In a recent preprint, Haonan Zhang shows that if $\nu\in M_p(Y_n)$ (where $Y_n$ is a Sekine Finite Quantum Group), then the convolution powers, $\nu^{\star k}$, converges if

$\nu(e_{(0,0)})>0$.

The algebra of functions $F(Y_n)$ is a multimatrix algebra:

$F(Y_n)=\left(\bigoplus_{i,j\in\mathbb{Z}_n}\mathbb{C}e_{(i,j)}\right)\oplus M_n(\mathbb{C})$.

As it happens, where $a=\sum_{i,j\in\mathbb{Z}_n}x_{(i,j)}e_{(i,j)}\oplus A$, the counit on $F(Y_n)$ is given by $\varepsilon(a)=x_{(0,0)}$, that is $\varepsilon=e^{(0,0)}$, dual to $e_{(0,0)}$.

To help with intuition, making the incorrect assumption that $Y_n$ is a classical group (so that $F(Y_n)$ is commutative — it’s not), because $\varepsilon=e^{(0,0)}$, the statement $\nu(e_{(0,0)})>0$, implies that for a real coefficient $x^{(0,0)}>0$,

$\nu=x^{(0,0)}\varepsilon+\cdots= x^{(0,0)}\delta^e+\cdots$,

as for classical groups $\varepsilon=\delta^e$.

That is the condition $\nu(e_{(0,0)})>0$ is a quantum analogue of $e\in\text{supp}(\nu)$.

Consider a random walk on a classical (the algebra of functions on $G$ is commutative) finite group $G$ driven by a $\nu\in M_p(G)$.

The following is a very non-algebra-of-functions-y proof that $e\in \text{supp}(\nu)$ implies that the convolution powers of $\nu$ converge.

Proof: Let $H$ be the smallest subgroup of $G$ on which $\nu$ is supported:

$\displaystyle H=\bigcap_{\underset{\nu(S_i)=1}{S_i\subset G}}S_i$.

We claim that the random walk on $H$ driven by $\nu$ is ergordic (see Theorem 1.3.2).

The driving probability $\nu\in M_p(G)$ is not supported on any proper subgroup of $H$, by the definition of $H$.

If $\nu$ is supported on a coset of proper normal subgroup $N$, say $Nx$, then because $e\in \text{supp}(\nu)$, this coset must be $Ne\cong N$, but this also contradicts the definition of $H$.

Therefore, $\nu^{\star k}$ converges to the uniform distribution on $H$ $\bullet$

Apart from the big reason — that this proof talks about points galore — this kind of proof is not available in the quantum case because there exist $\nu\in M_p(G)$ that converge, but not to the Haar state on any quantum subgroup. A quick look at the paper of Zhang shows that some such states have the quantum analogue of $e\in\text{supp}(\nu)$.

So we have some questions:

• Is there a proof of the classical result (above) in the language of the algebra of functions on $G$, that necessarily bypasses talk of points and of subgroups?
• And can this proof be adapted to the quantum case?
• Is the claim perhaps true for all finite quantum groups but not all compact quantum groups?

Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, will take place on Friday 30 November in the usual lecture venue of B214.

Here is a provisional sample test. If we don’t cover something by 20 November inclusive it won’t be on the test: this is so you will have two full tutorials before the test.

I will give you a copy of the sample today, Friday 9 November. The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

Week 8

In Week 8 we finished talking about equations and started studying quadratics.

Week 9

In Week 9 we will finish talking about quadratics and begin studying exponents.

Assessment 1 – Results

I will have the assignments with me tomorrow if you want to see your work.

Assessment 2

Assessment 2 is on p.136. It has a hand-in time of 16:00 Monday 26 November.

As suggested in class, I would advise you to — if possible — complete this assignment early if you can, freeing up time in your tutorial to get work done on Chapter 3: Probability & Statistics.

Week 8

We looked at the Poisson distribution, the Normal distribution, and started discussing Sampling.

Week 9

We will complete Chapter 3 by looking at Sampling and Hypothesis Testing. We may have an extra tutorial during one of the lectures.

Test 2

The 15% Test 2 will take place at 16:00 on Monday 26 October, Week 11, in B263. There is a sample test in the notes, p.146. Chapter 3: Differentiation is going to be examined. A Summary of Vectors (p.144): you will want to know this stuff very well. You will be given a copy of these tables

I strongly advise you that attending tutorials alone will not be sufficient preparation for this test and you will have to devote extra time outside classes to study aka do exercises.

Week 8

We looked at Implicit Differentiation and Partial Differentiation. If you are interested in a very “mathsy” approach to curves you can look at this.

Week 9

We will look at applications of partial differentiation to differentials and error analysis. We might start Chapter 4 on (Further) Integration. A good revision of integration/antidifferentiation may be found here.

Study

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

Student Resources

This strategy is by no means optimal nor exhaustive. It is for students who are struggling with basic integration and anti-differentiation and need something to help them start calculating straightforward integrals and finding anti-derivatives.

TL;DR: The strategy to antidifferentiate a function $f$ that I present is as follows:

1. Direct
2. Manipulation
3. $u$-Substitution
4. Parts

Quantum Subgroups

Let $C(G)$ be a the algebra of functions on a finite or perhaps compact quantum group (with comultiplication $\Delta$) and $\nu\in M_p(G)$ a state on $C(G)$. We say that a quantum group $H$ with algebra of function $C(H)$ (with comultiplication $\Delta_H$) is a quantum subgroup of $G$ if there exists a surjective unital *-homomorphism $\pi:C(G)\rightarrow C(H)$ such that:

$\displaystyle \Delta_H\circ \pi=(\pi\circ \pi)\circ \Delta$.

The Classical Case

In the classical case, where the algebras of functions on $G$ and $H$ are commutative,

$\displaystyle \pi(\delta_g)=\left\{\begin{array}{cc}\delta_g & \text{ if }g\in H \\ 0 & \text{ otherwise}\end{array}\right..$

There is a natural embedding, in the classical case, if $H$ is open (always true for $G$ finite) (thanks UwF) of $\imath: C(H) \xrightarrow\, C(G)$,

$\displaystyle \sum_{h\in H}a_h \delta_h \mapsto \sum_{g\in G} a_g \delta_g$,

with $a_g=a_h$ for $h\in G$, and $a_g=0$ otherwise.

Furthermore, $\pi$ is has the property that

$\pi\circ\imath\circ \pi=\pi$,

which resembles $\pi^2=\pi$.

In the case where $\nu$ is a probability on a classical group $G$, supported on a subgroup $H$, it is very easy to see that convolutions $\nu^{\star k}$ remain supported on $H$. Indeed, $\nu^{\star k}$ is the distribution of the random variable

$\xi_k=\zeta_k\cdots \zeta_2\cdot \zeta_1$,

where the i.i.d. $\zeta_i\sim \nu$. Clearly $\xi_k\in H$ and so $\nu^{\star k}$ is supported on $H$.

We can also prove this using the language of the commutative algebra of functions on $G$, $C(G)$. The state $\nu\in M_p(G)$ being supported on $H$ implies that

$\nu\circ\imath\circ \pi=\nu\imath\pi=\nu$.

Consider now two probabilities on $G$ but supported on $H$, say $\mu,\,\nu$. As they are supported on $H$ we have

$\mu=\mu\imath\pi$ and $\nu=\nu\imath\pi$.

Consider

$(\mu\star \nu)\imath\pi=(\mu\otimes \nu)\circ \Delta\circ \imath\pi$

$=((\mu\imath\pi)\otimes(\nu\imath\pi))\circ \Delta\circ\imath\pi =(\mu\imath\otimes \nu\imath)(\pi\circ \pi)\Delta\circ\imath\pi$

$=(\mu\imath\otimes\nu\imath)(\Delta_H\circ \pi\circ \imath\circ \pi)=(\mu\imath\otimes\nu\imath)(\Delta_H\circ \pi)$

$(\mu\imath\otimes \nu\imath)\circ (\pi\circ \pi)\circ\Delta(\mu\imath\pi\otimes \nu\imath\pi)\circ\Delta$

$=(\mu\otimes\nu)\circ\Delta=\mu\star \nu$,

that is $\mu\star \nu$ is also supported on $H$ and inductively $\nu^{\star k}$.

Some Questions

Back to quantum groups with non-commutative algebras of functions.

• Can we embed $C(H)$ in $C(G)$ with a map $\imath$ and do we have $\pi\circ \imath\circ \pi=\pi$, giving the projection-like quality to $\pi$?
• Is $\nu\circ\imath\circ \pi=\nu$ a suitable definition for $\nu$ being supported on the subgroup $H$.

If this is the case, the above proof carries through to the quantum case.

• If there is no such embedding, what is the appropriate definition of a $\nu\in M_p(G)$ being supported on a quantum subgroup $H$?
• If $\pi$ does not have the property of $\pi\circ \imath\circ \pi=\pi$, in this or another definition, is it still true that $\nu$ being supported on $H$ implies that $\nu^{\star k}$ is too?

Edit

UwF has recommended that I look at this paper to improve my understanding of the concepts involved.

Week 7

In Week 7 we started delving more into algebra and started talking about equations.

Week 8

In Week 8 we will finish talking about equations and start studying quadratics.

Assessment 1 – Results

I will have the assignments with me tomorrow and next Friday if you want to see your work.

Assessment 2

Assessment 2 is on p.136. It has a hand-in time of 16:00 Monday 26 November.

Week 7

We finished looking at Chapter 2 by looking at the Three Term Taylor Method for approximating solutions of ordinary differential equations.

We started Chapter 3 (Probability and Statistics) by looking at some general concepts in probability and then we looked at random variables with a binomial distribution.

Week 8

We will look at the Poisson distribution and perhaps the Normal distribution.

Week 7

We looked at Parametric Differentiation and Related Rates

Week 8

We will look at Implicit Differentiation and Partial Differentiation. If you are interested in a very “mathsy” approach to curves you can look at this.

Test 2

On Chapter 3, not until Week 11: perhaps Monday 26 November.

Study

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

Assessment 1 – Results

I am starting corrections today and will get the results to you as soon as I can. I cannot give an accurate day at this stage: it could be Monday but just as easily could be a few days after this – I can’t make any promises.

Assessment 2

Assessment 2 is on p.136. It has a hand-in date of Monday 26 November and we have already covered everything that will be asked and so you have over five weeks to complete the assignment.

MicDrop Project

On Monday you will be sent a 15 minute survey that you will take on a mobile internet device — such as your mobile phone — during Monday’s lecture.

This survey is part of a larger project the Mathematics Department is undertaking —  Mathematics in Context: Developing Relevancy-Orientated Problems — in an effort to improve our teaching.

If you do not have an internet ready device you may leave class early.

Maths Classes will be going full steam ahead on Monday 22 October as well as Wednesday, Thursday, Friday 1, 2, 3 November. I will call the next two weeks by Week 7.

Week 6

In Week 6 we finished looking at cantilvers and then summarised what we learnt about beams. We had one lecture as a tutorial but then looked at numerical approximations to solutions of differential equations that we cannot solve exactly.

After the storm last year I recorded some examples. If you missed some classes this week you could do worse than watch this cantilever example and this summary of beams to catch up

Week 7

In Week 7 we will look at the Three Term Taylor Method and begin Chapter 3 on Probability and Statistics.