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In May 2017, shortly after completing my PhD and giving a talk on it at a conference in Seoul, I wrote a post describing the outlook for my research.

I can go through that post paragraph-by-paragraph and thankfully most of the issues have been ironed out. In May 2018 I visited Adam Skalski at IMPAN and on that visit I developed a new example (4.2) of a random walk (with trivial -dependence) on the Sekine quantum groups with upper and lower bounds sharp enough to prove the non-existence of the cutoff phenomenon. The question of developing a walk on showing cutoff… I now think this is unlikely considering the study of Isabelle Baraquin and my intuitions about the ‘growth’ of (perhaps if cutoff doesn’t arise in somewhat ‘natural’ examples best not try and force the issue?). With the help of Amaury Freslon, I was able to improve to presentation of the walk (Ex 4.1) on the dual quantum group . With the help of others, it was seen that the quantum total variation distance is equal to the projection distance (Prop. 2.1). Thankfully I have recently proved the Ergodic Theorem for Random Walks on Finite Quantum Groups. This did involve a study of subgroups (and quasi-subgroups) of quantum groups but normal subgroups of quantum groups did not play so much of a role as I expected. Amaury Freslon extended the upper bound lemma to compact Kac algebras. Finally I put the PhD on the arXiv and also wrote a paper based on it.

Many of these questions, other questions in the PhD, as well as other questions that arose around the time I visited Seoul (e.g. what about random transpositions in ?) were answered by Amaury Freslon in this paper. Following an email conversation with Amaury, and some communication with Uwe Franz, I was able to write another post outlining the state of play.

This put some of the problems I had been considering into the categories of Solved, to be Improved, More Questions, and Further Work. Most of these have now been addressed. That February 2018 post gave some direction, led me to visit Adam, and I got my first paper published.

After that paper, my interest turned to the problem of the Ergodic Theorem, and in May I visited Uwe in Besancon, where I gave a talk outlining some problems that I wanted to solve. The main focus was on proving this Ergodic Theorem for Finite Quantum Groups, and thankfully that has been achieved.

What I am currently doing is learning my compact quantum groups. This work is progressing (albeit slowly), and the focus is on delivering a series of classes on the topic to the functional analysts in the UCC School of Mathematical Sciences. The best way to learn, of course, is to teach. This of course isn’t new, so here I list some problems I might look at in short to medium term. Some of the following require me to know my compact quantum groups, and even non-Kac quantum groups, so this study is not at all futile in terms of furthering my own study.

I don’t really know where to start. Perhaps I should focus on learning my compact quantum groups for a number of months before tackling these in this order?

- My proof of the Ergodic Theorem leans heavily on the finiteness assumption but a lot of the stuff in that paper (and there are many partial results in that paper also) should be true in the compact case too. How much of the proof/results carry into the compact case? A full Ergodic Theorem for Random Walks on Compact Quantum Groups is probably quite far away at this point, but perhaps partial results under assumptions such as (co?)amenability might be possible. OR try and prove ergodic theorems for specific compact quantum groups.
- Look at random walks on quantum homogeneous spaces, possibly using Gelfand Pair theory. Start in finite and move into Kac?
- Following Urban, study convolution factorisations of the Haar state.
- Examples of non-central random walks on compact groups.
- Extending the Upper Bound Lemma to the non-Kac case. As I speak, this is beyond what I am capable of. This also requires work on the projection and quantum total variation distances (i.e. show they are equal in this larger category)

Finally cracked this egg.

Preprint here.

I thought I had a bit of a breakthrough. So, consider the algebra of a functions on the dual (quantum) group . Consider the projection:

.

Define by:

.

Note

.

Note so is a partition of unity.

I know that corresponds to a quasi-subgroup but not a quantum subgroup because is not normal.

This was supposed to say that the result I proved a few days ago that (in context), that corresponded to a quasi-subgroup, was as far as we could go.

For , note

,

is a projection, in fact a group like projection, in .

Alas note:

That is the group like projection associated to is subharmonic. This *should* imply that nearby there exists a projection such that for all … also is subharmonic.

This really should be enough and I should be looking perhaps at the standard representation, or the permutation representation, or … but I want to find the projection…

Indeed …and .

The punchline… the result of Fagnola and Pellicer holds when the random walk is is irreducible. This walk is not… back to the drawing board.

I have constructed the following example. The question will be does it have periodicity.

Where is the permutation representation, , and , is given by:

.

This has (duh), , and otherwise .

The above is still a cyclic partition of unity… but is the walk irreducible?

The easiest way might be to look for a subharmonic . This is way easier… with it is easy to construct non-trivial subharmonics… not with this . It is straightforward to show there are no non-trivial subharmonics and so is irreducible, periodic, but is not a quantum subgroup.

It also means, in conjunction with work I’ve done already, that I have my result:

**Definition** Let be a finite quantum group. A state is *concentrated on a cyclic coset of a proper quasi-subgroup* if there exists a pair of projections, , such that , is a group-like projection, and there exists () such that .

## (Finally) The Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group is ergodic if and only if the driving probability is not concentrated on a proper quasi-subgroup, nor on a cyclic coset of a proper quasi-subgroup.

The end of the previous Research Log suggested a way towards showing that can be associated to an idempotent state . Over night I thought of another way.

Using the Pierce decomposition with respect to (where ),

.

The corner is a hereditary -subalgebra of . This implies that if and for , .

Let . We know from Fagnola and Pellicer that and .

By assumption in the background here we have an irreducible and periodic random walk driven by . This means that for all projections , there exists such that .

Define:

.

Define:

.

The claim is that the support of , is equal to .

We probably need to write down that:

.

Consider for any . Note

that is each is supported on . This means furthermore that .

Suppose that the support . A question arises… is ? This follows from the fact that and is hereditary.

Consider a projection . We know that there exists a such that

.

This implies that , say (note ):

By assumption . Consider

.

For this to equal one every must equal one but .

Therefore is the support of .

Let . We have shown above that for all . This is an idempotent state such that is its support (a similar argument to above shows this). Therefore is a group like projection and so we denote it by and !

Today, for finite quantum groups, I want to explore some properties of the relationship between a state , its density (), and the support of , .

I also want to learn about the interaction between these object, the stochastic operator

,

and the result

,

where is defined as (where by ).

.

An obvious thing to note is that

.

Also, because

That doesn’t say much. We are possibly hoping to say that .

## Quasi-Subgroups that are not Subgroups

Let be a finite quantum group. We associate to an idempotent state a *quasi-*subgroup . This nomenclature must be included in the manuscript under preparation.

As is well known from the GNS representation, positive linear functionals can be associated to closed left ideals:

.

In the case of a quasi-subgroup, , my understanding is that by looking at we can tell if is actually a subgroup or not. Franz & Skalski show that:

Let be a quasi-subgroup. TFAE

- is a subgroup
- is a two-sided or self-adjoint or invariant ideal of

I want to look again at the Kac & Paljutkin quantum group and see how the Pal null-spaces and fail these tests. Both Franz & Gohm and Baraquin should have the necessary left ideals.

### The Pal Null-Space

The following is an idempotent probability on the Kac-Paljutkin quantum group:

.

Hence:

.

If were two-sided, . Consider and

.

We see problems also with when it comes to the adjoint and also . It is not surprise that the adjoint AND the antipode are involved as they are related via:

.

In fact, for finite or even Kac quantum groups, .

Can we identity the support ? I think we can, it is (from Baraquin)

.

This does not commute with :

.

The other case is similar.

Back before Christmas I felt I was within a week of proving the following:

Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group is ergodic if and only if is not concentrated on a proper quasi-subgroup, nor the coset of a ?normal ?-subgroup.

The first part of this conjecture says that if is concentrated on a quasi-subgroup, then it stays concentrated there. Furthermore, we can show that if the random walk is *reducible *that the Césaro limit gives a quasi-subgroup on which is concentrated.

The other side of the ergodicity coin is periodicity. In the classical case, it is easy to show that if the driving probability is concentrated on the coset of a proper normal subgroup , that the convolution powers jump around a cyclic subgroup of .

One would imagine that in the quantum case this might be easy to show but alas this is not proving so easy.

I am however pushing hard against the other side. Namely, that if the random walk is periodic *and irreducible, *that the driving probability in concentrated on some quasi-normal quasi-subgroup!

The progress I have made depends on work of Fagnola and Pellicer. They show that if the random walk is irreducible and periodic that there exists a partition of unity such that is concentrated on .

This cyclic nature suggests that might be equal to for some and perhaps:

and perhaps there is an isomorphism . Unfortunately I have been unable to progress this.

What is clear is that the ‘supports’ of the behave very much like the cosets of proper normal subgroup .

As the random walk is assumed irreducible, we know that for any projection , there exists a such that .

Playing this game with the Haar element, , note there exists a such that .

Let . I have proven that if , then the convolution powers of converge. Convergence is to an idempotent. This means that converges to an idempotent , and so we have a quasi-subgroup corresponding to it, say .

The question is… does coincide with ?

If yes, is there any quotient structure by a quasi-subgroup? Is there a normal quasi-subgroup that allows such a structure?

Is a subgroup? Could it be a normal subgroup?

As nice as it was to invoke the result that if is in the support of , then the convolution powers of converge, by looking at those papers which cite Fagnola and Pellicer we see a paper that gives the same result without this neat little lemma.

In the case of a finite classical group , we can show that if we have i.i.d. random variables , that if , for a coset of a proper normal subgroup , that the random walk on driven by , the random variables:

,

exhibits a periodicity because

.

This shows that a necessary condition for ergodicity of a random walk on a finite classical group driven by is that the support of 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.

*Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups, *Journal of Fourier Analysis and Applications, doi: 10.1007/s00041-019-09670-4 (earlier preprint available here)

**Abstract**

*A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses Fourier analysis on the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The Fourier analysis of quantum groups is remarkably similar to that of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group as well as on the truly quantum groups of Sekine.*

Slides of a talk given at the Irish Mathematical Society 2018 Meeting at University College Dublin, August 2018.

**Abstract** *Four generalisations are used to illustrate the topic. The generalisation from finite “classical” groups to finite quantum groups is motivated using the language of functors (“classical” in this context meaning that the algebra of functions on the group is commutative). The generalisation from random walks on finite “classical” groups to random walks on finite quantum groups is given, as is the generalisation of total variation distance to the quantum case. Finally, a central tool in the study of random walks on finite “classical” groups is the Upper Bound Lemma of Diaconis & Shahshahani, and a generalisation of this machinery is used to find convergence rates of random walks on finite quantum groups.*

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