Spring 2025

Lie Groups and Lie Algebras

Listed in: Mathematics and Statistics, as MATH-405


Chris Elliott (Section 01)


Lie groups and Lie algebras appear naturally in the study of symmetries of geometric objects. Lie algebras carry local information and can be studied using tools from linear algebra. Finite dimensional Lie groups can be constructed using techniques from calculus and group theory.

This course serves as a first introduction to Lie groups and Lie algebras. We will examine the structure of finite dimensional matrix Lie groups, the exponential and differentiation maps, as well as compact Lie groups. We will also study Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity and root systems, and the classification of semisimple Lie algebras. This classification is not only a fundamental result in Lie Theory, but is also an archetype of classifications that appear in other areas of math. More amazingly, this classification is embodied in simple combinatorial pictures called Dynkin diagrams, which underlie surprisingly disparate fields, such as geometric group theory, quiver representation theory, and string theory.

How to handle overenrollment: Preference will be given to seniors.

Students who enroll in this course will likely encounter and be expected to engage in the following intellectual skills, modes of learning, and assessment: Problem sets, In-class quizzes or exams, Take-home exams, In-class group work.

Course Materials


2023-24: Not offered
Other years: Offered in Fall 2022, Spring 2025