6510.001, Fall 2000
the University of North Texas.
INSTRUCTOR: Charles Conley, GAB 475, (940) 565 - 3326
OFFICE HOURS: Monday and Wednesday, 11:30-12:30 and 2:00-3:00
CLASS MEETS: Monday and Wednesday, 12:30 - 1:50, GAB 318
TEXT: Introduction to Lie Algebras and Representation Theory, J. Humphreys
This text is a little bit too concise for our purposes, so I will try to hand out notes as we go along. I am open to suggestions for different texts: I would like something that avoids Lie groups, assumes only a minimal knowledge of linear algebra, and does many examples in detail. There are many choices, for example the well-known books by Varadarajan (published by Springer), Jacobson (Dover), and Serre.
PLAN: My aim is to teach this class as concretely as possible, so as to make it accessible to those who are not completely comfortable with abstract algebra. For this reason, I will prove most of the main theorems only in the setting of one of the main examples to which they are applied. I will assume only elementary linear algebra, and will do everything over the fields C and R. We will begin with the definitions of Lie algebras and their representations, and give some examples. Then we will look at nilpotent and solvable Lie algebras, concentrating on the examples of the upper triangular nilpotent matrices, and the upper triangular matrices. We will prove the theorems of Lie and Engel for these examples, and discuss the Cartan criterion for semisimplicity.
From this point on we will study semisimple Lie algebras, concentrating on sl(n,C), the complex n by n matrices of trace 0, especially the cases n=2 and n=3. We will begin with the representations of sl(2,C), and then study the "fine structure" of sl(n,C), i.e., its Cartan subalgebras, root system, and Weyl group (which in this case is the permutation group on n letters). Finally, we will study the representations of sl(n,C), especially those of sl(3,C). We will conclude with an algebraic proof of the Weyl character formula, certainly for sl(2,C), and if there is time, for sl(3,C), and maybe even sl(n,C).