Math 735 - Lie Groups, University of South Carolina

Instructor: Frank Thorne, LeConte 317O, thorne [at] math [dot] sc [dot] edu.

Office hours: TBD.

## What are Lie groups?

A Lie group is a group that is simultaneously a differentiable manifold, such that these structures are compatible. These objects enjoy rich geometric and algebraic structure, are the subject of the deepest part of representation theory, and also have connections to finite group theory, number theory, algebraic geometry .... there is scarcely an area of mathematics which doesn't come into contact with Lie groups.

The first part of the course will introduce the basic definitions and introduce the most famous and important examples, namely the "classical groups": the general linear and special linear groups, orthogonal groups, symplectic groups, unitary groups. We will discuss how they are defined, where they come up, and talk a little about their geometry. We will also say some about the structure theory of Lie groups.

We will then define the Lie algebra: the tangent space to the Lie group at the identity, together with a bilinear gadget that describes how two paths through the identity interact with each other. We will study the structure theory of Lie algebras and prove a surprising theorem: this small amount of structure is enough to essentially determine the Lie group from which it came.

The remainder of the course will cover additional topics such as representation theory, root systems, Dynkin diagrams, classification theorems, algebraic groups, and more. If there is additional material which you would like to see covered, please let me know.

## Course objectives and learning outcomes:

Successful students will: master the mathematical theory outlined above. Or, more precisely, the (very) small portion of it which is possible to reach in a one-semester course.

## Course Textbooks:

The course textbook will be Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, second edition. At least at first, I plan to follow the presentation there fairly closely while also bringing in some side material from other books (particularly Stillwell's Naive Lie Theory). Mastering the material in Hall's book will suffice for the comprehensive exams.

Other interesting books include:

• Savage, Introduction to Lie Groups, online lecture notes.

• Stillwell, Naive Lie Theory. An overview of the basics of the subject, with lots of cool geometric examples and fascinating discussion.

• Tapp, Matrix Groups for Undergraduates. Concise, rigorous, covers the basics, easy. Kind of like Stillwell's older brother that dresses in button-downs and khakis.

• Rossmann, Lie Groups. A nice presentation. Also suitable for beginners.

• Bump; Procesi; Knapp (all books with titles beginning with "Lie Groups" and continuing in the latter two cases). Some other books I'm not personally familiar with but were highly recommended by others. Bump and Knapp are more advanced than the books above. Procesi adopts a rather unconventional approach (the book is long and looks extremely interesting).

## Course Requirements:

• (70%) Written homework assignments will be assigned at most weekly.

• (20%) Take-home final exam, which will be similar to this course's component of the Ph.D. comprehensive exam.

Students who have advanced to Ph.D. candidacy are excused from the exam (and may additionally be excused from the homework at the request of your thesis advisor).

• (10%) Regular attendance is required at least one of: the department colloquium, the algebra seminar, the student algebra seminar, and the number theory seminar. Truant students will be given large quantities of calculus homeworks to grade.

(Shamelessly stolen from Matt Ballard.)

• A: You have a strong chance of passing the comprehensive exam if you prepare reasonably. This also represents a strong foundation for learning further related topics.
• B+: With additional effort, you should be able to pass the comprehensive exam if you prepare. This represents a partial foundation for learning further related topics.
• B: You have demonstrated some mastery of the subject material, but should put in a lot of additional effort if you want to pass the comprehensive exam or write a thesis in a related area.
• < B: You apparently spent the entire semester playing Pokemon Go.

### Homework Assignments:

Homework 1, due Wednesday, September 14.

Homework 2, due Friday, September 30: Ch. 2, 2, 4, 7, 9, 10; Ch. 3, 2, 3, 12. Also verify the that the Lie bracket [X, Y] = XY - YX satisfies the Jacobi identity. You should do this once in your life, so if you have ever proved this no need to do it again.

Homework 3, due Wednesday, October 26: Ch. 4, 1-4, 12.

## Lecture Notes:

Lectures 1-3: introduction, examples of Lie groups, restriction of scalars.

## I love this subject. What should I study next?

If you enjoyed learning about Lie groups, I would recommend learning about related topics such as algebraic geometry, representation theory, algebraic groups, and differential geometry. My work touches on these topics, but faculty members whose work is even more relevant include Ralph Howard, Jesse Kass, Matt Ballard, and Alex Duncan. If you want to meet these people, show up to the algebra seminar, come early for the pretalk, join me in the peanut gallery, and ask lots of questions.