Instructor: Frank Thorne, LeConte 317O, thorne [at] math [dot] sc [dot] edu.
Office hours: TBD.
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.
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.
Other interesting books include:
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).
(Shamelessly stolen from Matt Ballard.)
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.
Lectures 1-3: introduction, examples of Lie groups, restriction of scalars.
Lectures 26-27 (including review).
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.