Lie Groups, Lie Algebras, and Representations

Math 496 - Fall 2015


instructor	phone #	office	e-mail
Renato Feres	5-6752	Cupples I, 17	feres@math.wustl.edu

Section information: Classes meet on Tuesdays and Thursdays from 2:30PM to 4:00PM at Cupples I 218. I'll tell you about office hours on the first day of classes.

Subject: This is an introduction to Lie theory, with emphasis on matrix Lie groups, particularly the so-called classical groups, and their Lie algebras. Lie theory is a central subject in mathematics and an important tool in physics and engineering. In addition to the mathematical theory, I hope to cover some applications of the theory (as time permits and depending on students' interests) to classical and quantum physics.

Text: There is no official text, but the following will be useful:

Lie Groups: an introduction through linear groups by Wulf Rossmann, Oxford University Press, 2002;
Lie Groups, Lie Algebras, and Representations: an elementary introduction (2nd edition) by Brian Hall, Springer, 2015.

I initially thought of adopting the first text (the bookstore may have some copies to sell) but recently I found that the Wash U. library has an electronic copy of the second text (look for the second edition, which is significantly different than the first). You should be able to freely download it. My present idea is to follow this text relatively closely.

Tentative list of topics:

The classical (matrix) groups
The exponential map
Lie algebras
The Lie groups - Lie algebras correspondence
Basic representation theory
The Baker-Campbell-Hausdorff formula
Representations of SL(3,C), SU(3), and other related groups
Unitary representations, compact, semisimple groups (a short overview only)
Selected applications, likely from physics, depending on students' interests.

Coursework: There will be weekly homework assignments and a final presentation at the end of the semester on a topic of your choosing (related to Lie theory and applications). I'll tell you more details about the presentation in class.

Grades: Your final grade will be based on your homework assignments and the final presentation. We'll discuss the details in class.

Homework assignments: