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*This post follows on from this one. The purpose of posts in this category is for me to learn more about the research being done in quantum groups. This post looks at this paper of Schmidt.*

# Preliminaries

## Compact Matrix Quantum Groups

The author gives the definition and gives the definition of a (left, quantum) group action.

### Definition 1.2

Let be a compact matrix quantum group and let be a . An (left) *action *of on is a unital *-homomorphism that satisfies the analogue of , and the Podlés density condition:

.

## Quantum Automorphism Groups of Finite Graphs

Schmidt in this earlier paper gives a slightly different presentation of . The definition given here I understand:

### Definition 1.3

The *quantum automorphism group* of a finite graph with adjacency matrix is given by the universal -algebra generated by such that the rows and columns of are partitions of unity and:

.

_______________________________________

The difference between this definition and the one given in the subsequent paper is that in the subsequent paper the quantum automorphism group is given as a quotient of by the ideal given by … ah but this is more or less the definition of universal -algebras given by generators and relations :

where presumably all works out OK, and it can be shown that is a suitable ideal, a Hopf ideal. I don’t know how it took me so long to figure that out… Presumably the point of quotienting by (a presumably Hopf) ideal is so that the quotient gives a subgroup, in this case via the surjective *-homomorphism:

.

_______________________________________

## Compact Matrix Quantum Groups acting on Graphs

### Definition 1.6

Let be a finite graph and a compact matrix quantum group. An action of on is an action of on (coaction of on ) such that the associated magic unitary , given by:

,

commutes with the adjacency matrix, .

By the universal property, we have via the surjective *-homomorphism:

, .

## Theorem 1.8 (Banica)

Let , and , be an action, and let be a linear subspace given by a subset . The matrix commutes with the projection onto if and only if

### Corollary 1.9

The action preserves the eigenspaces of :

*Proof: *Spectral decomposition yields that each , or rather the projection onto it, satisfies a polynomial in :

,

as commutes with powers of

# A Criterion for a Graph to have Quantum Symmetry

### Definition 2.1

Let . Permutations are *disjoint *if , and vice versa, for all .

In other words, we don’t have and permuting any vertex.

### Theorem 2.2

Let be a finite graph. If there exists two non-trivial, disjoint automorphisms , such that and , then we get a surjective *-homomorphism . In this case, we have the quantum group , and so has quantum symmetry.

*Warning: This is written by a non-expert (I know only about finite quantum groups and am beginning to learn my compact quantum groups), and there is no attempt at rigour, or even consistency. Actually the post shows a wanton disregard for reason, and attempts to understand the incomprehensible and intuit the non-intuitive. Speculation would be too weak an adjective.*

## Groups

A group is a well-established object in the study of mathematics, and for the purposes of this post we can think of a group as the set of symmetries on some kind of space, given by a set together with some additional structure . The elements of *act *on as bijections:

,

such that , that is the structure of the space is invariant under .

For example, consider the space , where the set is , and the structure is the cardinality. Then the set of all of the bijections is a group called .

A set of symmetries , a group, comes with some structure of its own. The identity map , is a symmetry. By transitivity, symmetries can be composed to form a new symmetry . Finally, as bijections, symmetries have inverses , .

Note that:

.

A group can carry additional structure, for example, compact groups carry a topology in which the composition and inverse are continuous.

## Algebra of Functions

Given a group together with its structure, one can define an algebra of complex valued functions on , such that the multiplication is given by a commutative pointwise multiplication, for :

.

Depending on the class of group (e.g. finite, matrix, compact, locally compact, etc.), there may be various choices and considerations for what algebra of functions to consider, but on the whole it is nice if given an algebra of functions we can reconstruct .

Usually the following *transpose *maps will be considered in the structure of , for some tensor product such that , and , is the group multiplication:

See Section 2.2 to learn more about these maps and the relations between them for the case of the complex valued functions on *finite *groups.

## Quantum Groups

Quantum groups, famously, do not have a single definition in the same way that groups do. All definitions I know about include a coassociative (see Section 2.2) comultiplication for some tensor product (or perhaps only into a multiplier algebra ), but in general that structure alone can only give a quantum *semi*group.

Here is a non-working (quickly broken?), meta-definition, inspired in the usual way by the famous Gelfand Theorem:

A quantum group is given by an algebra of functions satisfying a set of axioms such that:

- whenever is noncommutative, is a
virtualobject,every commutative algebra of functions satisfying is an algebra of functions on aset-of-pointsgroup, andwhenever commutative algebras of functions , asset-of-pointsgroups.

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 hope of gleaning information for the study of aperiodic random walks on (finite) quantum groups, which I am struggling with here, by couching Freslon’s Proposition 3.2, in the language of Fagnola and Pellicer, and in the language of my (failed) attempts (see here, here, and here) to find the necessary and sufficient conditions for a random walk to be aperiodic. It will be necessary to extract the irreducibility and aperiodicity from Freslon’s rather ‘unilateral’ result.

Well… I have an inkling that because dual groups satisfy what I would call the condition of abelianness (under the ‘quantisation’ functor), all (quantum) subgroups are normal… this is probably an obvious thing to write down (although I must search the literature) to ensure it is indeed known (or is untrue?). **Edit: **Wang had it already, see the last proposition here.

Let be a the algebra of functions on a finite classical (as opposed to quantum) group . This has the structure of both an algebra and a coalgebra, with an appropriate relationship between these two structures. By taking the dual, we get the *group algebra**,* . The dual of the pointwise-multiplication in is a coproduct for the algebra of functions on the dual group … this is all well known stuff.

Recall that the set of probabilities on a finite quantum group is the set of states , and this lives in the dual, and the dual of is , and so probabilities on are functions on . To be positive is to be positive definite, and to be normalised to one is to have .

The ‘simplicity’ of the coproduct,

,

means that for ,

,

so that, inductively, is equal to the (pointwise-multiplication power) .

The Haar state on is equal to:

,

and therefore necessary and sufficient conditions for the convergence of is that is *strict. *It can be shown that for any that . Strictness is that this is a strict inequality for , in which case it is obvious that .

Here is a finite version of Freslon’s result which holds for discrete groups.

### Freslon’s Ergodic Theorem for (Finite) Group Algebras

*Let be a probability on the dual of finite group. The random walk generated by is ergodic if and only if is not-concentrated on a character on a non-trivial subgroup .*

Freslon’s proof passes through the following equivalent condition:

*The random walk on driven by is *not *ergodic if is *bimodular*with respect to a non-trivial subgroup , in the sense that*

.

Before looking at the proof proper, we might note what happens when is abelian, in which case is a classical group, the set of characters on .

To every positive definite function , we can associate a probability such that:

.

This is Bochner’s Theorem for finite abelian groups. This implies that positive definite functions on finite abelian groups are exactly convex combinations of characters.

Freslon’s condition says that to be not ergodic, must be a character on a non-trivial subgroup . Such characters can be extended in ways.

Therefore, if is not ergodic, .

For , we have

,

dividing both sides by yields:

.

As , and , this implies that is supported on characters such that, for all :

,

such that . The set of such is the annihilator of in , and it is a subgroup. Therefore is concentrated on the coset of a normal subgroup (as all subgroups of an abelian group are normal).

This, via Pontragin duality, is not looking at the ‘support’ of , but rather of . Although we denote , and when is abelian, is a group (unnaturally, of characters) isomorphic to . Is it the case though that,

gives the same object in as

?

Well… of course this is true because .

We could proceed to look at Fagnola & Pellicer’s work but first let us prove Freslon’s result, hopefully in the finite case the analysis disappears…

*Proof: *Assume that is not strict and let

.

There exists a unitary representation and a unit vector such that

Cauchy-Schwarz implies that

.

If is not strict there is an such that this is an inequality and so is colinear to , it follows that .

This implies for and :

,

and so is closed under multiplication. Also and so and so is a subgroup. It follows that is a character on , which is not trivial because is not strict.

I don’t really need to go through the third equivalent condition. If coincides with a character on a subgroup , for

,

and so is not strict

Now let us look at the language of Fagnola and Pellicer. What is a projection in ? First note the involution in is . The second multiplication is the convolution. This means projections in the algebra are symmetric with respect to the group inverses and they are also idempotents. They are actually equal to Haar states on finite subgroups.

I think periodicity is also wrapped up in Freslon’s result as I think all subgroups of dual groups are normal. Perhaps, oddly, not being concentrated on a subgroup means that the positive function (probability on the dual) is one on that subgroup…

Well… let us start with irreducible. Suppose fails to be ergodic because it is irreducible. This means there is a projection such that that (and support less than ?)

Let us look at the first condition:

.

What now is the support of ? Well… some work I have done offline shows that the special projections, the group-like projections, correspond to to for a subgroup of . If is reducible, it is concentrated on such a quasi-subgroup, and this means that coincides with a *trivial *character on . In terms of Fagnola Pellicer, .

Now let us tackle aperiodicity. It is going to correspond, I think, with being concentrated on a ‘coset’ of a non-trivial character on …

Well, we can show that if is periodic, there is a subset such that for all . We can use Freslon’s proof to show that is in a subgroup on which .

Now what I want to do is put this in the language of ‘inclusion’ matrices… but the inclusions for cocommutative quantum groups are trivial so no go…

We can reduce Freslon’s conditions down to irreducible and aperiodic: not coinciding with a trivial character, and not coinciding with a character.

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