Self-similar quantum groups

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.


Introduction
Self-similar groups are a class of groups acting faithfully on an infinite rooted homogeneous tree X * .In particular, given an automorphism g ∈ Aut(X * ) and a vertex w ∈ X * , by identifying wX * with g(w)X * , we get an automorphism g| w ∈ Aut(X * ) which is uniquely determined by the identity The automorphism g| w is called the restriction of g by w, and a subgroup G ≤ 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 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 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 Date: February 6, 2023.Brownlowe was supported by the Australian Research Council grant DP200100155, and both authors were supported by the Sydney Mathematical Research Institute.
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 A X of the homogeneous rooted tree X * , and then identifying the quantum analogue of the restriction maps g → g| w for g ∈ Aut(X * ), w ∈ X * .We then define a self-similar quantum group to be any quantum subgroup A of A X for which the restriction maps factor through the quotient map A X → A. We characterise self-similar quantum groups in terms of a certain homomorphism A ⊗ C(X) → C(X) ⊗ 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 A X which we prove is the quantum automorphism group of the homogeneous rooted tree X * .The C * -algebra A X is a noncommutative, infinitedimensional 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 ⊗ C(X) → C(X) ⊗ A, mimicking the fact that classical self-similar action are governed by the maps 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 A d of A X , which are the quantum automorphism groups of the finite subtrees X [d] of X * of depth d.To each quantum subgroup P of A d , we construct a quantum subgroup A 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 A 1 of A X ), and we prove that every A P coming from a quantum subgroup P of A 1 is canonically isomorphic to the free wreath product A P * w P.

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, Φ) where A is a unital C * -algebra and Φ : We call Φ the comultiplication and ( 1) is called coassociativity.Remark 2.2.It is proved in [14] that (A, Φ) is a compact quantum group if and only if there is a family of matrices {a λ = (a λ i,j ) ∈ M d λ (A) : λ ∈ Λ} for some indexing set Λ such that (2) a λ and its transpose (a λ ) T are invertible elements of M d λ (A) for every λ ∈ Λ, (3) the * -subalgebra A of A generated by the entries {a λ i,j : Example 2.3.A key example for us is Wang's quantum permutation groups (A s (n), Φ) from [11].Here, n is a positive integer, and A s (n) is the universal C * -algebra generated by elements a ij , 1 ≤ i, j ≤ n, satisfying Definition 2.5.Let (A, Φ) be a compact quantum group.A Woronowicz ideal is an ideal I of A such that Φ(I) ⊆ ker(q ⊗ q), where q is the quotient map A → A/I.Then (A/I, Φ ′ ), where Φ ′ : A/I → A/I ⊗ A/I satisfies Φ ′ • q = (q ⊗ q) • Φ is a compact quantum group called a quantum subgroup of (A, Φ).Definition 2.6.A (left) coaction of a compact quantum group (A, Φ) on a unital C *algebra B is a unital * -homomorphism α : We refer to (1) as the coaction identity and (2) is known as the Podleś condition.

Quantum automorphisms of a homogeneous rooted tree
In this section we introduce a compact quantum group 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 ≥ 0 we write X n for all the words in X of length n, and we then the tree X * can be identified with n≥ X n , where X 0 = {∅} and ∅ is the root of the tree.Definition 3.1.Let X be a finite set and let (A, Φ) be a compact quantum group.An action of A on the homogeneous rooted tree X * is a system of left coactions, such that for any m < n the diagram We now define the main object of interest in this section, the C * -algebra 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 QAut Π of a locally finite connected graph Π.A straightforward argument shows that A X is QAut Π for Π 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 A X to be the universal C * -algebra generated by elements {a u,v : u, v ∈ X n , n ≥ 0} subject to the following relations: (1) (iii) Repeated applications of (3) from Definition 3.2 show that for all u, u ′ , v, v ′ , w ∈ X n , n ∈ N, we have and that for all , n ∈ N, and x, y ∈ X we have We will freely use these two identities without comment throughout the rest of the paper.
Theorem 3.4.The C * -algebra A X is a compact quantum group with comultiplication for all u, v ∈ X n and n ≥ 1 Proof.To see that ∆ exists, it's enough to show that the elements So by the universal property of A X there is a homomorphism ∆ : For coassociativity, we have Finally, we show that that the set of matrices satisfy the conditions of Definition 2.2.Conditions (1) and (3) are clear.For (2) we show that given any n ≥ 1 the matrix a n is invertible with inverse given by (a n ) T .Given u, v ∈ X n we have We now show that (A X , ∆) 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
is an action of a compact quantum group (A, Φ) on X * then there is a quantum group homomorphism π : Proof.For any n ≥ 1, the elements for each w ∈ X n are mutually orthogonal projections and satisfy and so each γ n satisfies the coaction identity.
For a fixed v ∈ X n we have

Multiplying by any element
) and hence the required density is satisfied.Finally, fix m < n and w ∈ X m .Then and so the collection γ = (γ n ) ∞ n=1 defines an action of (A X , ∆) on the homogeneous rooted tree The coaction identity for α n says that We claim that the collection {b u,v : u, v ∈ X n , n ≥ 0} ⊆ 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 ≥ 1.Since for any n ≥ 1, α n defines a coaction of (A, Φ) on C(X n ), [11,Theorem 3.1] says that the elements {b u,v : u, v ∈ 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 ∈ X n we have For any u ∈ X n and x ∈ X we have and hence It follows that b ux,vy ≤ b u,v for any x, y ∈ X.Therefore, for any u, v ∈ X n and x ∈ X we have Therefore, the universal property of A X provides a homomorphism π : 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 , which is non-commutative and infinite dimensional.Define the matrix Then these elements satisfy the relations in Definition 3.2 and hence there is a surjective homomorphism A X → B. Since B is non-commutative and infinite-dimensional so is A X .
Remark 3.8.The group 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 and since the orbit of any u ∈ X * is finite, each of these open subgroups is closed and hence compact.Cosets of these subgroups are of the form Then {G u,v : u, v ∈ X * } is a basis of compact open sets for the topology on Aut(X * ).It follows that the indicator functions f u,v := 1 Gu,v span a dense subset of C(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(Aut(X * )) then implies that it is the abelianisation of A X .

Self-similarity
If g ∈ Aut(X * ) and x ∈ X, the restriction g| x is the unique element of Aut(X * ) satisfying g That is, whenever g ∈ G and x ∈ X, the restriction g| x is an element of G.With the topology inherited from Aut(X * ), the restriction map G → G : g → 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 Aut(X * ) is self-similar.
To have a reasonable notion of self-similarity for quantum subgroups of A X , we need to understand how restriction manifests itself in the function algebra C(Aut(X * )).Given x ∈ X and u, v ∈ X n we have and hence the corresponding indicator functions satisfy This formula motivates the following result.Proposition 4.1.For each x ∈ X there is a homomorphism ρ We illustrate the formula for a restriction map in Figure 1  Proof of Proposition 4.1.Fix x ∈ X.We show that the elements A similar calculation shows z∈X b uz,vy = b u,v .Hence there is a homomorphism ρ x with the desired formula.
Remark 4.2.We define ρ ∅ to be the identity homomorphism A X → A X , and for w = shows that for all u, v ∈ X n we have a zu,wv .
Remark 4.3.A similar argument to the one in the proof of Proposition 4.1 shows that for each x ∈ X there is a homomorphism σ where κ is the coinverse.
We can now state the main definition of the paper.
Definition 4.4.We call ρ w the restriction by w.A quantum subgroup A of A X is selfsimilar if for each x ∈ X the restriction ρ x factors through the quotient map q : A X → A; that is, if there exists a homomorphism ρ x : A → A such that the diagram To motivate the main result of this section, let G be a group.To construct a selfsimilar action of G on X * , it suffices to have a function f : G × X → X × G such that f (e, x) = (x, e) for all x ∈ X, and such that the following diagram commutes This data allows us to define an action of G on X * , which is self-similar with g • x and g| x the unique elements of X and G satisfying (g • 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 id A for the identity homomorphism on a C * -algebra A, and for n ≥ 1 write 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, Φ) is a compact quantum group equipped with a unital * - and Then (A, Φ) acts on the homogeneous rooted tree X * and moreover the image of A X , under the homomorphism π : A X → A from Proposition 3.6, is a self-similar compact quantum group.
Proof.We begin by defining an action of (A, Φ) on Then α 1 is clearly unital and the coaction identity and Podleś condition for α 1 follow from (4.2) and (4.3).Now inductively define ) for n ≥ 1, where we are supressing the canonical isomorphism C(X n+1 ) ∼ = C(X) ⊗ C(X n ).Again, α n+1 is clearly unital whenever α n is.If we assume α n satisfies the coaction identity, then and so α n+1 also satisfies the coaction identity.Since α 1 is a coaction, we see that α n satisfies the coaction identity for any n ≥ 1.
To see that each α n satisfies the Podleś condition, we argue by induction.We know it is satisfied for n = 1.Suppose for some n ≥ 1 that Fix a spanning element a ⊗ p u ⊗ p x ∈ A ⊗ C(X n+1 ) where u ∈ X n and x ∈ X.By the inductive hypothesis we can approximate By definition of α n , for any f ∈ C(X n ) we have So we can write (4.4) as Since ψ is unital, we have which can be approximated using the induction hypothesis by Finally, applying the Podleś condition for α 1 we can approximate Combining these approximations we can write where As in the proof of Proposition 3.6 , for any n ≥ 1 and u, v ∈ X n we will let b u,v ∈ A be the unique elements satifsying We know from the same proof that for any n ≥ 1 and v ∈ X n we have So we have that (α n ) ∞ n=1 defines an action of (A, Φ) on X * .
Finally, let π : A X → A be the homomorphism from Proposition 3.6.We have π(a u,v ) = b u,v for any u, v ∈ X n and n ≥ 1.For each x ∈ X define a homomorphism ρx : where a ∈ A. For any u ∈ X n we have On the other hand, we know and by comparing tensor factors we see that ρx (b u,v ) = y∈X b yu,xv .Hence, the diagram ρx commutes, and so π(A X ) ⊆ A is a self-similar quantum group.
Proposition 4.6.The following are equivalent (1) (A, Φ) is a quantum self-similar group, and (2) (A, Φ) is a quantum subgroup of (A X , ∆) and there is a homomorphism ψ : C(X)⊗ A → A ⊗ C(X) satisfying the hypotheses of Theorem 4.5.
Proof.Theorem 4.5 is the implication (2) =⇒ (1).To see (1) =⇒ (2) suppose (A, Φ) is a quantum self-similar group.By definition there is a surjective quantum group morphism q : A X → A. It is routine to check that there is a homomorphism ψ : Given u, v ∈ X n we have and so ψ satisfies (4.2).For (4.3) notice that for any q(a) ∈ A and z ∈ X we have )) takes a generator a u,v to the indicator function f u,v defined in Remark 3.8.For a function f ∈ C(Aut(X * )) and x ∈ X the restriction homomorphism ρx satisfies ρx (f )(g) = f (g| x ), for any g ∈ G.
5. Finitely constrained self-similar quantum groups 5.1.Classical finitely constrained self-similar groups.Fix d ≥ 1, and let X [d] = k≤d X k be the finite subtree of X * of depth d.The group of automorphism Aut(X [d] ) is a quotient of Aut(X * ), and the quotient map is given by restriction to the finite subtree.We write r d : Aut(X * ) → Aut(X [d] ) for this restriction map.
Fix a subgroup P ≤ Aut(X [d] ).Define By the properties of restriction, if g, h ∈ G P , then for any w ∈ X * 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 A d ⊆ A X generated by the elements {a u,v : |u| = |v| ≤ d}.Since ∆ : ) of continuous functions on the finite group Aut(X [d] ).
Definition 5.1.Suppose P is a quantum subgroup of A d , where P = A d /I.Denote by q I : A d → P the quotient map; so I = ker(q I ).We denote by J the smallest closed 2-sided ideal of A X generated by {ρ w (I) : w ∈ X * }, and by A P the quotient A P := A X /J.In the next result we prove that A P is a self-similar quantum group, and we call it a finitely constrained self-similar quantum group.
Proposition 5.2.Each A P is a self-similar quantum group.
To prove Proposition 5.2 we need two lemmas.Recall that for g, h ∈ Aut(X * ), w ∈ X * we have (gh)| w = g| h•w h| w .
In the first lemma, we establish an analogous relationship between the comultiplication ∆ on A X and the restriction maps ρ w .
Lemma 5.3.For any n ≥ 1, w ∈ X n and a ∈ A X we have Proof.Let a u,v be a generator of A X , with |u| = |v| = k ≥ 0. Then To see that this formula extends to A X it's enough to show that for any w ∈ X * the map is linear and multiplicative.Linearity is clear, and multiplicativity follows from the orthogonality of the projections 1 ⊗ a y,w and 1 ⊗ a z,w for y = z.
Lemma 5.4.Consider the quotient maps q I : A d → A d /I and q J : A X → A X /J.Then for any n ≥ 1 and y, w ∈ X n ker(q I ⊗ q I ) ⊆ ker((q J • ρ y ) ⊗ (q J • ρ w )).
Proof.By definition of J we have I ⊆ J • ρ w for any w ∈ X * .Therefore there is a commuting diagram as required.
Proof of Proposition 5.2.To see that A P is a compact quantum group, it suffices to show that J is a Woronowicz ideal.In other words, we need to show that ∆(J) ⊂ ker(q J ⊗ q J ) where q J : A X → A X /J =: A P is the quotient map.Since J is generated as an ideal by w∈X * ρ w (I) it's enough to show that (q J ⊗ q J )(∆ • ρ w (i)) = 0 for any i ∈ I and w ∈ X * .Because I is a Woronowicz ideal we know that ∆(i) ∈ ker(q I ⊗ q I ).Then by Lemmas 5.3 and 5.4 we have Finally, A P is self-similar since by definition of J we have ρ w (J) ⊂ J for any w ∈ X * .
5.3.Free wreath products.It is well known that for any d ≥ 1 the group Aut(X [d+1] ) is isomorphic to the wreath product Aut(X [d] ) ≀ Sym(X).Since Aut(X * ) is the inverse limit over d of the groups Aut(X [d] ), it can be thought as the infinitely iterated wreath product . . .≀ Sym(X) ≀ Sym(X).It follows that Aut(X * ) ∼ = Aut(X * ) ≀ Sym(X).More generally, it is shown in [4] that if P ≤ Sym(X) = Aut(X [1] ), then the finitely constrained self-similar group G P is the infinitely iterated wreath product . . .≀ P ≀ 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 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 A s (n).In this section we formally extend this definition to take free wreath products by any quantum subgroup of A s (n), and we prove that the finitely constrained self-similar quantum group A P induced from a quantum subgroup P of A s (n) is a free wreath product by P. We begin by recalling the definition of the free wreath product from [3]; note that we use our notation A 1 instead of A s (|X|).Definition 5.5.Let X be a set of at least two elements.Let (A, Φ) be a compact quantum group, and P a quantum subgroup of A 1 .For each x ∈ X, we denote by ν x the inclusion of A in the free product C * -algebra ( * x∈X A) * P. The free wreath product of A by P is the quotient of ( * x∈X A) * P by the two-sided ideal generated by the elements ν x (a)q I (a x,y ) − q I (a x,y )ν x (a), x, y ∈ X, a ∈ A.
The resulting C * -algebra is denoted by A * X,w P, and the quotient map is denoted by q w .If X is understood, we typically just write A * w P. Theorem 5.6.Let (A, Φ) be a compact quantum group, and P a quantum subgroup of A 1 .The free wreath product A * w P from Definition 5.5 is a compact quantum group with comultiplication Φ w satisfying Φ w (q w (q I (a x,y ))) = z∈X q w (q I (a x,z )) ⊗ q w (q I (a z,y )) (5.1) for each x, y ∈ X and a ∈ A.
Proof.Since I is a Woronowicz ideal, we have ∆| I ⊆ ker(q I ⊗ q I ), and so the map (q I ⊗ q I ) • ∆ A 1 descends to a map φ : P → P ⊗ P ⊆ (( * x∈X A) * P) ⊗2 .
Then (q w ⊗ q w ) • φ : P → (A * w P) ⊗2 satisfies (q w ⊗ q w ) • φ(q I (a x,y )) = z∈X q w (q I (a x,z )) ⊗ q w (q I (a z,y )) for all x, y ∈ X.
We claim that φ x is a homomorphism.To see this, let {a λ = (a λ i,j ) ∈ M d λ (A) : λ ∈ Λ} is be a family of matrices satisfying ( 1 and then since Since A is dense in A, it follows that φ x is a homomorphism on A. The universal property of ( * x∈X A) * P now gives a homomorphism Φ : ( * x∈X A) * P → (A * w P) ⊗2 satisfying Φ(q I (a x,y )) = z∈X q w (q I (a x,z )) ⊗ q w (q P (a z,y )), Φ(ν x (a)) = z∈X (q w ⊗ q w ) (ν x ⊗ ν z )(Φ(a))(q I (a x,z ) ⊗ 1) .
It follows that Φ(ν x (a)q I (a x,y )) = Φ(q I (a x,y )ν x (a)) for each a ∈ A, x, y ∈ X, and hence Φ descends to the desired Φ w : A * w P → (A * w P) ⊗2 .
We know that (1) is satisfied for the matrix a X .For each z ∈ X we have z ′ ∈X q w (q I (a x,z a x,z ′ )) ⊗ q w (q I (a z ′ ,y )) = 1≤k≤d λ q w (ν x (a λ i,k )) ⊗ q w (ν z (a λ k,j )) (q w (q I (a x,z )) ⊗ q w (q I (a z,y ))) = 1≤k≤d λ q w (ν x (a λ i,k )q I (a x,z )) ⊗ q w (ν z (a λ k,j )q I (a z,y )).
We also have (a X (a X ) T ) x,y = z∈X a X x,z a X y,z = q w q I z∈X a x,z a y,z = δ 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 (λ,X) : λ ∈ Λ} ∪ {a X } span a dense subset of A * X,w P. For each x, y ∈ 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 ∈ X, λ ∈ Λ and 1 ≤ i, j ≤ d λ we have y∈X a (λ,X) (i,x),(j,y) = q w ν x (a λ i,j )q I y∈X a x,y = q w (ν x (a λ i,j )), and so each q w (ν x (a λ i,j )) is in the span of the entries.The claim follows, and so (3) holds.
Theorem 5.7.Let A P be a finitely-constrained self-similar quantum group in the sense of Definition 5.1.There is a unital quantum group isomorphism π : A P → A P * w P satisfying (5.3) π(q J (a xu,yv )) = q w q I (a x,y )ν x (q J (a u,v )) for all x, y ∈ X, u, v ∈ X m , m ≥ 0.
Proof.We define b ∅,∅ to be the identity of A P * w P, and for each x, y ∈ X, u, v ∈ X m , m ≥ 0, b xu,yv := q w q I (a x,y )ν x (q J (a u,v )) .We claim that this gives a family of projections satisfying (1)-(3) of Definition 3.2.Condition (1) holds by definition.We have b * xu,yv = q w ν x (q J (a * u,v ))q I (a * x,y ) = q w ν x (q J (a u,v ))q I (a x,y ) = q w q I (a x,y )ν x (q J (a u,v )) = b xu,yv and b 2 xu,yv = q w q I (a x,y )ν x (q J (a u,v ))q I (a x,y )ν x (q J (a u,v )) = q w q I (a 2 x,y )ν x (q J (a 2 u,v )) = b xu,yv .So (2) holds.For each w ∈ X we have z∈X b xuw,yvz = z∈X q w q I (a x,y )ν x (q J (a uw,vz )) = q w q I (a x,y )ν x q J z∈X a uw,vz = q w q I (a x,y )ν x (q J (a u,v )) = b xu,yv , and z∈X b xuz,yvw = z∈X q w q I (a x,y )ν x (q J (a uz,vw )) = q w q I (a x,y )ν x q J z∈X a uz,vw = q w q I (a x,y )ν x (q J (a u,v )) = b xu,yv , and hence (3) holds.This proves the claim, and hence the universal property of A X now gives a homomorphism π : A X → A P * w P satisfying π(a xu,yv ) = q w q I (a x,y )ν x (q J (a u,v )) , for all x, y ∈ X, u, v ∈ X m , m ≥ 0.
We now claim that J is contained in ker π.To see this, fix w ∈ X n , with w = w 1 w ′ for w 1 ∈ X, w ′ ∈ X n−1 .We first prove the claim that for each where k ≥ 1 and each pair u i , v i ∈ X m i for some m i ≥ 0, we have (5.4) π(ρ w (x k )) = y∈X q w q I (a y,w 1 )ν y (q J (ρ w ′ (x k ))) .

It follows that
π(ρ w (x k+1) ) = π(ρ w (x k )) π(ρ w (a u k+1 ,v k+1 )) = y∈X q w q I (a y ′ ,w 1 )ν y (q J (ρ w ′ (x k a u k+1 ,v k+1 ))) , 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 A X , it follows that π(ρ w (a)) = y∈X q w q I (a y,w 1 )ν y (q J (ρ w ′ (a))) for all a ∈ A X .Now, if a ∈ I, then ρ w ′ (a) ∈ J = ker q J , and hence the above equations shows that π(ρ w (a)) = 0. Hence ρ w (a) ∈ ker π for all w ∈ X n and a ∈ I, and hence J ⊆ ker π.This means π descends to a homomorphism π : A P → A P * w P satisfying π(q J (a xu,yv )) = q w q I (a x,y )ν x (q J (a u,v )) for all x, y ∈ X, u, v ∈ X m , m ≥ 0. We now show that π is an isomorphism by finding an inverse.For each x ∈ X consider the homomorphism q J • σ x : A X → A P , where σ x is the homomorphism from Remark 4.3.Since σ x = κ • ρ x • κ, and we know from [10, Remark 2.10] that κ(J) ⊆ J, it follows that q J • σ x descends to a homomorphism φ x : A P → A P satisfying φ x (q J (a u,v )) = q J (σ x (a u,v )) = y∈X q J (a xu,yv ), for all u, v ∈ X m , m ≥ 0.
Each φ x , and the map q I (a) → q J (a) from P to A P , now allow us to apply the universal property of the free product ( * x∈X A P ) * P to get a homomorphism φ : ( * x∈X A P ) * P → A P satisfying φ • ν x = φ x for each x ∈ X, and φ(q I (a)) = q J (a) for all a ∈ A 1 ⊆ A X .We claim that φ ν x (q J (a u,v ))q I (a x,y ) − q I (a x,y )ν x (q J (a u,v ) = 0, for each x ∈ X, u, v ∈ X m , m ∈ N. We have φ ν x (q J (a u,v ))q I (a x,y ) − q I (a x,y )ν x (q J (a u,v ) = φ x (q J (a u,v ))q J (a x,y ) − q G (a x,y )φ x (q J (a u,v )) = y∈X q J (a xu,yv )q J (a x,y ) − y ′ ∈X q J (a x,y )q J (a xu,y ′ v ) = q J (a xu,yv ) − q J (a xu,yv ) = 0.
It follows that φ descends to a homomorphism φ : A P * w P → A P satisfying φ(q w (ν x (q J (a u,v )))) = q J (σ x (a u,v )) = y∈X q J (a xu,yv ) for all x ∈ X, u, v ∈ X m , m ≥ 0, and φ(q w (q I (a x,y ))) = q J (a x,y ) for all x, y ∈ X.
We claim that π and φ are mutually inverse.For x, y ∈ X, u, v ∈ X m , m ≥ 0, we have φ(π(q J (a xu,yv ))) = φ q w q I (a x,y )ν x (q J (a u,v )) = q J (a x,y ) y∈X q J (a xu,yv ) = q J (a xu,yv ), and it follows that φ • π is the identity on A P .For x ∈ X, u, v ∈ X m , m ≥ 0, we have π(φ(q w (ν x (q J (a u,v ))))) = π y∈X q J (a xu,yv ) = y∈X q w q I (a x,y )ν x (q J (a u,v )) = q w q I y∈X a x,y ν x (q J (a u,v )) = q w (ν x (q J (a u,v ))), and for all x, w ∈ X we have π(φ(q w (q I (a x,y )))) = π(q J (a x,y )) = q w (q I (a x,y )).
b u,v = b u,v w∈X n y∈X b ux,wy = y∈X b ux,vy .Likewise for any y ∈ X we have b u,v = x∈X b ux,vy and (3) holds.

vy = z∈X a uz,vx .
For each d ∈ N we denote by A d the subalgebra of A X generated by {a u,v : u, v ∈ X d }.Note that A 1 is the Wang's quantum permutation group and a i , c k ∈ A. Thus α n+1 satisfies the Podleś condition and so by induction α n satisfies the Podleś condition for every n ≥ 1.It remains to show that α n