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What started about ten months ago as a technical question to an expert, led to a talk, and led to me producing this weird production here.

Now that it is complete, although I really like all its contents (well except for the note to reader and introduction I spilled out very hastily), I can see on reflection it represents rather than a cogent piece of mathematics, almost a log of all the things I have learnt in the process of writing it. It also includes far too much speculation and conjecture. So I am going to post it here and get to work on editing it down to something a little more useful and cogent.

EDIT: Edited down version here.

I am currently (slowly) working on an essay/paper where I expand upon the ideas in this talk. In this post I will try and explain in this framework why there is no quantum cyclic group, no quantum , and ask why there is no quantum alternating group.

### Quantum Permutations Basics

Let be a unital -algebra. We say that a matrix is a *magic unitary *if each entry is a projection , and each row and column of is a partition of unity, that is:

.

It is necessarily the case (but not for *-algebras) that elements along the same row or column are orthogonal:

and .

Shuzou Wang defined *the algebra of continuous functions on *the *quantum permutation group on symbols* to be the universal -algebra generated by an magic unitary . Together with (leaning heavily on the universal property) the *-homomorphism:

,

and the fact that and are invertible (), the quantum permutation group is a compact matrix quantum group.

Any compact matrix quantum group generated by a magic unitary is a *quantum permutation group* in that it is a quantum subgroup of *the *quantum permutation group. There are finite quantum groups (finite dimensional algebra of functions) which are not quantum permutation groups and so Cayley’s Theorem does not hold for quantum groups. I *think *this is because we can have quantum groups which act on algebras such as rather than — the algebra of functions equivalent of the finite set .

This is all basic for quantum group theorists and probably unmotivated for everyone else. There are traditional motivations as to why such objects should be considered algebras of functions on *quantum *groups:

- find a presentation of an algebra of continuous functions on a group, , as a commutative universal -algebra. Study the the same object
*liberated*by dropping commutativity. Call this the quantum or free version of , . - quotient by the commutator ideal, that is we look at the
*commutative*algebra generated by an magic unitary. It is isomorphic to , the algebra of functions on (classical) . - every commutative algebra of continuous functions on a compact matrix quantum group is the algebra of functions on a (classical) compact matrix group, etc.

Here I want to take a very different direction which while motivationally rich might be mathematically poor.

### Weaver Philosophy

Take a quantum permutation group and represent the algebra of functions as bounded operators on a Hilbert space . Consider a norm-one element as a quantum permutation. We study the properties of the quantum permutation by making a series of measurements using self-adjoint elements of .

Suppose we have a finite-spectrum, self-adjoint measurement . It’s spectral decomposition gives a partition of unity . The measurement of with gives the value with probability:

,

and we have the expectation:

.

What happens if the measurement of with yields (which can only happen if )? Then we have some wavefunction collapse of

.

Now we can keep playing the game by taking further measurements. Notationally it is easier to describe what is happening if we work with projections (but straightforward to see what happens with finite-spectrum measurements). At this point let me quote from the essay/paper under preparation:

*Suppose that the “event” has been observed so that the state is now . Note this is only possible if is non-null in the sense that*

*The probability that measurement produces , and , is:*

*Define now the event , said “given the state , is measured to be after is measured to be “. Assuming that is non-null, using the expression above a probability can be ascribed to this event:*

*Inductively, for a finite number of projections , and :*

*In general, and so*

*and this helps interpret that and are not simultaneously observable. However the sequential projection measurement is “observable” in the sense that it resembles random variables with values in . Inductively the sequential projection measurement resembles a -valued random variable, and*

*If and do commute, they share an orthonormal eigenbasis, and it can be interpreted that they can “agree” on what they “see” when they “look” at , and can thus be determined simultaneously. Alternatively, if they commute then the distributions of and are equal in the sense that*

*it doesn’t matter what order they are measured in, the outputs of the measurements can be multiplied together, and this observable can be called .*

Consider the (classical) permutation group or moreover its algebra of functions . The elements of can be represented as bounded operators on , and the algebra is generated by a magic unitary where:

.

Here (‘unrepresented’) that asks of … do you send ? One for yes, zero for no.

Recall that the product of commuting projections is a projection, and so as is commutative, products such as:

,

There are, of, course, such projections, they form a partition of unity themselves, and thus we can build a measurement that will identify a random permutation and leave it equal to some after measurement. This is the essence of classical… all we have to do is enumerate and measure using:

.

A *quantum *permutation meanwhile is impossible to pin down in such a way. As an example, consider the Kac-Paljutkin quantum group of order eight which can be represented as . Take . Then

.

If you think for a moment this cannot happen classically, and the issue is that we cannot know simultaneously if and … and if we cannot know this simultaneously we cannot pin down to a single element of .

### No Quantum Cyclic Group

Suppose that is a quantum permutation (in ). We can measure where the quantum permutation sends, say, one to. We simply form the self-adjoint element:

.

The measurement will produce some … but if is supposed to represent some “quantum cyclic permutation” then we already know the values of from , and so, after measurement,

, .

The significance of the intersection is that whatever representation of we have, we find these subspaces to be -invariant, and can be taken to be one-dimensional.

I believe this explains why there is no quantum cyclic group.

#### Question 1

Can we use a similar argument to show that there is no quantum version of any abelian group? Perhaps using together with the structure theorem for finite abelian groups?

### No Quantum

Let be represented as bounded operators on a Hilbert space . Let . Consider the random variable

.

Assume without loss of generality that then measuring with gives with probability , and the quantum permutation projects to:

.

Now consider (for *any *, using the fact that and the rows and columns of are partitions of unity:

(*)

Now suppose, again without loss of generality, that measurement of with produces , then we have projection to . Now let us find the Birkhoff slice of this. First of all, as has just been observed it looks like:

In light of (*), let us find . First let us normalise correctly to

So

Now use (*):

,

and as maps to doubly stochastic matrices we find that is equal to the permutation matrix .

Not convincing? Fair enough, here is proper proof inspired by the above:

Let us show . Fix a Hilbert space representation and let .

The basic idea of the proof is, as above, to realise that once a quantum permutation is observed sending, say, , the fates of and are entangled: if you see you know that .

This is the conceptional side of the proof.

Consider which is equal to both:

.

This is the manifestation of, if you know , then two and one are entangled. Similarly we can show that and .

Now write

.

Similarly,

Which is equal to , that is and commute.

### Question 2

Is it true that if every quantum permutation in a can be fully described using some combination of -measurements, then the quantum permutation group is classical? I believe this to be true.

### Quantum Alternating Group

Freslon, Teyssier, and Wang state that *there is no quantum alternating group*. Can we use the ideas from above to explain why this is so? Perhaps for .

A possible plan of attack is to use the number of fixed points, , and perhaps show that commutes with . If you know these two simultaneously you nearly know the permutation. Just for completeness let us do this with :

tr(u)\x(1) | 1 | 2 | 3 | 4 |

0 | – | (12)(34) | (13)(24) | (14)(23) |

1 | (234),(243) | (134),(143) | (124),(142) | (123),(132) |

4 | e | – | – | – |

The problem is that we cannot assume that that the spectrum of is , and, euh, the obvious fact that it doesn’t actually work.

What is more promising is

x(1)\x(2) | 1 | 2 | 3 | 4 |

1 | – | e | (234) | (243) |

2 | (12)(34) | – | (123) | (124) |

3 | (132) | (134) | – | (13)(24) |

4 | (142) | (143) | (14)(23) | – |

However while the spectrums of x(1) and x(2) are cool (both in ), they do not commute.

### Question 3

Are there some measurements that can identify an element of and via a positive answer to Question 3 explain *why* there is no quantum ? Can this be generalised to .

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.

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