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 chapter 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 Fridays. Most of problems will be from the textbook. You will have at least a week before the deadline. You should type your homework in TeX and upload it on the gradescope. Each problem will have equal weights toward final grades. 

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)