There are many different angles to take in introducing this material as Lie groups and Lie algebras are present in a diverse range of topics, including physics (Gauge theory, Yang-Mills equations) and number theory (Langlands program). The tact we will take, especially early on in the course, will be on the use of symmetry groups in solving certain ODE and PDE - this was in fact a motivation for Sophius Lie to study Lie groups. This is a good place to start because it is not very abstract but clearly indicates the usefulness of symmetry/presence of Lie group and provides historical context.

The first part of the class will have the added bonus of not using the language of differential geometry/manifolds to give students less familiar with these topics time to brush up. After getting some experience and motivation with Lie groups in a concrete setting we will transition to a more abstract/traditional approach. If there is time and depending on the interests of the class we will say something about symmetric spaces from a different source to finish.

Instructor: Alex Mramor, amramor1 'at' math.jhu.edu

Office hours 3-5 Thursdays at Krieger hall 311

The course is scheduled to meet Tuesdays and Thursdays from 9 to 10:15 in Kreiger 204. Since the class size is small depending on everyones schedule we may try to meet at a different time and have lecture in my office.

The text for the start of the class will be Symmetry Methods for Differential Equations, by Peter Hydon, ISBN: 9780511623967.

After spending a few weeks or so with Hydon, we will discuss manifold theorem out of John Lee's Introduction to differential geometry (the first edition since its the one I own - I sent a link), and later on to An Introduction to Lie Groups and Lie Algebras, by Alexander Kirilov Jr., ISBN: 0521889693.

Depending on the needs of the students we might also spend a short amount of time between Hydon and Krylov on some preliminaries in manifolds. There are many good books on the subject but if you need to get the definitions quickly searching online for lecture notes might be the better option - feel free to ask me if you are unsure.

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This week we will cover chapter 1 of Hydon (and possibly more). No homework will be assigned.

This week we wrapped up chapter 2 of Hydon on Tuesday and will cover chapter 3 on Thursday.

This week there is homework (assigned 2/4, due 2/11): fill out the details (as in, give a complete exposition) of examples 2.10, 2.11, 2.15, 2.17 and do problems 2.2, 2.5 out of Chapter 2 of Hydon.

These weeks we covered more or less chapters 3-5 of Hydon. On thursday 2/20 we will start discussing some differential geometry for our more advanced foray into Lie groups and algebras. Our goal is to discuss the correspondence between Lie groups and Lie algebras - this will take 2 or 3 lectures.

Homework 2 (assigned 2/19, due 3/3): do problems 5.1, 5.5, 5.7, 6.1

We've been romping around "Introduction to Smooth Manifolds, 1st edition" by John Lee. Pulling material from chapter 4,8, 17, 18, 19, and 20. So far (before spring break) we were able to prove Frobenius Theorem for integral submanifolds of distributions which then quickly gives the "Lie group- Lie algebra" correspondence. In the last lecture (3/10) - before JHU decided to end in person classes due to the Corona virus - we introduced the exponential map. Then we were able to discuss the Baker-Cambell-Hausdorff formula, which is a formula for z in terms of iterated Lie brackets of x and y in the equation e^xe^y = e^z - a great corollary being that locally a Lie group is determined by its algebra's bracket (a proof not using Frobenius theorem as in Lee).

We have just a couple more things to say for now at the level of Lie groups before we exclusively focus on Lie algebras. In particular I want to cover the closed subgroup theorem and say something about the adjoint map (on the level of Lie groups) since its pushforward will give us the adjoint representation, which is of central importance in the representation theory side of the course. To whet your appetite and forshadow some of whats to come, take a look at this sketch of Ado's theorem:

Ado's theorem

which you'll remember was important for us giving the direction Lie algebra -> Lie group in the correspondence.

Homework 3 (assigned 3/11, not for turning in): Since there still seemed to be some difficulty with manifold theory (like when you may apply the inverse function theorem, ahem), I'm going to assign an unusual/large homework. I want you to do as much of it as you can - think of it as a concrete suggestion on how to bring your manifolds knowledge up to par for the needs of understanding the Lie group-Lie algebra correspondence better in a finite amount of work:

Read chapters 1,2, 3,4,7,8, 17, 18, 19, 20 (for ch 20 don't worry about the quotient Lie group stuff too much for the time being) and do problems: 1-5, 2-5, 2-11,2-12, 3-1, 3-3, 3-6, 3-7, 3-8, 4-5, 4-15, 4-16, 4-17, 4-18, 7-3, 7-5, 8-4, 8-5, 8-11, 8-19, 8-23, 17-1, 17-2, exercise 20.2, 20-2, 20-6

We are starting the repsentation theory part of the course and transitioning to Kirilov, at last.

Homework 4 (assigned 3/26, due 4/2): do problems 3.1, 3.7 and fill in the details in examples 4.12, 4.14 (namely, explain why Hom_G is the space of intertwining operators), and prove lemma 4.20.

So far we covered chapter 3, 4, 5, and part of 6 out of Kirilov; highlights being: the Haar measure, Cartan's criterion, reductive Lie algebras and the Levi decomposition theorem, culminating in the complete reducibility theorem for representations over semisimple Lie algebras. Next week we will continue on with chapter 6, discussing toral algebras and root systems. This will leave plenty of time for chapter 7!

Homework 5 (assigned 4/10, due 4/20): do problems 3.6, 3.19, 4.4, 4.5, 5.1, 5.2, 5.3

We wrapped up the course with covering chapters 6 and 7 out of Kirilov, culminating of course in the classification of Dynkin diagrams in the simply laced (i.e. only one line per edge) case. Assuming all Dynkin diagrams are classified then one can classify all root systems (abstractly defined), from which one may ``build'' a corresponding Lie algebra (the Serre relations section). Since we may always associate the root system of a semisimple Lie algebra with an abstract root system and hence a Dynkin diagram, this gives a classification of semisimple Lie algebras.

Considering that the Lie algebra of a comapct Lie group was reductive so a direct sum of its center and semisimple parts, this give a local ``classification'' of compact Lie groups by Lie's theorems discussed a while back (the Frobenius theorem stuff) as a connect sum of R^n (with group law given by addition) plus whatever the semisimple part gives. Note that the global picture is much more complicated however, because there might be many connected components/nontrivial topology to the group.