As the title suggests, this course will survey the theory of matrix Lie groups, i.e., those Lie groups which have a faithful representation by matrices.  The emphasis will be on examples and computational tools rather than on proofs of deep theorems.  Most of the deep theorems in the subject will be mentioned, occasionally with some comments on what's involved in the proofs.  Easy results will be proved.
       Lie groups "pop up" in many areas of mathematics.  The goal of this course is to provide acquaintance with tools and major results in the subject. Essentially every reasonable Lie group is a matrix Lie group and those that are not (e.g., the universal covering group of SL(2,R)) are locally isomorphic to a matrix Lie group.  So getting comfortable with Lie theory and being able to use it effectively comes down to knowing the ins and outs of handling large collections of matrices (better, operators on a finite dimensional real vector space) simultaneously.

Time and Place: Monday, Wednesday and Friday, 12:00-1:00, Lopata 229

Instructor: Edward N. Wilson
             Office:  Cupples I, Room 18 (in the basement)
             Office Hours:  MWF 1:00-2:00 and by appointment
             Office Tel:    935-6729 (has voice-mail)
             E-mail:        enwilson@math.wustl.edu

Prerequisites: Math 429-30, Math 411-412 OR Math 417 and Math 412 are the minimum pre-requisites.  At least a smidgen of knowledge about Lebesgue integrals will be helpful
(e.g., the quick treatment  done in Math 412 for the past two years).  For the benefit of those who have taken Math 441-2, there will occasionally be fast mention of results from manifold theory, but manifold techniques and proofs will be avoided.

Textbook/Reference Books:  There won't be a textbook since all of the relevant books I know are either too elementary, too specialized, or too advanced and also, are so laden with the theoretical development that examples get only passing mention.  That said, the relatively new book by Anthony Knapp, Beyond an Introduction to Lie Groups, comes closer than any book I know to being an eminently suitable all-purpose text for a course of this kind.  Knapp's book is currently being sold by Amazon for slightly under $50, in contrast to Amazon's price of around $100 for another recent all-purpose text by Brian Hall.  Much of Lie group theory rests on Lie algebra theory.  The book by James Humphreys, with a title something like Lie Algebras and Their Representations, is strongly recommended.  I'll mention in class a list of a dozen or so other standard books on Lie theory, all of which contain very good material.  But this list won't come close to doing justice to the enormous literature on Lie theory, literally many hundreds of books written for various purposes (topology, harmonic analysis, manifold theory, representation theory, applications to crystallography, quantum mechanics, biology, finite group theory, K-theory, other cohomology theories, graph theory, spectral theory, differential equations, dynamical systems,...).

Topic Outline: 1.  We'll begin with a review of Hoffman-Kunze style linear algebra, sketching the proofs of some deep results either not mentioned in most undergraduate texts or not  normally covered in Math 429.
                2.  Lie theory revolves around the exponential map.  We'll go over the easy and deep properties of the exponential map on matrices and operators. Some of the definitions and proofs will be in the context of Banach algebras--partly to have a framework later on to discuss why infinite dimensional representation theory differs so sharply from the finite dimensional theory.
                3.  Definitions of matrix Lie groups with a summary of the pertinent theorems framing the subject.  In the review of linear algebra, we'll go over the polar decomposition, Iwasawa decomposition, and Bruhat decompositions for the group of all invertible n x n real matrices.  At this point, we'll give some examples of matrix groups to which these structural theorems generalize.
                4.  A close look at the group of 2 x 2 real matrices with determinant 1 sets the stage for finite dimensional representation theory.  We'll spend some time disecting this group and its representations as well as adapting the results to closely related low dimensional Lie groups.
                5.  The Peter-Weyl theorem for compact groups is not only a lovely theorem in its own right but a key part of coming to grips with what's involved with harmonic analysis on non-commutative, not necessarily compact groups.  We'll prove the theorem modulo a needed functional analysis tool and discuss general ideas in harmonic analysis.
                6.  Topics 1-5 are intended to provide enough general background to make educated guesses on general theorems.  Proofs of these theorems rest heavily on Lie algebra theory, which is just code language for fancy linear algebra.  We'll run through the proofs of most of the standard results on Lie algebras, cutting short the treatment if the semester is nearly over.
                7.  Topic 6 will provide the tools for the highest weight theory by which all finite dimensional representations of connected semi-simple Lie groups are classified and the Dynkin diagram classification of such groups.  We'll go over as much of this material as time permits.

Exams/Homework: There won't be any exams.  Instead, homework exercises will be suggested from time to time.  Some of these exercises will be considerably harder than others.  It's not anticipated that all students will do all of the suggested exercises.  But not doing any of the exercises will make taking the course a waste of time since the only way to come to grips with Lie theory is to work through exercises and examples.

Grading: Grades will be based mostly on homework performance with class participation also factored in.  There won't be a fixed formula for determining a letter grade since success on homework exercises will rest to some extent on mathematical maturity (how many fancy tools students may know about, how much experience they have had with subtle linear algebra arguments, etc.)  Suffice it to say that everyone who puts forth a good faith effort and demonstrates "minimal" understanding of the material will earn no less than a B and it's anticipated that lots of A grades will be assigned.

Academic Integrity: The nature of the course makes the usual bombast about acting with integrity "or else..." unnecessary.  But,in this course as well as all others, it's important to follow common guidelines.  Thus, it's O.K. to consult with others or to consult reference books on homework questions, but, if so, write a brief line or two at the top of your paper naming the people or books consulted.