# Projects

## Project

### Assessment

You will need to:

• produce one written project (7-10 pages, A4). This will contribute 30% of your final grade.

• give one presentation (10 minutes + 5 minutes questions) based on your written project. This will contribute 20% of your final grade.

• as a warm-up for the final presentation, you will also be asked to present one of your solutions to the weekly homeworks. This will be worth 10% of the final grade.

It is likely that your presentation will need to be online.

The deadline for the written project will be 17th January 2022 (first Monday of Term 2).

The presentations will happen early during Term 2 (not yet timetabled).

I am happy to offer you each a maximum of 1-hour of one-to-one support on this project (most likely 2 half-hour meetings) during term 1. I will advertise slots for these meetings during term.

### Format

• The written project should be prepared using LaTeX.

• Increasing/decreasing margins or decreasing/increasing font-size by a ridiculous amount to fit the page-limit is strongly discouraged. The page limit (7-10 pages) is there to (a) encourage conciseness, (b) make sure you've done enough to demonstrate engagement with the learning outcomes.

• The page count does not include the bibliography: you can cite as many relevant sources as you want. I recommend you use BibTeX for references. The American Mathematical Society's MR Lookup Page is a fantastic resource for looking up the BibTeX code for books and papers.

### Mark scheme

I will mark projects using the university's "aggregate grade" system; see the Part II handbook for information about this system and what level of engagement with the learning objectives will gain you the various aggregate grades. The learning outcomes are listed below.

### Intended learning outcomes

• Read and explore an idea or circle of ideas related to the MATH426 course but going beyond the main sequence of lectures.

• Construct a mathematical narrative which explains these ideas to your assessor and your peers, in written and oral form. You should aim to be comprehensible to someone who has taken the same course, but may have focused on a different project.

• Demonstrate judgement about the importance and relevance of material you include.

• Illustrate your explanation with carefully-chosen examples.

• Demonstrate an ability to unpack and interrogate proofs rather than simply copying them out or paraphrasing them. This means:

• Extracting and summarising the key new idea or construction at the heart of a proof.

• Reworking proofs with extra simplifying assumptions to allow the key idea to shine through.

• Identifying points in a proof that need amplifying in more detail for you to understand them.

• Correctly evaluating which parts of a proof are easy or routine and which require more thought or new ideas.

### Notes

Here are some of the phrases used in the assessment criteria in the Part II handbook, and how I choose to interpret them.

• "deployment of considered judgement relating to key issues, concepts and procedures". Suppose you're writing your project and the book you're reading mentions some theorem whose relevance you don't understand; it seems like you can get away without mentioning it to tell the story you want to tell. Nonetheless, you feel uneasy: maybe there's a subtle point you're missing. You include it anyway. When I read your essay, this sticks out like a sore thumb because I also can't see the relevance of it. You would not have demonstrated "considered judgement" about what to include.

• "clearly grounded on a close familiarity with a wide range of supporting evidence". You tried reading the book I recommended for your project but it made no sense. You spent a few hours in the library looking at other books about Lie theory on the shelf and you found a couple of books which explain things in a way you like (though nothing which explains everything by itself). These books leave some details out, but give references to other books or papers, which you chase up (even if you don't understand what's written very well). In the end, your essay synthesises what you read from these different books and gives appropriate references for the material you felt was important to mention but chose to omit. You would have certainly met this criterion.

## Project descriptions

Here are some of the possible projects you could do for MATH426. If there is something you'd like to focus on which is not listed, please let me know! Your project should be driven primarily by your own interests.

### BCH formula

The Baker-Campbell-Hausdorff formula asserts that if $X$ and $Y$ are matrices the logarithm of $\exp(X)\exp(Y)$ as an infinite sum of terms involving $X$ , $Y$ , and iterated bracket operations between them. The full, explicit formula is less important than the existence of such a formula. The goal of this project is to prove that the formula exists (without proving the explicit formula itself).

There is a very nice 2-page proof due to Eichler however, it is quite terse and leaves a to the imagination of the reader. Here are some questions to bear in mind while you are reading and assimilating the proof.

• What does Eichler mean when he talks about "Lie polynomials"? What does the notation $\sim$ mean (e.g. in Eq. 2)?

• Can you figure out how his argument is going in the $n=3$ case? The $n=4$ case? Maybe that will help you to understand what he's going on about for general $n$ .

• He claims that the proof "gives a recursive scheme for...computation" of $F_{n}$ . Can you see how to turn what he has written into a calculation of $F_{3}$ ?

If you don't like Eichler's proof, there are messier (longer and more explicit) proofs in Hall's online notes and Knapp.

You could start working on this project at any point in term: it only requires you to know about the exponential map for matrices; it is purely "elementary" (which is not the same as "easy"). I would warn you that although a lot of people tried this project in the past very successfully, it might be hard to make this project "your own" (rather than Eichler's) and distinguish yourself in some way.

### Lie's theorem/Engel's theorem

In this course, we have focused almost exclusively on compact Lie groups and semisimple Lie algebras. The one example that went beyond this was the Heisenberg group (mentioned in some of the question sheets), but there is a whole world of "nilpotent" and "solvable" Lie algebras we haven't touched. The theorems of Lie and Engel concern these Lie algebras and their representations. For example, Lie's theorem tells us that any representation of a solvable Lie algebra can be made upper triangular by a suitable choice of basis. The goal of this project would be to understand what the terms "nilpotent" and "solvable" mean in this context, and to give a proof of one or both of these theorems. This is related to Qs.10 and 11 on Sheet 1.

Good references include Knapp, starting around p.40; Fulton and Harris, Chapter 9; and Kirillov, Chapter 5.

This project is largely orthogonal to what we do in class, so you could start working on it as soon as you know what a Lie algebra is (early in term).

### Classify 3-dimensional Lie algebras up to isomorphism

For low-dimensional vector spaces, it is feasible to classify all possible Lie brackets up to isomorphism. This is worked out in some detail in Fulton and Harris, Chapter 10 (with some details left as exercises). The goal of this project is to understand and explain this classification.

Beyond the definition of a Lie algebra (week 2) this project doesn't require much material from the course, so you could start work on it very early in term.

### Lie groups as manifolds

A Lie group is a manifold $G$ with a group structure (multiplication and inversion maps which are smooth in local coordinates).

In this project you can explore this in more detail, starting by understanding the definition of a manifold. The ultimate goal of the project could be:

• Explain how to associate a Lie algebra to a Lie group (not just a matrix group). This will involve you learning about vector fields, and about exponentiation and Lie bracket of vector fields. I recommend Warner's book or Bump's book (chapters 5-8).

• Explain why $GL(n,\mathbf{R})$ is a Lie group and why a topologically closed subgroup of a Lie group is a Lie group (which then explains why our course was about Lie groups!). One of the additional videos (constructing local exponential charts on a matrix group) is relevant because it contains the idea for proving that a topologically closed subgroup of a Lie group is a manifold. Adams's book would also be useful for this.

This project only really relies on material from weeks 1-3, so you could start work on it quite early in term.

### Lie's theorem on homomorphisms

Recall Lie's theorem on homomorphisms:

Theorem:

Let $G_{1}$ and $G_{2}$ be Lie groups with Lie algebras $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ respectively. If $G_{1}$ is simply-connected then any Lie algebra homomorphism $\theta\colon\mathfrak{g}_{1}\to\mathfrak{g}_{2}$ gives rise to a Lie group homomorphism $f\colon G_{1}\to G_{2}$ such that $f(\exp X)=\exp(\theta(X))$ .

There is a very brief sketch proof of this on pages 77--79 of Carter-Segal-MacDonald. Can you understand this sketch proof well enough to fill in some of the details? I also have a worksheet with hints if you get lost/stuck.

This project relies on the foundational material from weeks 1-3, but nothing later, so you could start work on this project quite early in term.

### Spin groups

The rotation groups $SO(n)$ in dimension $n\geq 3$ are not simply-connected: they have a simply-connected double cover called $Spin(n)$ . We studied the example $Spin(3)\cong SU(2)$ in a video (the 2-to-1 map $SU(2)\to SO(3)$ ). In general, the spin groups can be constructed as follows:

• First construct the Clifford algebra on $\mathbf{R}^{n}$ .

• Inside the "even part" of the Clifford algebra, take the invertible elements with "unit norm".

Your task is to explain this construction, and why it gives a double-cover of $SO(n)$ .

Nice references include:

• Curtis, Chapter 10.

• Fulton and Harris, Chapter 20.

• This is largely orthogonal to what we do in class, so you could start working on this project quite early in term.

### Topology of Lie groups

The video on the topology of Lie groups explains some very sketchy arguments for why various groups are simply-connected or for why certain non-compact groups deformation retract onto their maximal compact subgroups. Two possible projects include filling in the details. Note that these projects will require you to know some topology (maybe you are learning some for your dissertation) but most of the relevant course content is covered by the end of week 3, so you could start doing this midway through term.

#### SU(n) is simply-connected

First, learn a bit more about the fundamental group, and the proof that the $k$ -dimensional sphere is simply-connected if $k\geq 2$ . Then learn about sphere bundles: enough to see that $SU(3)$ is a bundle over $S^{5}$ with fibre $SU(2)=S^{3}$ , and more generally that $SU(n)$ is a bundle over $S^{2n-1}$ with fibre $SU(n-1)$ . Learn about fibrations and the homotopy lifting property, and why sphere bundles are fibrations. It's true in general that:

Lemma:

The total space of a fibre bundle is simply-connected if the base and fibre are both simply-connected.

Can you see why using homotopy lifting? From this, prove that $SU(n)$ is simply-connected.

Some resources include my notes about the fundamental group (which should be enough for an introduction to the fundamental group and the simply-connectedness of spheres) and Hatcher's book "Algebraic topology", which discusses fibrations in Chapter 4. The "homotopy long exact sequence of a fibration" is a big generalisation of the lemma above, so if you struggle to prove it for yourself, you should be able to extract a proof by looking at the proof of the homotopy long exact sequence of a fibration.

#### Deforming to a compact subgroup

Learn about the polar decomposition of a matrix in $GL(n,\mathbf{C})$ . Use this to show that $GL(n,\mathbf{C})$ can be retracted back to $U(n)$ . You can also find the maximal compact subgroups of some other noncompact groups and show that they retract in a similar way.

A great reference for this is Hall Section 2.5. There is also a very terse summary of how to construct the polar decomposition in Carter-Segal-MacDonald Part II, Chapter 4, but I'm looking for quite a lot more detail!

### Universal enveloping algebra and Casimirs

If we're working with a Lie algebra $\mathfrak{g}$ of matrices, and we have two elements $X,Y\in\mathfrak{g}$ , the product $XY$ makes sense as a matrix even if it doesn't live in $\mathfrak{g}$ . In fact, given any Lie algebra $\mathfrak{g}$ , we can form its universal enveloping algebra (UEA), $U(\mathfrak{g})$ which is an associative algebra containing $\mathfrak{g}$ as a subspace, in which you can multiply elements, and such that the Lie bracket is given by $[X,Y]=XY-YX$ . Importantly, whenever you have a representation $f$ of $\mathfrak{g}$ , you get for free a representation of $U(\mathfrak{g})$ .

The centre of the UEA (consisting of elements which commute with everything) is particularly important because if you have a representation $f\colon\mathfrak{g}\to\mathfrak{gl}(V)$ then $f$ of a central element in the UEA will commute with all the matrices in your representation. One important element which exists for any semisimple Lie algebra is the Casimir element.

In this essay, you should discuss the definition and construction of the UEA, then you could take it in several different directions:

• Prove the Poincaré-Birkhoff-Witt theorem which tells us a basis for the UEA. See for example Knapp, Chapter III.

• Include some examples or applications of Casimirs. For example:

• in Fulton and Harris, Chapter 25, they use the Casimir to prove a formula (Freudenthal's multiplicity formula) for the multiplicities of irreducible representations.

• In Kirillov (Section 6.3) he gives a proof of complete reducibility using the Casimir element, avoiding the unitarian trick.

This project doesn't require much beyond what we do in weeks 4-5, so you could start it midway through term.

### Character theory

The character of a representation $R\colon G\to GL(n,\mathbf{C})$ is the function $\chi\colon G\to\mathbf{C}$ given by $\chi(g)=\mathrm{Tr}(R(g))$ . Assuming $G$ is compact (so we have the existence of a Haar integral $\int_{G}\cdot dg$ on $G$ ), you can prove some wonderful results, for example Schur orthogonality:

Theorem:

The characters $\chi_{1},\chi_{2}$ for irreps $R_{1},R_{2}$ are orthonormal with respect to the inner product $\langle\chi_{1},\chi_{2}\rangle=\int_{G}\overline{\chi_{1}(g)}\chi_{2}(g)dg$ . In other words, $\langle\chi_{1},\chi_{2}\rangle=\begin{cases}0\mbox{ if }R_{1}\not\cong R_{2}% \\ 1\mbox{ if }R_{1}\cong R_{2}\end{cases}$ .

This means that if you have a representation with character $\chi$ , you can figure out its decomposition into irreducibles $R_{i}$ : it will be $\bigoplus_{i}R_{i}^{\oplus\langle\chi,\chi_{i}\rangle}.$

This project could do the following:

• Prove Schur orthogonality.

• Calculate the characters of finite-dimensional irreducible representations of $U(1)$ , and show that decomposition of a representation into irreducibles corresponds to Fourier expansion of its character.

• Discuss (at least state) the Weyl integration formula, which explains how to evaluate integrals like $\int_{G}\overline{\chi_{1}}(g)\chi_{2}(g)dg$ by restricting to a maximal torus.

• Calculate the characters of finite-dimensional irreducible representations of $SU(2)$ and then use this and the Weyl integration formula to redo some of the "decompose into irreducibles" examples from the course using characters.

Good references include:

• Knapp, Chapter IV for the basics, Chapter VIII for the Weyl integration formula.

• C. Teleman's notes, Section 19 onwards.

• This project doesn't rely on much beyond what we do in week 4-5, so you could start it midway through term. It is highly relevant material for those who are doing representation theory of finite groups in term 2.

### Classification of Dynkin diagrams

The final video of the course gives a sketch proof of how to classify the Dynkin diagrams of compact semisimple groups. Expand this sketch to give a full proof.

Useful references include:

Kirillov only covers the "simply-laced" case where there are no double- or triple-edges. Feel free to focus on this case if it improves your understanding of the proof or the quality of your exposition.

This project would be best for over the holidays, as Dynkin diagrams don't come up until week 10.

### 3 possible projects about the Lie algebra g2

We have encountered the root system of the group $G_{2}$ . One way to construct this group (or its Lie algebra) is as the group of automorphisms (respectively Lie algebra of derivations) of the octonion algebra. In any of the following projects, you can learn more about this Lie algebra, the smallest of the exceptional Lie algebras.

#### Project 1 (tricky!)

Show that the Lie algebra of derivations of the octonions $\mathbf{O}$ has Dynkin diagram of type $G_{2}$ (you might need to use a computer algebra system like Sage to help you). I don't have any book to tell you what to do here, and I haven't tried it myself so it might be tricky, but for those brave enough to try it, here's what I suggest you do

• Learn what a derivation is.

• Learn what the octonions are (e.g. read a bit of J. Baez "The octonions").

• Start with a general 8-by-8 matrix $M$ defining a linear map $\mathbf{O}\to\mathbf{O}$ and figure out the conditions on the matrix entries that ensure this is a derivation; you should be left with 14 free variables.

• Find a 2-dimensional subalgebra of these matrices that commute with one another to be your maximal torus; this is probably the trickiest step as it requires some guesswork, and you have no guarantee that your chosen matrices are actually the Lie algebra of a maximal torus ("Cartan subalgebra"). This amounts to saying that your matrices should both be diagonalisable.

• Find the weight spaces for the action of these matrices acting using the adjoint representation.

• Compute the dual Killing form to determine the angles between the weights.

• verify that you have indeed got the root system $G_{2}$ .

#### Project 2

Reconstruct the $\mathfrak{g}_{2}$ Lie algebra from its Dynkin diagram. See Fulton and Harris, chapters 21 and 22 for an in-depth discussion of this. This should also give you an idea how the Dynkin diagram determines the corresponding Lie algebra more generally. This would be a good project to do over the holidays, after the relevant material has been covered in week 10.

#### Project 3

Show that the Lie algebra with Dynkin diagram of type $G_{2}$ has a 7-dimensional representation $V$ and an invariant element in $\Lambda^{3}V^{*}$ ($\Lambda$ denotes the exterior power). See Fulton and Harris, Chapter 22.3 for a discussion. This turns out to be related to point (1) about the octonions. This would be a good project to do some time after week 8 when similar examples for $SU(3)$ have been covered.

### Representation theory of the symplectic groups

By the end of the course, we will have seen weight diagrams for representations of $SU(2)$ and $SU(3)$ and used these to decompose tensor products of such representations. The symplectic groups were introduced on one of the question sheets, and they also have a nice representation theory. You could explain the possible weight diagrams and do some calculations for $\mathfrak{sp}(4)$ where the diagrams are 2-dimensional. A good reference for this is Fulton and Harris "Representation theory", Chapter 16. In particular, you could try to show that:

• the second exterior power of the standard representation contains an invariant element (the "symplectic form")

• there is a 5-dimensional representation $W$ such that $Sym^{2}(W)$ contains an invariant element.

• You could also explain what this has to do with the relationship between $\mathfrak{sp}(4)$ and $\mathfrak{so}(5)$ .

This would make a good project to work on over the holiday, using what you learned in the latter part of the course (weeks 8-10).

### Exceptional isomorphisms

The Lie algebras $SO(6)$ and $SU(4)$ have the same root system. This is because their Lie algebras are the same. This is one of a small list of unexpected isomorphisms of Lie algebras called the "exceptional isomorphisms". In this project you could verify some or all the exceptional isomorphisms by calculating the root diagrams of the corresponding Lie algebras (already $SO(6)$ and $SU(4)$ would be a good example). Good references for this include Fulton and Harris, Chapters 15 and 19.1.

This would be a good project to work on over the holiday, after the relevant material was covered (last week of term).

### Maximal tori are unique up to conjugation

The optional videos for Week 9 explain why maximal tori exist in compact groups. In this project, I want you to explain the proof that any two maximal tori are conjugate. There is a nice proof on pages 90-92 in Adams's book. This will require you to know or learn a little bit about cohomology, in particular, Lefschetz's fixed point theorem. However, you can treat cohomology and the fixed point theorem as a "black box" (as long as you explain roughly what it's about).

I highly recommend trying to work out what the proof is saying in the example of $SU(2)$ first.

Here are some questions you should be asking yourself whilst reading the proof:

• What is $G/T$ ? For example, when $G=SU(2)$ and $T$ is the subgroup of diagonal elements?

• Why are we interested in finding fixed points of the map $xT\mapsto gxT$ ? Can you figure out what this map is for some elements $g\in SU(2)$ when $G=SU(2)$ and $T$ is the subgroup of diagonal elements?

• What does he mean by a "generator" of the torus? (It might help to watch the videos and read more of Chapter 4 of Adams's book). When $g_{0}$ is a generator of the torus, why is the set of fixed points what he claims it is? Can you calculate $N(T)/T$ when $G=SU(2)$ ?

This is a tough project which will need to be tackled after week 10 when the relevant material is covered.

### Hydrogen

The beautiful orbital structure of the hydrogen atom comes about from the fact that the electrical potential is symmetric under the action of $SO(3)$ (rotations about the nucleus). For anyone who has been through the pain of solving Schrödinger's equation for the hydrogen atom, this may come as a surprise, though the very fact that you can separate variables should have been a hint that there is some underlying symmetry. This example is discussed in Kirillov, Section 4.9, and explaining it well would make a good project that you could start any time after Week 6, but I recommend it only if you know a bit of quantum mechanics already.