# MA3407 Introduction to Lie Theory

### Official Course Matters

### Web Page

I set up the web page for this course here.

To get Blackboard access, you need to

- have an active NTNU user account,
- have paid the semester fees for fall 2020, and
- have registered for the course (check at StudentWeb).

### Contact and Reference Lie Group

When you have a question about this course, and you cannot find it answered here, just send an email to me.

Do not forget to use your student (`@stud.ntnu.no`

) account for this.

The reference Lie group for this course consists of Oskar (`oskarg@`

) and Tallak (`tallakm@`

).

### Classes

The current plan is that the class meets twice a week:

- Wednesdays from 10:15 to 12:00 in EL6
- Thursdays from 14:15 to 16:00 in H1

The first class was on Wednesday, August 19th, in EL6.

### Course Notes

I have started to write notes that reflect the course contents,

and the latest stable version is here: MA3407notes.pdf.

I will update the file regularly as the course progresses.

### Classes

**Wed 19 Aug: Class 1**

attendance form for EL6

Plan: We'll meet for the first time and discuss the new hygiene regulations.

I will then introduce the course before we start for real with the first topic: groups.

There is no need to prepare.

Update: I explained Sections 1.1 and 1.2 in the notes.

**Thu 20 Aug: Class 2**

attendance form for H1

Plan: Cover Sections 1.4 and 1.3 of the notes in that order.

If you'd like to prepare, do some of the exercises for Sections 1.1 and 1.2,

and take a look at the appendix on category language, as long as you can stand it.

Test question: do you feel comfortable with the statement that groups form a category?

**Wed 26 Aug: Class 3**

attendance form for EL6

Plan: Discuss commutators in groups, commutator subgroups, and abelianization.

How to prepare: the notes are already online. Take a look. Bring questions.

Take a look at the exercises to get a sense of when you might be ready to tackle them.

**Thu 27 Aug: Class 4**

attendance form for H1

Plan: The lower central series, the associated graded, and examples.

How to prepare: the notes are already online. Take a look. Bring questions.

If you'd like to experiment with a computer: find a way to use GAP.

This should work on the workstations for students. It is also included in SageMath.

**Wed 2 Sep: Class 5**

digital registration for classes

Plan: Define Lie algebras and give basic examples from groups and associative algebras.

How to prepare: there are some notes from last week that I need to finish first. I will put new ones, for this week, online soon.

Use Google or one of our references to find and read one definition of Lie algebras before you come to class.

**Thu 3 Sep: Class 6**

Plan: Finish the plan for Wednesday, if necessary. Then, exercises, exercises, exercises.

No new material, just consolidation and rethinking of what we have seen so far.

How to prepare: Try some of the exercises. Make up your mind which ones you would like to present or be presented.

**Wed 9 Sep: Class 7**

Plan: derivations for various sorts of algebras.

They always form Lie algebras.

How to prepare: the notes for this lecture are already online.

**Thu 10 Sep: Class 8**

Plan: universal enveloping algebras and representations.

How to prepare: the notes for this lecture are already online.

**Wed 16 Sep: Class 9**

Plan: The Poincaré-Birkhoff-Witt theorem

How to prepare: the notes for this lecture are already online.

**Thu 17 Sep: Class 10**

Plan: Hopf algebras and groups

**Wed 23 Sep: Class 11**

Plan: Hopf algebras and Lie algebras

**Thu 24 Sep: Class 12**

Plan: Free Lie algebras

See below for the links to Alissa Crans's videos on Catalan numbers.

**Wed 30 Sep: Class 13**

Plan: I will prove the formula for the dimension of the homogenous parts of the free Lie algebras.

We'll do quite a few computations with it to get a feeling for these.

Also: free groups revisited

We will see that the associated graded Lie algebra of a free group is free.

How to prepare: the notes for this lecture are already online.

Though I will explain what I need about Möbius inversion, it might be a good idea to review it before we meet.

**Thu 1 Oct: Class 14**

Plan: Exponentials

We'll see exp and log and the Baker-Campbell-Hausdorff Theorem.

How to prepare: the notes for this lecture are already online.

**Wed 7 Oct: Class 15** and **Thu 8 Oct: Class 16**

Plan: The algebraic part of the course is over, and it's time for Revision Week!

We'll review the notes, look at the exercises, etc.

How to prepare: Look at the notes and the exercises for the entire first part.

Bring questions, solutions, etc.

**Wed 14 Oct: Class 17**

Plan: We'll start the course's geometric part by looking at some differential equations and their symmetries.

How to prepare: look back at what you learned about differential equations in Calculus 3.

It might be helpful to review some Galois theory, although that is not strictly necessary; I will only use it for motivation.

**Thu 15 Oct: Class 18**

Plan: more differential equations, more symmetries.

Galois, Picard-Vessiot, and Liouville extensions of differential fields

**Wed 21 Oct: no class **

**Thu 22 Oct: no class **

**Wed 28 Oct: Class 19**

Plan: review the basics of affine geometry and see the most basic examples of linear groups.

How to prepare: review what we have discussed about functions and Hopf algebras from Sections 9 and 12 in the notes.

**Thu 29 Oct: Class 20**

Plan: prove that affine groups and linear groups are basically the same.

**Wed 4 Nov: Class 21**

Plan: derivations (once again!) and tangent spaces in affine geometry

**Thu 5 Nov: Class 22**

Plan: Lie algebras of algebraic groups

**Wed 11 Nov: Class 23**

Classes have been moved online: Zoom

Plan: Group actions, representations, and Tannaka duality. I.

Notes: ~~Lie Theory 01.pdf~~ (The notes have been working into the regular course notes.)

**Thu 12 Nov: Class 24**

Link: Zoom

Plan: Group actions, representations, and Tannaka duality. II.

Notes: ~~Lie Theory 02.pdf~~ (The notes have been working into the regular course notes.)

**Wed 18 Nov: Class 25**

Link: Zoom

Plan: Revision, Q&A, exercises. I.

Notes: Lie Theory 03.pdf (on Exercises 59-65)

**Thu 19 Nov: Class 26**

Link: Zoom

The password is almost as usual, but without the "a" in the end (my fault).

Plan: Revision, Q&A, exercises. II.

Notes: Lie Theory 04.pdf (on Exercises 64-69)

### A Dozen Texts to Read (or Videos to Watch)

As I said in class, my intention is to list twelve (more or less) short texts.

Try to read half of them–your choice.

I am happy to get feedback on your reading. For instance:

What have you learned from the text?
What didn't you understand?
What would you like to find out next?

- M. Hairer. Oral examinations / presentations.

Mathematics Research Centre, University of Warwick.

- F. Dyson. Birds and Frogs.

Notices Amer. Math. Soc. 56 (2009) 212–223.

The analogy evoked in there is much older of course, even within mathematics.

For instance, a famous algebraic topologist Adams compared hares to tortoises from page 23 on:

J.F. Adams. Stable homotopy theory. Springer-Verlag, 1964.

- A.J. Coleman. The greatest mathematical paper of all time.

Math. Intelligencer 11 (1989) 29–38.

This is*not*the greatest mathematical paper of all time.

It is*about*the greatest mathematical paper of all time, at least in the author's view.

I won't give away here which paper he is referring to.

- I. Stewart. Symmetry in applied mathematics.

In: The Princeton Companion to Applied Mathematics.

Edited by Nicolas J. Higham, Mark R. Dennis, Paul Glendinning, Paul A. Martin, Fadil Santosa and Jared Tanner.

Princeton University Press, Princeton, NJ, 2015.

- A. Crans. A Surreptitious Sequence: The Catalan Numbers.

There is a short video and a long video on YouTube.

- P. Nyland. Bass-Serre Theory.

This a trial lecture on YouTube.

- S. Mac Lane. Structure in Mathematics.

Philosophia Mathematica 4 (1996) 174–183.

- S. Garibaldi. What is...a Linear Algebraic Group?

Notices Amer. Math. Soc. 57 (2010) 1125–1126.

### Exams

The exams will be oral.

The exam period is from Tuesday, 24 November on.

The exam dates are as early as possible for you and me; 24 November is not possible for me.

Please contact me by email if you haven't been assigned a slot, yet.

I will send out reminders during the last week of the term.