:

There will be weekly homework assignments due on Fridays, with problems primarily chosen from the textbook. You will always have at least one week to complete each assignment. Please type your solutions using LaTeX and submit them via Gradescope. Each problem set will carry equal weight toward your final grade.

Alternatively, if you have a specific topic of interest related to Lie algebras, you may choose to complete 7 problem sets and write a term paper. If you wish to pursue this option, please discuss your proposed paper topic with me in advance.

In both cases, NO late pset will be accepted, however, the lowest grade pset will be dropped.

Exams:

There will be 2 exams: October 14, December 2

Grades

Homework (15%), Exam 1 (40%), Exam 2 (40%), Attendance/Participation (5%)

Office Hours

TBD (in person 2-442)

DAS Statement:

MIT is committed to the principle of equal access. Students who need disability accommodations are encouraged to speak with Disability and Access Services (DAS), prior to or early in the semester so that accommodation requests can be evaluated and addressed in a timely fashion. If you have a disability and are not planning to use accommodations, it is still recommended that you meet with DAS staff to familiarize yourself with their services and resources. Contact: das-student@mit.edu | (617) 253-1674.