Quantum isometry group of dual of finitely generated discrete groups- $\textrm{II}$

As a contribution of the programme of Goswami and Mandal (2014), we carry out explicit computations of $\mathbb{Q}(\Gamma,S)$, the quantum isometry group of the canonical spectral triple on $C_{r}^{*}(\Gamma)$ coming from the word length function corresponding to a finite generating set S, for several interesting examples of $\Gamma$ not covered by the previous work Goswami and Mandal (2014). These include the braid group of 3 generators, $\mathbb{Z}_4^{*n}$ etc. Moreover, we give an alternative description of the quantum groups $H_s^{+}(n,0)$ and $K_n^{+}$ (studied in Banica and Skalski (2012), Banica and Skalski (2013)) in terms of free wreath product. In the last section we give several new examples of groups for which $\mathbb{Q}(\Gamma)$ turn out to be doubling of $C^*(\Gamma)$.


Introduction
It is a very important and interesting problem in the theory of quantum groups and noncommutative geometry to study 'quantum symmetries' of various classical and quantum structures. S.Wang initiated by studying quantum permutation group of finite sets and quantum automorphism groups of finite dimensional algebras. Later on, a number of mathematicians including Wang, Banica, Bichon and others ( [22], [1], [12]) developed a theory of quantum automorphism groups of finite dimensional C * algebras as well as quantum isometry groups of finite metric spaces and finite graphs. In [17] Goswami extended such constructions to the set up of possibly infinite dimensional C * algebras, and more interestingly, that of spectral triples a la Connes [14], by defining and studying quantum isometry groups of spectral triples. This led to the study of such quantum isometry groups by many authors including Goswami, Bhowmick, Skalski, Banica, Bichon, Soltan, Das, Joardar and others. In the present paper, we are focusing on a particular class of spectral triples, namely those coming from the word-length metric of finitely generated discrete groups with respect to some given symmetric generating set. There have been several articles already on computations and study of the quantum isometry groups of such spectral triples, e.g [11], [21], [16], [6], [4] and references therein. In [18] together with Goswami we also studied the quantum isometry groups of such spectral triples in a systematic and unified way. Here we compute Q(Γ, S) for more examples of groups including braid groups, Z 4 * Z 4 · · · * Z 4 n copies etc.
The paper is organized as follows. In Section 2 we recall some definitions and facts related to compact quantum groups, free wreath product with quantum permutation group and quantum isometry group of spectral triples defined by Bhowmick and Goswami in [9]. This section also contains the doubling procedure of a compact quantum group say Q, which is denoted by D(Q). In Section 3 we compute Q(Γ, S) for braid group with 3 generators, as a C * algebra it turns out to be four direct copies of the group C * algebra ( in fact, it is precisely doubling of doubling of the group C * algebra as a Hopf algebra). Section 4 contains an interesting description of the quantum groups H + s (n, 0) and K + n (studied in [6], [4]) in terms of free wreath product. Moreover, Q(Γ, S) is computed for Γ = Z 4 * Z 4 · · · * Z 4 n copies . In the last section we present more examples of groups as in ( [16], [21], Section 5 of [18]) where Q(Γ, S) turn out to be the doubling of C * (Γ).

Preliminaries
First of all, we fix some notational conventions which will be useful for the rest of the paper. Throughout the paper, the algebraic tensor product and the spatial (minimal) C * tensor product will be denoted by ⊗ and⊗ respectively. We'll use the leg-numbering notation, consider the multiplier algebra M(K(H)⊗Q), it has two natural embeddings into M(K(H)⊗Q⊗Q). The first one is obtained by extending the map x → x ⊗ 1. The second one is obtained by composing this map with the flip on the last two factors. We will write ω 12 and ω 13 for the images of an element ω ∈ M(K(H)⊗Q) under these two maps respectively.

Compact quantum groups and free wreath product
Let us recall some definitions about compact quantum groups, its action on a C * algebra and free wreath product with quantum permutation group. Definition 2.1 A Compact quantum group (CQG for short) is a pair (Q, ∆) where Q is a unital C * algebra where ∆ : Q → Q⊗Q is a unital C * homomorphism satisfying two conditions : 1.(∆ ⊗ id)∆ = (id ⊗ ∆)∆ (co-associativity ). 2. Each of the linear spans of ∆(Q)(1 ⊗ Q) and that of ∆(Q)(Q ⊗ 1) is norm dense in Q⊗Q.
Here << ., . >> is the C * valued inner product and H⊗Q denotes the completion of H ⊗ Q with respect to the natural Q valued inner product . Given such a unitary representation we have a unitary elementŨ belonging to M(K(H)⊗Q)

Definition 2.3
We say that CQG (Q, ∆) acts on a unital C * algebra B if there is a unital C * homomorphism (called action) α : B → B⊗Q satisfying the following : Given two CQG's Q 1 , Q 2 the free product Q 1 ⋆ Q 2 admits the natural CQG structure equipped with the following universal property (for more details see in [23]). Proposition 2.4 (i) The canonical injections, say i 1 , i 2 from Q 1 and Q 2 to Q 1 ⋆ Q 2 is CQG morphism. (ii) Given any CQG C and morphisms π 1 : Q 1 → C and π 2 : Q 2 → C there always exists a unique morphism denoted by π := π 1 * π 2 from Q 1 ⋆ Q 2 to C satisfying π • i k = π k for k = 1, 2. Now we recall the definition of free wreath product by the quantum permutation group (see in [13]). We denote by ν i the canonical homomorphism ν i : Q → Q * N .
Definition 2.5 The quantum permutation group denoted by (C(S + N ), ∆) is the universal C * algebra generated by N 2 elements t ij such that the matrix ((t ij )) is unitary and The coproduct ∆ is given by ∆(t ij ) = Σ N k=1 t ik ⊗ t kj and it admits the CQG structure. For further details see in [22]. Definition 2.6 Let Q be a compact quantum group and N > 1. The free wreath product of Q by the quantum permutation group C(S + N ) is the quotient of the C * algebra Q * N ⋆ C(S + N ) by the two sided ideal generated by the elements where ((t ij )) is the matrix coefficients of the quantum permutation group C(S + N ) (see in [22]) . This is denoted by Q ⋆ w C(S + N ). Furthermore, it admits the CQG structure. The comultiplication satisfies (2) ).

The definition of Q(Γ, S)
First of all, we are defining the quantum isometry group of spectral triples defined by Bhowmick and Goswami in [9].  Now we discuss the special case of our purpose. Let Γ be a finitely generated discrete group with generating set S = {a 1 , a −1 1 , a 2 , a −1 2 , · · a k , a −1 k }. We make the convention of choosing the generating set to be symmetric, i.e. a i ∈ S implies a −1 i ∈ S ∀ i . In case some a i has order 2, we include only a i , i.e. not count it twice. Define length function on the group as l(g) = min {r ∈ N, g = h 1 h 2 · · · h r } where h i ∈ S i.e. for each h i = a j or a −1 j for some j. Notice that S = {g ∈ Γ, l(g) = 1}, using this length function we can define a metric on Γ by d(a, b) = l(a −1 b) ∀ a, b ∈ Γ. This is called the word metric. Now consider the algebra C * r (Γ), which is the C * completion of the group ring CΓ viewed as a subalgebra of B(l 2 (Γ)) in the natural way via left regular representation. We define the Dirac operator D Γ (δ g ) = l(g)δ g . In general, D Γ is an unbounded operator.
Here, δ g is the vector in l 2 (Γ) which takes value 1 at the point g and 0 at all other points. Natural generators of the algebra CΓ (images in the left regular representation ) will be denoted by λ g , i.e. λ g (δ h ) = δ gh . It is easy to check that (CΓ, l 2 (Γ), D Γ ) is a spectral triple. Now take A = C * r (Γ), A ∞ = CΓ, H = l 2 (Γ) and D = D Γ as before, δ e is cyclic separating vector for CΓ then QISO + (CΓ, l 2 (Γ), D Γ ) exists by Theorem 2.8 . As the object depends on the generating set of Γ it is denoted by Q(Γ, S). Most of the time we denote it by Q(Γ) if S is understood from the context. Now as in [11] Its action α (say) on where the matrix [q γ,γ ′ ] γ,γ ′ ∈S is called the fundamental unitary in M card(S) (Q (Γ, S)). Now we fix some notational conventions which will be useful in later sections. Note that the action α is of the form

From this we get the unitary corepresentation
The coefficients A ij and A * ij 's generate a norm dense subalgebra of Q(Γ, S). In fact, it is easy to see that Q(Γ, S) is the CQG generated by A ij as above subject to the relation U is a unitary and α given above * homomorphism on C * r (Γ). we also note that the antipode of Q(Γ, S) maps u ij to u * ji .

Q(Γ) as the doubling of certain quantum groups
In this subsection we briefly recall the doubling procedure of a compact quantum group from [16], [20]. Let (Q, ∆) be a CQG with a CQG-automorphism θ such that θ 2 = id. The doubling of this CQG, say (D(Q),∆) is given by D(Q) := Q⊕Q (direct sum as a C * algebra), and the coproduct is defined by the following, where we have denoted the injections of Q onto the first and second coordinate in D(Q) by ξ and η respectively, i.e. ξ(a) = (a, 0), η(a) = (0, a), (a ∈ Q).
Below we give some sufficient conditions for the quantum isometry group to be the doubling of the certain algebras. For this, it is convenient to use a slightly different notational convention: Proposition 2.9 Let Γ be a group with k generators {a 1 , a 2 , · · a k } (say) and θ be an automorphism of order 2 of the group algebra which gives a permutation σ on the set {1, 2, ··, 2k − 1, 2k}. Now assume the following : Then Q(Γ) is doubling of the group algebra (i.e. Q(Γ) ∼ = D(C * (Γ))) corresponding to the given automorphism θ. Moreover the fundamental unitary takes the following form The proof is given in [18]. Now we give a sufficient condition for Q(Γ) to be D(D(C * (Γ))).

Proposition 2.10
Let Γ be a group with k generators {a 1 , a 2 , · · a k } (say) and θ 1 , θ 2 , θ 3 be the automorphisms of order 2 of the group algebra which gives the permutations σ 1 , σ 2 , σ 3 respectively on the set {1, 2, ··, 2k − 1, 2k}. Now assume the following : Then Q(Γ) is doubling of D(C * (Γ)) corresponding to the given automorphisms. Moreover the fundamental unitary takes the following form The proof is very similar to the Proposition 2.9, thus omitted. We end the discussion of Section 2 with the following easy observation.

Proposition 2.11
If U V = 0 for two normal elements in a C * algebra then The proof is very straightforward, hence omitted. This will be helpful in later sections.

QISO computation for the braid group
In this section we will compute the quantum isometry group of the braid group with 3 generators. The group has the presentation Γ =< a, b, c|ac = ca, aba = bab, cbc = bcb > .
Theorem 3.1 Let Γ be the braid group with above presentation. Then Q(Γ, S) will be D(D(C * (Γ)), this means as a C * algebra it is same as four direct copies of group C * algebra.

Proof:
First of all we deduce the following relations among the generators, We also get same relations replacing a by c. Using the condition α(λ aba ) = α(λ bab ) and comparing on both sides the coefficients of λ aba , λ a This completes the proof. ✷ 4 Alternative description of the Quantum groups H + s (n, 0), K + n and computing the QISO of free copies of Z 4 We recall the quantum groups H + s (n, 0), K + n which are discussed in [6], [4] and [5]. K + n is the universal C * algebra generated by the unitary matrix ((u ij )) which is described in Subsection 2.2 subject to the conditions given below. 1. Each u ij is normal, partial isometry. 0) is the universal C * algebra satisfying the above conditions and moreover, u * ij = u s−1 ij . In this section we are giving another description of these objects in terms of free wreath product motivated from the fact H + n ∼ = C * (Z 2 ) * w C(S + n ) (see in [7]). First of all, we compute the quantum isometry group of n free copies of Z 4 . The group is presented as follows: Γ =< a 1 , a 2 , · · a n |o(a i ) = 4 ∀ i > Now the fundamental unitary is of the form . . . . . .

First of all, our aim is to show
Using the condition α(λ a 3

Thus, using the above relations we can find
For any k we can prove this result, which means Using this we have . Thus, we can also conclude all A ij 's are normal using the above equations. Now consider the transpose of the above matrix, we denote the entries (u ij ), then from the co-associativity condition we can easily deduce the co-product given by ∆(u ij ) = Σ 2n k=1 u ik ⊗ u kj . Let us define by using the above relations. This means U i and U i+1 satisfy the following relations: .∀ i Moreover, the coproduct is given by . t ij satisfy all the conditions of the quantum permutation groups. Furthermore, ∆(t ij ) = Σ n k=1 t ik ⊗ t kj . Now it can be easily checked that

Remark 4.1
The quantum groups H + s (n, 0), K + n can be described also such way. For finite s > 2 admit the quantum group structure as in [11]. The above facts will follow by essentially the same arguments of the previous theorem.
Corollary 4.2 Using the above theorem, remark and the result of [7] we can conclude that for every finite s

Remark 4.3
If we consider Γ = Z n * Z n where n is finite, then Q(Γ) is doubling of the quantum group Q(Z n ) * Q(Z n ). This means as a C * algebra. In particular for n = 2, Q(Γ) becomes doubling of the group algebra as Q(Z n ) ∼ = C * (Z 2 ) and C * (Z 2 ) * C * (Z 2 ) ∼ = C * (Z 2 ⋆ Z 2 ).
5 Examples of (Γ, S) for which Q(Γ) ∼ = D(C * (Γ)) We already observed in [20] that, if there exists a non trivial automorphism of order 2 which preserves the generating set, then D(C * (Γ)) ( [20], [16]) will be always a quantum subgroup of Q(Γ). In ( [11], [16], [21])the authors could show that Q(Γ) coincides with doubled group algebra for some examples. In Section 5 of [18] together with Goswami we also present few examples of groups where this happens. Our aim in this section is to give few more examples of such groups.
Using Lemma 5.3 of [18] its fundamental unitary is of the form Now the action is defined as, This gives the following reduction : Moreover, using the relations between the generators one can find Now using the above relations we can easily show that G * G, H * H are central projections of the desired algebra, hence Q(Γ, S) is isomorphic to D(C * (Γ)) by proposition 2.9 corresponding to the automorphism The group is presented as Γ =< a, b, c|ba = ab, bc = cb, a 2 = b 2 = c 2 = e >. Here S = {a, b, c}. The action is given by, Write the fundamental unitary as Our aim is to show D = B = F = H = 0.
Applying α(λ a 2 ) = λ e ⊗ 1 Q comparing the coefficients of λ ac , λ ca on both sides we have AC = CA = 0. Using the antipode one can get AG = GA = 0. Applying the same process with b, c we can deduce, Further, using the condition α(λ ab ) = α(λ ba ) comparing the coefficients of λ ac , λ ca on both sides we can get AF = DC, CD = F A. Applying κ we have HA = GB, AH = BG. Proceeding the same argument with α(λ cb ) = α(λ bc ) one can find, DK = GF, KD = F G, KB = HC, BK = CH.

Lamplighter Group
The group is presented as Γ =< a, t|a 2 = [t m at −m , t n at −n ] = e > where m, n ∈ Z. Fundamental unitary is of the form Now the aim is to show B = C = D = 0. Using the condition α(λ a 2 ) = α(λ e ) = λ e ⊗ 1 Q we deduce D 2 = 0, this implies B 2 = C 2 = 0 applying antipode. Further, we know DD * + EE * + F F * = 1, this gives us DEE * + DF F * = D as D 2 = 0. If we can show DE = DF = 0 then we will be able to prove our first claim i.e, D = 0. Using group relations we deduce t (m−n) at −(m−n) a = at (m−n) at −(m−n) [where m, n ∈ Z]. In particular, t −1 ata = atat −1 , tat −1 a = atat −1 , this gives us at = tat −1 ata. Now using the condition α(λ at ) = α(λ tat −1 ata ) comparing the coefficient of λ t 2 on both sides we have BE = 0 because there are no terms with coefficient λ t 2 on the right hand side as D 2 = BF = BE * = F B = E * B = 0. Applying the antipode one can get DE = 0. Similarly using the relation α(λ at −1 ) = α(λ t −1 atat −1 a ) following the same argument we deduce BF * = 0, DF = 0. Hence, we find that D = 0. This gives us B = C = 0 using the antipode. Thus, the fundamental unitary is reduced to the form From the relation of the group one can easily get, AE = EAE * AEA, E * A = AE * AEAE * , AE * = E * AEAE * A.
Thus we have, hence EE * is a central projection. Similarly, F F * is a central projection. Now we can define the map from C * {A, E, F } to C * (Γ) ⊕ C * (Γ) such as A → (λ a ⊕ λ a ), E → (λ t ⊕ 0), F → (0 ⊕ λ t −1 ). This gives the isomorphism between these two algebras, which is also a CQG isomorphism and by Proposition 2.9 corresponding to the automorphism a → a, t → t −1 we can conclude that Q(Γ) ∼ = D(C * (Γ)) . ✷