I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Manuals

The manuals are  available in the Copy Centre (at a cost of €14) and are required for Monday’s lecture.

## Test 1

The 15% Test 1 will be held in the Melbourne Hall, 16:00 Monday 14 October, Week 6. Ye all have lectures at 15:00 but I will request from your lecturers that ye be left go on time.  There is a sample test in the notes.

## Telegram App

If this is something you might be interested in, please download the Telegram app and set yourself up with an @ handle.

## Diagnostic Test

If you haven’t already, you are invited to take the following ‘Diagnostic Test’:

click here

This ‘Test’ does not go towards your grade, but allows me to give you some feedback on where you are in terms of material you have seen before that will be used in this module.

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.

## Week 2

We continued working with the dot product and then introduced the cross product.

## Week 3

We will look at the applications of vectors to work and moments. We might begin Chapter 2: Matrices.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 1

We had one lecture and after listening to me go on about the importance of mathematics to your programme we started the first chapter on Sets and Relations by looking at some number sets. We saw something new with the concept of the power set of a set. We also took the quick test.

## Week 2

In Week 2 we will start the first chapter proper. We will see something new with the concept of the power set of a set, and set identities, and we’ll explore Cartesian Products and perhaps introduce relations.

## Telegram App

I am exploring the possibility of setting up a classroom group chat that would have the functionality of WhatsApp without having your phone number made public.

If this is something you might be interested in, please download the Telegram app and set yourself up with an @ handle.

## Tutorials

Tutorials start properly in Week 2.

• COMP1C-X: Tuesday at 15:00 in B241L
• COMP1C-Y: Wednesday at 12:00 in B143

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Manuals

The manuals are available in the Copy Centre. Please purchase ASAP. More information has been sent via email.

## Telegram App

I am exploring the possibility of setting up a classroom group chat that would have the functionality of WhatsApp without having your phone number made public.

If this is something you might be interested in, please download the Telegram app and set yourself up with an @ handle.

## FAO: Erasmus Students — Calculators

The Student Resources tab above contains some information about calculators.

Here is a list of some allowed and not allowed calculators.

If you have to purchase a calculator, my recommendation is that you purchase something like a Casio fx-83GT PLUS. This might be available in the CIT shop.

## Tutorials

Tutorials, which are absolutely vital, start next week. There may be a split but this might not occur until Week 3.

## Week 1

In week one we had one and a half classes. One half class was given over to a general overview of MATH7019 and we spent about an hour introducing the topic of Curve Fitting including Lagrange Interpolation.

## Week 2

We will start talking about Least Squares curve fitting.

## Quick Test: Academic Learning Centre

I would urge anyone having any problems with material that isn’t being addressed in the tutorials to use the Academic Learning Centre. If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, you need to be aware of this resource. The timetable should be up some time next week.

You will get best results if you come to the helpers there with specific questions. Next week, some students will receive slips detailing areas of maths that they should brush up on.

## Assessment 1

Assessment 1 will have a hand-in date the Friday of Week 5, 11 October. The Assignment is in the manual but I must also send on your personal data sets next week.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

## Student Resources

In the case of a finite classical group $G$, we can show that if we have i.i.d. random variables $\zeta_i\sim\nu\in M_p(G)$, that if $\text{supp }\nu\subset Ng$, for $Ng$ a coset of a proper normal subgroup $N\rhd G$, that the random walk on $G$ driven by $\nu$, the random variables:

$\xi_k=\zeta_k\cdots \zeta_1$,

exhibits a periodicity because

$\xi_k\in Ng^{k}$.

This shows that a necessary condition for ergodicity of a random walk on a finite classical group $G$ driven by $\nu\in M_p(G)$ is that the support of $\nu$ not be concentrated on the coset of a proper normal subgroup.

I had hoped that something similar might hold for the case of random walks on finite quantum groups but alas I think I have found a barrier.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Diagnostic Test

If you haven’t already, you are invited to take the following ‘Diagnostic Test’:

click here

This ‘Test’ does not go towards your grade, but allows me to give you some feedback on where you are in terms of material you have seen before that will be used in this module.

## Telegram App

I am exploring the possibility of setting up a classroom group chat that would have the functionality of WhatsApp without having your phone number made public.

If this is something you might be interested in, please download the Telegram app and set yourself up with an @ handle.

## Manuals

The manuals are  available in the Copy Centre (at a cost of €14) and should be purchased as soon as possible.

## Tutorials — Subject to Change — Keep an Eye on Your Timetable

Tutorial for BioEng2A: Wednesdays at 10:00 in B149

Tutorial for BioEng2B: Mondays at 17:00 in B189 (starts this Monday 16 September)

Tutorial for SET2: Mondays at 9:00 in B180 (starts Monday 23 September)

## Week 1

We began our study of Chapter 2, Vector Algebra. We looked at how to both visualise vectors and describe them algebraically. We learned how to find the magnitude  and direction of a vector.

## Week 2

We will continue working with the vectors and hopefully learn how to add them and scalar multiply them, about displacement vectors,  the vector product known as the dot product and perhaps then introduce the cross product.

## Test 1

The test will probably be the Monday of Week 5. Official notice will be given in Week 3. There is a sample test in the notes.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Student Resources

Slides of a talk given at Munster Groups 2019, WIT.

Abstract: It is a folklore theorem that necessary and sufficient conditions for a random walk on a finite group to converge in distribution to the uniform distribution – “to random” – are that the driving probability is not concentrated on a proper subgroup nor the coset of a proper normal subgroup. This is the Ergodic Theorem for Random Walks on Finite Groups. In this talk we will outline the rarely written down proof, and explain why, for example, adjacent transpositions can never mix up a deck of cards. From here we will, in a very leisurely and natural fashion, introduce (and motivate the definition of) finite quantum groups, and random walks on them. We will see how the group algebra of a finite group is the algebra of functions on a finite quantum group. Freslon has very recently proved the Ergodic Theorem in this setting, and we present ongoing work towards an Ergodic Theorem in the more general finite quantum group setting; a result that would generalise both the folklore and group algebra Ergodic Theorems.

### Introduction

Every finite quantum group has finite dimensional algebra of functions:

$\displaystyle F(G)=\bigoplus_{j=1}^m M_{n_j}(\mathbb{C})$.

At least one of the factors must be one-dimensional to account for the counit $\varepsilon:F(G)\rightarrow \mathbb{C}$, and if this factor is denoted $\mathbb{C}e_1$, the counit is given by the dual element $e^1$. There may be more and so reorder the index $j\mapsto i$ so that $n_i=1$ for $i=1,\dots,m_1$, and $n_i>1$ for $i>m_1$:

$\displaystyle F(G)=\left(\bigoplus_{i=1}^{m_1} \mathbb{C}e_{i}\right)\oplus \bigoplus_{i=m_1+1}^m M_{n_i}(\mathbb{C})=:A_1\oplus B$,

Denote by $M_p(G)$ the states of $F(G)$. The pure states of $F(G)$ arise as pure states on single factors.

In the case of the Kac-Paljutkin and Sekine quantum groups, the convolution powers of pure states exhibit a periodicity of sorts. Recall for these quantum groups that $B$ consists of a single matrix factor.

In these cases, for pure states of the form $e^i$, that is supported on $A_1$ (and we can say a little more than is necessary), the convolution remains supported on $A_1$ because

$\Delta(A_1)\subset A_1\otimes A_1+B\otimes B$.

If we have a pure state $\nu$ supported on $B=M_{\sqrt{\dim B}}(\mathbb{C})$, then because

$\Delta(B)\subset A_1\otimes B+B\otimes A_1$,

then $\nu\star\nu$ must be supported on, because of $\Delta(A_1)\subset A_1\otimes A_1+B\otimes B$, $A_1$.

Inductively all of the $\nu^{\star 2k}$ are supported on $A_1$ and the $\nu^{\star 2k+1}$ are supported on $B$. This means that the convolutions powers of a pure state, in these cases, cannot converge to the Haar state.

The question is, do the results above about the image of $A_1$ and $B$ under the coproduct hold more generally? I believe that the paper of Kac and Paljutkin shows that this is the case whenever $B$ consists of a single factor… but does it hold more generally?

To find out we go back and do some sandboxing with the paper of Kac and Paljutkin. Which is a pleasure because that paper is beautiful. The blue stuff is my own scribbling.

## Finite Ring Groups

Let $G$ be a finite quantum group with notation on the algebra of functions as above. Note that $A_1$ is commutative. Let

$p=\sum_{i=1}^{m_1}e_i$,

which is a central idempotent.

### Lemma 8.1

$S(p)=p$.

Proof: If $S(\mathbf{1}_G-p)p\neq 0$, then for some $i>m_1$, and $f\in M_{n_i}(\mathbb{C})$, the mapping $f\mapsto S(f)p$ is a non-zero homomorphism from $M_{n_i}(\mathbb{C})$ into commutative $A_1$ which is impossible.

If $S(\mathbf{1}_G-p)p=g\neq 0$, then one of the $S(I_{n_i})\in A_1\oplus B$, with ‘something’ in $A_1$. Using the centrality and projectionality of $p$, we can show that the given map is indeed a homomorphism.

It follows that $S(p)p=p\Rightarrow S(S(p)p)=S(p)=S(p)p=S(p)$, and so $p=S(p)$ $\bullet$

### Lemma 8.2

$(p\otimes p)\Delta(p)=p\otimes p$

Proof: Suppose that $(p\otimes p)\Delta(f)=b$ for some non-commutative $f\in M_{n_i}(\mathbb{C})$. This means that there exists an index $k$ such that $f_{(1)_k}\otimes f_{(2)_k}\in A_1\otimes A_1$. Then for that factor,

$f\mapsto \Delta(f)(p\otimes p)$

is a non-null homomorphism from the non-commutative into the commutative.

We see that $(p\otimes p)\Delta(f)=0$ for all $f\in B$. Putting $a=\mathbf{1}-p$ we get the result $\bullet$

The following says that $p$ is a group-like projection. We know from previous work that if a state is supported on a group-like projection that it will remain supported on it. In particular, any state supported on $A_1$ will remain there.

### Lemma 8.3

$(p\otimes \mathbf{1}_G)\Delta(p)=p\otimes p=(\mathbf{1}_G\otimes p)\Delta(p)$.

Proof: Since $\Delta$ is a homomorphism, $\Delta(p)$ is an idempotent in $F(G)\otimes F(G)$I  do not understand nor require the rest of the proof.

### Lemma 8.4

$A_1=F(G_1)$ is the algebra of functions on finite group with elements $i=1,\dots,m_1$, and we write $e_i=\delta_i$. The coproduct is given by $(p\otimes p)\Delta$.

We have:

$(p\otimes p)\Delta(e_i)=\sum_{t\in G_1}\delta_{it^{-1}}\otimes \delta_t$,

$S(\delta_i)=\delta_{i^{-1}}$,

$\varepsilon(e_i)=\delta_{i,1}$,

as $e_1=\delta_e$.

The element $\Delta(\delta_i)$ is a sum of four terms, lying in the subalgebras:

$A_1\otimes A_1,\,B\otimes B,\,A_1\otimes B,\,B\otimes A_1$.

We already know what is going on with the first summand. Denote the second by $P_i$. From the group-like-projection property, the last two summands are zero, so that

$\Delta(\delta_i$=\sum_{t\in G_1}\delta_{t}\otimes \delta_{t^{-1}i}+P_i\$.

Since the $\delta_i$ are symmetric ($\delta_i^*=\delta_i$) mutually orthogonal idempotents, $P_i$ has similar properties:

$P_i^*=P_i,\,P_i^2=P_i,\,P_iP_j=0$

for $i\neq j$.

At this point Kac and Paljutkin restrict to $B=M_{n_{i+1}}(\mathbb{C})$, that is there is only one summand. Here we try to keep arbitrarily (finitely) many summands in $B$.

Let the summand $M_{n_i}(\mathbb{C})$ have matrix units $E_{mn}^i$, where $m,n=1,\dots,n_i$Kac and Paljutkin now do something which I think is a little dodgy, but basically that the integral over $G$ is equal on each of the $\delta_i$, equal on each of the $E_{mm}^i$, and then zero off the diagonal.

It does follow from above that each $P_i\in B\otimes B$ is a projection.

Now I am stuck!

This sandbox is going to take from a variety of sources, mostly Shuzhou Wang.

## C*-Ideals

Let $J\subset C(X)$ be a closed (two-sided) ideal in a non-commutative unital $C^*$-algebra $C(X)$. Such an ideal is self-adjoint and so a non-commutative $C^*$-algebra $J=C(S)$. The quotient map is given by $\pi:C(X)\rightarrow C(X)/C(S)$, $f\mapsto f+J$, where $f+J$ is the equivalence class of $f$ under the equivalence relation:

$f\sim_{J} g\Rightarrow g-f\in C(S)$.

Where we have the product

$(f+J)(g+J)=fg+J$,

and the norm is given by:

$\displaystyle\|f+J\|=\sup_{j\in C(S)}\|f+j\|$,

the quotient $C(X)/ C(S)$ is a $C^*$-algebra.

Consider now elements $j_1,\,j_2\in C(S)$ and $f_1,\, f_2\in C(X)$. Consider

$j_1\otimes f_1+f_2\otimes j_2\in C(S)\otimes C(X)+C(X)\otimes C(S)$.

The tensor product $\pi\otimes \pi:C(X)\otimes C(X)\rightarrow (C(X)/C(S))\otimes (C(X)/ C(S))$. Now note that

$(\pi\otimes\pi)(j_1\otimes f_1+f_2\otimes j_2)=(0+J)\otimes(f_1+J)+$

$(f_2+J)\otimes(0+J)=0$,

by the nature of the Tensor Product ($0\otimes a=0$). Therefore $C(X)\otimes C(S)+C(S)\otimes C(X)\subset \text{ker}(\pi\otimes\pi)$.

### Definition

A WC*-ideal (W for Woronowicz) is a C*-ideal $J=C(S)$ such that $\Delta(J)\subset \text{ker}(\pi\otimes\pi)$, where $\pi$ is the quotient map $C(G)\rightarrow C(G)/C(S)$.

Let $F(G)$ be the algebra of functions on a classical group $G$. Let $H\subset G$. Let $J$ be the set of functions which vanish on $H$: this is a C*-ideal. The kernal of $\pi:F(G)\rightarrow F(G)/J$ is $J$.

Let $\delta_s\in J$ so that $s\not\in H$. Note that

$\displaystyle\Delta(\delta_s)=\sum_{t\in G}\delta_{st^{-1}}\otimes\delta_t$

and so

$\displaystyle(\pi\otimes \pi)\Delta(\delta_s)=\sum_{t\in G}\pi(\delta_{st^{-1}})\otimes \pi(\delta_t)$.

Note that $\pi(\delta_t)=0+J$ if $t\not\in H$. It is not possible that both $st^{-1}$ and $t$ are in $H$: if they were $st^{-1}\cdot t\in H$, but $st^{-1}\cdot t=s$, which is not in $H$ by assumption. Therefore one of $\pi(\delta_{st^{-1}})$ or $\pi(\delta_t)$ is equal to zero and so:

$(\pi\otimes\pi)\Delta(\delta_s)=0$,

and so by linearity, if $f$ vanishes on a subgroup $H$,

$\Delta(f)\subset \text{ker}(\pi\otimes\pi)$.

In this way, WC*-ideals generalise functions which vanish on distinguished subgroups. In fact, without checking all the details, I imagine that first isomorphism theorem can show that $F(G)/ J=F(H)$. Let $\pi_H:F(G)\rightarrow F(H)$ be the ring homomorphism

$\displaystyle\pi_H\left(\sum_{t\in G}a_t\delta_t\right)=\sum_{t\in H}a_t\delta_t$.

Then $\text{ker}\,\pi_H=J$, $\text{im}\,\pi_H=F(H)$, and so we have the isomorphism of rings, which presumably carries forward to the algebras of functions level…

Just some notes on section 1 of this paperFlags and notes are added but mistakes are mine alone.

#### Definition

Let $C(G)$ be the algebra of continuous functions on a compact matrix quantum group. Such an object is given by a matrix $u=\{u_{ij}\}_{i,j=1}^N$ which generates $C(G)$ as a C*-algebra. Furthermore, there exists a C*-algebra homomorphism $\Delta:C(G)\rightarrow C(G)\otimes C(G)$ such that

$\displaystyle \Delta(u_{ij})=\sum_{k=1}^N u_{ik}\otimes u_{kj}$,

and both $u$ and $u^T$ are invertible in $M_N(C(G))$.

Any subgroup $G\subset \text{GL}(N,\mathbb{C})$ is such an object, with the $u_{ij}\in C(G)$ given by $u_{ij}(g)=g_{ij}\in\mathbb{C}$. Furthermore

$\mathrm{C}_{\text{comm}}\langle u_{ij}\rangle \cong C(G)$.

We say that $\rho=(\rho_{ij})_{i,j=1}^{d_\rho}\in M_{d_{\rho}}(C(G))$ is a representation if it is invertible and

$\displaystyle \Delta(\rho_{ij})=\sum_{k=1}^{d_\rho}\rho_{ik}\otimes\rho{kj}$.

The transpose $\rho^T=(\rho_{ji})_{i,j=1}^N\in M_{d_{\rho}}(C(G))$ is also invertible and so we have:

#### Proposition

The C*algebra generated by the $\rho_{ij}$ is also the algebra of continuous functions on a compact matrix quantum group.

## Background

I am trying to prove an Ergodic Theorem for Random Walks on Finite Quantum Groups. A random walk on a quantum group driven by $\nu\in M_p(G)$ is ergodic if the convolution powers $(\nu^{\star k})_{k\geq 0}$ converge to the Haar state $\int_G$.

The classical theorem for finite groups:

#### Ergodic Theorem for Random Walks on Finite Groups

A random walk on a finite group $G$ driven by a probability $\nu\in M_p(G)$ is ergodic if and only if $\nu$ is not concentrated on a proper subgroup nor the coset of a proper normal subgroup.

Not concentrated on a proper subgroup gives irreducibility. A random walk is irreducible if for all $g\in G$, there exists $k\in\mathbb{N}$ such that $\nu^{\star k}(\{g\})>0$.

Not concentrated on the coset of a proper normal subgroup gives aperiodicity. Something which should be equivalent to aperiodicity is if

$p:=\gcd\{k>0:\nu^{\star k}(e)>0\}$

is equal to one (perhaps via invariance $\mathbb{P}[\xi_{i+1}=t|\xi_{i}=s]=\mathbb{P}[\xi_{i+1}=th|\xi_{i}=sh]$).

If $\nu$ is concentrated on the coset a proper normal subgroup $N\rhd G$, specifically on $Ng\neq Ne$, then we have periodicity ($Ng\rightarrow Ng^2\rightarrow \cdots\rightarrow Ng^{o(g)}=Ne\rightarrow Ng\rightarrow \cdots$), and $p=o(g)$, the order of $g$.

In Markov chain theory, ergodicity is equivalent to irreduciblity and aperiodicity.

The theorem in the quantum case should look like:

#### Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group $G$ driven by a state $\nu\in M_p(G)$ is ergodic if and only if “X”.

## Irreducibility

At the moment I have some part of X;the irreducibility bit. As is well known since Pal (1996), it is possible to have a probability not concentrated on a quantum subgroup be reducible. This led Franz & Skalski to generalise quantum subgroups to group-like-projections, which I will say correspond to quasi-subgroups following Kasprzak & Sołtan.

I have shown that if $\nu$ is concentrated on a proper quasi-subgroup $S$, in the sense that $\nu(P_S)=1$ for a group-like-projection $P_S$, that so are the $\nu^{\star k}$. The analogue of irreducible is that for all $q$ projections in $F(G)$, there exists $k\in\mathbb{N}$ such that $\nu^{\star k}(q)>0$. If $\nu$ is concentrated on a quasi-subgroup $S$, then for all $k$, $\nu^{\star k}(Q_S)=0$, where $Q_S=\mathbf{1}_G -P_S$.

I have also shown on the other hand that if the random walk is reducible that it must be concentrated on a proper quasi-subgroup. Franz & Skalski show that group-like-projections also have a correspondence with idempotent states. The Césaro Means

$\displaystyle \nu_n:=\frac{1}{n}\sum_{k=1}^n\nu^{\star k}$,

converge to an idempotent state $\nu_\infty$. If $\nu^{\star k}(q)=0$ for all $k$ then the $\nu_{\infty}(q)=0$ also, so that $\nu_\infty\neq \int_G$ (as the Haar state is faithful). I was able to prove that $\nu$ is supported on the quasi-subgroup given by the idempotent $\nu_\infty$.

I believe this result — irreducible if and only if not concentrated on a quasi-subgroup — holds more generally than just in finite quantum groups.