Math 538 Fall 2023
Lie algebras

Lectures: Wednesday and Friday 10:00am-11:30am 1518 North University Building

Instructor: Thomas Lam,

Office Hours: W 9-9:50am, 11:30-12:30am and F 9-9:50am in EH2834.

Course description: Lie algebras arise naturally in mathematics and physics, and are fundamental from many perspectives. Lie algebras are fascinating in their own right, and the study of finite dimensional Lie algebras leads to interesting combinatorial structures, such as root systems, Dynkin diagrams, and Coxeter groups. This course should be valuable to those interested in representation theory and the study of algebraic and Lie groups, and should also be useful to those whose interests lie in areas such as combinatorics, geometry, and physics. In this course, we will study the basic theory of Lie algebras, with the majority of our focus on the complex semisimple case. We intend to cover most of the content of Humphrey's book (Introduction to Lie Algebras and Representation Theory), including structure theorems for Lie algebras, classifications of root systems, and highest weight representation theory. This course will be taught at the advanced undergraduate, or introductory graduate level, and should be accessible to students with experience in abstract algebra and linear algebra.

Prerequesites: Familiarity with linear algebra is essential. One year of abstract algebra, for example at the level of 493-494, will be assumed.

Grading: There will be problem sets roughly every two weeks. There will be no exams.
Grades will be calculated from: Problem Sets (90%) and Attendance (10%).

Required textbook:
Introduction to Lie Algebras and Representation Theory, by J.E. Humphreys.
This book is available electronically through the UM library.

Homework policy: Homework must be written in LaTeX and is due at the start of class. Late homework is generally not accepted. There are no makeups for missed or late homework; the lowest homework score will be dropped in the final calculation.
You are allowed (and encouraged) to work with other students on the problem sets, but you must write the solutions yourself and include the names of those you worked with when you hand in your homework. You are not allowed to post homework problems on question websites such as mathoverflow or stackexchange. If you use a solution you find in a book, online, or elsewhere, you must acknowledge the source.

Problem Sets:
Pset 1
Pset 2
Pset 3 (due date extended to Oct 20)
Pset 4
Pset 5
Pset 6

Academic Misconduct: The University of Michigan community functions best when its members treat one another with honesty, fairness, respect, and trust. The college promotes the assumption of personal responsibility and integrity, and prohibits all forms of academic dishonesty and misconduct. All cases of academic misconduct will be referred to the LSA Office of the Assistant Dean for Undergraduate Education. Being found responsible for academic misconduct will usually result in a grade sanction, in addition to any sanction from the college. For more information, including examples of behaviors that are considered academic misconduct and potential sanctions, please see

Disabilities: The University of Michigan recognizes disability as an integral part of diversity and is committed to creating an inclusive and equitable educational environment for students with disabilities. Students who are experiencing a disability-related barrier should contact Services for Students with Disabilities; 734-763-3000 or For students who are connected with SSD, accommodation requests can be made in Accommodate. If you have any questions or concerns please contact your SSD Coordinator or visit SSD’s Current Student webpage. SSD considers aspects of the course design, course learning objects and the individual academic and course barriers experienced by the student. Further conversation with SSD, instructors, and the student may be warranted to ensure an accessible course experience.

Syllabus (subject to change):
Defintion and examples of Lie algebras. Engel's Theorem and Lie's Theorem. Jordan decomposition and Killing Form. Weyl's Theorem. Root systems and classification. Weyl groups and Cartan matrices. Classification of semisimple Lie algebras. Universal enveloping algebra. Highest weight modules. Weyl character formula.

List of lectures: