# MA3407 Introduction to Lie Theory

### Web Page

The web page for this course will be set up soon, either here or in Blackboard.

To get Blackboard access, you need to

- have an active NTNU user account,
- have paid the semester fees for spring 2019, and
- have registered for the course (check at StudentWeb).

### Class Hours

Mondays and Thursdays, 10:15-12:00, F3 Gamle Fysikk

### Course Content

The course gives a basic introduction to Lie algebras and their connections to various aspects of group theory: discrete groups, algebraic groups, and (of course) Lie groups. The main focus will be on the examples given by matrices because the general theory can often be reduced to these by means of representation theory. Possible specific topics are Lie algebras, universal enveloping algebras, free, nilpotent, solvable, and semi-simple Lie algebras; Lie groups, vector fields and integration, one-parameter groups and the exponential map, homogeneous spaces, Clifford algebras, and spinor groups.

### Textbook

You do not have to buy a textbook to follow this course.

As said in class, the first part of the course is inspired by Serre's notes.

bibliography.pdf (draft, watch out for updates)

### Prerequisites

The course does not assume any topology, but some analysis (in several variables), and the more algebra, the better. We will start with groups and move on from there.

### Exams

We shall have oral exams in the period between 20 May and 7 June. More on this later.

### Reference Group

Sigurd Gaukstad (`sigurgau`

)

Torgeir Aambø (`torgeaam`

)

# Part I: Lie algebras

### Meeting 1 (scheduled for 7.1.)

The first class meeting will be on Monday, January 7th.

How to prepare? Make sure that you can define what a *group* is. I will ask for it!

### Meeting 2 (scheduled for 10.1.)

Luxury: I have typed some notes on groups that cover (roughly) the material that we have discussed in the first meeting and things that we will consider in the second meeting.

Please take a look. Let me know of any typos that you find. Try the exercises.

01_groups.pdf (again: draft, watch out for updates)

### Meeting 3 (scheduled for 14.1.)

I proved some identities in groups, in particular, the Hall-Witt identity. Then I defined commutators of subgroups and the abelianization of a group and computed some examples.

Here is some GAP code that I used to compute examples in class.

LowCenSer := function( G, n ) if n = 1 then return G; else return CommutatorSubgroup( G, LowCenSer( G, n-1 ) ); fi; end; AssGra := function( G ) local L,n; L:=[]; n:=1; while Size( LowCenSer(G,n)/LowCenSer(G,n+1) )>1 do Append( L, [LowCenSer(G,n)/LowCenSer(G,n+1)] ); n := n+1; od; return L; end; G := DihedralGroup(8); G := QuaternionGroup(8); List( AssGra( G ), Q -> StructureDescription( Q ) );

### Meeting 4 (scheduled for 17.1.)

The plan is to define iterated commutators in a group and to define the Lie algebra associated to them. Then we will do examples.

How to prepare? Make sure that you can work out the examples from the end of Meeting 3, which were S(3), D(8), and Q(8).

Once we have understood how to work out examples by hand, it's boring, and we can leave it to computers. The GAP System is pretty good at doing that, and if time permits, I will show you how to use it for this purpose.

### Meeting 5 (scheduled for 21.1.)

We'll see a definition of Lie algebras, do more examples coming from groups, and then examples coming from rings. We'll also construct universal examples, as we did for groups.

How to prepare? Make sure that you are comfortable with tensor products.

### Meeting 6 (scheduled for 24.1.)

I will review the construction of the (universal) enveloping algebra. (If you found it too quick on Monday: bring and ask your questions!) We can then work out so some formal examples and steer towards the Poincaré–Birkhoff–Witt theorem.

How to prepare? Try to work out the enveloping algebra U*L* in the case when the Lie algebra *L* is abelian or free. Don't use the construction that I gave in class for that, but the universal property. We'll do this in class, but it's better if you have tried it before.

### Meeting 7 (scheduled for 28.1.)

I finished the proof of the Poincaré–Birkhoff–Witt theorem and showed how to recover the Lie algebra from its universal enveloping algebra (as the primitives) in characteristic 0.

### Meeting 8 (scheduled for 31.1.)

We'll spend the next couple of classes studying free Lie algebras. As a warmup, we'll discuss free monoids, free abelian monoids, free magmas, and their linearizations.

How to prepare? Given two elements *x* and *y* in a Lie algebra, how many different triple brackets can you produce? How many different quadruple brackets?

### Meeting 9 (scheduled for 4.2.)

We continue the story started in Meeting 8.

Here is the GAP code that I used.

dimLie := function( r, d ) return Sum( DivisorsInt( d ), c-> MoebiusMu( c ) * r^( d/c ) ) / d; end;; Display( List( [1..6], r-> List( [1..6], d -> dimLie( r, d ) ) ) );

### Meeting 10 (scheduled for 7.2.)

We'll see a proof that the associated graded of a free group is a free Lie algebra.

### Meeting 11 (scheduled for 11.2.)

I discussed exponentials and the formula of Baker, Campbell, Dynkin, and Hausdorff.

### Meeting 12 (scheduled for 14.2.)

I gave two outlooks: First, dwelling on the derivations theme, there are structures related to Lie algebras, such as Leibniz algebras and Poisson algebras. Second, I indicated what one can expect from studying representations of Lie algebras such as \(\mathfrak{sl}_2\mathbb{C}\).

# Part II: Lie groups

### Meeting 13 (scheduled for 18.2.)

I will start the second part of the course: Lie groups.

I have updated the bibliography (see link above) a bit.

If you can't find Segal's chapter in [CSM], just send me an email.

### Meeting 14 (scheduled for 21.2.)

I will continue with many more examples of Lie groups.

How to prepare? Work on the exercises(.pdf) that I mentioned on Monday.

### Meeting 15 (scheduled for 25.2.)

The first half was spent on a thorough discussion of split extensions and their description as semi-direct products. The second half was spent on quaternions and how they can be used to give another description of \(SU(2)\), and the local isomorphisms \(SU(2)\to SO(3)\) as well as \(SU(2)\times SU(2)\to SO(4)\to SO(3)\times SO(3)\).

### Meeting 16 (scheduled for 28.2.)

We'll discuss Minkowski geometry: Lorentz and Poincare groups.

### Meeting 17 (scheduled for 4.3.)

We'll study the Lie groups that appear as automorphisms groups of some complex curves: conformal geometry.

### Meeting 18 (scheduled for 7.3.)

We'll discuss the exponential map \(M_n\mathbb{C}\to GL_n\mathbb{C}\).

### Meeting 19 (scheduled for 11.3.)

We showed that closed subgroups of \(GL_n\mathbb{C}\) are submanifolds via the exponential map.

### Meeting 20 (scheduled for 14.3.)

We'll define the Lie algebras of closed subgroups of \(GL_n\mathbb{C}\) and study the functoriality of this construction.

### Meeting 21 (scheduled for 18.3.)

We'll classify abelian Lie groups.

### Meeting 22 (scheduled for 21.3.)

No meeting in class. Instead, I ask you to read (at least) one of the following short notes and send me an email with three questions that you have about it.

Source: The Princeton companion to mathematics. Edited by Timothy Gowers, June Barrow-Green and Imre Leader. Princeton University Press, Princeton, NJ, 2008. xxii+1034 pp.

As usual, your email should be sent from your `stud.ntnu.no`

account. Indicate in the greeting that it is addressed to me.

### Meeting 23 (scheduled for 25.3.)

Representations in general. Simple and irreducible representations. Representations of the circle group (and tori). The Heisenberg group as an example of a non-linear Lie group.

### Meeting 24 (scheduled for 28.3.)

More representation theory: characters.

### Meeting 25 (scheduled for 1.4.)

Representations of some finite groups.

Display(CharacterTable(SymmetricGroup(3)));

Representations of \(SU(2)\).

### Meeting 26 (scheduled for 4.4.)

Representations of \(SU(2)\), continued.

### Meeting 27 (scheduled for 8.4.)

Representations of \(SU(2)\), revisited. Representations of \(SU(3)\).

### Meeting 28 (scheduled for 11.4.)

Representations of \(SU(3)\), continued. Infinite-dimensional examples: unitary groups of Hilbert spaces and diffeomorphism groups.

### Exams

The exams will take place in the last week of May (27.5.-31.5.) with time slots daily between 13:00 and 16:00.

If you would like to take the exam, please email me to tell me

- which days of these won't work for you and
- which book you are using to prepare.

Deadline: 3.5.

On 2.5., we will use the usual class time and place to prepare.