Coxeter groups and Lie algebras — WS 2020/21

Time: Tuesday and Thursday, 14:15 – 15:45
Place: Virtual, on BBB. Room 3414 "Spassky".
Lecturer: Johan Commelin
More info: See below for a description and literature.

Exercises

All exercises are collected in one file (with 1 section per week): exercises.pdf

Lectures

Week 1 (Tuesday)

Introduction (10:06) [mp4] [pdf]
A brief overview of the course.
(Sorry for the bad sound quality, I used the wrong settings.)
Algebraic motivation (14:52) [mp4] [pdf]
A little experiment with commutator brackets naturally leads to the definition of a Lie algebra.
(Sorry for the bad sound quality, I used the wrong settings.)
Geometric motivation (33:25) [mp4] [pdf]
Investigating the structure on the tangent space at the identity element of a Lie group, leads to the definition of a Lie algebra.
Combinatorial motivation (14:55) [mp4] [pdf]
The classification of semisimple Lie algebras by certain graph-theoretic objects (called Dynkin diagrams) is elegant and surprising. Knowing what's ahead is a motivation for understanding the path towards this classification.

Week 1 (Thursday)

Basic definitions (13:01) [mp4] [pdf]
Definitions of representation, ideal, subalgebra.
{Lower/upper central, derived} series (10:07) [mp4] [pdf]
Several series of ideals that we will use in the definition of nilpotent and solvable Lie algebras.
Nilpotent and solvable Lie algebras (10:26) [mp4] [pdf]
Nilpotent and solvable Lie algebras are defined in terms of the series from the preceding lecture. They are important for the “rough” classification that is our first goal. They are also crucial in analyzing semisimple Lie algebras at a later stage.
Warning: The exercise on p.3 contains a mistake. The upper central series should be equal to the entire Lie algebra (not 0) for some k. This is fixed in the pdf, but not in the video.
Engel's theorem (15:49) [mp4] [pdf]
Engel's theorem shows, roughly speaking, that every nilpotent Lie algebra is a subalgebra of strictly upper-triangular matrices.

Week 2 (Tuesday)

Lie's theorem (22:55) [mp4] [pdf]
Lie's theorem shows, roughly speaking, that every solvable Lie algebra is a subalgebra of upper-triangular matrices.
Simple/semisimple Lie algebras (13:59) [mp4] [pdf]
The definition of simple and semisimple Lie algebras. We also look at irreducible representations.
The Killing form (13:15) [mp4] [pdf]
The Killing form is a natural bilinear form that can be defined on every Lie algebra. It is used in a powerful criterion for determining whether a Lie algebra is solvable or semisimple: Cartan's criterion, which we will prove in the next lecture.
Cartan's criterion (26:00) [mp4] [pdf]
Cartan's criterion is a method for determining whether a Lie algebra is solvable using the Killing form. A different variant gives a criterion for semisimplicity. In this lecture we prove Cartan's criterion. This involves some technical lemmas, but the end result is worth it.

Week 2 (Thursday)

Three-dimensional Lie algebras (23:20) [mp4] [pdf]
We study Lie algebras with dimension 3. Over an algebraically closed field of characteristic 0, we see that every simple Lie algebra of dimension 3 is isomorphic to sl_2.
Representations of sl_2 (36:01) [mp4] [pdf]
We classify the irreducible representations of sl_2. There is, up to isomorphism, a unique irreducible representation in each dimension 1, 2, 3, etc. In this classification, we assume to results that have not yet been proven: “Complete Reducibility”, and “Preservation of the Jordan decomposition”. We will prove these at a later stage.

Week 3 (Tuesday)

This week we take a step back. In the last two weeks, the pace was very high. My apologies for that. From now on, the pace should be slower. We will see a bit of new content this week, but it would also be good to use the extra time to review material from last week. On Tuesday, we look at the Jordan decomposition. Next time (Thurday), we will look at multilinear algebra: tensor products, symmetric powers, alternating powers.

Recap: Jordan decomposition [pdf]
[Due to illness, there is no video recording.]
The Jordan decomposition is a fundamental result in linear algebra: it shows that every endomorphism of a vector space (over an algebraically closed field) can be written as the sum of a semisimple (in matrix language: diagonalizable) and a nilpotent endomorphism.

Week 3 (Thursday)

Tensor products (20:19) [mp4] [pdf]
Tensor products are construction in linear algebra. In this lecture, we look at the definition, and we see how they make it easier to work with bilinear maps.
Symmetric/exterior powers (27:17) [mp4] [pdf]
Symmetric powers and exterior powers are variations on the theme of tensor products. They are useful when working with symmetric/alternating bilinear maps.

Week 4 (Tuesday)

Tensor products of representations (16:47) [mp4] [pdf]
In this lecture we study the tensor product of two irreducible (finite-dimensional) representations of sl_2.
The basics of sl_3 (19:27) [mp4] [pdf]
We look at the basic structure of the Lie algebra sl_3. The approach is similar to the case of sl_2, but some of the techniques need to be modified. This will be a useful guide for when we study an arbitrary simple Lie algebra.

Week 4 (Thursday)

Irreducible representations of sl_3 I (30:23) [mp4] [pdf]
Our first steps in understanding the irreducible representations of sl_3. We see that this naturally leads to the concepts of weights and roots. We end with the definition of a heighest weight.
Irreducible representations of sl_3 II (21:50) [mp4] [pdf]
In this lecture, we see that every irreducible representation is generated by the image of a heighest weight vector under the action of the negative root spaces.

Week 5 (Tuesday)

Irreducible representations of sl_3 III (18:14) [mp4] [pdf]
We determine what shape the set of weights of an irreducible representation can have: a convex hexagon preserved by 3 natural symmetries.
Irreducible representations of sl_3 IV (09:16) [mp4] [pdf]
We calculate the weight diagram of some low-dimensional representations of sl_3.

Week 5 (Thursday)

Classical Lie algebras (30:14) [mp4] [pdf]
The definition of the classical Lie algebras. We have already seen sl_n. In this lecture we define so_n and sp_2n.

Week 6 (Tuesday)

Complete reducibility, preparations (25:02) [mp4] [pdf]
This video contains several lemmas that will be helpful when we prove that every finite-dimensional representation of a semisimple Lie algebra is completely reducible.
Casimir operators (27:10) [mp4] [pdf]
In this lecture we introduce the notion of invariant elements, and apply this to invariant bilinear forms to produce Casimir operators. The Casimir operators are a crucial ingredient in the proof of complete reducibility. Warning: In the final lemma, the first sentence should say dim(g) = n, not dim(V) = n. This is corrected in the pdf, but not in the video.

Week 6 (Thursday)

Complete reducibility (19:28) [mp4] [pdf]
Today we finish the proof of complete reducibility. Because both video on Tuesday were quite long, this time we have only one video.

Week 7 (Tuesday)

Absolute Jordan decomposition (30:04) [mp4] [pdf]
In this lecture we prove the preservation of the Jordan decomposition.

Week 7 (Thursday)

Detailed roadmap [pdf]
Unfortunately, there is no video for this lecture. My apologies. The lecture notes contain a roadmap that outlines the steps that we need to take to complete the classification of semisimple Lie algebras and their finite-dimensional representations. The roadmap is inspired by our study of sl_2 and sl_3. You are encouraged to review those lectures, and match the steps that we took there with the steps in this roadmap.

Week 8 (Tuesday)

Cartan subalgebras [pdf]
In this lecture we introduce the notion of Cartan subalgebra, and of regular/generic elements of a Lie algebra. We prove the existence of Cartan subalgebras, by constructing them as centraliser of a regular element.

Week 8 (Thursday)

Towards root systems [pdf]
We use Cartan algebras from the preceding lecture to study the roots of semisimple Lie algebras. We derive a lot of properties of the set of roots. In the next lecture, we will capture all these properties in the concept of a so-called root system.

Week 9 (Tuesday)

Root system [pdf]
In this lecture, we continue our investigation of the roots of a semisimple Lie algebra. We show that we can nicely package the results with the notion of a root system. Finally, we meet the Weyl group of a root system.

Week 9 (Thursday)

Root system (basics) [pdf]
We derive some fundamental restrictions on the geometric shape of a root system. This lecture is short, but contains several calculations that you should check in detail for yourself!

Week 10 (Tuesday)

Simple roots and Weyl chambers [pdf]
We study bases of root systems, and the associated notions of simple roots, and positive/negative roots. We also see Weyl chambers, and how the Weyl group acts on them.

Week 10 (Thursday)

The action of the Weyl group [pdf]
We prove several properties of the action of the Weyl group on the set of Weyl chambers (resp. bases of the root system).

Irreducible root systems [pdf]
We look at irreducible root systems, and most notably we prove that there are at most two different root lenghts in an irreducible root system.

Week 11 (Tuesday)

Classification of root systems [pdf]
In this lecture we prove one of the main theorems of this course: the classification of root systems, and hence of semisimple Lie algebras. To really complete the classification, we need to show that we can go back from Dynkin diagrams/root systems to semisimple Lie algebras. This is something that we will do in the future.

Week 11 (Thursday)

Constructing root systems and Lie algebras [pdf]
Given a Dynkin diagram (or Coxeter matrix), we construct a root system with that Dynkin diagram. We also construct a semisimple Lie algebra from a root system R, but the proof that this Lie algebra has root system R> is omitted.

Week 12 (Tuesday)

Abstract theory of weights [pdf]
We have seen several times that representations can be decomposed into weight spaces. In this lecture, we study the weight lattice, and derive properties of so-called saturated sets. We will use this later in the classification of representations of semisimple Lie algebras.

Week 12 (Thursday)

Universal enveloping algebra [pdf]
The universal enveloping algebra is an associative algebra attached to a Lie algebra, satisfying a suitable universal property. It is a powerful tool in studying the representations of a semisimple Lie algebras.
Standard cyclic modules [pdf]
We introduce maximal vectors and standard cyclic modules. We prove that irreducible standard cyclic modules of a given weight are unique up to isomorphism. Existence of such modules is a topic for next time.

Week 13 (Tuesday)

Existence of finite-dimensional representations [pdf]
We finish the classification of finite-dimensional representations of finite-dimensional semisimple Lie algebras: For every weight, we construct an irreducible standard cyclic module of that weight. This representation is finite-dimensional if and only if the weight is dominant and integral (hence a weight of the root system).

Week 13 (Thursday)

Freudenthal's multiplicity formula [pdf]
In this final lecture, we prove Freudenthal's multiplicity formula. This formula computes the dimension of the weight spaces of irreducible representations in terms of the highest weight of the representation and the root system of the Lie algebra.

Description

Coxeter groups and Lie algebras are central notions in so-called Lie theory. They appear naturally in the study of representation theory of (certain) infinite groups, and have applications in various other fields of mathematics such as differential geometry, algebraic geometry and number theory. In this course we will learn about the basic properties of Coxeter groups and reflection groups, root systems, and Lie algebras. We will see how these concepts interact with each other, and finally learn about the marvellous classification in terms of Dynkin diagrams: a certain type of decorated graphs that naturally fall apart into four infinite lists and a handful of ``exceptional'' examples.

Literature

N. Bourbaki, Éléments de Mathématique, Groupes et algèbres de Lie, Springer, 2007.
W. Fulton and J. Harris, Representation theory: a first course, Springer 1991.
W. Soergel, Lecture notes “Spiegelungsgruppen und Wurzelsysteme”.