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Giving a talk 17:00, September 1 2020:

See here for more.

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 $G$ be a compact matrix quantum group and let $C(X)$ be a $\mathrm{C}^*-algebra$. An (left) action of $G$ on $X$ is a unital *-homomorphism $\alpha: C(X)\rightarrow C(X)\otimes C(G)$ that satisfies the analogue of $g_2(g_1x)=(g_2g_1)x$, and the Podlés density condition:

$\displaystyle \overline{\text{span}(\alpha(C(X)))(\mathbf{1}_X\otimes C(G))}=C(X)\otimes C(G)$.

## Quantum Automorphism Groups of Finite Graphs

Schmidt in this earlier paper gives a slightly different presentation of $\text{QAut }\Gamma$. The definition given here I understand:

### Definition 1.3

The quantum automorphism group of a finite graph $\Gamma=(V,E)$ with adjacency matrix $A$ is given by the universal $\mathrm{C}^*$-algebra $C(\text{QAut }\Gamma)$ generated by $u\in M_n(C(\text{QAut }\Gamma))$ such that the rows and columns of $u$ are partitions of unity and:

$uA=Au$.

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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 $C(S_n^+)$ by the ideal given by $\mathcal{I}=\langle Au=uA\rangle$… ah but this is more or less the definition of universal $\mathrm{C}^*$-algebras given by generators $E$ and relations $R$:

$\displaystyle \mathrm{C}^*(E,R)=\mathrm{C}^*( E)/\langle \mathcal{R}\rangle$

$\displaystyle \Rightarrow \mathrm{C}^*(E,R)/\langle I\rangle=\left(\mathrm{C}^*(E)/R\right)/\langle I\rangle=\mathrm{C}*(E)/(\langle R\rangle\cap\langle I\rangle)=\mathrm{C}^*(E,R\cap I)$

where presumably $\langle R\rangle \cap \langle I \rangle=\langle R\cap I\rangle$ all works out OK, and it can be shown that $I$ 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 $\text{QAut }\Gamma\leq S_{|V|}^+$ via the surjective *-homomorphism:

$C(S_n^+)\rightarrow C(S_n^+)/\langle uA=Au\rangle=C(\text{QAut }\Gamma)$.

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## Compact Matrix Quantum Groups acting on Graphs

### Definition 1.6

Let $\Gamma$ be a finite graph and $G$ a compact matrix quantum group. An action of $G$ on $\Gamma$ is an action of $G$ on $V$ (coaction of $C(G)$ on $C(V)$) such that the associated magic unitary $v=(v_{ij})_{i,j=1,\dots,|V|}$, given by:

$\displaystyle \alpha(\delta_j)=\sum_{i=1}^{|V|} \delta_i\otimes v_{ij}$,

commutes with the adjacency matrix, $uA=Au$.

By the universal property, we have $G\leq \text{QAut }\Gamma$ via the surjective *-homomorphism:

$C(\text{QAut }\Gamma)\rightarrow C(G)$, $u\mapsto v$.

### Theorem 1.8 (Banica)

Let $X_n=\{1,\dots,n\}$, and $\alpha:F(X_n)\rightarrow F(X_n)\otimes C(G)$, $\alpha(\delta_j)=\sum_i\delta_i\otimes v_{ij}$ be an action, and let $F(K)$ be a linear subspace given by a subset $K\subset X_n$. The matrix $v$ commutes with the projection onto $F(K)$ if and only if $\alpha(F(K))\subseteq F(K)\otimes C(G)$

### Corollary 1.9

The action $\alpha: F(V)\rightarrow F(V)\otimes C(\text{QAut }\Gamma)$ preserves the eigenspaces of $A$:

$\alpha(E_\lambda)\subseteq E_\lambda\otimes C(\text{QAut }\Gamma)$

Proof: Spectral decomposition yields that each $E_\lambda$, or rather the projection $P_\lambda$ onto it, satisfies a polynomial in $A$:

$\displaystyle P_\lambda=\sum_{i}c_iA^i$

$\displaystyle \Rightarrow P_\lambda A=\left(\sum_i c_i A^i\right)A=A P_\lambda$,

as $A$ commutes with powers of $A$ $\qquad \bullet$

# A Criterion for a Graph to have Quantum Symmetry

### Definition 2.1

Let $V=\{1,\dots,|V|\}$. Permutations $\sigma,\,\tau: V\rightarrow V$ are disjoint if $\sigma(i)\neq i\Rightarrow \tau(i)=i$, and vice versa, for all $i\in V$.

In other words, we don’t have $\sigma$ and $\tau$ permuting any vertex.

### Theorem 2.2

Let $\Gamma$ be a finite graph. If there exists two non-trivial, disjoint automorphisms $\sigma,\tau\in\text{Aut }\Gamma$, such that $o(\sigma)=n$ and $o(\tau)=m$, then we get a surjective *-homomorphism $C(\text{QAut }\Gamma)\rightarrow C^*(\mathbb{Z}_n\ast \mathbb{Z}_m)$. In this case, we have the quantum group $\widehat{\mathbb{Z}_n\ast \mathbb{Z}_m}\leq \text{QAut }\Gamma$, and so $\Gamma$ 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 $G$ as the set of symmetries on some kind of space, given by a set $X$ together with some additional structure $D(X)$. The elements of $G$  act on $X$ as bijections:

$G \ni g:X\rightarrow X$,

such that $D(X)=D(g(X))$, that is the structure of the space is invariant under $g$.

For example, consider the space $(X_n,|X_n|)$, where the set is $X_n=\{1,2,\dots,n\}$, and the structure is the cardinality. Then the set of all of the bijections $X_n\rightarrow X_n$ is a group called $S_n$.

A set of symmetries $G$, a group, comes with some structure of its own. The identity map $e:X\rightarrow X$, $x\mapsto x$ is a symmetry. By transitivity, symmetries $g,h\in G$ can be composed to form a new symmetry $gh:=g\circ h\in G$. Finally, as bijections, symmetries have inverses $g^{-1}$, $g(x)\mapsto x$.

Note that:

$gg^{-1}=g^{-1}g=e\Rightarrow (g^{-1})^{-1}=g$.

A group can carry additional structure, for example, compact groups carry a topology in which the composition $G\times G\rightarrow G$ and inverse ${}^{-1}:G\rightarrow G$ are continuous.

## Algebra of Functions

Given a group $G$ together with its structure, one can define an algebra $A(G)$ of complex valued functions on $G$, such that the multiplication $A(G)\times A(G)\rightarrow A(G)$ is given by a commutative pointwise multiplication, for $s\in G$:

$(f_1f_2)(s)=f_1(s)f_2(s)=(f_2f_1)(s)$.

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 $A(G)$ we can reconstruct $G$.

Usually the following transpose maps will be considered in the structure of $A(G)$, for some tensor product $\otimes_\alpha$ such that $A(G\times G)\cong A(G)\otimes_\alpha A(G)$, and $m:G\times G\rightarrow G$, $(g,h)\mapsto gh$ is the group multiplication:

\begin{aligned} \Delta: A(G)\rightarrow A(G)\otimes_{\alpha}A(G)&,\,f\mapsto f\circ m,\,\text{the comultiplication} \\ S: A(G)\rightarrow A(G)&,\, f\mapsto f\circ {}^{-1},\,\text{ the antipode} \\ \varepsilon: A(G)\rightarrow \mathbb{C}&,\, f\mapsto f\circ e,\,\text{ the counit} \end{aligned}

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 $\Delta: A(G)\rightarrow A(G)\otimes_\alpha A(G)$ for some tensor product $\otimes_\alpha$ (or perhaps only into a multiplier algebra $M(A(G)\otimes_\alpha A(G))$), but in general that structure alone can only give a quantum semigroup.

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

A quantum group $G$ is given by an algebra of functions $A(G)$ satisfying a set of axioms $\Theta$ such that:

• whenever $A(G)$ is noncommutative, $G$ is a virtual object,
• every commutative algebra of functions satisfying $\Theta$ is an algebra of functions on a set-of-points group, and
• whenever commutative algebras of functions $A(G_1)\cong_{\Theta} A(G_2)$, $G_1\cong G_2$ as set-of-points groups.

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 $n$-dependence) on the Sekine quantum groups $Y_n$ with upper and lower bounds sharp enough to prove the non-existence of the cutoff phenomenon. The question of developing a walk on $Y_n$ showing cutoff… I now think this is unlikely considering the study of Isabelle Baraquin and my intuitions about the ‘growth’ of $Y_n$ (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 $\widehat{S_n}$. 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 $S_n^+$?) 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?

1. 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.
2. Look at random walks on quantum homogeneous spaces, possibly using Gelfand Pair theory. Start in finite and move into Kac?
3. Following Urban, study convolution factorisations of the Haar state.
4. Examples of non-central random walks on compact groups.
5. 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 $\widehat{S_3}$. Consider the projection:

$\displaystyle p_0=\frac12\delta^e+\frac12\delta^{(12)}\in F(\widehat{S_3})$.

Define $u\in M_p(\widehat{S_3})$ by:

$u(\delta^\sigma)=\langle\text{sign}(\sigma)1,1\rangle=\text{sign}(\sigma)$.

Note

$\displaystyle T_u(p_0)=\frac12\delta^e-\frac12 \delta^{(12)}:=p_1$.

Note $p_1=\mathbf{1}_{\widehat{S_3}}-p_0=\delta^0-p_0$ so $\{p_0,p_1\}$ is a partition of unity.

I know that $p_0$ corresponds to a quasi-subgroup but not a quantum subgroup because $\{e,(12)\}$ is not normal.

This was supposed to say that the result I proved a few days ago that (in context), that $p_0$ corresponded to a quasi-subgroup, was as far as we could go.

For $H\leq G$, note

$\displaystyle p_H=\frac{1}{|H|}\sum_{h\in H}\delta^h$,

is a projection, in fact a group like projection, in $F(\widehat{G})$.

Alas note:

$\displaystyle T_u(p_{\langle(123)\rangle})=p_{\langle (123)\rangle}$

That is the group like projection associated to $\langle (123)\rangle$ is subharmonic. This should imply that nearby there exists a projection $q$ such that $u^{\star k}(q)=0$ for all $k\in\mathbb{N}$… also $q_{\langle (123)\rangle}:=\mathbf{1}_{\widehat{S_3}}-p_{\langle(123)\rangle}$ is subharmonic.

This really should be enough and I should be looking perhaps at the standard representation, or the permutation representation, or $S_3\leq S_4$… but I want to find the projection…

Indeed $u(q_{(123)})=0$…and $u^{\star 2k}(q_{\langle (123)\rangle})=0$.

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 $\rho:S_n\rightarrow \text{GL}(\mathbb{C}^3)$ is the permutation representation, $\rho(\sigma)e_i=e_{\sigma_i}$, and $\xi=(1/\sqrt{2},-1/\sqrt{2},0)$, $u\in M_p(G)$ is given by:

$u(\sigma)=\langle\rho(\sigma)\xi,\xi\rangle$.

This has $u(\delta^e)=1$ (duh), $u(\delta^{(12)})=-1$, and otherwise $u(\sigma)=-\frac12 \text{sign}(\sigma)$.

The $p_0,\,p_1$ above is still a cyclic partition of unity… but is the walk irreducible?

The easiest way might be to look for a subharmonic $p$. This is way easier… with $\alpha_\sigma=1$ it is easy to construct non-trivial subharmonics… not with this $u$. It is straightforward to show there are no non-trivial subharmonics and so $u$ is irreducible, periodic, but $p_0$ is not a quantum subgroup.

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

Definition Let $G$ be a finite quantum group. A state $\nu\in M_p(G)$ is concentrated on a cyclic coset of a proper quasi-subgroup if there exists a pair of projections, $p_0\neq p_1$, such that $\nu(p_1)=1$, $p_0$ is a group-like projection, $T_\nu(p_1)=p_0$ and there exists $d\in\mathbb{N}$ ($d>1$) such that $T_\nu^d(p_1)=p_1$.

## (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 $p_0$ can be associated to an idempotent state $\int_S$. Over night I thought of another way.

Using the Pierce decomposition with respect to $p_0$ (where $q_0:=\mathbf{1}_G-p_0$),

$F(G)=p_0F(G)p_0+p_0F(G)q_0+q_0F(G)p_0+q_0F(G)q_0$.

The corner $p_0F(G)p_0$ is a hereditary $\mathrm{C}^*$-subalgebra of $F(G)$. This implies that if $0\leq b\in p_0F(G)p_0$ and for $a\in F(G)$, $0\leq a\leq b\Rightarrow a\in p_0F(G)p_0$.

Let $\rho:=\nu^{\star d}$. We know from Fagnola and Pellicer that $T_\rho(p_0)=p_0$ and $T_\rho(p_0F(G)p_0)=p_0F(G)p_0$.

By assumption in the background here we have an irreducible and periodic random walk driven by $\nu\in M_p(G)$. This means that for all projections $q\in 2^G$, there exists $k_q\in\mathbb{N}$ such that $\nu^{\star k_q}(q)>0$.

Define:

$\displaystyle \rho_n=\frac{1}{n}\sum_{k=1}^n\rho^{\star k}$.

Define:

$\displaystyle n_0:=\max_{\text{projections, }q\in p_0F(G)p_0}\left\{k_q\,:\,\nu^{\star k_q}(q)> 0\right\}$.

The claim is that the support of $\rho_{n_0}$, $p_{\rho_{n_0}}$ is equal to $p_0$.

We probably need to write down that:

$\varepsilon T_\nu^k=\nu^{\star k}$.

Consider $\rho^{\star k}(p_0)$ for any $k\in\mathbb{N}$. Note

\begin{aligned}\rho^{\star k}(p_0)&=\varepsilon T_{\rho^{\star k}}(p_0)=\varepsilon T^k_\rho(p_0)\\&=\varepsilon T^k_{\nu^{\star d}}(p_0)=\varepsilon T_\nu^{kd}(p_0)\\&=\varepsilon(p_0)=1\end{aligned}

that is each $\rho^{\star k}$ is supported on $p_0$. This means furthermore that $\rho_{n_0}(p_0)=1$.

Suppose that the support $p_{\rho_{n_0}}. A question arises… is $p_{\rho_{n_0}}\in p_0F(G)p_0$? This follows from the fact that $p_0\in p_0F(G)p_0$ and $p_0F(G)p_0$ is hereditary.

Consider a projection $r:=p_0-p_{\rho_{n_0}}\in p_0F(G)p_0$. We know that there exists a $k_r\leq n_0$ such that

$\nu^{\star k_r}(p_0-p_{\rho_{n_0}})>0\Rightarrow \nu^{\star k_r}(p_0)>\nu^{\star k_r}(p_{\rho_{n_0}})$.

This implies that $\nu^{\star k_r}(p_0)>0\Rightarrow k_r\equiv 0\mod d$, say $k_r=\ell_r\cdot d$ (note $\ell_r\leq n_0$):

\begin{aligned}\nu^{\star \ell_r\cdot d}(p_0)&>\nu^{\star \ell_r\cdot d}(p_{\rho_{n_0}})\\\Rightarrow (\nu^{\star d})^{\star \ell_r}(p_0)&>(\nu^{\star d})^{\star \ell_r}(p_{\rho_{n_0}})\\ \Rightarrow \rho^{\star \ell_r}(p_0)&>\rho^{\star \ell_r}(p_{\rho_{n_0}})\\ \Rightarrow 1&>\rho^{\star \ell_r}(p_{\rho_{n_0}})\end{aligned}

By assumption $\rho_{n_0}(p_{\rho_{n_0}})=1$. Consider

$\displaystyle \rho_{n_0}(p_{\rho_{n_0}})=\frac{1}{n_0} \sum_{k=1}^{n_0}\rho^{\star k}(p_{\rho_{n_0}})$.

For this to equal one every $\rho^{\star k}(p_{\rho_{n_0}})$ must equal one but $\rho^{\star \ell_r}(p_{\rho_{n_0}})<1$.

Therefore $p_0$ is the support of $\rho_{n_0}$.

Let $\rho_\infty=\lim \rho_n$. We have shown above that $\rho^{\star k}(p_0)=1$ for all $k\in\mathbb{N}$. This is an idempotent state such that $p_0$ is its support (a similar argument to above shows this). Therefore $p_0$ is a group like projection and so we denote it by $\mathbf{1}_S$ and $\int_S=d\mathcal{F}(\mathbf{1}_S)$!

Today, for finite quantum groups, I want to explore some properties of the relationship between a state $\nu\in M_p(G)$, its density $a_\nu$ ($\nu(b)=\int_G ba_\nu$), and the support of $\nu$, $p_{\nu}$.

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

$\displaystyle T_\nu=(\nu\otimes I)\circ \Delta$,

and the result

$T_\nu(a)=S(a_\nu)\overline{\star}a$,

where $\overline{\star}$ is defined as (where $\mathcal{F}:F(G)\rightarrow \mathbb{C}G$ by $a\mapsto (b\mapsto \int_Gba)$).

$\displaystyle a\overline{\star}b=\mathcal{F}^{-1}\left(\mathcal{F}(a)\star\mathcal{F}(b)\right)$.

An obvious thing to note is that

$\nu(a_\nu)=\|a_\nu\|_2^2$.

Also, because

\begin{aligned}\nu(a_\nu p_\nu)&=\int_Ga_\nu p_\nu a_\nu=\int_G(a_\nu^\ast p_\nu^\ast p_\nu a_\nu)\\&=\int_G(p_\nu a_\nu)^\ast p_\nu a_\nu\\&=\int_G|p_\nu a_\nu|^2\\&=\|p_\nu a_\nu\|_2^2=\|a_\nu\|^2\end{aligned}

That doesn’t say much. We are possibly hoping to say that $a_\nu p_\nu=a_\nu$.

## Quasi-Subgroups that are not Subgroups

Let $G$ be a finite quantum group. We associate to an idempotent state $\int_S$quasi-subgroup $S$. 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:

$\displaystyle N_{\rho}:=\left\{ f\in F(G):\rho(|f|^2)=0\right\}$.

In the case of a quasi-subgroup, $S\subset G$, my understanding is that by looking at $N_S:=N_{\int_S}$ we can tell if $S$ is actually a subgroup or not. Franz & Skalski show that:

Let $S\subset G$ be a quasi-subgroup. TFAE

• $S\leq G$ is a subgroup
• $N_{\int_S}$ is a two-sided or self-adjoint or $S$ invariant ideal of $F(G)$
• $\mathbf{1}_Sa=a\mathbf{1}_S$

I want to look again at the Kac & Paljutkin quantum group $\mathfrak{G}_0$ and see how the Pal null-spaces $N_{\rho_6}$ and $N_{\rho_7}$ fail these tests. Both Franz & Gohm and Baraquin should have the necessary left ideals.

### The Pal Null-Space $N_{\rho_6}$

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

$\displaystyle \rho_6(f)=2\int_{\mathfrak{G}_0}f\cdot (e_1+e_4+E_{11})$.

Hence:

$N_{\rho_6}=\langle e_1,e_3,E_{12},E_{22}\rangle$.

If $N_{\rho_6}$ were two-sided, $N_{\rho_6}F(\mathfrak{G}_0)\subset N_{\rho_6}$. Consider $E_{21}\in F(\mathfrak{G}_0)$ and

$E_{12}E_{21}=E_{11}\not\in N_{\rho_6}$.

We see problems also with $E_{12}$ when it comes to the adjoint $E_{12}^{\ast}=E_{21}\not\in N_{\rho_6}$ and also $S(E_{12})=E_{21}\not\in N_{\rho_6}$. It is not surprise that the adjoint AND the antipode are involved as they are related via:

$S(S(f^\ast)^\ast)=f$.

In fact, for finite or even Kac quantum groups, $S(f^\ast)=S(f)^\ast$.

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

$p_{\rho_6}=e_1+e_4+E_{11}$.

This does not commute with $F(G)$:

$E_{21}p_{\rho_6}=E_{21}\neq 0=p_{\rho_6}E_{21}$.

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 $G$ is ergodic if and only if $\nu$ is not concentrated on a proper quasi-subgroup, nor the coset of a ?normal ?-subgroup.

The first part of this conjecture says that if $\nu$ 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 $\nu$ 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 $N\lhd G$, that the convolution powers jump around a cyclic subgroup of $G/N$.

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 $\{p_0,p_1,\dots,p_{d-1}\}$ such that $\nu^{\star k}$ is concentrated on $p_{k\mod d}$.

This cyclic nature suggests that $p_0$ might be equal to $\mathbf{1}_N$ for some $N\lhd G$ and perhaps:

$\Delta(p_i)=\sum_{j=0}^{d-1}p_{i-j}\otimes p_j,$

and perhaps there is an isomorphism $G/N\cong C_d$. Unfortunately I have been unable to progress this.

What is clear is that the ‘supports’ of the $p_i$ behave very much like the cosets of proper normal subgroup $N\lhd G$.

As the random walk is assumed irreducible, we know that for any projection $q\in 2^G$, there exists a $k_q\in \mathbb{N}$ such that $\nu^{\star k_q}(q)\neq 0$.

Playing this game with the Haar element, $\eta\in 2^G$, note there exists a $k_\eta\in\mathbb{N}$ such that $\nu^{k_\eta}(\eta)>0$.

Let $\overline{\nu}=\nu^{\star k_\eta}$. I have proven that if $\mu(\eta)>0$, then the convolution powers of $\mu\in M_p(G)$ converge. Convergence is to an idempotent. This means that $\overline{\nu}^{\star k}$ converges to an idempotent $\overline{\nu}_\infty$, and so we have a quasi-subgroup corresponding to it, say $\overline{p}$.

The question is… does $\overline{p}$ coincide with $p_0$?

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

Is $\overline{p}$ a subgroup? Could it be a normal subgroup?

As nice as it was to invoke the result that if $e$ is in the support of $\nu$, then the convolution powers of $\nu$ 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.