Deformations of spectral triples and their quantum isometry groups via monoidal equivalences

In this paper, we propose a new procedure to deform spectral triples and their quantum isometry groups. The deformation data are a spectral triple $(\mathcal A,\mathcal H, D)$, a compact quantum group $\mathbb G$ acting algebraically and by orientation-preserving isometries on $(\mathcal A,\mathcal H,D)$ and a unitary fiber functor $\psi$ on $\mathbb G$. The deformation procedure is a genuine generalization of the cocycle deformation of Goswami and Joardar.


Introduction
An important source of examples of non-commutative manifolds in the sense of A. Connes (spectral triples, [8]) relies on 2-cocycle deformations. For instance, the so-called 'isospectral deformations' ( [9]) of compact spin manifolds admitting an action of a torus (or an action of the abelian group R d ) may be seen as a by-product of Rieffel's machinery which, given a C * -or Fréchet-algebra A on which R d acts, produces a one-parameter continuous field of C * -algebras {A θ } θ∈R with A 0 = A. The cocycle involved in this case is the usual Moyal 2-cocycle in R d . When A is the algebra underlying a spectral triple (A, H, D), and the action of R d lifts to an isometric action on H, i.e. an action commuting with D, Rieffel machinery produces a new (family of) spectral triple(s) (A θ , H, D). The paradigm there consists in the noncommutative torus within its metric version.
In the present work, we generalize the deformation procedure through quantum group 2-cocycles (Goswami-Joardar, [14]) which is a way to produce new spectral triples from a given one. Our procedure is based on the notion of monoidal equivalence (introduced by Bichon , de Rijdt and Vaes, [6]) of (some subgroup of) its quantum isometry group ( [13]). The generalized procedure here leads to examples that cannot be obtained by 2-cocycle deformations.
The paper is structured as follows. In the first section we recall some basic material and in the second, we describe the deformation procedure. In the third section we show that 2-cocycle deformations are particular cases of our deformation procedure. Moreover not all examples are coming from 2-cocycles: in the fourth section, we give such an example that is not a 2-cocycle deformation, proving our procedure is a proper generalization of the one by Goswami and Joardar. Finally in the last section, we prove that the quantum isometry group of the deformed spectral triple is a certain deformation of the quantum isometry group of the original spectral triple.
Before we end this introduction, we will clarify some notation. Given a Hilbert space H, the inner product ·, · is linear in the second variable. Moreover, for ξ, η ∈ H, ξ * is the functional H → C : η → ξ, η and ξη * the rank one operator H → H : ζ → ξ η, ζ . We will denote by B(H) resp. K(H) the bounded resp. compact operators on H and for a bounded or unbounded operator D on H, σ(D) will be used to denote its spectrum. Given a C * -algebra A, the multiplier algebra of A will be denoted by M(A) and for a subset B of A, we define B to be the linear span of B, [B] the closed linear span, S(B) the * -algebra generated by the elements of B and C * (B) the C * -subalgebra of A generated by the elements of B. Furthermore, we use ω ξ,η to denote the linear functional which maps a ∈ B(H) to ξ, aη where ξ, η ∈ H, having linearity in the inner product in the second variable.
An algebraic tensor product will be denoted by ⊙ while the minimal C * -algebraic tensor product and a tensor product of Hilbert spaces is denoted by ⊗. We will also use the legnumbering notation in three and multiple tensor products: for a ∈ A ⊗ A, we let a 12 = a ⊗ 1 A , a 23 = 1 A ⊗ a 23 and a 13 = (id ⊗τ )(a ⊗ 1 A ), all three elements in For a Hopf algebra H, the coproduct, counit and antipode will be denoted by ∆, ε and S resp. We also use the Sweedler notation ∆(h) = h (1) ⊗ h (2) . A left, resp. right H-comodule is a vector space A endowed with a linear map α : A → H ⊙ A resp. α : A → A ⊙ H satisfying (∆ ⊗ id)α = (id ⊗α)α resp. (α ⊗ id)α = (id ⊗∆)α. If A is an algebra and α is multiplicative, it is called a coaction of H on A and A is called an H-comodule algebra. If A and B are a right resp. left H-comodule algebra with resp. coactions α and β, A ⊡ H B will denote the algebra {z ∈ A ⊙ B|(α ⊗ id)(z) = (id ⊗β)(z)}.

Compact quantum groups and Monoidal equivalences
We start this section with a short overview of the theory of compact quantum groups. The theory is essentially developed in [24], [26] and also explained in [15]. implementing coassociativity and the cancellation properties. Moreover there exists a unique state h on C(G) which is left and right invariant in the sense that (id ⊗h)∆(x) = h(x)1 C(G) = (h ⊗ id)∆(x) for all x ∈ C(G) ( [24,26,15]).This state is called the Haar state of G. In the classical case that C(G) = C(G) for a classical compact group G, the Haar state is the state on C(G) obtained by integrating along the Haar measure.

Compact quantum groups and representations
It is well known that, like compact groups, compact quantum groups have a rich representation theory ( [24,26,15]). A right unitary representation of a compact quantum group G = (C(G), ∆) on a Hilbert space H is a unitary element U of M(K(H) ⊗ C(G)) satisfying (id ⊗∆)U = U 12 U 13 . Analgously, a left unitary representation of G on H is a unitary element U of M(C(G) ⊗ K(H)) satisfying (∆ ⊗ id)U = U 13 U 23 . In this paper all representations will be right representations unless indicated otherwise. The dimension of H is called the dimension of the representation. Identifying M(K(H) ⊗ C(G)) with B(H ⊗ C(G)), the C * -algebra of C(G)-linear adjointable maps on the Hilbert-C * -module H ⊗ C(G), we will also see representations as maps u : H → H ⊗ C(G) : ξ → U(ξ ⊗ 1 C(G) ) satisfying that u(ξ), u(η) C(G) = ξ, η 1 C(G) , (u ⊗ id)u = (id ⊗∆)u and [u(ξ)(1 ⊗ a) : ξ ∈ H, a ∈ C(G)] = H ⊗ C(G).
Moreover, there is the notion of tensor product of representations: if U and V are representations of a quantum group G = (C(G), ∆) on Hilbert spaces H 1 , H 2 respectively, the tensor product U ⊗ V of U and V is defined as U ⊗ V = U 13 V 23 ∈ M(K(H 1 ⊗ H 2 ) ⊗ C(G)). Furthermore, we call a representation U of G on H irreducible if Mor(U, U) = C1 B(H) where Mor(U 1 , U 2 ) := {S ∈ B(H 2 , H 1 )|(S ⊗ 1 C(G) )U 2 = U 1 (S ⊗ 1 C(G) )} for representations U 1 and U 2 on H 1 resp. H 2 . An important result states that every irreducible representation is finite dimensional and that every unitary representation is unitarily equivalent to a direct sum of finite dimensional irreducible representations. Finally, for every irreducible unitary representations, there exist the notion of contragredient representation ( [26], [15]).
For a compact quantum group G, we denote by Irred(G) the set of equivalence classes of irreducible representations of G and for x ∈ Irred(G), we will always take a unitary representative U x ∈ B(H x ) ⊗ C(G). By ε, we will denote the class of the trivial representation 1 C(G) .
Also for a compact quantum group G = (C(G), ∆) and an equivalence class x ∈ Irred(G), we will denote by (ω ξ,η ⊗ id C(G) )U x a matrix coefficient where ξ, η ∈ H and define O(G) to be the linear span of matrix coefficients of all irreducible (hence finite dimensional) representations of G: O(G) = (ω ξ,η ⊗ id C(G) )U x |x ∈ Irred(G), ξ, η ∈ H x , even more, the matrix coefficients of the irreducible representations form a basis of O(G). Note that O(G) is a unital dense * -subalgebra of C(G) which has, endowed with the restriction of ∆ to O(G), the structure of a Hopf * -algebra. This is a very nontrivial result obtained in [26], see also [15]. Also, for a ). Let G be a compact quantum group. The reduced C * -algebra C r (G) is defined as the norm closure of O(G) in the GNS-representation with respect to the Haar state h of G.The Remark 1.3. Note that, for a given compact quantum group G, we have surjective morphisms between the different completions of O(G): C u (G) → C(G) → C r (G). We will think of all these algebras as describing the same quantum group.
, ∆ H ) be compact quantum groups equipped with their universal C * -norms. Suppose moreover that there exists a surjective map θ : Then we callĜ the dual quantum group which has the structure of a discrete quantum group (see [20] for the definition and results).
Using the notation V = ⊕ x∈Irred(G) U x , we can define the dual comultiplication 1.2 Actions of compact quantum groups and the spectral subalgebra Definition 1.6 ([18]). Let B be a unital C * -algebra and G = (C(G), ∆) a compact quantum group.
Analogously, a left action is a unital * -morphism β ′ : B → C(G) ⊗ B satisfying the analogous conditions. We say that the action is ergodic if One can choose to call the map in this definition 'a coaction' as it is a coaction of the C * -algebra C(G) on B. However, we choose to call it an action of the compact quantum group in order to be compatible with the classical case: if C(G) = C(G) and B = C(X) with G a classical compact group and X a compact space, it is an action of G on X. One can prove that in the case of ergodic actions, there is a unique invariant state on B ( [7]), which we will denote by ω.
Note that the most evident example is a quantum group acting on itself by comultiplication. In that situation, one can check that ω = h.
Using the intimate link between the ergodic action of a compact quantum group on a unital C * -algebra and the representations of the quantum group, one has the following result. Proposition 1.7 ([7]). Let B be a unital C * -algebra and β : B → C(G) ⊗ B a left action of G on B. Define for every x ∈ Irred(G), Then the spaces B x with x ∈ Irred(G) are called the spectral subspaces of B and is a dense unital * -subalgebra of B which we will call the spectral subalgebra of B with respect to β. Moreover β |B is an algebraic coaction of the Hopf * -algebra (O(G), ∆) on B.
In remark 1.3 we saw that a compact quantum group can be described using different C * -algebras, having the same underlying (dense) Hopf * -subalgebra. Similarly here, given an action β : B → B⊗C(G) of G on B, passing through B we can associate to it its universal and reduced C * -completions B u and B r , and we have surjective morphisms: B u → B → B r . Definition 1.9 ([6]). Let G 1 = (C(G 1 ), ∆ 1 ) and G 2 = (C(G 2 ), ∆ 2 ) be two compact quantum groups. G 1 and G 2 are called monoidally equivalent if there exists a bijection ϕ : Irred(G 1 ) → Irred(G 2 ) which satisfies ϕ(ε G1 ) = ε G2 together with linear isomorphisms:

Monoidal equivalences between compact quantum groups
whenever the formulas make sense. The collection of maps is called a monoidal equivalence.
Note that this is indeed the usual definition of equivalence between strict monoidal categories, but adapted to the concrete case of the category of representations of a compact quantum group. Definition 1.10 ( [6]). Let G = (C(G), ∆) be a compact quantum group. A unitary fiber functor is a collection of maps ψ such that • for every x ∈ Irred(G), there is a finite dimensional Hilbert space H ψ(x) , • there are linear maps which satisfy the equations (1.1) of definition 1.9.
Remark 1.11 ([6]). To define a unitary fiber functor it suffices to attach to every x ∈ Irred(G) a finite dimensional Hilbert space H ψ(x) (H ε = C) and to define the linear maps together with a non-degenerateness condition In fact, the notions of unitary fiber functor and monoidal equivalence are equivalent, which is stated in the following proposition, taken from Proposition 3.12 in [6]. Proposition 1.12. Let G 1 be a compact quantum group and ψ a unitary fiber functor on it. Then there exist a unique universal compact quantum group G 2 with underlying Hopf algebra 2. the matrix coefficients of the U ψ(x) , x ∈ Irred(G 1 ) form a linear basis of O(G 2 ).
Moreover, the set {U ψ(x) |x ∈ G 1 } forms a complete set of irreducible representations of G 2 and the unitary fiber functor ψ on G 1 will induce a monoidal equivalence ϕ : The following theorems of Bichon et al. will be crucial in our main result. They explain what extra structure a monoidal equivalence induces.
The first theorem follows from Theorem 3.9 and Proposition 3.13 of [6]. 6]). 1 Let G 1 be a compact quantum group and let ψ be a unitary fiber functor on G 1 . Denote with ϕ : G 1 → G 2 the monoidal equivalence induced by ψ (see previous proposition).
1. There exists a unique unital * -algebra B equipped with a faithful state ω and unitary elements (a) X y 13 X z 23 (ϕ(S) ⊗ 1) = (S ⊗ 1)X x for all S ∈ Mor(y ⊗ z, x), (b) the matrix coefficients of the X x form a linear basis of B, 2. There exist unique commuting coactions β 1 : B → O(G 1 ) ⊙ B and β 2 : for all x ∈ Irred(G). Moreover, 3. The state ω is invariant under β 1 and β 2 . Denoting by B r the C * -algebra generated by B in the GNS-representation associated with ω and denoting by B u the universal enveloping C *algebra of B, the Hopf algebraic coactions β 1 and β 2 admit unique extensions to actions of the compact quantum groups on B r , resp. B u . These actions are reduced, resp. universal and they are ergodic and of full quantum multiplicity (see [6] for the definition).
In what follows, we will call B the G 1 − G 2 -bi-Galois object associated with ϕ.
In the spirit of this theorem, we can introduce the notion of isomorphism of unitary fiber functors, which will be equivalent to the isomorphism of the associated bi-Galois objects. Definition 1.15 (Def. 3.10 in [6]). Let ψ and ψ ′ be two unitary fiber functors on a compact quantum group G. We say they are isomorphic if there exist unitaries u Proposition 1.16. Let ψ and ψ ′ be two unitary fiber functors on a compact quantum group G. Let B ψ ′ and B ψ be the associated bi-Galois objects with respective coactions β ψ , β ′ ψ . Then ψ and ψ ′ are isomorphic as unitary fiber functors if and only if there exists a * -isomorphism λ : There is even more, De Rijdt and Vander Vennet proved in [11] that there exists a bijection between actions of monoidal equivalent compact quantum groups. Indeed, let G 1 and G 2 be two compact quantum groups, ϕ : G 1 → G 2 be a monoidal equivalence between them. Let B, β 1 , β 2 , X x be as in the previous theorem. Suppose moreover that we have a C * -algebra D 1 and an action α 1 : D 1 → D 1 ⊗ C(G 1 ) of G 1 on D 1 . Using the dense Hopf * -algebras, we have a coaction α 1 : D 1 → D 1 ⊙ O(G 1 ) of O(G 1 ) on D 1 and we can define the * -algebra: Moreover, in [11], the authors prove that the same construction with the inverse monoidal equivalence ϕ −1 will give D 1 again up to isomorphism. Theorem 1.17. Given the data above, there exists an action α 2 = (id ⊗β 2 ) |D 2 on D 2 . Moreover, if α 1 is ergodic, α 2 is ergodic as well.

Deformation procedure for spectral triples
Before we start with the description of the deformation procedure, we recapitulate the notion of spectral triples and that of CQG acting on spectral triples.  In [4,13] Bhowmick and Goswami described how compact quantum groups can act isometrically and orientation-preserving on a non-commutative manifold, i.e. a spectral triple. This definition is a very strong one: it ensures the existence of a universal object in the category of all compact quantum groups acting by orientation-preserving isometries. However, in some cases the second condition is to weak: the quantum group representation on H may behave badly with respect to the algebra A in the sense that the induced action of the CQG on A is not a CQG-action on the C * -closure of A. This is in some situations a disadvantage. Therefore, we note the following proposition of Goswami, found in [14].

Spectral triples and compact quantum groups acting on them
In what follows, we will always work with compact quantum groups acting algebraically on the algebra A.

Deformation procedure for spectral triples
In this subsection we will describe the actual deformation procedure for spectral triples. The deformation data to start with are: The unitary fiber functor will induce a new compact quantum group G 2 and a * -algebra B with left resp. right coaction of O(G 1 ) resp. O(G 2 ). Using this, one can deform the data one by one to obtain a new, deformed, spectral triple on which G 2 acts in an appropriate way.
To be more precise, consider the following: 1. As ψ is a unitary fiber functor on G 1 , following theorem 1.13 there exists a compact quantum group G 2 and a monoidal equivalence ϕ : G 1 → G 2 . We will call G 2 the deformed quantum group.
2. Let (B, ω) be the * -algebra and faithful invariant state associated to ϕ with the coactions We start by introducing the deformed data and proving some basic facts about them.
Proposition 2.6. Defining L 2 (B) to be the GNS representation of B with respect to ω and Λ : B → L 2 (B) the GNS map, we have: Proof.
1. As ω is faithful on B, Λ is injective and hence β ′ 1 is well defined on Λ(B). Using that β 1 is a well defined coaction of O(G 1 ) and that ω is β 1 -invariant, β ′ 1 can be extended to a unitary representation on L 2 (B).

Let ξ be an element in
is a vector subspace of the tensor product Hilbert space where V λ is the eigenspace of λ ∈ σ(D).
Motivated by the first fact, we will call H ϕ(x) the deformation of H x for x ∈ Irred(G 1 ). Proof.
Hence we can define the following maps: Using that β ′ 1 is ergodic (Proposition 2.6(2)), one can check that g x (z) ⊗ Λ(1 B ) = X x * z which ensures that f x and g x are inverse to each other. Finally, using that X x is unitary, it is easy to see that f x and g x are also unitary.
2. Note first that as D has compact resolvent, there exist a sequence of real eigenvalues (λ n ) n with finite dimensional eigenspaces and such that lim n→∞ λ n = ∞.
where we used the first statement of this proposition. This last direct sum of finite dimensional of compact resolvent.
Proof. As D has compact resolvent, its restriction D λ to V λ is multiplication with λ for every λ in the spectrum. Therefore D λ ⊗ id can be restricted to Taking the direct sum we get an unbounded operatorD on H ⊠ Hence it is of compact resolvent by proposition 2.7(3) and selfadjoint as D is selfadjoint. Moreover, as . Moreover,Ã acts by bounded operators onH: for z ∈Ã, we haveL z :H →H : v → zv by multiplication on B and action of A on H as a bounded operator onH.
Proof. The first statement is an application of theorem 1.17. For the second, note thatÃ ⊂ A ⊙ B and A ⊙ B acts by bounded operators on H ⊗ L 2 (B). Hence it suffices to prove thatÃ leavesH invariant. Indeed, we have for a ∈Ã, ξ ∈H 13 . Proof. Combining all the previous propositions, it suffices to prove that the commutator ofD with an element a ∈Ã is bounded. For that, we will first prove thatÃ leaves the domain ofD invariant and secondly we will proof that the commutator ofD and an arbitrary a ∈Ã is bounded. Let z be an arbitrary element in A ⊙ B and let ξ be an arbitrary vector in dom(D ⊗ id). We will prove zξ ∈ dom(D ⊗ id). As ξ ∈ dom(D ⊗ id), there exists a sequence ξ n in dom(D) ⊙ L 2 (B) such that simultaneously ξ n → ξ and (D ⊗id)ξ n → (D ⊗id)ξ for n → ∞. Note that as A leaves the domain of D invariant, A⊙ B leaves the core dom(D)⊙ L 2 (B) of D ⊗ id invariant and hence zξ n ∈ dom(D)⊙ L 2 (B) for all n. Moreover, as A has bounded commutator with D, one can prove that [D ⊗ id, z] is bounded on dom(D) ⊙ L 2 (B) and (D ⊗ id)z(ξ n ) n is a Cauchy sequence and thus converging. As zξ n is an element of the core converging to zξ and ((D ⊗id)z(ξ n )) n converges, we know that zξ ∈ dom(D ⊗id) Finally, we prove thatDz − zD is indeed bounded on the domain ofD. Let ξ ∈ dom(D) arbitrary and take a sequence ξ n → ξ in dom(D) ⊙ L 2 (B). Then we know from above, that simultaneously acts algebraically and by orientation-preserving isometries on (Ã,H,D) withŨ.
Proof. Using the coaction β 2 : B → B ⊙ O(G 2 ) and the CQG-action β 2 : B u → B u ⊗ C(G 2 ), one can construct, along the lines of Lemma 5 in [7] and the discussion above it, a representatioñ U 0 ∈ M(K(L 2 (B)) ⊗ C(G 2 )) such that Moreover, we know this is a unitary representation and furthermore, Then it suffices to prove thatŨ commutes with the Dirac operator of the deformed spectral triple and that there is a coaction of O(G 2 ) onÃ. AsD is the restriction of D ⊗ id L 2 (B) andŨ is the restriction of id H ⊗Ũ 0 , it follows directly that they commute. Using theorem 1.17, we know that, given the coaction Then there exist a spectral triple (Ã,H,D), a compact quantum group G 2 = (C(G 2 ), ∆ 2 ) monoidally equivalent with G 1 and a unitary representationŨ of G 2 onH such that the monoidal equivalence is associated to ψ and G 2 acts algebraically and by orientation-preserving isometries on the new spectral triple withŨ.
Denoting B to be the G 1 − G 2 -bi-Galois object, one has In what follows, we will call this deformation procedure 'monoidal deformation'.
To end this section, we will show that via the inverse monoidal equivalence on the deformed quantum group and spectral triple, one can obtain the original data again.
Theorem 2.13. Let (A, H, D) be a spectral triple, G 1 a compact quantum group acting algebraically and by orientation-preserving isometries on (A, H, D). Let ψ be a unitary fiber functor, inducing a monoidal equivalence ϕ : is isomorphic with (A, H, D) as spectral triples (definition 2.2).
Proof. From proposition 1.18, one obtains the following * -isomorphisms: which are all compatible with the coaction of C(G 2 ). Furthermore, recall the unitaries for x ∈ Irred(G 1 ) of proposition 2.7. Note that these unitaries intertwine the representations of G 2 on the two Hilbert spaces. We then also have and combining them, we have a unitary: Denoting by X and Z resp. ⊕ x∈Irred(G1) X x and ⊕ x∈Irred(G1) Z ϕ(x) (where we take the direct sum over and hence ). This concludes the proof. QED

Cocycle deformation of spectral triples
In this section we will fix a spectral triple (A, H, D), a quantum group G acting algebraically on it by orientation-preserving isometries and a unitary fiber functor ψ on G which satisfies dim(H x ) = dim(H ψ(x) ) for every x ∈ Irred(G). Unitary fiber functors which satisfy this condition will be called to be dimension-preserving and monoidal deformation via a dimension-preserving unitary fiber functor, a dimension-preserving monoidal deformation. Bichon et al. proved in [6] that dimension-preserving unitary fiber functors are in one-to-one correspondence with 2-cocycles on the dual quantum group. Using this, we will prove that dimension-preserving monoidal deformation is equivalent to the cocycle deformation introduced in [14]. In this section we will frequently use slight adaptations of the work of Bichon et al. [6].

Cocycles on the dual of a compact quantum group
Let G be a compact quantum group.
Denoting for x ∈ Irred(G), p x to be the projection c 0 (Ĝ) → B(H x ), we will say a cocycle is normalized if (p ε ⊗ id)Ω = p ε ⊗ id and (id ⊗p ε )Ω = id ⊗p ε . From now on we will always assume 2-cocycles to be normalized.
Let Ω be a normalized unitary 2-cocycle onĜ and denote Then there exists a unique unitary fiber functor ψ Ω on G such that for all S ∈ Mor(y ⊗ z, x) and T ∈ Mor(x ⊗ y ⊗ z, a) and x, y , z ∈ Irred(G). Moreover it is dimensionpreserving.
Proof. The proof follows directly as our ψ satisfies the conditions of remark 1.11. That it is dimensionpreserving, follows directly by construction. QED Using this unitary fiber functor, one can make a new compact quantum group G Ω = (C(G Ω ), ∆ Ω ) [6] and a monoidal equivalence ϕ : G → G Ω along the lines of proposition 1.12. Note that the dual quantum group will be (c 0 (Ĝ Ω ),∆ Ω ) where where∆ Ω (a) = Ω∆(a)Ω * . Proof. The proof is a slightly adapted version of the proof of proposition 4.5 in [6]. QED This theorems tells us that every dimension-preserving monoidal equivalence comes from a cocycle. The next step to prove that a dimension-preserving monoidal deformation of a spectral triple is a cocycle deformation is to introduce the algebraic notion of a 2-cocycle. We will proof that every 2-cocycle on the dual of a compact quantum group induces an algebraic 2-cocycle on the compact quantum group and that the monoidal deformation is equivalent to a cocycle deformation of the spectral triple as was introduced by Goswami in [14].

Algebraic 2-cocycle deformation of a spectral triple
We will start with defining the algebraic counterpart of a 2-cocycle on the dual of a compact quantum group. In algebraic literature (for example Schauenburg [19]), the definition and theorems are stated for Hopf algebras. We make slight adaptations to Hopf * -algebras. 1. An (algebraic) dual 2-cocycle on H is a linear map σ : (2) , b (2) ).

If H is a Hopf
In the rest of the section, when we use 2-cocycles on Hopf * -algebras, we will always assume them to be unitary.

QED
In this paragraph we give a slightly adapted version of a result of Goswami and Joardar in [14].
Proof. (a) Denote the the coaction α = ad U of O(G) on A # σ −1 C by α(a) = a (0) ⊗ a (1) . Let N be a dense subspace of H such that U(N ) ⊂ N ⊙ O(G) and on that subspace, let U(ξ) = ξ (0) ⊗ ξ (1) . Then we can define, for a ∈ A # σ −1 C: In section 4.3 of [14] in it is proved that π σ (a) is bounded for all a ∈ A # σ −1 C and that π σ is a well defined * -morphism.

Linking dimension-preserving monoidal equivalences with algebraic cocycles
In proposition 3.3, we proved that there is an equivalence between dimension-preserving unitary fiber functors on a compact quantum group G and cocycles on the dualĜ. In the following theorem 3.10, we will prove that there is also an equivalence between cocycles onĜ and (algebraic) dual cocycles on O(G). Moreover, we will show in theorem 3.11 that the bi-Galois object B associated with the monoidal equivalence induced by the fiber functor, will be of the form B = O(G) σ −1 #C.
Theorem 3.10. Let G be a compact quantum group. If Ω is a unitary 2-cocycle on the dualĜ, the formula defines a unique ( Remark that, as Ω * is the inverse of Ω, we see that σ ′ associated with Ω * is the convolution inverse of σ. We will denote it with σ −1 and we have Theorem 3.11. Let G be a compact quantum group with a dimension-preserving unitary fiber functor ψ. Let B be the bi-Galois object associated to ψ with coaction β 1 : B → O(G 1 ) ⊙ B, let Ω be the unitary 2-cocycle on the dualĜ associated to ψ ∼ = ψ Ω and σ the algebraic dual 2-cocycle equivalent with Ω (proposition 3.10). Then there exists a * -algebra isomorphism Proof. Denoting ϕ : G → G Ω to be the monoidal equivalence associated to ψ, we can find unitaries u x = H x → H ϕ(x) , as dim(ϕ(x)) = dim(x) for all x ∈ Irred(G). Fixing a x ∈ Irred(G), we can define (where we take the direct sum over all classes, all of them with multiplicity one). Note that the matrix coefficients of the X x constitute a basis of B by theorem 1.13. As the u x are unitaries, also the matrix coefficients of the Y x (let's call them b x i j ) and hence of Y ′ form a basis of B. As both the (u x i j ) i j,x and (b x i j ) i j,x are bases of O(G) resp. B, we have a vector space isomorphism which is compatible with the coactions (i.e. (id ⊗χ)∆ = β 1 • χ). Moreover, one can prove that, analogously as in the proof proposition 4.5 of [6], Y ′ satisfies the equation where we used theorem 3.10. Hence, also χ(u x i j )χ(u y st ) = k,l χ(u x i k u y sl )σ −1 (u x kj , u y l t ), which means Finally, to check that χ is a * -algebra isomorphism, note that by the previous equation, we also have and hence by unitarity of the U x and the Y x , which implies where V (a) = σ(S −1 (a (2) ), a (1) ) as before. This proves the last statement. QED

Dimension-preserving monoidal deformation is isomorphic to algebraic 2cocycle deformation
In this last paragraph of section 3, we state and prove the main result of this section: the Goswami-Joardar cocycle deformation amounts to our deformation with a dimension-preserving monoidal equivalence. Remember that B is the bi-Galois object associated to the fiber functor ψ, L 2 (B) the GNS-space with respect tot the invariant state ω = (h ⊗ id)β 1 and the deformed Dirac operatorD from section 2. We give the proof via some propositions. Proof.
1. Recall the unitaries u x : H x → H ϕ(x) from the proof of theorem 3.11 and the mutually inverse unitaries and from the proof of proposition 2.7 point 1. Therefore, defining φ where in both cases we take the sum over the irreducible representations appearing in the decomposition of U) such that φ(ξ) = Y (ξ ⊗ 1) for ξ ∈ H. Y is unitary and hence φ is the desired isomorphism of Hilbert spaces.

Constructing a non-dimension-preserving example
In this section, we will construct an example of a monoidal deformation coming from a non-dimensionpreserving monoidal equivalence. We will use the spectral triple on the Podleś spheres ( [17]) defined in [10] and SU q (2), which acts on it in the appropriate way.

Monoidal equivalences on SU q (2)
We look at orthogonal quantum groups and SU q (2) in particular. Moreover, A o (F ) = (C(A o (F )), U) is a compact matrix quantum group (as defined in [24]). They are called universal orthogonal quantum groups.
As the matrices F are not in one to one correspondence with the universal quantum groups (i.e. different F 's can define the same universal quantum groups), it is necessary (but not so hard) to classify the quantum groups A o (F ). This has been done in [6].
Therefore, we will describe a fundamental domain for ∼ as is done in [6].

Proposition 4.3.
A fundamental domain of ∼ is given by the following classes of matrices: Remark 4.4. Note that for F ∈ GL(2, C), up to equivalence, there only exists matrices of the form  Note that this last statement indeed implies that the only orthogonal quantum groups coming from matrices of dimension 2, are the quantized versions of SU(2).
We state some results obtained by de Rijdt et al. in [6] (Corollary 5.4 and Theorem 5.5).
where the U i are the unitary representations of A o (F i ), which matrix coefficients generate the quantum groups.
• the monoidal equivalence preserves the dimensions if and only if n 2 = n 1 . In this case, we denote the unitary 2-cocycle by Ω(F 2 ). The Ω(F 2 ) describe up to equivalence all unitary 2-cocycles on the dual of A o (F 1 ).

Remark 4.8.
In [2] Banica shows that the irreducible representations of A o (F ) can be labeled by N (say r k , k ∈ N). Moreover, for dim(F ) = n, he states that dim(r k ) = (x k+1 − y k+1 )/(x − y ) where x and y are solutions of X 2 − nX + 1 = 0 for n ≥ 3 and dim(r k ) = k + 1 for n = 2. Hence, it is easy to show by induction that if ϕ is a monoidal equivalence between SU q (2) and A o (F ) with dim(F ) ≥ 4, then dim(ϕ(r k )) > dim(r k ) = k + 1 for every irreducible representation r k with k ≥ 1.
Moreover, looking at the concrete orthogonal quantum group SU q (2), it is possible to classify all compact quantum groups which are monoidally equivalent with SU q (2): indeed applying the result of the last paragraph to the specific situation of F = F q , we know exactly what the quantum groups are which are monoidal equivalent with SU q (2).

Proposition 4.9 ([6]
). Let 0 < q ≤ 1. For every even natural number n with 2 ≤ n ≤ q + 1/q, there exists a monoidal equivalence on SU q (2) such that the multiplicity of the fundamental representation Let 0 > q ≥ −1. Then for every natural number n with 2 ≤ n ≤ |q + 1/q|, there exists a monoidal equivalence on SU q (2) such that the multiplicity of the fundamental representation is n.

Monoidal deformation of the Podleś sphere
In section 3, we proved that our monoidal deformation of spectral triples is a generalization of the cocycle deformation, developed in [14]. In this subsection, we will give a concrete example to prove that our construction is a genuine generalization: we will construct a monoidal deformation of the Podleś sphere (with spectral triple of Dabrowski, Landi, Wagner and D'Andrea [10]) which is not a 2-cocycle deformation. First we recapitulate the definition of the Podles sphere S 2 q,c and the spectral triple on it. Then we will use the results of subsection 4.1 to apply the construction of section 2.

The Podleś sphere, its spectral triple and its quantum isometry group
The Podles sphere was initially constructed by Podleś in [18] as follows. Let q ∈ (0, 1) and t ∈ (0, 1), hence c = t −1 − t > 0. We define O(S 2 q,c ) to be the * -algebra generated by elements A, B which satisfy the relations You can see that for q = 1, we have A * = A, AB = BA, B * B = BB * = A − A 2 + c1 and this is the classical sphere: putting A = z + 1/2, B = x + i y , r 2 = c + 1/4, we indeed have The associated quantum space is called the Podles sphere S 2 q,c . Note first that for q ∈ (0, 1), setting we see that the definition in [10] with {x 0 , x −1 , x 1 } is equivalent to the original definition of Podleś given above. Moreover, defining , one can prove that the unital * -subalgebra of C(SU q (2)) generated byÃ and B is isomorphic to O(S 2 q,c ) where c = t −1 − t, sending A toÃ and B toB. Doing as above, we have 3 equivalent descriptions of the Podles sphere.
The spectral triple on S 2 q,c we will use, is the spectral triple developed by Dabrowski, D'Andrea, Landi and Wagner in [10]. The spectral triple uses the representation theory of SU q (2) described by Banica in [2]. To be compatible with [10], we use their notation. For each n in {0, 1/2, 1, . . .}, there exists a unique irreducible representation D n (r 2n in Banica's notation) of dimension 2n + 1. For example , we have Denoting d n k,l to be the k, l -matrix coefficient of D n , one can prove that it is easy to see that ∆ SUq (2) induces a unitary representation U of SU q (2) on H. By [10] the spectral triple is equivariant with respect to this representation and hence, SU q (2) acts algebraically and by orientation-preserving isometries on (O(S 2 q,c ), H, D). We will use this representation and the monoidal equivalences of subsection 4.1 to deform this spectral triple.

Monoidal deformation of the Podleś sphere
To conclude this section, we construct a non-dimension-preserving example. Now we know that there is a well defined spectral triple (O(S 2 q,c ), H, D) on which SU q (2) acts algebraically and by orientationpreserving isometries. Furthermore, we know from proposition 4.9 what the monoidal equivalences of SU q (2) are and we know that those monoidal equivalences are non-dimension-preserving by remark 4.8. Putting all this together, we can apply the construction described in section 2 to get the following theorem.

Deformation of the quantum isometry group
The goal of this last section is to prove that the deformation (in the sense of theorem 2.12) of the quantum isometry group of a spectral triple (defined by Bhowamick and Goswami) is the quantum isometry group of the deformed spectral triple. We start by recalling some concepts and results of [4].
Definition 5.1 (Definition 2.7 in [4]). An R-twisted spectral triple (of compact type) is given by a triple (A, H, D) and an operator R on H where 1. (A, H, D) is a compact spectral triple, 2. R is a positive (possibly unbounded) invertible operator such that R commutes with D Remark 5.2. We note that in Definition 2.7 in [4], there is a third condition in the definition of R-twisted spectral triple. However in remark 2.11 of [4], the authors state that this third condition is not necessary. Therefore, we gave the definition above.
Such an operator R is linked with the preservation of a non-commutative analogue of a volume form.
for all x ∈ E D , where τ R (x) = Tr(Rx) and where E D is the * -subalgebra of B(H) generated by the rank-one operators of the form ηξ * , η, ξ eigenvectors of D.
In what follows we will denote by Q R (A, H, D) (or just Q R ) the category of all compact quantum groups acting by R-twisted volume-and orientation-preserving isometries with as morphisms the morphisms of quantum groups which are compatible with the representations on H.
Moreover, one can prove (as is done in [12]) that for every compact quantum group acting by orientation-presering isometries, there exists an operator R such that the quantum group is an elements of Q R . Now Goswami and Bhowmick proved in [4] that there exists a universal object in Q(A, H, D). For notational convenience, we will write QISO 0 R if there is no confusion possible about the spectral triple. However, in general α U0 may not be faithful even if U 0 is so. Therefore one has the following definition.
Definition 5.5 (Definition 2.16 in [4]). Let C = C * ({(f ⊗ id)α U0 (a) | a ∈ A, f ∈ A * }) be the C * -subalgebra of C(QISO 0 R ) generated by elements of the form (f ⊗ id)α U0 (a), a ∈ A. Then C is a Woronowicz C * -subalgebra of QISO 0 R and the compact quantum group is called the quantum group of R-twisted volume-and orientation-preserving isometries or simply quantum isometry group.
In subsection 5.3, we will prove that if (A, H, D) is an R-twisted spectral triple and is a monoidal equivalence, then there exists an operatorR such that (Ã,H,D) is anR-spectral triple and G 2 = QISOR(Ã,H,D). But before we do that, we describe, given a monoidal equivalence ϕ : G 1 → G 2 , how to construct a monoidal equivalence between certain Woronowicz-C * -subalgebras (subsection 5.1) resp. quantum supergroups (subsection 5.2) of G 1 and G 2 .

5.1
Inducing monoidal equivalences on Woronowicz-C * -subalgebras Definition 5.6 ([1]). Let G = (C(G), ∆) be a compact quantum group and A a C * -subalgebra of Then A is called a Woronowicz C * -subalgebra. We will write A = (A, ∆ | A ) for the quantum group.
It is good to remark that the notion of compact quantum quotient group introduced in [22] is a special case of a Woronowicz C * -subalgebra. However it is still unknown whether all Woronowicz C * -subalgebras are compact quantum quotient groups.
In this section let G = (C(G), ∆) be a CQG and A a Woronowicz C * -subalgebra of G. In order to define a unitary fiber functor on A, it is good to examine its representations. It is easy to see that every representation U of A on a Hilbert space H is a representation of G and that every representation V of G is a representation of A if and only if V ∈ M(K(H) ⊗ A). To distinguish, we will write U G for a representation U of A seen as representation of G. Moreover, we have the following proposition Analogously as before, we will write x G if we look at the equivalence class x ∈ Irred(A) seen as equivalence class in Irred(G). Using this proposition, the unitary fiber functor is easily made: let G 1 be a compact quantum group and ϕ : G 1 → G 2 a monoidal equivalence between them. Suppose moreover that A 1 is a Woronowicz subalgebra of G 1 . Then we can construct a unitary fiber functor on A 1 = (A 1 , ∆ | A 1 ) by restricting ϕ to the representations of A and proof it is a monoidal equivalence between A 1 and a compact quantum group A 2 such that C(A 2 ) is a Woronowciz C * -algebra of G 2 .
Proposition 5.8. Let G 1 be a compact quantum group, A 1 a Woronowicz C * -subalgebra of G 1 and ψ a unitary fiber functor on G 1 . Then there exists a unitary fiber functor ψ ′ on Proof. Let x ∈ Irred(A 1 ). Define H ψ ′ (x) to be H ψ(x G ) and ψ ′ (S) = ψ(S) for every S ∈ Mor(y 1 ⊗ . . . ⊗ y k , x 1 ⊗ . . . ⊗ x r ), y 1 , . . . , y k , x 1 , . . . , x r ∈ Irred(A 1 ). As ψ is a unitary fiber functor, ψ ′ will satisfy all the necessary conditions to be a unitary fiber functor as well. QED Denoting by ϕ : G 1 → G 2 the monoidal equivalence associated to ψ, we can see C(G 2 ) as the C *algebra generated (as vector space) by the coefficients of the U ϕ(x) , x ∈ Irred(G 1 ). Now we can define A 2 as the C * -algebra generated (as vector space) by the coefficients of the U ϕ(x G 1 ) , x ∈ Irred(A 1 ). Equivalently, and we also write Now it is clear that ψ ′ induces a monoidal equivalence ϕ ′ between A 1 and a compact quantum group with algebra A 2 .
is a compact quantum group. Moreover the monoidal equivalence ϕ ′ , induced by ψ is an equivalence between A 1 and A 2 .
Proof. Written differently, A 2 is the closed linear span of the elements u defined as above, we get: and as x ∈ Irred(A 1 ), we see that ∆ ′ 2 (A 2 ) ⊂ A 2 ⊗ A 2 . Now denote by ε ′ and S ′ the restrictions of the counit ε and antipode S of is indeed a compact quantum group. By construction of ϕ ′ , it is evident that it is a monoidal equivalence between A 1 and A 2 . QED Before we go the next paragraph, we want to explore how the G 1 − G 2 -bi-Galois object behaves with respect to the A 1 − A 2 -bi-Galois object .
Theorem 5.10. Let G 1 , G 2 , A be compact quantum groups such that C(A) is a Woronowicz C *subalgebra of C(G 1 ) and such that ϕ : Let ϕ ′ be the monoidal equivalence between A 1 and A 2 as defined above and define B ′ to be the A 1 − A 2 -bi-Galois object with γ 1 : Proof. From the original proof of theorem 1.13 (which is theorem 3.9 in [6]), we know that , H x ) * . By definition, we see that for x ∈ Irred(A 1 ), X x = X x G 1 . As β 1 resp. γ 1 are defined by (id ⊗β 1 )(X This concludes the proof. QED Remark 5.11. In the special case of compact quantum quotient groups, a compact quantum quotient group of G 1 will be monoidally equivalent with a compact quantum group which has as algebra a Woronowicz C * -subalgebra of G 2 . Whether that compact quantum group is a compact quantum quotient group as well is still unknown [22].

Inducing monoidal equivalences on supergroups
In this subsection we describe, given a monoidal equivalence ϕ : G 1 → G 2 , how to construct a monoidal equivalence between certain quantum supergroups of G 1 and G 2 . So, let G 1 and G 2 be two compact quantum groups and let ϕ : G 1 → G 2 be a monoidal equivalence. Moreover suppose G 1 is a compact quantum subgroup of a compact quantum group H 1 . As we have done in subsection 5.1 for Woronowicz C * -subalgebras, we will describe a method to construct a unitary fiber functor on H 1 from the monoidal equivalence ϕ.
Let π : C u (H 1 ) → C u (G 1 ) be the surjective morphism which is compatible with the quantum group structure. Now note that for a representation U of H on a Hilbert space H, (id H ⊗π)U is a representation of G 1 . Therefore, for x ∈ Irred(H 1 ) define x G1 to be the equivalence class of (id ⊗π)U x as representation of G 1 and If x 1 , . . . , x r , y 1 , . . . , y s are classes of irreducible representations of H 1 with U constitutes a unitary fiber functor ψ ′ on H 1 .
The proof follows directly by construction of H ψ ′ and ψ ′ (S). By theorem 1.12, there exists a compact quantum group H 2 and a monoidal equivalence ϕ ′ : H 1 → H 2 . In theorem 5.13 we will describe the bi-Galois object associated to ϕ and the compact quantum group H 2 explicitly.
Theorem 5.13. Let G 1 , G 2 , H 1 be compact quantum groups such that G 1 is a compact quantum subgroup of H 1 with surjective morphism π : C u (H 1 ) → C u (G 1 ). Let ϕ : G 1 → G 2 be a monoidal equivalence as above and let H 2 and ϕ ′ : H 1 → H 2 be the compact quantum group and monoidal equivalence induced by ϕ by propositions 5.12 and 1.12. Denoting by B the (G 1 -G 2 )-bi-Galois object associated to ϕ, byB the (G 2 -G 1 )-bi-Galois object associated to ϕ −1 and by B ′ the (H 1 -H 2 )-bi-Galois object associated to ϕ ′ , we have and using the right resp. left coactions (id ⊗π)∆ H1 : Proof. Let X x , x ∈ Irred(G 1 ) be the elements from theorem 1.13 associated to ϕ. Define for We claim that the Y x with the functional ω ′ = h H1 ⊗ ω (h H1 is the Haar state of H 1 ) satisfy the properties 1(a), 1(b) and 1(c) of theorem 1.13 applied to ϕ ′ . Indeed, we have for x, y , z ∈ Irred(H 1 ) and S ∈ Mor(y ⊗ z, x) Proposition 5.14. We have a surjective morphism of compact quantum groups π ′ : for every x ∈ Irred(H 1 ) implying that G 2 is a quantum subgroup of H 2 .
Proof. The map π ′ is well defined by (5.4) as the matrix coefficients of the V ϕ ′ (x) constitute a linear basis of O(H 2 ). Moreover, it is a linear surjection and it follows directly that it is coalgebra map. It suffices to prove that π ′ is an algebra map. Therefore, denoting by f : B the isomorphism of proposition 1.18 (applied to ϕ −1 : proving that π is multiplicative as composition of algebra maps. This concludes the proof. QED Finally we prove that the two monoidal equivalences ϕ and ϕ ′ make isomorphic deformed spectral triples. Proposition 5.15. Let G 1 , G 2 , H 1 be compact quantum groups such that G 1 is a compact quantum subgroup of H 1 with surjective morphism π : C u (H 1 ) → C u (G 1 ) and let ϕ : G 1 → G 2 be a monoidal equivalence as above. Let H 2 and ϕ ′ be the compact quantum group and monoidal equivalence induced by ϕ as in proposition 5.12. Suppose H 1 resp. G 1 act algebraically and by orientation preserving isometries with a unitary representation V resp. U on a spectral triple (A, H, D) such that U = (id ⊗π)V . Denoting by B the (G 1 -G 2 )-bi-Galois object associated to ϕ, byB the (G 2 -G 1 )-bi-Galois object associated to ϕ −1 and by B ′ the (H 1 -H 2 )-bi-Galois object associated to ϕ ′ , the deformed spectral triples and Proof. It is easy to see that the map is an isomorphism of * -algebras with inverse (id A ⊗ε H1 ⊗ id B ). Moreover, let φ : H ⊠ In paragrah subsection, we will investigate how the universal objects in the category Q R (A, H, D) behave with respect to our deformation procedure. is anR-twisted spectral triple on which G 2 acts byRtwisted volume-and orientation-preserving isometries. Moreover, applying the same construction to ϕ −1 , we obtain R again.
Proof. We can decompose H as for some Hilbert spaces W x where the direct sum is taken over all x ∈ Irred(G 1 ), all with multiplicity one. As D commutes with the representation U, D is of the form D = ⊕ x∈Irred(G1) id Hx ⊗D x where the D x are operators W x → W x . As G 1 acts by R-twisted volume-preserving isometries, for all x ∈ E D , where τ R (x) = Tr(Rx) and where E D is the * -subalgebra of B(H) generated by the rank-one operators of the form ηξ * , η, ξ eigenvectors of D. Therefore, also (τ R ⊗ h G1 )(α U (x)) = τ R (x) from which it follows (as in the proof of theorem 3.8 of [14]) that R must be of the form (described by Woronowicz [26]) and R x : W x → W x positive operators. As (A, H, D) is an R-twisted spectral triple, R and D commute and hence each D x commutes with R x for all x ∈ Irred(G 1 ). Now, in ThenR is again positive, and invertible and it commutes withD. Moreover, G 2 acts byR-twisted volume preserving isometries by the defining property of F ϕ(x) . It is clear that the inverse construction gives R again. QED Proof of theorem 5.17. By proposition 5.4, there exists a universal object QISO 0 R (A, H, D) in the category Q R of compact quantum groups acting by R-twisted volume-and orientation preserving isometries on (A, H, D). For notational convenience, we will denote this quantum group by QISO 0 R . Now, as ϕ : QISO 0 R → G 2 is a monoidal equivalence, G 2 acts algebraically and by orientation preserving isometries on ( This completes the proof. QED

Deformation of the quantum isometry group
In this paragraph we use subsection 5.1 and paragraph 5.3.1 to strengthen the result of theorem 5.17 to quantum isometry groups. where (Ã,H,D) is the spectral triple obtained by deformation with ϕ by theorem 2.12 andR the operator obtained from proposition 5.16.
Remark 5.20. One can make again remark 5.18 here.
This concludes the proof.

Deformation of the quantum isometry group of the Podleś sphere
In this last subsection, we use subsection 5.3 to find the quantum isometry group of the newly constructed spectral triple in theorem 4.10. Therefore we investigate first the quantum isometry group of the Podleś sphere.
In the classical situation, we know that SO (3) is a quotient group of SU(2), indeed SO(3) = SU(2)/{−1, 1}. In the quantum versions this is also true: we can prove that Z 2 is a normal quantum subgroup of SU q (2) and SU q (2)/Z 2 equals SO q (3). Now we will investigate monoidal equivalences of SO q (3) in order to apply theorem 5.19 to find the quantum isometry group of the spectral triples constructed in 4.10. We defined SO q (3) as coming from a Woronowicz-C * -subalgebra of SU q (2). Using the theorems of subsection 5.1, we will use the induction method to construct monoidal equivalences on SO q (3). Therefore fix a monoidal equivalence between SU q (2) and a suitable A o (F ′ ) with dim(F ′ ) ≥ 3. As SO q (3) = SU q (2)/Z 2 , we find a Woronowicz subalgebra R(F ′ ) of A o (F ′ ) such that SO q (3) is monoidally equivalent with R(F ′ ). Now Theorem 4.1 in [23], gives us a concrete description of R(F ′ ).
Theorem 5.23 (Theorem 4.1 in [23]). Let F ∈ GL(n, C) be such that F F = ±I n . Then every Woronowicz subalgebra of A o (F ) is a quantum quotient group. Moreover it has only one normal subgroup of order 2 with quantum quotient group C * (r 2m ) (where r 2m is the irreducible representation of dimension 2m).
Applying this theorem to F = F q , it affirms that SO q (3) is the only compact quantum quotient group of SU q (2). Applying it to F = F ′ , we get a concrete description of R(F ′ ). By remark 4.8, it can be seen that the induced monoidal equivalence is not dimension-preserving and hence not a 2-cocycle deformation (by proposition 3.2).
Combining all of this, we get Theorem 5.24. Let F ∈ GL(n, C) be such that F F = ±I n and ϕ : SU q (2) → A o (F ) a monoidal equivalence with bi-Galois object B = A o (F q , F ). Define I(F ) to be the C * -algebra generated by the U i j U kl where U is the unitary in M n (A o (F )) satisfying the relation U = F UF −1 as in definition 4.1.
Define P (F q , F ) to be the * -algebra generated by the Y i j Y kl where Y is the unitary in M n2,n1 (C) ⊗ C u (A o (F 1 , F 2 )) described in theorem 4.7. Then there exists a monoidal equivalence ϕ ′ : SO q (3) → I(F ) with bi-Galois object B ′ = P (F q , F ) which is not dimension-preserving (by remark 4.8).
Now we are ready to characterize the quantum isometry groups of the spectral triples constructed in 4.10.