# Self-similar quantum groups

Nathan Brownlowe  Nathan Brownlowe, School of Mathematics and Statistics,
University of Sydney, NSW 2006, Australia<EMAIL_ADDRESS>and
David Robertson  David Robertson, School of Science and Technology, University
of New England, NSW 2351, Australia<EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of self-similarity for compact quantum groups. For a
finite set $X$, we introduce a $C^{*}$-algebra $\mathbb{A}_{X}$, which is the
quantum automorphism group of the infinite homogeneous rooted tree $X^{*}$.
Self-similar quantum groups are then certain quantum subgroups of
$\mathbb{A}_{X}$. Our main class of examples are called finitely-constrained
self-similar quantum groups, and we find a class of these examples that can be
described as quantum wreath products by subgroups of the quantum permutation
group.

###### Key words and phrases:

Brownlowe was supported by the Australian Research Council grant DP200100155,
and both authors were supported by the Sydney Mathematical Research Institute.

##  1\. Introduction

Self-similar groups are a class of groups acting faithfully on an infinite
rooted homogeneous tree $X^{*}$. In particular, given an automorphism
$g\in\operatorname{Aut}(X^{*})$ and a vertex $w\in X^{*}$, by identifying
$wX^{*}$ with $g(w)X^{*}$, we get an automorphism
$g|_{w}\in\operatorname{Aut}(X^{*})$ which is uniquely determined by the
identity

$g\cdot(wv)=(g\cdot w)g|_{w}\cdot v\quad\text{for all }v\in X^{*}.$

The automorphism $g|_{w}$ is called the restriction of $g$ by $w$, and a
subgroup $G\leq\operatorname{Aut}(X^{*})$ is self-similar if it is closed
under restrictions. Self-similar groups are a significant class of groups that
play an important role in geometric group theory, and have been a rich source
of groups displaying interesting phenomena. Most notably, the Grigorchuk group
[5] is a self-similar group which is an infinite, finitely generated periodic
group and provided the first example of a group with intermediate growth, as
well as the first known amenable group to not be elementary amenable.

When the group of automorphisms $\operatorname{Aut}(X^{*})$ is equipped with
the permutation topology, the closed self-similar groups are examples of
compact, totally disconnected groups, and hence are profinite groups. A
particular class of examples of interest are the self-similar groups of finite
type, which are subgroups of automorphisms of $X^{*}$ that act like elements
of a given finite group locally around every vertex. Grigorchuk introduced
this concept in [6], where he also showed that the closure of the Grigorchuk
group is a self-similar group of finite type. Note that these groups are
called finitely constrained self-similar groups in [9], and we will use that
terminology.

The theory of compact quantum groups is by now a very substantial part of the
wider field of quantum groups, and one which sits in the framework of operator
algebras. The theory started with Woronowicz’s introduction of the quantum
$\operatorname{SU}(2)$ group in [12]. Woronowicz then defined compact matrix
quantum groups in [13], before developing a general theory of compact quantum
groups in [14]. An important class of compact matrix quantum groups was
identified and studied by Wang through his quantum permutation groups in [11].
Wang was motivated by one of Connes’ questions from his noncommutative
geometry program: what is the quantum automorphism group of a space? Wang’s
work in [11] provided an answer for finite spaces; in particular, Wang
formally defined the notion of a quantum automorphism group, and then showed
that his quantum permutation group $A_{s}(n)$ is the quantum automorphism
group of the space with $n$ points. For three or fewer points this algebra is
commutative, and hence indicating no quantum permutations; but for four or
more points, remarkably the algebra is noncommutative and infinite-
dimensional.

Since the appearance of [11], follow-up work progressed in multiple
directions, including the results of Bichon in [2] in which he introduced
quantum automorphisms of finite graphs. These algebras are quantum subgroups
of the quantum permutation groups. Bichon used this construction to define the
quantum dihedral group $D_{4}$. Later still in [1], Banica and Bichon
classified all the compact quantum groups acting on four points; that is, all
the compact quantum subgroups of $A_{s}(4)$. Quantum automorphisms of infinite
graphs have recently been considered by Rollier and Vaes in [8], and by Voigt
in [15].

Our current work is the result of us asking the question: is there a
reasonable notion of self-similarity for quantum groups? We answer this
question in the affirmative for compact quantum groups. We do this by first
constructing the quantum automorphism group $\mathbb{A}_{X}$ of the
homogeneous rooted tree $X^{*}$, and then identifying the quantum analogue of
the restriction maps $g\mapsto g|_{w}$ for $g\in\operatorname{Aut}(X^{*})$,
$w\in X^{*}$. We then define a self-similar quantum group to be any quantum
subgroup $A$ of $\mathbb{A}_{X}$ for which the restriction maps factor through
the quotient map $\mathbb{A}_{X}\to A$. We characterise self-similar quantum
groups in terms of a certain homomorphism $A\otimes C(X)\to C(X)\otimes A$,
which can be thought of as quantum state-transition function. The main class
of examples we examine are quantum analogues of finitely constrained self-
similar groups. In our main theorem about these examples we describe a class
of finitely constrained self-similar groups as free wreath products by quantum
subgroups of quantum permutation groups.

We start with a small preliminaries section in which we collect all the
required definitions from the literature on compact quantum groups. In Section
3 we then identify a compact quantum group $\mathbb{A}_{X}$ which we prove is
the quantum automorphism group of the homogeneous rooted tree $X^{*}$. The
$C^{*}$-algebra $\mathbb{A}_{X}$ is a noncommutative, infinite-dimensional
$C^{*}$-algebra whose abelianisation is the algebra of continuous functions on
the automorphism group of the tree $X^{*}$. In Section 4 we introduce the
notion of self-similarity for compact quantum groups, and we characterise
self-similar quantum groups $A$ in terms of morphisms $A\otimes C(X)\to
C(X)\otimes A$, mimicking the fact that classical self-similar action are
governed by the maps $G\times X\to X\times G\colon(g,x)\mapsto(g\cdot
x,g|_{x})$.

In Section 5 we define finitely constrained self-similar quantum groups, which
are the quantum analogues of the classical finitely constrained self-similar
groups studied in [4, 9]. In particular, we consider subalgebras
$\mathbb{A}_{d}$ of $\mathbb{A}_{X}$, which are the quantum automorphism
groups of the finite subtrees $X^{[d]}$ of $X^{*}$ of depth $d$. To each
quantum subgroup $\mathbb{P}$ of $\mathbb{A}_{d}$, we construct a quantum
subgroup $A_{\mathbb{P}}$, which we prove is a self-similar quantum group. We
then build on the work of Bichon in [3] by constructing free wreath products
of compact quantum groups by quantum subgroups of the quantum permutation
group (which corresponds to the subalgebra $\mathbb{A}_{1}$ of
$\mathbb{A}_{X}$), and we prove that every $A_{\mathbb{P}}$ coming from a
quantum subgroup $\mathbb{P}$ of $\mathbb{A}_{1}$ is canonically isomorphic to
the free wreath product $A_{\mathbb{P}}\ast_{w}\mathbb{P}$.

##  2\. Preliminaries

In this section we collect some basics on compact quantum groups. We start
with Woronowicz’s definition of a compact quantum group [14].

######  Definition 2.1.

A _compact quantum group_ is a pair $(A,\Phi)$ where $A$ is a unital
$C^{*}$-algebra and $\Phi:A\to A\otimes A$ is a unital $*$-homomorphism such
that

  1. (1)

$(\Phi\otimes\operatorname{id})\Phi=(\operatorname{id}\otimes\Phi)\Phi$

  2. (2)

$\overline{(A\otimes 1)\Phi(A)}=A\otimes A=\overline{(1\otimes A)\Phi(A)}$.

We call $\Phi$ the _comultiplication_ and (1) is called _coassociativity_.

######  Remark 2.2.

It is proved in [14] that $(A,\Phi)$ is a compact quantum group if and only if
there is a family of matrices $\\{a^{\lambda}=(a^{\lambda}_{i,j})\in
M_{d_{\lambda}}(A):\lambda\in\Lambda\\}$ for some indexing set $\Lambda$ such
that

  1. (1)

$\Phi(a^{\lambda}_{i,j})=\displaystyle\sum_{k=1}^{d_{\lambda}}a^{\lambda}_{i,k}\otimes
a^{\lambda}_{k,j}$ for all $\lambda\in\Lambda$ and $1\leq i,j\leq
d_{\lambda}$,

  2. (2)

$a^{\lambda}$ and its transpose $(a^{\lambda})^{T}$ are invertible elements of
$M_{d_{\lambda}}(A)$ for every $\lambda\in\Lambda$,

  3. (3)

the $*$-subalgebra $\mathcal{A}$ of $A$ generated by the entries
$\\{a^{\lambda}_{i,j}:1\leq i,j\leq d_{\lambda},\lambda\in\Lambda\\}$ is dense
in $A$.

######  Example 2.3.

A key example for us is Wang’s quantum permutation groups $(A_{s}(n),\Phi)$
from [11]. Here, $n$ is a positive integer, and $A_{s}(n)$ is the universal
$C^{*}$-algebra generated by elements $a_{ij}$, $1\leq i,j\leq n$, satisfying

$\displaystyle a_{ij}^{2}=a_{ij}=a_{ij}^{*}\text{ for all $1\leq i,j\leq n$,}$
$\displaystyle\sum_{j=1}^{n}a_{ij}=1\text{ for all $1\leq i\leq n$,}$
$\displaystyle\sum_{i=1}^{n}a_{ij}=1\text{ for all $1\leq j\leq n$.}$

The comultiplication $\Phi$ satisfies
$\Phi(a_{ij})=\sum_{k=1}^{n}a_{ik}\otimes a_{kj}$ for all $1\leq i,j\leq n$.

######  Definition 2.4.

If $(A_{1},\Phi_{1})$ and $(A_{2},\Phi_{2})$ are compact quantum groups, then
a morphism $\pi$ from $(A_{1},\Phi_{1})$ to $(A_{2},\Phi_{2})$ is a
homomorphism of $C^{*}$-algebras $\pi\colon A_{1}\to A_{2}$ satisfying
$(\pi\otimes\pi)\circ\Phi_{1}=\Phi_{2}\circ\pi$.

######  Definition 2.5.

Let $(A,\Phi)$ be a compact quantum group. A Woronowicz ideal is an ideal $I$
of $A$ such that $\Phi(I)\subseteq\operatorname{ker}(q\otimes q)$, where $q$
is the quotient map $A\to A/I$. Then $(A/I,\Phi^{\prime})$, where
$\Phi^{\prime}\colon A/I\to A/I\otimes A/I$ satisfies $\Phi^{\prime}\circ
q=(q\otimes q)\circ\Phi$ is a compact quantum group called a quantum subgroup
of $(A,\Phi)$.

######  Definition 2.6.

A _(left) coaction_ of a compact quantum group $(A,\Phi)$ on a unital
$C^{*}$-algebra $B$ is a unital $*$-homomorphism $\alpha:B\to A\otimes B$
satisfying

  1. (1)

$(\operatorname{id}\otimes\alpha)\alpha=(\Phi\otimes\operatorname{id})\alpha$

  2. (2)

$\overline{\alpha(B)(A\otimes 1)}=A\otimes B$.

We refer to (1) as the _coaction identity_ and (2) is known as the _Podleś
condition_.

##  3\. Quantum automorphisms of a homogeneous rooted tree

In this section we introduce a compact quantum group $\mathbb{A}_{X}$ which we
prove is the quantum automorphism group of the infinite homogeneous rooted
tree $X^{*}$. We start with the notion of an action of a compact quantum group
on $X^{*}$. Note that for $n\geq 0$ we write $X^{n}$ for all the words in $X$
of length $n$, and we then the tree $X^{*}$ can be identified with
$\bigcup_{n\geq}X^{n}$, where $X^{0}=\\{\varnothing\\}$ and $\varnothing$ is
the root of the tree.

######  Definition 3.1.

Let $X$ be a finite set and let $(A,\Phi)$ be a compact quantum group. An
action of $A$ on the homogeneous rooted tree $X^{*}$ is a system

$\alpha=(\alpha_{n}:C(X^{n})\to A\otimes C(X^{n}))$

of left coactions, such that for any $m<n$ the diagram

${C(X^{m})}$${C(X^{n})}$${A\otimes C(X^{m})}$${A\otimes
C(X^{n})}$$\scriptstyle{i_{m,n}}$$\scriptstyle{\alpha_{m}}$$\scriptstyle{\alpha_{n}}$$\scriptstyle{\operatorname{id}\otimes
i_{m,n}}$

commutes, where $i_{m,n}:C(X^{m})\to C(X^{n})$ is the injective homomorphism
satisfying

$i_{m,n}(p_{w})=\sum_{w^{\prime}\in X^{n-m}}p_{ww^{\prime}}.$

We now define the main object of interest in this section, the $C^{*}$-algebra
$\mathbb{A}_{X}$, before proving that it is indeed a compact quantum group in
Theorem 3.4. At some point in the later stages of this project we became aware
of [8], and their notion of the quantum automorphism group
$\operatorname{QAut}\Pi$ of a locally finite connected graph $\Pi$. A
straightforward argument shows that $\mathbb{A}_{X}$ is
$\operatorname{QAut}\Pi$ for $\Pi$ the homogeneous rooted tree, but we include
the proof of Theorem 3.4 for completeness.

######  Definition 3.2.

Let $X$ be a finite set. Define $\mathbb{A}_{X}$ to be the universal
$C^{*}$-algebra generated by elements $\\{a_{u,v}:u,v\in X^{n},n\geq 0\\}$
subject to the following relations:

  1. (1)

$a_{\varnothing,\varnothing}=1$,

  2. (2)

for any $n\geq 0,u,v\in X^{n}$, $a_{u,v}^{*}=a_{u,v}^{2}=a_{u,v}$,

  3. (3)

for any $n\geq 0,u,v\in X^{n}$ and $x\in X$

$a_{u,v}=\sum_{y\in X}a_{ux,vy}=\sum_{z\in X}a_{uz,vx}.$

######  Remarks 3.3.

  * (i)

For each $d\in\mathbb{N}$ we denote by $\mathbb{A}_{d}$ the subalgebra of
$\mathbb{A}_{X}$ generated by $\\{a_{u,v}:u,v\in X^{d}\\}$. Note that
$\mathbb{A}_{1}$ is the Wang’s quantum permutation group $A_{s}(|X|)$ from
Example 2.3.

  * (ii)

We can interpret (3) as follows: each projection $a_{u,v}$ decomposes as an
$|X|\times|X|$ square of projections $\\{a_{ux,vy}:x,y\in X\\}$ with a magic
square type property where every row and column sums to $a_{u,v}$. For
example, if $X=\\{0,1,2\\}$ we have the following structure.

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  * (iii)

Repeated applications of (3) from Definition 3.2 show that for all
$u,u^{\prime},v,v^{\prime},w\in X^{n},n\in\mathbb{N}$, we have

$u\neq u^{\prime},v\neq v^{\prime}\implies
a_{u,w}a_{u^{\prime},w}=0=a_{w,v}a_{w,v^{\prime}},$

and that for all $u=u_{1}\cdots u_{n},v=v_{1}\cdots v_{n}\in
X^{n},n\in\mathbb{N}$, and $x,y\in X$ we have

$a_{x,y}a_{u,v}=a_{u,v}a_{x,y}=\begin{cases}a_{u,v}&\text{if
$u_{1}=x,v_{1}=y$}\\\ 0&\text{otherwise.}\end{cases}$

We will freely use these two identities without comment throughout the rest of
the paper.

######  Theorem 3.4.

The $C^{*}$-algebra $\mathbb{A}_{X}$ is a compact quantum group with
comultiplication $\Delta:\mathbb{A}_{X}\to\mathbb{A}_{X}\otimes\mathbb{A}_{X}$
satisfying

$\Delta(a_{u,v})=\sum_{w\in X^{n}}a_{u,w}\otimes a_{w,v},$

for all $u,v\in X^{n}$ and $n\geq 1$

###### Proof.

To see that $\Delta$ exists, it’s enough to show that the elements

$b_{u,v}:=\sum_{w\in X^{n}}a_{u,w}\otimes a_{w,v}$

for $u,v\in X^{n}$ and $n\geq 1$ satisfy Definition 3.2.

Firstly,
$b_{\varnothing,\varnothing}=\Delta(a_{\varnothing,\varnothing})=a_{\varnothing,\varnothing}\otimes
a_{\varnothing,\varnothing}=1\otimes 1$. For (2), we have

$b_{u,v}^{*}=\sum_{w\in X^{n}}a_{u,w}^{*}\otimes a_{w,v}^{*}=\sum_{w\in
X^{n}}a_{u,w}\otimes a_{w,v}=b_{u,v}$

and

$\displaystyle b_{u,v}^{2}$ $\displaystyle=\left(\sum_{w\in
X^{n}}a_{u,w}\otimes a_{w,v}\right)^{2}$ $\displaystyle=\sum_{w,z\in
X^{n}}a_{u,w}a_{u,z}\otimes a_{w,v}a_{z,v}$ $\displaystyle=\sum_{w\in
X^{n}}a_{u,w}^{2}\otimes a_{w,v}^{2}$ $\displaystyle=\sum_{w\in
X^{n}}a_{u,w}\otimes a_{w,v}$ $\displaystyle=b_{u,v}.$

For (3) fix $u,v\in X^{n}$ and $x\in X$. Then

$\displaystyle b_{u,v}$ $\displaystyle=\sum_{w\in X^{n}}a_{u,w}\otimes
a_{w,v}$ $\displaystyle=\sum_{w\in X^{n}}\sum_{z\in X}a_{ux,wz}\otimes
a_{w,v}$ $\displaystyle=\sum_{w\in X^{n}}\sum_{z\in
X}a_{ux,wz}\otimes\sum_{y\in X}a_{wz,vy}$ $\displaystyle=\sum_{y\in
X}\sum_{w\in X^{n+1}}a_{ux,w}\otimes a_{w,vy}$ $\displaystyle=\sum_{y\in
X}b_{ux,vy}.$

So by the universal property of $\mathbb{A}_{X}$ there is a homomorphism
$\Delta:\mathbb{A}_{X}\to\mathbb{A}_{X}\otimes\mathbb{A}_{X}$ such that

$\Delta(a_{u,v})=\sum_{w\in X^{n}}a_{u,w}\otimes a_{w,v}.$

For coassociativity, we have

$\displaystyle(\operatorname{id}\otimes\Delta)\circ\Delta(a_{u,v})$
$\displaystyle=\sum_{w\in X^{n}}a_{u,w}\otimes\Delta(a_{w,v})$
$\displaystyle=\sum_{w\in X^{n}}a_{u,w}\otimes\left(\sum_{z\in
X^{n}}a_{w,z}\otimes a_{z,v}\right)$ $\displaystyle=\sum_{z\in
X^{n}}\left(\sum_{w\in X^{n}}a_{u,w}\otimes a_{w,z}\right)\otimes a_{z,v}$
$\displaystyle=\sum_{z\in X^{n}}\Delta(a_{u,z})\otimes a_{z,v}$
$\displaystyle=(\Delta\otimes\operatorname{id})\circ\Delta(a_{u,v}).$

Finally, we show that that the set of matrices

$\\{a_{n}=(a_{u,v})_{u,v\in X^{n}}\in M_{X^{n}}(\mathbb{A}_{X}):n\geq 1\\}$

satisfy the conditions of Definition 2.2. Conditions (1) and (3) are clear.
For (2) we show that given any $n\geq 1$ the matrix $a_{n}$ is invertible with
inverse given by $(a_{n})^{T}$. Given $u,v\in X^{n}$ we have

$(a_{n}(a_{n})^{T})_{u,v}=\sum_{w\in
X^{n}}a_{u,w}a_{v,w}=\delta_{u,v}\sum_{w\in X^{n}}a_{u,w}=\delta_{u,v}1_{A}.$

Likewise, we can show $((a_{n})^{T}a_{n})_{u,v}=\delta_{u,v}1_{A}$ and hence
$(a_{n})^{T}=a_{n}^{-1}$ as required. ∎

######  Remark 3.5.

The canonical dense $*$-subalgebra of $\mathbb{A}_{X}$ is the $*$-subalgebra
generated by the projections $\\{a_{u,v}:u,v\in X^{n},n\geq 0\\}$. This is a
Hopf $*$-algebra with counit $\varepsilon\colon\mathbb{A}_{X}\to\mathbb{C}$
and coinverse $\kappa\colon\mathbb{A}_{X}\to\mathbb{A}_{X}$ satisfying
$\varepsilon(a_{u,v})=\delta_{u,v}$ and $\kappa(a_{u,v})=a_{v,u}$, for $u,v\in
X^{n}$, $n\in\mathbb{N}$.

We now show that $(\mathbb{A}_{X},\Delta)$ is the quantum automorphism group
(in the sense of [11, Definition 2.3]) of the homogeneous rooted tree.

######  Proposition 3.6.

There is an action $\gamma=(\gamma_{n})_{n=1}^{\infty}$ of $\mathbb{A}_{X}$ on
$X^{*}$. Moreover, if $\alpha=(\alpha_{n})_{n=1}^{\infty}$ is an action of a
compact quantum group $(A,\Phi)$ on $X^{*}$ then there is a quantum group
homomorphism $\pi:\mathbb{A}_{X}\to A$ such that
$(\pi\otimes\operatorname{id})\circ\gamma_{n}=\alpha_{n}$ for any $n\geq 1$.

###### Proof.

For any $n\geq 1$, the elements

$q_{w}:=\sum_{w^{\prime}\in X^{n}}a_{w,w^{\prime}}\otimes
p_{w^{\prime}}\in\mathbb{A}_{X}\otimes C(X^{n})$

for each $w\in X^{n}$ are mutually orthogonal projections and satisfy

$\sum_{w\in X^{n}}q_{w}=\sum_{w,w^{\prime}\in X^{n}}a_{w,w^{\prime}}\otimes
p_{w^{\prime}}=1\otimes 1$

Therefore there is a unital $*$-homomorphism
$\gamma_{n}:C(X^{n})\to\mathbb{A}_{X}\otimes C(X^{n})$ satisfying
$\gamma_{n}(p_{w})=q_{w}$. We have

$\displaystyle(\Delta\otimes\operatorname{id})\gamma_{n}(p_{w})$
$\displaystyle=\sum_{w^{\prime}\in X^{n}}\Delta(a_{w,w^{\prime}})\otimes
p_{w^{\prime}}$ $\displaystyle=\sum_{w^{\prime},z\in X^{n}}a_{w,z}\otimes
a_{z,w^{\prime}}\otimes p_{w}^{\prime}$ $\displaystyle=\sum_{z\in
X^{n}}a_{w,z}\otimes\alpha_{n}(p_{z})$
$\displaystyle=(\operatorname{id}\otimes\gamma_{n})\gamma_{n}(p_{w}),$

and so each $\gamma_{n}$ satisfies the coaction identity.

For a fixed $v\in X^{n}$ we have

$\sum_{u\in X^{n}}\gamma_{n}(p_{u})(a_{u,v}\otimes 1)=\sum_{u,w\in
X^{n}}a_{u,w}a_{u,v}\otimes p_{w}=\sum_{u\in X^{n}}a_{u,v}\otimes
p_{v}=1\otimes p_{v}.$

Multiplying by any element $a\otimes 1\in\mathbb{A}_{X}\otimes 1$ shows that
$\gamma_{n}(C(X^{n}))(\mathbb{A}_{X}\otimes 1)$ contains the elements
$a\otimes p_{v}$ of $\mathbb{A}_{X}\otimes C(X^{n})$ and hence the required
density is satisfied.

Finally, fix $m<n$ and $w\in X^{m}$. Then

$\displaystyle(\operatorname{id}\otimes i_{m,n})\gamma_{m}(p_{w})$
$\displaystyle=\sum_{z\in X^{m}}a_{w,z}\otimes i_{m,n}(p_{z})$
$\displaystyle=\sum_{z\in X^{m}}\sum_{z^{\prime}\in X^{n-m}}a_{w,z}\otimes
p_{zz^{\prime}}$ $\displaystyle=\sum_{z\in X^{m}}\sum_{z^{\prime}\in
X^{n-m}}\sum_{w^{\prime}\in X^{n-m}}a_{ww^{\prime},zz^{\prime}}\otimes
p_{zz^{\prime}}$ $\displaystyle=\sum_{w^{\prime}\in
X^{n-m}}\alpha_{n}(p_{ww^{\prime}})$
$\displaystyle=\gamma_{n}(i_{m,n}(p_{w})),$

and so the collection $\gamma=(\gamma_{n})_{n=1}^{\infty}$ defines an action
of $(\mathbb{A}_{X},\Delta)$ on the homogeneous rooted tree $X^{*}$.

Now suppose $(\alpha_{n})_{n=1}^{\infty}$ is an action of a compact quantum
group $(A,\Phi)$ on $X^{*}$. Let $b_{\varnothing,\varnothing}:=1\in A$ and for
$n\geq 1$ and $u,v\in X^{n}$ define $b_{u,v}\in A$ to be the unique elements
satisfying

$\alpha_{n}(p_{u})=\sum_{v\in X^{n}}b_{u,v}\otimes p_{v}.$

The coaction identity for $\alpha_{n}$ says that

(3.1) $\Phi(b_{u,v})=\sum_{w\in X^{n}}b_{u,w}\otimes b_{w,v}$

for any $u,v\in X^{n}$.

We claim that the collection $\\{b_{u,v}:u,v\in X^{n},n\geq 0\\}\subseteq A$
satisfies Definition 3.2. Condition (1) is by definition. For (2) and (3), we
appeal to the universal property of the quantum permutation groups
$A_{s}(|X|^{n})$ for $n\geq 1$. Since for any $n\geq 1$, $\alpha_{n}$ defines
a coaction of $(A,\Phi)$ on $C(X^{n})$, [11, Theorem 3.1] says that the
elements $\\{b_{u,v}:u,v\in X^{n}\\}$ satisfy conditions (3.1)–(3.3) of [11,
Section 3]. Condition (3.1) is precisely (2). Conditions (3.1) and (3.2) say
that for any $v\in X^{n}$ we have

$\sum_{u\in X^{n}}b_{u,v}=1_{A}=\sum_{w\in X^{n}}b_{v,w}.$

For any $u\in X^{n}$ and $x\in X$ we have

$p_{ux}\leq\sum_{y\in X}p_{uy}=i_{n,n+1}(p_{u}),$

and hence

$\displaystyle\sum_{v\in X^{n}}\sum_{y\in X}b_{ux,vy}\otimes p_{vy}$
$\displaystyle=\alpha_{n+1}(p_{ux})$
$\displaystyle\leq\alpha_{n+1}(i_{n,n+1}(p_{u}))$
$\displaystyle=(\operatorname{id}_{A}\otimes i_{n,n+1})\alpha_{n}(p_{u})$
$\displaystyle=\sum_{v\in X^{n}}\sum_{y\in X}b_{u,v}\otimes p_{vy}.$

It follows that $b_{ux,vy}\leq b_{u,v}$ for any $x,y\in X$. Therefore, for any
$u,v\in X^{n}$ and $x\in X$ we have

$b_{u,v}=b_{u,v}\left(\sum_{w\in X^{n}}\sum_{y\in
X}b_{ux,wy}\right)=\sum_{y\in X}b_{ux,vy}.$

Likewise for any $y\in X$ we have $b_{u,v}=\sum_{x\in X}b_{ux,vy}$ and (3)
holds.

Therefore, the universal property of $\mathbb{A}_{X}$ provides a homomorphism
$\pi:\mathbb{A}_{X}\to A$ satisfying $\pi(a_{u,v})=b_{u,v}$. It follows from
(3.1) that $(\pi\otimes\pi)\circ\Delta=\Phi\otimes\pi$ and so $\pi$ is a
compact quantum group homomorphism. The identity
$(\pi\otimes\operatorname{id})\circ\gamma_{n}=\alpha_{n}$ is immediate. ∎

######  Proposition 3.7.

For $|X|\geq 2$ the $C^{*}$-algebra $\mathbb{A}_{X}$ is non-commutative and
infinite dimensional.

###### Proof.

Without loss of generality, assume $X=\\{0,1\\}$. Let $B$ be the universal
unital $C^{*}$-algebra generated by two (non-commuting) projections $p$ and
$q$. It is known from [7] that $B\cong C^{*}(\mathbb{Z}_{2}*\mathbb{Z}_{2})$,
which is non-commutative and infinite dimensional. Define the matrix

$(b_{u,v})_{u,v\in X^{2}}=\begin{pmatrix}p&1_{B}-p&0&0\\\ 1_{B}-p&p&0&0\\\
0&0&q&1_{B}-q\\\ 0&0&1_{B}-q&q\\\ \end{pmatrix}\in M_{4}(B).$

Define $b_{\varnothing,\varnothing}=b_{0,0}=b_{1,1}=1_{B}$,
$b_{0,1}=b_{1,0}=0$ and for $u,v\in X^{2}$ and $w,w^{\prime}\in X^{*}$ define
$b_{uw,vw^{\prime}}:=\delta_{w,w^{\prime}}b_{u,v}$. Then these elements
satisfy the relations in Definition 3.2 and hence there is a surjective
homomorphism $\mathbb{A}_{X}\to B$. Since $B$ is non-commutative and infinite-
dimensional so is $\mathbb{A}_{X}$.

∎

######  Remark 3.8.

The group $\operatorname{Aut}(X^{*})$ of automorphisms of a homogeneous rooted
tree $X^{*}$ is compact totally disconnected Hausdorff group under the
permutation topology. A neighbourhood basis of the identity is given by the
family of subgroups

$\\{G_{u}:=\\{g\in\operatorname{Aut}(X^{*}):g\cdot u=u\\}:u\in X^{*}\\},$

and since the orbit of any $u\in X^{*}$ is finite, each of these open
subgroups is closed and hence compact. Cosets of these subgroups are of the
form $G_{u,v}:=\\{g\in G:g\cdot v=u\\}$. Then $\\{G_{u,v}:u,v\in X^{*}\\}$ is
a basis of compact open sets for the topology on $\operatorname{Aut}(X^{*})$.
It follows that the indicator functions $f_{u,v}:=1_{G_{u,v}}$ span a dense
subset of $C(\operatorname{Aut}(X^{*}))$. It is easily checked that the
elements $f_{u,v}$ satisfy (1)–(3) of Definition 3.2 and the universal
property of $C(\operatorname{Aut}(X^{*}))$ then implies that it is the
abelianisation of $\mathbb{A}_{X}$.

##  4\. Self-similarity

If $g\in\operatorname{Aut}(X^{*})$ and $x\in X$, the restriction $g|_{x}$ is
the unique element of $\operatorname{Aut}(X^{*})$ satisfying

$g\cdot(xw)=(g\cdot x)g|_{x}\cdot w\quad\text{for all $w\in X^{*}$}.$

A subgroup $G\leq\operatorname{Aut}(X^{*})$ is called _self-similar_ if $G$ is
closed under taking restrictions. That is, whenever $g\in G$ and $x\in X$, the
restriction $g|_{x}$ is an element of $G$. With the topology inherited from
$\operatorname{Aut}(X^{*})$, the restriction map $G\to G\colon g\mapsto
g|_{x}$ is continuous. If $G$ is any group acting on $X^{*}$ by automorphisms,
we call the action _self-similar_ if the image of $G$ in
$\operatorname{Aut}(X^{*})$ is self-similar.

To have a reasonable notion of self-similarity for quantum subgroups of
$\mathbb{A}_{X}$, we need to understand how restriction manifests itself in
the function algebra $C(\operatorname{Aut}(X^{*}))$. Given $x\in X$ and
$u,v\in X^{n}$ we have

$\displaystyle\\{g:g|_{x}\cdot u=v\\}$ $\displaystyle=\left(\bigcup_{y\in
X}\\{g:g\cdot x=y\\}\right)\cap\\{g:g|_{x}\cdot u=v\\}$
$\displaystyle=\bigcup_{y\in X}\left(\\{g:g\cdot x=y\\}\cap\\{g:g|_{x}\cdot
u=v\\}\right)$ $\displaystyle=\bigcup_{y\in X}\\{g:g\cdot(xu)=yv\\},$

and hence the corresponding indicator functions satisfy

$1_{\\{g:g|_{x}\cdot u=v\\}}=\sum_{y\in X}1_{\\{g:g\cdot(xu)=yv\\}}.$

This formula motivates the following result.

######  Proposition 4.1.

For each $x\in X$ there is a homomorphism
$\rho_{x}:\mathbb{A}_{X}\to\mathbb{A}_{X}$ satisfying

(4.1) $\rho_{x}(a_{u,v})=\sum_{y\in X}a_{yu,xv},$

for all $u,v\in X^{n}$.

We illustrate the formula for a restriction map in Figure 1 by considering
$X=\\{0,1,2\\}$ and looking at what the restriction map $\rho_{1}$ does to the
projection $a_{1,2}$.

$\mapsto$

Figure 1. $\rho_{1}(a_{1,2})=a_{01,12}+a_{11,12}+a_{21,12}$

###### Proof of Proposition 4.1.

Fix $x\in X$. We show that the elements

$\\{b_{u,v}:=\rho_{x}(a_{u,v}):u,v\in X^{n},n\geq 1\\}$

satisfy the conditions of Definition 3.2. For (1) we have

$b_{\varnothing,\varnothing}=\rho_{x}(a_{\varnothing,\varnothing})=\sum_{y\in
X}a_{y,x}=1.$

For (2), we have

$b_{u,v}^{*}=\left(\sum_{y\in X}a_{yu,xv}\right)^{*}=\sum_{y\in
X}a_{yu,xv}^{*}=\sum_{y\in X}a_{yu,xv}=b_{u,v}$

and

$\displaystyle b_{u,v}^{2}=\left(\sum_{y\in
X}a_{yu,xv}\right)^{2}=\sum_{y,z\in X}a_{yu,xv}a_{zu,xv}=\left(\sum_{y\in
X}a_{yu,xv}\right)=b_{u,v}.$

For (3), fix $y\in X$. Then

$\sum_{z\in X}b_{uy,vz}=\sum_{z\in X}\sum_{w\in X}a_{wuy,xvz}=\sum_{w\in
X}a_{wu,xv}=b_{u,v}.$

A similar calculation shows $\sum_{z\in X}b_{uz,vy}=b_{u,v}$. Hence there is a
homomorphism $\rho_{x}$ with the desired formula. ∎

######  Remark 4.2.

We define $\rho_{\varnothing}$ to be the identity homomorphism
$\mathbb{A}_{X}\to\mathbb{A}_{X}$, and for $w=w_{1}\cdots w_{n}\in X^{n}$ we
define $\rho_{w}$ to be the composition
$\rho_{w_{1}}\circ\cdots\circ\rho_{w_{n}}$. A routine calculation shows that
for all $u,v\in X^{n}$ we have

$\rho_{w}(a_{u,v})=\sum_{z\in X^{n}}a_{zu,wv}.$

######  Remark 4.3.

A similar argument to the one in the proof of Proposition 4.1 shows that for
each $x\in X$ there is a homomorphism
$\sigma_{x}\colon\mathbb{A}_{X}\to\mathbb{A}_{X}$ satisfying

$\sigma_{x}(a_{u,v})=\sum_{y\in X}a_{xu,yv}$

for all $u,v\in X^{n}$, $n\in\mathbb{N}$. It is straightforward to see that
$\sigma_{x}=\kappa\circ\rho_{x}\circ\kappa$, where $\kappa$ is the coinverse.

We can now state the main definition of the paper.

######  Definition 4.4.

We call $\rho_{w}$ the restriction by $w$. A quantum subgroup $A$ of
$\mathbb{A}_{X}$ is _self-similar_ if for each $x\in X$ the restriction
$\rho_{x}$ factors through the quotient map $q:\mathbb{A}_{X}\to A$; that is,
if there exists a homomorphism $\widetilde{\rho_{x}}:A\to A$ such that the
diagram

${\mathbb{A}_{X}}$${\mathbb{A}_{X}}$${A}$${A}$$\scriptstyle{\rho_{x}}$$\scriptstyle{q}$$\scriptstyle{q}$$\scriptstyle{\widetilde{\rho_{x}}}$

commutes.

To motivate the main result of this section, let $G$ be a group. To construct
a self-similar action of $G$ on $X^{*}$, it suffices to have a function
$f\colon G\times X\to X\times G$ such that $f(e,x)=(x,e)$ for all $x\in X$,
and such that the following diagram commutes

${G\times G\times X}$${G\times X}$${G\times X\times G}$${X\times G\times
G}$${X\times
G}$$\scriptstyle{m_{G}\times\operatorname{id}_{X}}$$\scriptstyle{\operatorname{id}_{G}\times
f}$$\scriptstyle{f}$$\scriptstyle{f\times\operatorname{id}_{G}}$$\scriptstyle{\operatorname{id}_{X}\times
m_{G}}$

This data allows us to define an action of $G$ on $X^{*}$, which is self-
similar with $g\cdot x$ and $g|_{x}$ the unique elements of $X$ and $G$
satisfying $(g\cdot x,g|_{x}):=f(g,x)$.

Our next result is a compact quantum group analogue of the above result. We
will be working with multiple different identity homomorphisms and units. For
clarity we adopt the following notational conventions: we write
$\operatorname{id}_{A}$ for the identity homomorphism on a $C^{*}$-algebra
$A$, and for $n\geq 1$ write $\operatorname{id}_{n}$ for the identity
homomorphism on the commutative $C^{*}$-algebra $C(X^{n})$. Likewise, $1_{A}$
will denote the unit of $A$, $1$ and $1_{n}$ will denote the units of $C(X)$
and $C(X^{n})$ respectively.

######  Theorem 4.5.

Suppose $(A,\Phi)$ is a compact quantum group equipped with a unital
$*$-homomorphism $\psi:C(X)\otimes A\to A\otimes C(X)$ satisfying

(4.2)
$(\Phi\otimes\operatorname{id}_{1})\psi=(\operatorname{id}_{A}\otimes\psi)(\psi\otimes\operatorname{id}_{A})(\operatorname{id}_{1}\otimes\Phi)$

and

(4.3) $\overline{\psi(C(X)\otimes 1_{A})(A\otimes 1)}=A\otimes C(X).$

Then $(A,\Phi)$ acts on the homogeneous rooted tree $X^{*}$ and moreover the
image of $\mathbb{A}_{X}$, under the homomorphism $\pi:\mathbb{A}_{X}\to A$
from Proposition 3.6, is a self-similar compact quantum group.

###### Proof.

We begin by defining an action of $(A,\Phi)$ on $X^{*}$. Identify $C(X)$ with
$C(X)\otimes 1_{A}\subseteq C(X)\otimes A$ and let
$\alpha_{1}:=\psi|_{C(X)\otimes 1_{A}}$. Then $\alpha_{1}$ is clearly unital
and the coaction identity and Podleś condition for $\alpha_{1}$ follow from
(4.2) and (4.3). Now inductively define
$\alpha_{n+1}:=(\psi\otimes\operatorname{id}_{n})(\operatorname{id}_{1}\otimes\alpha_{n}):C(X^{n+1})\to
A\otimes C(X^{n+1})$ for $n\geq 1$, where we are supressing the canonical
isomorphism $C(X^{n+1})\cong C(X)\otimes C(X^{n})$. Again, $\alpha_{n+1}$ is
clearly unital whenever $\alpha_{n}$ is. If we assume $\alpha_{n}$ satisfies
the coaction identity, then

$\displaystyle(\Phi\otimes\operatorname{id}_{n+1})\alpha_{n+1}$
$\displaystyle=(\Phi\otimes\operatorname{id}_{n+1})(\psi\otimes\operatorname{id}_{n})(\operatorname{id}_{1}\otimes\alpha_{n})$
$\displaystyle=(\operatorname{id}_{A}\otimes\psi\otimes\operatorname{id}_{n})(\psi\otimes\operatorname{id}_{A}\otimes\operatorname{id}_{n})(\operatorname{id}_{1}\otimes\Phi\otimes\operatorname{id}_{n})(\operatorname{id}_{1}\otimes\alpha_{n})$
$\displaystyle=(\operatorname{id}_{A}\otimes\psi\otimes\operatorname{id}_{n})(\psi\otimes\operatorname{id}_{A}\otimes\operatorname{id}_{n})(\operatorname{id}_{1}\otimes\operatorname{id}_{A}\otimes\alpha_{n})(\operatorname{id}_{1}\otimes\alpha_{n})$
$\displaystyle=(\operatorname{id}_{A}\otimes\psi\otimes\operatorname{id}_{n})(\operatorname{id}_{A}\otimes\operatorname{id}_{1}\otimes\alpha_{n})(\psi\otimes\operatorname{id}_{n})(\operatorname{id}_{1}\otimes\alpha_{n})$
$\displaystyle=(\operatorname{id}_{A}\otimes\alpha_{n+1})\alpha_{n+1},$

and so $\alpha_{n+1}$ also satisfies the coaction identity. Since $\alpha_{1}$
is a coaction, we see that $\alpha_{n}$ satisfies the coaction identity for
any $n\geq 1$.

To see that each $\alpha_{n}$ satisfies the Podleś condition, we argue by
induction. We know it is satisfied for $n=1$. Suppose for some $n\geq 1$ that

$\overline{\alpha_{n}(C(X^{n}))(A\otimes 1_{n})}=A\otimes C(X^{n}).$

Fix a spanning element $a\otimes p_{u}\otimes p_{x}\in A\otimes C(X^{n+1})$
where $u\in X^{n}$ and $x\in X$. By the inductive hypothesis we can
approximate

$a\otimes p_{u}\sim\sum_{i}\alpha_{n}(f_{i})(a_{i}\otimes 1_{n}),$

where $f_{i}\in C(X^{n})$ and $a_{i}\in A$. Then

(4.4) $a\otimes p_{u}\otimes p_{x}\sim\sum_{i}(\alpha_{n}(f_{i})\otimes
1)(1_{A}\otimes 1_{n}\otimes p_{x})(a_{i}\otimes 1_{n+1}).$

By definition of $\alpha_{n}$, for any $f\in C(X^{n})$ we have

$\displaystyle\alpha_{n}(f)\otimes 1$
$\displaystyle=((\psi\otimes\operatorname{id}_{n-1})\dots(\operatorname{id}_{n-1}\otimes\psi)(f\otimes
1_{A}))\otimes 1$
$\displaystyle=(\psi\otimes\operatorname{id}_{n})\dots(\operatorname{id}_{n-1}\otimes\psi\otimes\operatorname{id}_{1})(f\otimes
1_{A}\otimes 1)$
$\displaystyle=(\psi\otimes\operatorname{id}_{n})\dots(\operatorname{id}_{n-1}\otimes\psi\otimes\operatorname{id}_{1})(\operatorname{id}_{n}\otimes\psi)(f\otimes
1\otimes 1_{A})$ $\displaystyle=\alpha_{n+1}(f\otimes 1).$

So we can write (4.4) as

$\sum_{i}\alpha_{n+1}(f_{i}\otimes 1)(1_{A}\otimes 1_{n}\otimes
p_{x})(a_{i}\otimes 1_{n+1}).$

Since $\psi$ is unital, we have

$1_{A}\otimes 1_{n}\otimes p_{x}=(\psi\otimes\operatorname{id}_{n})(1\otimes
1_{A}\otimes 1_{n-1}\otimes p_{x}),$

which can be approximated using the induction hypothesis by

$\displaystyle(\psi\otimes\operatorname{id}_{n})(1\otimes 1_{A}\otimes
1_{n-1}\otimes p_{x})$
$\displaystyle\sim(\psi\otimes\operatorname{id}_{n})\left(1\otimes\sum_{j}\alpha_{n}(g_{j})(b_{j}\otimes
1_{n})\right)$
$\displaystyle=(\psi\otimes\operatorname{id}_{n})\left(\sum_{j}(\operatorname{id}_{1}\otimes\alpha_{n})(1\otimes
g_{j})(1\otimes b_{j}\otimes 1_{n})\right)$
$\displaystyle=\sum_{j}\alpha_{n+1}(1\otimes
g_{j})(\psi\otimes\operatorname{id}_{n})(1\otimes b_{j}\otimes 1_{n}).$

Finally, applying the Podleś condition for $\alpha_{1}$ we can approximate

$\psi(1\otimes b_{j})\sim\sum_{k}\alpha_{1}(h_{k})(c_{k}\otimes
1)=\sum_{k}\psi(h_{k}\otimes 1_{A})(c_{k}\otimes 1),$

so

$\displaystyle(\psi\otimes\operatorname{id}_{n})(1\otimes b_{j}\otimes 1_{n})$
$\displaystyle\sim\sum_{k}(\psi(h_{k}\otimes 1_{A})\otimes 1_{n})(c_{k}\otimes
1_{n+1})$ $\displaystyle=\sum_{k}\alpha_{n+1}(h_{k}\otimes 1_{n})(c_{k}\otimes
1_{n+1}).$

Combining these approximations we can write

$a\otimes p_{u}\otimes p_{x}\sim\sum_{i,j,k}\alpha_{n+1}((f_{i}\otimes
1)(h_{k}\otimes g_{j}))(c_{k}a_{i}\otimes 1_{n+1}),$

where $f_{i},g_{j}\in C(X^{n}),h_{k}\in C(X)$ and $a_{i},c_{k}\in A$. Thus
$\alpha_{n+1}$ satisfies the Podleś condition and so by induction $\alpha_{n}$
satisfies the Podleś condition for every $n\geq 1$.

It remains to show that $\alpha_{n}\circ i_{m,n}=(\operatorname{id}_{A}\otimes
i_{m,n})\circ\alpha_{m}$ for any $m<n$. As in the proof of Proposition 3.6 ,
for any $n\geq 1$ and $u,v\in X^{n}$ we will let $b_{u,v}\in A$ be the unique
elements satifsying

$\alpha_{n}(p_{u})=\sum_{v\in X^{n}}b_{u,v}\otimes p_{v}.$

We know from the same proof that for any $n\geq 1$ and $v\in X^{n}$ we have

$\sum_{u\in X^{n}}b_{u,v}=1_{A}.$

If $m<n$, for any $u\in X^{m}$ we have

$\displaystyle\alpha_{n}\circ i_{m,n}(p_{u})$
$\displaystyle=(\psi\otimes\operatorname{id}_{n-1})\dots(\operatorname{id}_{m-1}\otimes\psi\otimes\operatorname{id}_{n-m})(\operatorname{id}_{m}\otimes\alpha_{n-m})(i_{m,n}(p_{u}))$
$\displaystyle=\sum_{w\in
X^{n-m}}(\psi\otimes\operatorname{id}_{n-1})\dots(\operatorname{id}_{m-1}\otimes\psi\otimes\operatorname{id}_{n-m})(\operatorname{id}_{m}\otimes\alpha_{n-m})(p_{u}\otimes
p_{w})$ $\displaystyle=\sum_{w,z\in
X^{n-m}}(\psi\otimes\operatorname{id}_{n-1})\dots(\operatorname{id}_{m-1}\otimes\psi\otimes\operatorname{id}_{n-m})(p_{u}\otimes
b_{w,z}\otimes p_{z})$ $\displaystyle=\sum_{z\in
X^{n-m}}(\psi\otimes\operatorname{id}_{m-1})\dots(\operatorname{id}_{m-1}\otimes\psi)\left(p_{u}\otimes\sum_{w\in
X^{n-m}}b_{w,z}\right)\otimes p_{z}$ $\displaystyle=\sum_{z\in
X^{n-m}}(\psi\otimes\operatorname{id}_{m-1})\dots(\operatorname{id}_{m-1}\otimes\psi)(p_{u}\otimes
1_{A})\otimes p_{z}$ $\displaystyle=\sum_{z\in
X^{n-m}}\alpha_{m}(p_{u})\otimes p_{z}$ $\displaystyle=\sum_{v\in
X^{m}}b_{u,v}\otimes\sum_{z\in X^{n-m}}p_{v}\otimes p_{z}$
$\displaystyle=\sum_{v\in X^{m}}b_{u,v}\otimes i_{m,n}(p_{v})$
$\displaystyle=(\operatorname{id}_{A}\otimes i_{m,n})\circ\alpha_{m}(p_{u}).$

So we have that $(\alpha_{n})_{n=1}^{\infty}$ defines an action of $(A,\Phi)$
on $X^{*}$.

Finally, let $\pi:\mathbb{A}_{X}\to A$ be the homomorphism from Proposition
3.6. We have $\pi(a_{u,v})=b_{u,v}$ for any $u,v\in X^{n}$ and $n\geq 1$. For
each $x\in X$ define a homomorphism $\tilde{\rho_{x}}:A\to A$ by

$\psi(1\otimes a)=\sum_{x\in X}\tilde{\rho_{x}}(a)\otimes p_{x},$

where $a\in A$. For any $u\in X^{n}$ we have

$\alpha_{n+1}(1\otimes p_{u})=\sum_{y\in X}\alpha_{n+1}(p_{yu})=\sum_{v\in
X^{n}}\sum_{x,y\in X}b_{yu,xv}\otimes p_{x}\otimes p_{v}.$

On the other hand, we know
$\alpha_{n+1}=(\psi\otimes\operatorname{id}_{n})(\operatorname{id}_{1}\otimes\alpha_{n})$
and

$(\psi\otimes\operatorname{id}_{n})(\operatorname{id}_{1}\otimes\alpha_{n})(1\otimes
p_{u})=\sum_{v\in X^{n}}\psi(1\otimes b_{u,v})\otimes p_{v}=\sum_{v\in
X^{n}}\sum_{x\in X}\tilde{\rho_{x}}(b_{u,v})\otimes p_{x}\otimes p_{v},$

and by comparing tensor factors we see that
$\tilde{\rho_{x}}(b_{u,v})=\sum_{y\in X}b_{yu,xv}$. Hence, the diagram

${\mathbb{A}_{X}}$${\mathbb{A}_{X}}$${\pi(\mathbb{A}_{X})}$${\pi(\mathbb{A}_{X})}$$\scriptstyle{\rho_{x}}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\tilde{\rho_{x}}}$

commutes, and so $\pi(\mathbb{A}_{X})\subseteq A$ is a self-similar quantum
group. ∎

######  Proposition 4.6.

The following are equivalent

  1. (1)

$(A,\Phi)$ is a quantum self-similar group, and

  2. (2)

$(A,\Phi)$ is a quantum subgroup of $(\mathbb{A}_{X},\Delta)$ and there is a
homomorphism $\psi:C(X)\otimes A\to A\otimes C(X)$ satisfying the hypotheses
of Theorem 4.5.

###### Proof.

Theorem 4.5 is the implication (2) $\Longrightarrow$ (1). To see (1)
$\Longrightarrow$ (2) suppose $(A,\Phi)$ is a quantum self-similar group. By
definition there is a surjective quantum group morphism $q:\mathbb{A}_{X}\to
A$. It is routine to check that there is a homomorphism $\psi:C(X)\otimes A\to
A\otimes C(X)$ satisfying

$\psi(p_{x}\otimes q(a_{u,v}))=\sum_{y\in X}q(a_{xu,yv})\otimes p_{y}.$

Given $u,v\in X^{n}$ we have

$\displaystyle(\Phi\otimes\operatorname{id}_{1})\psi(p_{x}\otimes q(a_{u,v}))$
$\displaystyle=\sum_{y\in X}\Phi(q(a_{xu,yv}))\otimes p_{y}$
$\displaystyle=\sum_{w\in X^{n}}\sum_{y,z\in X}q(a_{xu,zw})\otimes
q(a_{zw,yv})\otimes p_{y}$
$\displaystyle=(\operatorname{id}_{A}\otimes\psi)\left(\sum_{w\in
X^{n}}\sum_{z\in X}q(a_{xu,zw})\otimes p_{z}\otimes q(a_{w,v})\right)$
$\displaystyle=(\operatorname{id}_{A}\otimes\psi)(\psi\otimes\operatorname{id}_{A})\left(\sum_{w\in
X^{n}}p_{x}\otimes q(a_{u,w})\otimes q(a_{w,v})\right)$
$\displaystyle=(\operatorname{id}_{A}\otimes\psi)(\psi\otimes\operatorname{id}_{A})(\operatorname{id}_{1}\otimes\Phi)(p_{x}\otimes
q(a_{u,v})),$

and so $\psi$ satisfies (4.2). For (4.3) notice that for any $q(a)\in A$ and
$z\in X$ we have

$\displaystyle q(a)\otimes p_{z}$ $\displaystyle=(1\otimes p_{z})(q(a)\otimes
1)$ $\displaystyle=\left(\sum_{x\in X}q(a_{x,z})\otimes
p_{z})\right)(q(a)\otimes 1)$ $\displaystyle=\left(\sum_{x,w\in
X}q(a_{x,w})\otimes p_{w})(q(a_{x,z})\otimes 1)\right)(q(a)\otimes 1)$
$\displaystyle=\sum_{x\in X}\psi(p_{x}\otimes 1)(q(a_{x,z}a)\otimes 1).\qed$

######  Example 4.7.

If $G$ is a closed subgroup of $\operatorname{Aut}(X^{*})$ which is self-
similar, then $C(G)$ is a commutative self-similar quantum group. The quotient
map $\mathbb{A}_{X}\to C(Aut(X^{*}))$ takes a generator $a_{u,v}$ to the
indicator function $f_{u,v}$ defined in Remark 3.8. For a function $f\in
C(\operatorname{Aut}(X^{*}))$ and $x\in X$ the restriction homomorphism
$\tilde{\rho_{x}}$ satisfies $\tilde{\rho_{x}}(f)(g)=f(g|_{x})$, for any $g\in
G$.

##  5\. Finitely constrained self-similar quantum groups

###  5.1. Classical finitely constrained self-similar groups

Fix $d\geq 1$, and let $X^{[d]}=\bigcup_{k\leq d}X^{k}$ be the finite subtree
of $X^{*}$ of depth $d$. The group of automorphism
$\operatorname{Aut}(X^{[d]})$ is a quotient of $\operatorname{Aut}(X^{*})$,
and the quotient map is given by restriction to the finite subtree. We write
$r_{d}:\operatorname{Aut}(X^{*})\to\operatorname{Aut}(X^{[d]})$ for this
restriction map.

Fix a subgroup $P\leq\operatorname{Aut}(X^{[d]})$. Define

$G_{P}:=\\{g\in\operatorname{Aut}(X^{*}):r_{d}(g|_{w})\in P\mbox{ for all
}w\in X^{*}\\}.$

By the properties of restriction, if $g,h\in G_{P}$, then for any $w\in X^{*}$

$r_{d}((gh)|_{w})=r_{d}(g|_{h\cdot w}h|_{w})=r_{d}(g|_{h\cdot
w})r_{d}(h|_{w})\in P.$

Likewise, $r_{d}(g^{-1}|_{w})=r_{d}(g|_{g^{-1}\cdot w})^{-1}\in P$. Hence
$G_{P}$ is a self-similar group, called a _finitely constrained self-similar
group_. More details for these groups can be found in [9].

###  5.2. Finitely constrained self-similar quantum groups

Consider the subalgebra $\mathbb{A}_{d}\subseteq\mathbb{A}_{X}$ generated by
the elements $\\{a_{u,v}:|u|=|v|\leq d\\}$. Since
$\Delta:\mathbb{A}_{d}\to\mathbb{A}_{d}\otimes\mathbb{A}_{d}$ the subalgebra
$\mathbb{A}_{d}$ is a quotient quantum group. The abelianisation of
$\mathbb{A}_{d}$ is the algebra $C(\operatorname{Aut}(X^{[d]})$ of continuous
functions on the finite group $\operatorname{Aut}(X^{[d]})$.

######  Definition 5.1.

Suppose $\mathbb{P}$ is a quantum subgroup of $\mathbb{A}_{d}$, where
$\mathbb{P}=\mathbb{A}_{d}/I$. Denote by $q_{I}:\mathbb{A}_{d}\to\mathbb{P}$
the quotient map; so $I=\operatorname{ker}(q_{I})$. We denote by $J$ the
smallest closed 2-sided ideal of $\mathbb{A}_{X}$ generated by
$\\{\rho_{w}(I):w\in X^{*}\\}$, and by $A_{\mathbb{P}}$ the quotient
$A_{\mathbb{P}}:=\mathbb{A}_{X}/J$. In the next result we prove that
$A_{\mathbb{P}}$ is a self-similar quantum group, and we call it a finitely
constrained self-similar quantum group.

######  Proposition 5.2.

Each $A_{\mathbb{P}}$ is a self-similar quantum group.

To prove Proposition 5.2 we need two lemmas. Recall that for
$g,h\in\operatorname{Aut}(X^{*}),w\in X^{*}$ we have

$(gh)|_{w}=g|_{h\cdot w}h|_{w}.$

In the first lemma, we establish an analogous relationship between the
comultiplication $\Delta$ on $\mathbb{A}_{X}$ and the restriction maps
$\rho_{w}$.

######  Lemma 5.3.

For any $n\geq 1,w\in X^{n}$ and $a\in\mathbb{A}_{X}$ we have

$(\Delta\circ\rho_{w})(a)=\sum_{y\in X^{n}}(1\otimes
a_{y,w})(\rho_{y}\otimes\rho_{w})(\Delta(a)).$

###### Proof.

Let $a_{u,v}$ be a generator of $\mathbb{A}_{X}$, with $|u|=|v|=k\geq 0$. Then

$\displaystyle(\Delta\circ\rho_{w})(a_{u,v})$
$\displaystyle=\Delta\left(\sum_{\alpha\in X^{n}}a_{\alpha u,wv}\right)$
$\displaystyle=\sum_{y\in X^{n}}\sum_{\beta\in X^{k}}\sum_{\alpha\in
X^{n}}a_{\alpha u,y\beta}\otimes a_{y\beta,wv}$ $\displaystyle=\sum_{y\in
X^{n}}\sum_{\beta\in X^{k}}\rho_{y}(a_{u,\beta})\otimes a_{y\beta,wv}$
$\displaystyle=\sum_{y\in X^{n}}\sum_{\beta\in
X^{k}}\rho_{y}(a_{u,\beta})\otimes a_{y,w}\rho_{w}(a_{\beta,v})$
$\displaystyle=\sum_{y\in X^{n}}(1\otimes
a_{y,w})(\rho_{y}\otimes\rho_{w})\left(\sum_{\beta\in X^{k}}a_{u,\beta}\otimes
a_{\beta,v}\right)$ $\displaystyle=\sum_{y\in X^{n}}(1\otimes
a_{y,w})(\rho_{y}\otimes\rho_{w}(\Delta(a_{u,v})).$

To see that this formula extends to $\mathbb{A}_{X}$ it’s enough to show that
for any $w\in X^{*}$ the map

$a\mapsto\sum_{y\in X^{n}}(1\otimes
a_{y,w})(\rho_{y}\otimes\rho_{w})(\Delta(a))$

is linear and multiplicative. Linearity is clear, and multiplicativity follows
from the orthogonality of the projections $1\otimes a_{y,w}$ and $1\otimes
a_{z,w}$ for $y\neq z$. ∎

######  Lemma 5.4.

Consider the quotient maps $q_{I}:\mathbb{A}_{d}\to\mathbb{A}_{d}/I$ and
$q_{J}:\mathbb{A}_{X}\to\mathbb{A}_{X}/J$. Then for any $n\geq 1$ and $y,w\in
X^{n}$

$\operatorname{ker}(q_{I}\otimes
q_{I})\subseteq\operatorname{ker}((q_{J}\circ\rho_{y})\otimes(q_{J}\circ\rho_{w})).$

###### Proof.

By definition of $J$ we have $I\subseteq J\circ\rho_{w}$ for any $w\in X^{*}$.
Therefore there is a commuting diagram

${\mathbb{A}_{d}}$${\mathbb{A}_{X}}$${\mathbb{A}_{d}/I}$${\mathbb{A}_{X}/\operatorname{ker}(q_{J}\circ\rho_{w})}$$\scriptstyle{q_{I}}$$\scriptstyle{q_{J}\circ\rho_{w}}$$\scriptstyle{\pi_{w}}$

Then if $c\in\operatorname{ker}(q_{I}\otimes q_{I})$ we have

$(q_{J}\circ\rho_{y})\otimes(q_{J}\circ\rho_{w})(c)=(\pi_{y}\circ
q_{I})\otimes(\pi_{w}\circ q_{I})(c)=\pi_{y}\otimes\pi_{w})\circ(q_{I}\otimes
q_{I})(c)=0,$

as required. ∎

###### Proof of Proposition 5.2.

To see that $A_{\mathbb{P}}$ is a compact quantum group, it suffices to show
that $J$ is a Woronowicz ideal. In other words, we need to show that
$\Delta(J)\subset\operatorname{ker}(q_{J}\otimes q_{J})$ where
$q_{J}:\mathbb{A}_{X}\to\mathbb{A}_{X}/J=:A_{\mathbb{P}}$ is the quotient map.
Since $J$ is generated as an ideal by $\bigcup_{w\in X^{*}}\rho_{w}(I)$ it’s
enough to show that

$(q_{J}\otimes q_{J})(\Delta\circ\rho_{w}(i))=0$

for any $i\in I$ and $w\in X^{*}$. Because $I$ is a Woronowicz ideal we know
that $\Delta(i)\in\operatorname{ker}(q_{I}\otimes q_{I})$. Then by Lemmas 5.3
and 5.4 we have

$\displaystyle(q_{J}\otimes q_{J})(\Delta\circ\rho_{w}(i))$
$\displaystyle=(q_{J}\otimes q_{J})\left(\sum_{y\in X^{n}}(1\otimes
a_{y,w})(\rho_{y}\otimes\rho_{w})(\Delta(i))\right)$ $\displaystyle=\sum_{y\in
X^{n}}(1\otimes q_{J}(a_{y,w}))(q_{J}\circ\rho_{y}\otimes
q_{J}\circ\rho_{w})(\Delta(i))$ $\displaystyle=0.$

Finally, $A_{\mathbb{P}}$ is self-similar since by definition of $J$ we have
$\rho_{w}(J)\subset J$ for any $w\in X^{*}$. ∎

###  5.3. Free wreath products

It is well known that for any $d\geq 1$ the group
$\operatorname{Aut}(X^{[d+1]})$ is isomorphic to the wreath product
$\operatorname{Aut}(X^{[d]})\wr\operatorname{Sym}(X)$. Since
$\operatorname{Aut}(X^{*})$ is the inverse limit over $d$ of the groups
$\operatorname{Aut}(X^{[d]})$, it can be thought as the infinitely iterated
wreath product $\ldots\wr\operatorname{Sym}(X)\wr\operatorname{Sym}(X)$. It
follows that
$\operatorname{Aut}(X^{*})\cong\operatorname{Aut}(X^{*})\wr\operatorname{Sym}(X)$.
More generally, it is shown in [4] that if
$P\leq\operatorname{Sym}(X)=\operatorname{Aut}(X^{[1]})$, then the finitely
constrained self-similar group $G_{P}$ is the infinitely iterated wreath
product $\ldots\wr P\wr P$. In this section we prove in Theorem 5.7 an
analogue of this result for finitely constrained self-similar quantum groups.

In [3], Bichon constructs a free wreath product of a compact quantum group by
the quantum permutation group $\mathbb{A}_{s}(n)$. Bichon also comments in
Remark 2.4 of [3] that there is a natural analogue of this construction for
free wreath products by quantum subgroups of $\mathbb{A}_{s}(n)$. In this
section we formally extend this definition to take free wreath products by any
quantum subgroup of $\mathbb{A}_{s}(n)$, and we prove that the finitely
constrained self-similar quantum group $A_{\mathbb{P}}$ induced from a quantum
subgroup $\mathbb{P}$ of $A_{s}(n)$ is a free wreath product by $\mathbb{P}$.
We begin by recalling the definition of the free wreath product from [3]; note
that we use our notation $\mathbb{A}_{1}$ instead of $A_{s}(|X|)$.

######  Definition 5.5.

Let $X$ be a set of at least two elements. Let $(A,\Phi)$ be a compact quantum
group, and $\mathbb{P}$ a quantum subgroup of $\mathbb{A}_{1}$. For each $x\in
X$, we denote by $\nu_{x}$ the inclusion of $A$ in the free product
$C^{*}$-algebra $(\ast_{x\in X}A)\ast\mathbb{P}$. The free wreath product of
$A$ by $\mathbb{P}$ is the quotient of $(\ast_{x\in X}A)*\mathbb{P}$ by the
two-sided ideal generated by the elements

$\nu_{x}(a)q_{I}(a_{x,y})-q_{I}(a_{x,y})\nu_{x}(a),\ x,y\in X,a\in A.$

The resulting $C^{*}$-algebra is denoted by $A\ast_{X,w}\mathbb{P}$, and the
quotient map is denoted by $q_{w}$. If $X$ is understood, we typically just
write $A\ast_{w}\mathbb{P}$.

######  Theorem 5.6.

Let $(A,\Phi)$ be a compact quantum group, and $\mathbb{P}$ a quantum subgroup
of $\mathbb{A}_{1}$. The free wreath product $A\ast_{w}\mathbb{P}$ from
Definition 5.5 is a compact quantum group with comultiplication $\Phi_{w}$
satisfying

(5.1) $\displaystyle\Phi_{w}(q_{w}(q_{I}(a_{x,y})))$ $\displaystyle=\sum_{z\in
X}q_{w}(q_{I}(a_{x,z}))\otimes q_{w}(q_{I}(a_{z,y}))$ (5.2)
$\displaystyle\Phi_{w}(q_{w}(\nu_{x}(a)))$ $\displaystyle=\sum_{z\in
X}(q_{w}\otimes
q_{w})\big{(}(\nu_{x}\otimes\nu_{z})(\Phi(a))(q_{I}(a_{x,z})\otimes
1)\big{)},$

for each $x,y\in X$ and $a\in A$.

###### Proof.

Since $I$ is a Woronowicz ideal, we have
$\Delta|_{I}\subseteq\operatorname{ker}(q_{I}\otimes q_{I})$, and so the map
$(q_{I}\otimes q_{I})\circ\Delta_{\mathbb{A}_{1}}$ descends to a map

$\phi\colon\mathbb{P}\to\mathbb{P}\otimes\mathbb{P}\subseteq((\ast_{x\in
X}A)\ast\mathbb{P})^{\otimes 2}.$

Then $(q_{w}\otimes
q_{w})\circ\phi\colon\mathbb{P}\to(A\ast_{w}\mathbb{P})^{\otimes 2}$ satisfies

$(q_{w}\otimes q_{w})\circ\phi(q_{I}(a_{x,y}))=\sum_{z\in
X}q_{w}(q_{I}(a_{x,z}))\otimes q_{w}(q_{I}(a_{z,y}))\quad\text{for all }x,y\in
X.$

For each $x\in X$, consider the continuous linear map $\phi_{x}\colon
A\to(A\ast_{w}\mathbb{P})^{\otimes 2}$ given by

$\phi_{x}(a)=(q_{w}\otimes q_{w})\left(\sum_{z\in
X}(\nu_{x}\otimes\nu_{z})(\Phi(a))(q_{I}(a_{x,z})\otimes 1)\right).$

We claim that $\phi_{x}$ is a homomorphism. To see this, let
$\\{a^{\lambda}=(a^{\lambda}_{i,j})\in
M_{d_{\lambda}}(A):\lambda\in\Lambda\\}$ is be a family of matrices satisfying
(1)–(3) of Definition 2.2, and $\mathcal{A}$ be the $*$-subalgebra of $A$
spanned by the entries $a^{\lambda}_{i,j}$. Let $a,b\in\mathcal{A}$ and use
Sweedler’s notation to write $\Phi(a)=a_{(1)}\otimes a_{(2)}$ and
$\Phi(b)=b_{(1)}\otimes b_{(2)}$. We have

$\phi_{x}(a)\phi_{x}(b)=\sum_{z,z^{\prime}\in
X}q_{w}\left(\nu_{x}(a_{(1)})q_{I}(a_{x,z})\nu_{x}(b_{(1)})q_{I}(a_{x,z^{\prime}})\right)\otimes
q_{w}(\nu_{z}(a_{(2)})\nu_{z^{\prime}}(b_{(2)})),$

and then since

$\displaystyle
q_{w}\left(\nu_{x}(a_{(1)})q_{I}(a_{x,z})\nu_{x}(b_{(1)})q_{I}(a_{x,z^{\prime}})\right)$
$\displaystyle=q_{w}\left(\nu_{x}(a_{(1)}b_{(1)})q_{I}(a_{x,z}a_{x,z^{\prime}})\right)$
$\displaystyle=\delta_{z,z^{\prime}}q_{w}\left(\nu_{x}(a_{(1)}b_{(1)})q_{I}(a_{x,z})\right),$

we have

$\displaystyle\phi_{x}(a)\phi_{x}(b)$ $\displaystyle=\sum_{z\in
X}q_{w}\left(\nu_{x}(a_{(1)}b_{(1)})q_{I}(a_{x,z})\right)\otimes
q_{w}(\nu_{z}(a_{(2)})\nu_{z}(b_{(2)}))$ $\displaystyle=(q_{w}\otimes
q_{w})\left(\sum_{z\in X}(\nu_{x}\otimes\nu_{z})(a_{(1)}b_{(1)}\otimes
a_{(2)}b_{(2)})(q_{I}(a_{x,z})\otimes 1)\right)$ $\displaystyle=(q_{w}\otimes
q_{w})\left(\sum_{z\in
X}(\nu_{x}\otimes\nu_{z})(\Phi(ab))(q_{I}(a_{x,z})\otimes 1)\right)$
$\displaystyle=\phi_{x}(ab).$

Since $\mathcal{A}$ is dense in $A$, it follows that $\phi_{x}$ is a
homomorphism on $A$.

The universal property of $(\ast_{x\in X}A)\ast\mathbb{P}$ now gives a
homomorphism $\widetilde{\Phi}\colon(\ast_{x\in
X}A)\ast\mathbb{P}\to(A\ast_{w}\mathbb{P})^{\otimes 2}$ satisfying

$\displaystyle\widetilde{\Phi}(q_{I}(a_{x,y}))$ $\displaystyle=\sum_{z\in
X}q_{w}(q_{I}(a_{x,z}))\otimes q_{w}(q_{\mathbb{P}}(a_{z,y})),$
$\displaystyle\widetilde{\Phi}(\nu_{x}(a))$ $\displaystyle=\sum_{z\in
X}(q_{w}\otimes
q_{w})\big{(}(\nu_{x}\otimes\nu_{z})(\Phi(a))(q_{I}(a_{x,z})\otimes
1)\big{)}.$

For each $a\in\mathcal{A},x,y\in X$ we have

$\displaystyle\widetilde{\Phi}(\nu_{x}(a)q_{I}(a_{x,y}))$
$\displaystyle=\sum_{z,z^{\prime}\in
X}q_{w}\left(\nu_{x}(a_{(1)})q_{I}(a_{x,z})q_{I}(a_{x,z^{\prime}})\right)\otimes
q_{w}(\nu_{z}(a_{(2)})q_{I}(a_{z^{\prime},y}))$
$\displaystyle=\sum_{z,z^{\prime}\in
X}q_{w}\left(q_{I}(a_{x,z})\nu_{x}(a_{(1)})q_{I}(a_{x,z^{\prime}})\right)\otimes
q_{w}(q_{I}(a_{z^{\prime},y})\nu_{z}(a_{(2)}))$
$\displaystyle=\widetilde{\Phi}(q_{I}(a_{x,y})\nu_{x}(a)).$

It follows that
$\widetilde{\Phi}(\nu_{x}(a)q_{I}(a_{x,y}))=\widetilde{\Phi}(q_{I}(a_{x,y})\nu_{x}(a))$
for each $a\in A,x,y\in X$, and hence $\widetilde{\Phi}$ descends to the
desired $\Phi_{w}\colon A\ast_{w}\mathbb{P}\to(A\ast_{w}\mathbb{P})^{\otimes
2}$.

We now claim that
$(\operatorname{id}\otimes\Phi_{w})\circ\Phi_{w}=(\Phi_{w}\otimes\operatorname{id})\circ\Phi_{w}$.
Since $\mathcal{A}$ is dense in $A$, to see that
$(\operatorname{id}\otimes\Phi_{w})\circ\Phi_{w}$ and
$(\Phi_{w}\otimes\operatorname{id})\circ\Phi_{w}$ agree on each
$q_{w}(\nu_{x}(A))$, it suffices to show that

$(\operatorname{id}\otimes\Phi_{w})\circ\Phi_{w}(q_{w}(\nu_{x}(a^{\lambda}_{i,j})))=(\Phi_{w}\otimes\operatorname{id})\circ\Phi_{w}(q_{w}(\nu_{x}(a^{\lambda}_{i,j}))),\,\text{for
all }\lambda\in\Lambda,1\leq i,j\leq d_{\lambda}.$

Routine calculations using (5.1) and (5.2) show that both sides of this
equation are equal to

$\sum_{z,z^{\prime}\in X}\sum_{1\leq k,l\leq d_{\lambda}}q_{w}^{\otimes
3}\big{(}\nu_{x}(a^{\lambda}_{i,k})q_{I}(a_{x,z})\otimes\nu_{z}(a^{\lambda}_{k,l})q_{I}(a_{z,z^{\prime}})\otimes\nu_{z^{\prime}}(a^{\lambda}_{l,j})\big{)},$

and hence are equal. So we have
$(\operatorname{id}\otimes\Phi_{w})\circ\Phi_{w}(q_{q}(\nu_{x}(A)))=(\Phi_{w}\otimes\operatorname{id})\circ\Phi_{w}(q_{q}(\nu_{x}(A)))$
for each $x\in X$. It is straightforward to check that evaluating both
$(\operatorname{id}\otimes\Phi_{w})\circ\Phi_{w}$ and
$(\Phi_{w}\otimes\operatorname{id})\circ\Phi_{w}$ at $q_{I}(a_{x,y})$ gives

$\sum_{z,z^{\prime}\in X}q_{w}^{\otimes 3}\left(q_{I}^{\otimes
3}\left(a_{x,z}\otimes a_{z,z^{\prime}}\otimes
a_{z^{\prime},y}\right)\right).$

Hence we have
$(\operatorname{id}\otimes\Phi_{w})\circ\Phi_{w}=(\Phi_{w}\otimes\operatorname{id})\circ\Phi_{w}$.

We now define the matrix $a^{X}$ by $a^{X}_{x,y}:=q_{w}(q_{I}(a_{x,y}))$, for
$x,y\in X$; and for each $\lambda\in\Lambda$, $1\leq i,j\leq d_{\lambda}$,
$x,y\in X$, the elements

$a^{(\lambda,X)}_{(i,x),(j,y)}:=q_{w}(\nu_{x}(a^{\lambda}_{i,j})q_{I}(a_{x,y}))\in
A\ast_{w}\mathbb{P},$

define matrices $a^{(\lambda,X)}=(a^{(\lambda,X)}_{(i,x),(j,y)})$. To finish
the proof we have to show that these matrices satisfy (1)–(3) of Definition
2.2.

We have

$\Phi_{w}(a^{(\lambda,X)}_{(i,x),(j,y)})=\Big{(}\sum_{z\in X}(q_{w}\otimes
q_{w})\big{(}(\nu_{x}\otimes\nu_{z})(\Phi(a^{\lambda}_{i,j}))(q_{I}(a_{x,z})\otimes
1)\big{)}\Big{)}(q_{w}\circ q_{I})^{\otimes 2}(\Delta(a_{x,y})).$

We know that (1) is satisfied for the matrix $a^{X}$. For each $z\in X$ we
have

$\displaystyle(q_{w}\otimes
q_{w})\big{(}(\nu_{x}\otimes\nu_{z})(\Phi(a^{\lambda}_{i,j}))(q_{I}(a_{x,z})\otimes
1)\big{)}(q_{w}\circ q_{I})^{\otimes 2}(\Delta(a_{x,y}))$
$\displaystyle\qquad=(q_{w}\otimes
q_{w})\Big{(}(\nu_{x}\otimes\nu_{z})\Big{(}\sum_{1\leq k\leq
d_{\lambda}}a_{i,k}^{\lambda}\otimes
a_{k,j}^{\lambda}\Big{)}(q_{I}(a_{x,z})\otimes 1)\Big{)}(q_{w}\circ
q_{I})^{\otimes 2}(\Delta(a_{x,y}))$ $\displaystyle\qquad=\sum_{1\leq k\leq
d_{\lambda}}\big{(}q_{w}(\nu_{x}(a_{i,k}^{\lambda}))\otimes
q_{w}(\nu_{z}(a_{k,j}^{\lambda}))\big{)}(q_{w}(q_{I}(a_{x,z}))\otimes
1)(q_{w}\circ q_{I})^{\otimes 2}(\Delta(a_{x,y}))$
$\displaystyle\qquad=\sum_{1\leq k\leq
d_{\lambda}}\big{(}q_{w}(\nu_{x}(a_{i,k}^{\lambda}))\otimes
q_{w}(\nu_{z}(a_{k,j}^{\lambda}))\big{)}\sum_{z^{\prime}\in
X}q_{w}(q_{I}(a_{x,z}a_{x,z^{\prime}}))\otimes q_{w}(q_{I}(a_{z^{\prime},y}))$
$\displaystyle\qquad=\sum_{1\leq k\leq
d_{\lambda}}\big{(}q_{w}(\nu_{x}(a_{i,k}^{\lambda}))\otimes
q_{w}(\nu_{z}(a_{k,j}^{\lambda}))\big{)}(q_{w}(q_{I}(a_{x,z}))\otimes
q_{w}(q_{I}(a_{z,y})))$ $\displaystyle\qquad=\sum_{1\leq k\leq
d_{\lambda}}q_{w}(\nu_{x}(a_{i,k}^{\lambda})q_{I}(a_{x,z}))\otimes
q_{w}(\nu_{z}(a_{k,j}^{\lambda})q_{I}(a_{z,y})).$

It follows that

$\displaystyle\Phi_{w}(a^{(\lambda,X)}_{(i,x),(j,y)})$
$\displaystyle=\sum_{z\in X}\sum_{1\leq k\leq
d_{\lambda}}q_{w}(\nu_{x}(a_{i,k}^{\lambda})q_{I}(a_{x,z}))\otimes
q_{w}(\nu_{z}(a_{k,j}^{\lambda})q_{I}(a_{z,y}))$ $\displaystyle=\sum_{z\in
X}\sum_{1\leq k\leq d_{\lambda}}a^{(\lambda,X)}_{(i,x),(k,z)}\otimes
a^{(\lambda,X)}_{(k,z),(j,y)},$

and so (1) holds for all matrices $a^{(\lambda,X)}$. To see that
$a^{(\lambda,X)}$ is invertible, we define $b^{(\lambda,X)}$ by

$b^{(\lambda,X)}_{(i,x),(j,y)}:=q_{w}(q_{I}(a_{y,x})\nu_{y}((a^{\lambda})^{-1}_{i,j})).$

Then we have

$\displaystyle(a^{(\lambda,X)}b^{(\lambda,X)})_{(i,x),(j,y)}$
$\displaystyle=\sum_{z\in X}\sum_{1\leq k\leq
d_{\lambda}}a^{(\lambda,X)}_{(i,x),(k,z)}b^{(\lambda,X)}_{(k,z),(j,y)}$
$\displaystyle=q_{w}\Big{(}\sum_{z\in X}\sum_{1\leq k\leq
d_{\lambda}}\nu_{x}(a^{\lambda}_{i,k})q_{I}(a_{x,z})q_{I}(a_{y,z})\nu_{y}((a^{\lambda})^{-1}_{k,j})\Big{)}$
$\displaystyle=\delta_{x,y}q_{w}\Big{(}\sum_{1\leq k\leq
d_{\lambda}}\nu_{x}(a^{\lambda}_{i,k})\Big{(}\sum_{z\in
X}q_{I}(a_{x,z})\Big{)}\nu_{x}((a^{\lambda})^{-1}_{k,j})\Big{)}$
$\displaystyle=\delta_{x,y}q_{w}\Big{(}\nu_{x}\Big{(}\sum_{1\leq k\leq
d_{\lambda}}a^{\lambda}_{i,k}(a^{\lambda})^{-1}_{k,j}\Big{)}\Big{)}$
$\displaystyle=\delta_{x,y}q_{w}(\nu_{x}((a^{\lambda}(a^{\lambda})^{-1})_{i,j}))$
$\displaystyle=\delta_{x,y}\delta_{i,j}1.$

A similar calculation shows that
$(b^{(\lambda,X)}a^{(\lambda,X)})_{(i,x),(j,y)}=\delta_{x,y}\delta_{i,j}1$,
and so $a^{(\lambda,X)}$ is invertible. Similar calculations also show that
$c^{(\lambda,X)}$ with entries

$c^{(\lambda,X)}_{(i,x),(j,y)}:=q_{w}(q_{I}(a_{x,y})\nu_{x}(((a^{\lambda})^{T})^{-1}_{i,j}))$

is the inverse of $(a^{(\lambda,X)})^{T}$.

We also have

$(a^{X}(a^{X})^{T})_{x,y}=\sum_{z\in
X}a^{X}_{x,z}a^{X}_{y,z}=q_{w}\left(q_{I}\left(\sum_{z\in
X}a_{x,z}a_{y,z}\right)\right)=\delta_{x,y}1.$

Similarly, $(a^{X})^{T}a^{X}$ is the identity. So $a^{X}$ and $(a^{X})^{T}$
are mutually inverse, and (2) is satisfied.

We now claim that the entries of the matrices
$\\{a^{(\lambda,X)}:\lambda\in\Lambda\\}\cup\\{a^{X}\\}$ span a dense subset
of $A\ast_{X,w}\mathbb{P}$. For each $x,y\in X$ we obviously have
$q_{w}(q_{I}(a_{x,y}))$ in this span since they are the entries of $a^{X}$.
For each $x\in X$, $\lambda\in\Lambda$ and $1\leq i,j\leq d_{\lambda}$ we have

$\sum_{y\in
X}a^{(\lambda,X)}_{(i,x),(j,y)}=q_{w}\left(\nu_{x}(a^{\lambda}_{i,j})q_{I}\left(\sum_{y\in
X}a_{x,y}\right)\right)=q_{w}(\nu_{x}(a^{\lambda}_{i,j})),$

and so each $q_{w}(\nu_{x}(a^{\lambda}_{i,j}))$ is in the span of the entries.
The claim follows, and so (3) holds. ∎

######  Theorem 5.7.

Let $A_{\mathbb{P}}$ be a finitely-constrained self-similar quantum group in
the sense of Definition 5.1. There is a unital quantum group isomorphism
$\pi\colon A_{\mathbb{P}}\to A_{\mathbb{P}}\ast_{w}\mathbb{P}$ satisfying

(5.3)
$\pi(q_{J}(a_{xu,yv}))=q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}$

for all $x,y\in X$, $u,v\in X^{m}$, $m\geq 0$.

###### Proof.

We define $b_{\varnothing,\varnothing}$ to be the identity of
$A_{\mathbb{P}}\ast_{w}\mathbb{P}$, and for each $x,y\in X$, $u,v\in
X^{m},m\geq 0$,

$b_{xu,yv}:=q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}.$

We claim that this gives a family of projections satisfying (1)–(3) of
Definition 3.2. Condition (1) holds by definition. We have

$\displaystyle b_{xu,yv}^{*}$
$\displaystyle=q_{w}\big{(}\nu_{x}(q_{J}(a_{u,v}^{*}))q_{I}(a_{x,y}^{*})\big{)}$
$\displaystyle=q_{w}\big{(}\nu_{x}(q_{J}(a_{u,v}))q_{I}(a_{x,y})\big{)}$
$\displaystyle=q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}$
$\displaystyle=b_{xu,yv}$

and

$b_{xu,yv}^{2}=q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}=q_{w}\big{(}q_{I}(a_{x,y}^{2})\nu_{x}(q_{J}(a_{u,v}^{2}))\big{)}=b_{xu,yv}.$

So (2) holds. For each $w\in X$ we have

$\displaystyle\sum_{z\in X}b_{xuw,yvz}$ $\displaystyle=\sum_{z\in
X}q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{uw,vz}))\big{)}$
$\displaystyle=q_{w}\Big{(}q_{I}(a_{x,y})\nu_{x}\Big{(}q_{J}\Big{(}\sum_{z\in
X}a_{uw,vz}\Big{)}\Big{)}\Big{)}$
$\displaystyle=q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}$
$\displaystyle=b_{xu,yv},$

and

$\displaystyle\sum_{z\in X}b_{xuz,yvw}$ $\displaystyle=\sum_{z\in
X}q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{uz,vw}))\big{)}$
$\displaystyle=q_{w}\Big{(}q_{I}(a_{x,y})\nu_{x}\Big{(}q_{J}\Big{(}\sum_{z\in
X}a_{uz,vw}\Big{)}\Big{)}\Big{)}$
$\displaystyle=q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}$
$\displaystyle=b_{xu,yv},$

and hence (3) holds. This proves the claim, and hence the universal property
of $\mathbb{A}_{X}$ now gives a homomorphism
$\widetilde{\pi}\colon\mathbb{A}_{X}\to A_{\mathbb{P}}\ast_{w}\mathbb{P}$
satisfying

$\widetilde{\pi}(a_{xu,yv})=q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)},$

for all $x,y\in X$, $u,v\in X^{m}$, $m\geq 0$.

We now claim that $J$ is contained in $\operatorname{ker}\widetilde{\pi}$. To
see this, fix $w\in X^{n}$, with $w=w_{1}w^{\prime}$ for $w_{1}\in
X,w^{\prime}\in X^{n-1}$. We first prove the claim that for each
$x_{k}:=a_{u_{1},v_{1}}\cdots a_{u_{k},v_{k}}$, where $k\geq 1$ and each pair
$u_{i},v_{i}\in X^{m_{i}}$ for some $m_{i}\geq 0$, we have

(5.4) $\widetilde{\pi}(\rho_{w}(x_{k}))=\sum_{y\in
X}q_{w}\big{(}q_{I}(a_{y,w_{1}})\nu_{y}(q_{J}(\rho_{w^{\prime}}(x_{k})))\big{)}.$

Let $k=1$. Then

$\displaystyle\widetilde{\pi}(\rho_{w}(a_{u_{1},v_{1}}))$
$\displaystyle=\sum_{y\in X}\sum_{\alpha\in X^{n-1}}\widetilde{\pi}(a_{y\alpha
u_{1},wv_{1}})$ $\displaystyle=\sum_{y\in X}\sum_{\alpha\in
X^{n-1}}q_{w}\big{(}q_{I}(a_{y,w_{1}})\nu_{y}(q_{J}(a_{\alpha
u_{1},w^{\prime}v_{1}}))\big{)}$ $\displaystyle=\sum_{y\in
X}q_{w}\Big{(}q_{I}(a_{y,w_{1}})v_{y}\Big{(}q_{J}\Big{(}\sum_{\alpha\in
X^{n-1}}a_{\alpha u_{1},w^{\prime}v_{1}}\Big{)}\Big{)}\Big{)}$
$\displaystyle=\sum_{y\in
X}q_{w}\big{(}q_{I}(a_{y,w_{1}})\nu_{y}(q_{J}(\rho_{w^{\prime}}(a_{u_{1},v_{1}})))\big{)},$

and so (5.4) holds for $k=1$. We now assume true for $x_{k}$, and prove for
$x_{k+1}$. Note that for $y,y^{\prime}\in X$ we have
$q_{I}(a_{y,w_{1}})q_{I}(a_{y^{\prime},w_{1}})=\delta_{y,y^{\prime}}q_{I}(a_{y,w_{1}})$,
and hence

$\displaystyle
q_{w}\big{(}q_{I}(a_{y,w_{1}})\nu_{y}(q_{J}(\rho_{w^{\prime}}(x_{k})))q_{I}(a_{y^{\prime},w_{1}})\nu_{y^{\prime}}(q_{J}(\rho_{w^{\prime}}(a_{u_{k+1},v_{k+1}})))\big{)}$
$\displaystyle\qquad\qquad=q_{w}\big{(}\nu_{y}(q_{J}(\rho_{w^{\prime}}(x_{k})))q_{I}(a_{y,w_{1}})q_{I}(a_{y^{\prime},w_{1}})\nu_{y^{\prime}}(q_{J}(\rho_{w^{\prime}}(a_{u_{k+1},v_{k+1}})))\big{)}$
$\displaystyle\qquad\qquad=\delta_{y,y^{\prime}}q_{w}\big{(}\nu_{y}(q_{J}(\rho_{w^{\prime}}(x_{k})))q_{I}(a_{y,w_{1}})\nu_{y}(q_{J}(\rho_{w^{\prime}}(a_{u_{k+1},v_{k+1}})))\big{)}$
$\displaystyle\qquad\qquad=\delta_{y,y^{\prime}}q_{w}\big{(}q_{I}(a_{y^{\prime},w_{1}})\nu_{y}(q_{J}(\rho_{w^{\prime}}(x_{k})))\nu_{y}(q_{J}(\rho_{w^{\prime}}(a_{u_{k+1},v_{k+1}})))\big{)}$
$\displaystyle\qquad\qquad=\delta_{y,y^{\prime}}q_{w}\big{(}q_{I}(a_{y^{\prime},w_{1}})\nu_{y}(q_{J}(\rho_{w^{\prime}}(x_{k}a_{u_{k+1},v_{k+1}})))\big{)}.$

It follows that

$\widetilde{\pi}(\rho_{w}(x_{k+1)})=\widetilde{\pi}(\rho_{w}(x_{k}))\widetilde{\pi}(\rho_{w}(a_{u_{k+1},v_{k+1}}))=\sum_{y\in
X}q_{w}\big{(}q_{I}(a_{y^{\prime},w_{1}})\nu_{y}(q_{J}(\rho_{w^{\prime}}(x_{k}a_{u_{k+1},v_{k+1}})))\big{)},$

and it follows that (5.4) holds for all $k$. Since linear combinations of
products of the form $x_{k}$ is a dense subalgebra of $\mathbb{A}_{X}$, it
follows that

$\widetilde{\pi}(\rho_{w}(a))=\sum_{y\in
X}q_{w}\big{(}q_{I}(a_{y,w_{1}})\nu_{y}(q_{J}(\rho_{w^{\prime}}(a)))\big{)}$

for all $a\in\mathbb{A}_{X}$. Now, if $a\in I$, then $\rho_{w^{\prime}}(a)\in
J=\operatorname{ker}q_{J}$, and hence the above equations shows that
$\widetilde{\pi}(\rho_{w}(a))=0$. Hence
$\rho_{w}(a)\in\operatorname{ker}\widetilde{\pi}$ for all $w\in X^{n}$ and
$a\in I$, and hence $J\subseteq\operatorname{ker}\widetilde{\pi}$. This means
$\widetilde{\pi}$ descends to a homomorphism $\pi\colon A_{\mathbb{P}}\to
A_{\mathbb{P}}\ast_{w}\mathbb{P}$ satisfying

$\pi(q_{J}(a_{xu,yv}))=q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}$

for all $x,y\in X$, $u,v\in X^{m}$, $m\geq 0$.

We now show that $\pi$ is an isomorphism by finding an inverse. For each $x\in
X$ consider the homomorphism $q_{J}\circ\sigma_{x}\colon\mathbb{A}_{X}\to
A_{\mathbb{P}}$, where $\sigma_{x}$ is the homomorphism from Remark 4.3. Since
$\sigma_{x}=\kappa\circ\rho_{x}\circ\kappa$, and we know from [10, Remark
2.10] that $\kappa(J)\subseteq J$, it follows that $q_{J}\circ\sigma_{x}$
descends to a homomorphism $\phi_{x}\colon A_{\mathbb{P}}\to A_{\mathbb{P}}$
satisfying

$\phi_{x}(q_{J}(a_{u,v}))=q_{J}(\sigma_{x}(a_{u,v}))=\sum_{y\in
X}q_{J}(a_{xu,yv}),$

for all $u,v\in X^{m},m\geq 0$.

Each $\phi_{x}$, and the map $q_{I}(a)\mapsto q_{J}(a)$ from $\mathbb{P}$ to
$A_{\mathbb{P}}$, now allow us to apply the universal property of the free
product $(\ast_{x\in X}A_{\mathbb{P}})\ast\mathbb{P}$ to get a homomorphism
$\widetilde{\phi}\colon(\ast_{x\in X}A_{\mathbb{P}})\ast\mathbb{P}\to
A_{\mathbb{P}}$ satisfying $\widetilde{\phi}\circ\nu_{x}=\phi_{x}$ for each
$x\in X$, and $\widetilde{\phi}(q_{I}(a))=q_{J}(a)$ for all
$a\in\mathbb{A}_{1}\subseteq\mathbb{A}_{X}$. We claim that

$\widetilde{\phi}\big{(}\nu_{x}(q_{J}(a_{u,v}))q_{I}(a_{x,y})-q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v})\big{)}=0,$

for each $x\in X$, $u,v\in X^{m}$, $m\in\mathbb{N}$. We have

$\displaystyle\widetilde{\phi}\big{(}\nu_{x}(q_{J}(a_{u,v}))q_{I}(a_{x,y})-q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v})\big{)}$
$\displaystyle\hskip
56.9055pt=\phi_{x}(q_{J}(a_{u,v}))q_{J}(a_{x,y})-q_{G}(a_{x,y})\phi_{x}(q_{J}(a_{u,v}))$
$\displaystyle\hskip 56.9055pt=\sum_{y\in
X}q_{J}(a_{xu,yv})q_{J}(a_{x,y})-\sum_{y^{\prime}\in
X}q_{J}(a_{x,y})q_{J}(a_{xu,y^{\prime}v})$ $\displaystyle\hskip
56.9055pt=q_{J}(a_{xu,yv})-q_{J}(a_{xu,yv})$ $\displaystyle\hskip
56.9055pt=0.$

It follows that $\widetilde{\phi}$ descends to a homomorphism $\phi\colon
A_{\mathbb{P}}\ast_{w}\mathbb{P}\to A_{\mathbb{P}}$ satisfying

$\phi(q_{w}(\nu_{x}(q_{J}(a_{u,v}))))=q_{J}(\sigma_{x}(a_{u,v}))=\sum_{y\in
X}q_{J}(a_{xu,yv})$

for all $x\in X$, $u,v\in X^{m},m\geq 0,$ and

$\phi(q_{w}(q_{I}(a_{x,y})))=q_{J}(a_{x,y})$

for all $x,y\in X$.

We claim that $\pi$ and $\phi$ are mutually inverse. For $x,y\in X$, $u,v\in
X^{m},m\geq 0,$ we have

$\phi(\pi(q_{J}(a_{xu,yv})))=\phi\big{(}q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}\big{)}=q_{J}(a_{x,y})\sum_{y\in
X}q_{J}(a_{xu,yv})=q_{J}(a_{xu,yv}),$

and it follows that $\phi\circ\pi$ is the identity on $A_{\mathbb{P}}$. For
$x\in X$, $u,v\in X^{m},m\geq 0,$ we have

$\displaystyle\pi(\phi(q_{w}(\nu_{x}(q_{J}(a_{u,v})))))$
$\displaystyle=\pi\Big{(}\sum_{y\in X}q_{J}(a_{xu,yv})\Big{)}$
$\displaystyle=\sum_{y\in
X}q_{w}\big{(}q_{I}(a_{x,y})\nu_{x}(q_{J}(a_{u,v}))\big{)}$
$\displaystyle=q_{w}\Big{(}q_{I}\Big{(}\sum_{y\in
X}a_{x,y}\Big{)}\nu_{x}(q_{J}(a_{u,v}))\Big{)}$
$\displaystyle=q_{w}(\nu_{x}(q_{J}(a_{u,v}))),$

and for all $x,w\in X$ we have

$\pi(\phi(q_{w}(q_{I}(a_{x,y}))))=\pi(q_{J}(a_{x,y}))=q_{w}(q_{I}(a_{x,y})).$

Hence $\pi\circ\phi$ is the identity on $A_{\mathbb{P}}\ast_{w}\mathbb{P}$,
and so $\pi$ is an isomorphism.

We now need to show that $\pi$ is a homomorphism of compact quantum groups,
which means that $\Delta_{w}\circ\pi=(\pi\otimes\pi)\circ\Delta_{J}$, where
$\Delta_{J}$ is the comultiplication on $A_{\mathbb{P}}$. For $x,y\in X$,
$u,v\in X^{m},m\geq 0$, we have

$\displaystyle(\pi\otimes\pi)\circ\Delta_{J}(q_{J}(a_{xu,yv}))$
$\displaystyle\qquad=\sum_{\alpha^{m+1}}\pi(q_{J}(a_{xu,\alpha}))\otimes\pi(q_{J}(a_{\alpha,yv}))$
$\displaystyle\qquad=\sum_{z\in X}\sum_{\beta\in
X^{m}}\pi(q_{J}(a_{xu,z\beta}))\otimes\pi(q_{J}(a_{z\beta,yv}))$
$\displaystyle\qquad=\sum_{z\in X}\sum_{\beta\in
X^{m}}q_{w}\big{(}q_{I}(a_{x,z})\nu_{x}(q_{J}(a_{u,\beta}))\big{)}\otimes
q_{w}\big{(}q_{I}(a_{z,y})\nu_{x}(q_{J}(a_{\beta,v}))\big{)}.$

We have

$\Delta_{w}\circ\pi(q_{J}(a_{xu,yv}))=\Delta_{w}(q_{w}(q_{I}(a_{x,y})))\Delta_{w}(q_{w}(\nu_{x}(q_{J}(a_{u,v}))),$

where

(5.5) $\Delta_{w}(q_{w}(q_{I}(a_{x,y})))=\sum_{z\in
X}q_{w}(q_{I}(a_{x,z}))\otimes q_{w}(q_{\mathbb{P}}(a_{z,y})),$

and

$\displaystyle\Delta_{w}(q_{w}(\nu_{x}(q_{J}(a_{u,v})))$
$\displaystyle=\sum_{z^{\prime}\in X}(q_{w}\otimes
q_{w})\big{(}(\nu_{x}\otimes\nu_{z^{\prime}})(\Delta_{J}(q_{J}(a_{u,v})))(q_{I}(a_{x,z^{\prime}})\otimes
1)\big{)}$ (5.6) $\displaystyle=\sum_{z^{\prime}\in X}\sum_{\beta\in
X^{m}}q_{w}\big{(}\nu_{x}(q_{J}(a_{u,\beta}))q_{I}(a_{x,z^{\prime}})\big{)}\otimes
q_{w}\big{(}\nu_{z^{\prime}}(q_{J}(a_{\beta,v}))\big{)}.$

A typical summand in the product of the expressions in (5.5) and (5.6) is

$\displaystyle
q_{w}\big{(}q_{I}(a_{x,z})\nu_{x}(q_{J}(a_{u,\beta}))q_{I}(a_{x,z^{\prime}})\big{)}\otimes
q_{w}\big{(}q_{\mathbb{P}}(a_{z,y})\nu_{z^{\prime}}(q_{J}(a_{\beta,v}))\big{)}$
$\displaystyle\qquad\qquad\qquad=q_{w}\big{(}\nu_{x}(q_{J}(a_{u,\beta}))q_{I}(a_{x,z})q_{I}(a_{x,z^{\prime}})\big{)}\otimes
q_{w}\big{(}q_{\mathbb{P}}(a_{z,y})\nu_{z^{\prime}}(q_{J}(a_{\beta,v}))\big{)}$
$\displaystyle\qquad\qquad\qquad=\delta_{z,z^{\prime}}q_{w}\big{(}\nu_{x}(q_{J}(a_{u,\beta}))q_{I}(a_{x,z})\big{)}\otimes
q_{w}\big{(}q_{\mathbb{P}}(a_{z,y})\nu_{z}(q_{J}(a_{\beta,v}))\big{)}$
$\displaystyle\qquad\qquad\qquad=\delta_{z,z^{\prime}}q_{w}\big{(}q_{I}(a_{x,z})\nu_{x}(q_{J}(a_{u,\beta}))\big{)}\otimes
q_{w}\big{(}q_{\mathbb{P}}(a_{z,y})\nu_{z}(q_{J}(a_{\beta,v}))\big{)}.$

Hence

$\displaystyle\Delta_{w}\circ\pi(q_{J}(a_{xu,yv}))$ $\displaystyle=\sum_{z\in
X}q_{w}(q_{I}(a_{x,z}))\otimes q_{w}(q_{\mathbb{P}}(a_{z,y}))$
$\displaystyle=\sum_{z\in X}\sum_{\beta\in
X^{m}}q_{w}\big{(}q_{I}(a_{x,z})\nu_{x}(q_{J}(a_{u,\beta}))\big{)}\otimes
q_{w}\big{(}q_{I}(a_{z,y})\nu_{x}(q_{J}(a_{\beta,v}))\big{)}$
$\displaystyle=(\pi\otimes\pi)\circ\Delta_{J}(q_{J}(a_{xu,yv})),$

and it follows that $\Delta_{w}\circ\pi=(\pi\otimes\pi)\circ\Delta_{J}$. ∎

######  Example 5.8.

An immediate consequence of Theorem 5.7 is that $A_{\mathbb{P}}$ is
noncommutative whenever $\mathbb{P}$ is a noncommutative quantum subgroup of
$\mathbb{A}_{1}$. A class of such examples comes from Banica and Bichon’s [1,
Theorem 1.1], in which they classify all the quantum subgroups $\mathbb{P}$ of
$\mathbb{A}_{1}$ for $|X|=4$; the corresponding list of quantum groups
$A_{\mathbb{P}}$ gives us a list of potentially interesting self-similar
quantum groups for further study.

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