# MA3407 Introduction to Lie Theory

### 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

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
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
Plan: Revision, Q&A, exercises. I.
Notes: Lie Theory 03.pdf (on Exercises 59-65)

Thu 19 Nov: Class 26
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.
What have you learned from the text? What didn't you understand? What would you like to find out next?

• 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.

### 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.