Quantum Isometry Groups of Noncommutative Manifolds Obtained by Deformation Using Dual Unitary 2-Cocycles

It is proved that the (volume and orientation-preserving) quantum isometry group of a spectral triple obtained by deformation by some dual unitary 2-cocycle is isomorphic with a similar twist-deformation of the quantum isometry group of the original (undeformed) spectral triple. This result generalizes similar work by Bhowmick and Goswami for Rieffel-deformed spectral triples in [Comm. Math. Phys. 285 (2009), 421-444, arXiv:0707.2648].


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. Indeed, symmetries of physical systems (classical or quantum) were conventionally modeled by group actions, and after the advent of quantum groups, group symmetries were naturally generalized to symmetries given by quantum group actions. In this context, it is natural to think of quantum automorphism or the full quantum symmetry groups of various mathematical and physical structures. The underlying basic principle of defining a quantum automorphism group of a given mathematical structure consists of two steps : first, to identify (if possible) the group of automorphisms of the structure as a universal object in a suitable category, and then, try to look for the universal object in a similar but bigger category by replacing groups by quantum groups of appropriate type. Initiated by S. Wang who defined and studied quantum permutation groups of finite sets and quantum automorphism groups of finite dimensional algebras, such questions were taken up by a number of mathematicians including Banica, Bichon (see, e.g. [4], [1], [26]), and more recently in the framework of Connes' noncommutative geometry ( [9]) by Goswami, Bhowmick, Skalski, Soltan, Banica and others who have extensively studied the quantum group of isometries (or quantum isometry group) defined in [12] (see also [7], [8], [2] etc.).
Most of the examples of noncommutative manifolds are obtained by deforming classical spectral triples. It was shown in [3] that the quantum isometry group of a Rieffel-deformed noncommutative manifold can be obtained by a similar deformation (Rieffel-Wang, see [30]) of the quantum isometry group of the original (undeformed) noncommutative manifold.
In the present paper our goal is to generalize Bhowmick-Goswami's results about Rieffel-deformation to any cocycle twisted spectral triple. Combining this with the fact (proved in [10]) that the quantum isometry group of a classical compact, connected, Riemannian manifold is the same as the classical isometry group of such manifold (i.e. there is no genuine quantum isometry for such manifold), we have been able to compute the quantum isometry group of non commutative manifolds obtained from classical manifolds using unitary 2-cocycles.
Let (A ∞ , H, D) be a spectral triple of compact type, R a positive, invertible operator commuting with D. Given a dual unitary 2-cocycle σ on a quantum subgroup (say Q) of the quantum isometry group QISO + R (A ∞ , H, D) (in the sense of [7]), we construct a spectral triple on the twisted algebra (A ∞ ) σ on the same Hilbert space H with the same Dirac operator D. Moreover, we construct a canonical 'twisted' operator R σ commuting with D using the Haar state of Q and the cocycle σ. The main result of this paper (Theorem 4.16) can now be stated: Theorem: The quantum isometry group QISO + R σ ((A ∞ ) σ , H, D) is isomorphic with QISO + R (A ∞ , H, D) σ i.e. the cocycle twist of the quantum isometry group of the original spectral triple.
In this paper we have also been able to relax the assumption of existence of a dense * -algebra for which the action is algebraic in case of Rieffel deformed manifolds.
Let us mention that after we wrote and uploaded to the internet a preliminary version of this article, Neshveyev and Tuset posted the article [20] in which they formulated a general theory of deformation of C * algebras by dual unitary 2-cocycles on locally compact quantum groups. Indeed, our set-up for von Neumann algebraic deformation is closely related to the restriction of their framework to the simpler case of compact quantum groups and is nothing but an alternative description of the von Neumann algebra generated by the C *algebraic deformation obtained in [20]. However, there are two important subtle differences between our set-up and that of [20]: first, we work with universal compact quantum groups rather than those in the reduced form as in [20], and second, we do not need C * -algebraic action of the quantum group. Moreover, the main emphasis of our article is on deformations of spectral triples and their quantum isometry groups, not just the study of deformed algebras. We plan to re-cast some of our results in the framework of [20] and also study deformed spectral triples more deeply in a forthcoming paper.
Let us conclude this section with a brief note on the notations to be used in the paper. For a subset A of a vector space V , SpA denotes the linear span of A. We use ⊕, ⊗ alg , and ⊗ to denote algebraic direct sum, algebraic tensor product and von Neumann algebraic tensor product respectively (see [24] for the details on tensor product). For a Hilbert space H, K(H) and B(H) will denote the C * algebra of compact operators and the C * algebra of bounded operators on H respectively. ⊗ stands for the exterior tensor product for two Hilbert bimodules (in particular a Hilbert space H and a C * algebra Q) and tensor product of Hilbert spaces. Spatial tensor products of C * algebras will be denoted by⊗. We use the standard Kronecker δ ij symbol to denote the function which is 1 if i = j and 0 if i = j. The left convolution of an element in a CQG with a linear functional f will be denoted by f * a(:= (f ⊗ id)∆(a)) and the right convolution will be denoted by a * f (:= (id ⊗ f )∆(a)) for a ∈ Q, ∆ is the coproduct of Q. We use the standard abbreviations for the strong and weak operator topologies, i.e. SOT and WOT respectively.
2 Compact quantum groups

Definition and representation of compact quantum groups
A compact quantum group (CQG for short) is a unital C * algebra Q with a coassociative coproduct (see [17]) ∆ from Q to Q⊗Q such that each of the linear spans of ∆(Q)(Q ⊗ 1) and that of ∆(Q)(1 ⊗ Q) is norm-dense in Q⊗Q. From this condition, one can obtain a canonical dense unital * -subalgebra Q 0 of Q on which linear maps κ and ǫ (called the antipode and the counit respectively) are defined making the above subalgebra a Hopf * algebra. In fact, we shall always choose this dense Hopf * -algebra to be the algebra generated by the 'matrix coefficients' of the (finite dimensional) irreducible non-degenerate representations (to be defined shortly) of the CQG. The antipode is an anti-homomorphism and also satisfies κ(a * ) = (κ −1 (a)) * for a ∈ Q 0 . A morphism from a CQG Q 1 to another CQG Q 2 is a unital * -homomorphism π such that ∆ 2 • π = (π ⊗ π) • ∆ 1 , where ∆ i denotes the coproduct of Q i (i = 1, 2). If π is surjective, we say that Q 2 is a quantum subgroup of Q 1 and write Q 2 < Q 1 . We shall use standard Swedler notation for Hopf algebras i.e. we shall write ∆(a) = a (1) ⊗ a (2) and It is known that there is a unique state h on a CQG Q (called the Haar state) which is bi-invariant in the sense that (id ⊗ h) • ∆(a) = (h ⊗ id) • ∆(a) = h(a)1 for all a ∈ Q. The Haar state need not be faithful in general, though it is always faithful on Q 0 at least. Given the Hopf * -algebra Q 0 , there can be several CQG's which have this * -algebra as the Hopf * -algebra generated by the matrix elements of finite dimensional representations. However there exists a largest such CQG Q u , called the universal CQG corresponding to Q 0 . It is obtained as the universal enveloping C * algebra of Q 0 . We also say that a CQG Q is universal if Q = Q u . For details the reader is referred to [14]. The C * -completion Q r of Q 0 in the norm of B(L 2 (h)) (GNS space associated to h) is a CQG and called the reduced quantum group corresponding to Q. If h is faithful then Q and Q r coincide. In general there will be a surjective CQG morphism from Q to Q r identifying the latter as a quantum subgroup of the former.
There is also a von Neumann algebraic framework of quantum groups suitable for development of the theory of locally compact quantum groups (see [15], [31] and references therein). In this theory, the von Neumann algebraic version of CQG is a von Neumann algebra M with a coassociative, normal, injective coproduct map ∆ from M to M⊗M and a faithful, normal, bi-invariant state ψ on M. Indeed, given a CQG Q, the weak closure Q r ′′ of the reduced quantum group in the GNS space of the Haar state is a von Neumann algebraic compact quantum group.
Let H be a Hilbert space. Consider the multiplier algebra M(K(H)⊗Q) which can also be identified with L(H ⊗ Q), the C * algebra of Q linear, adjointable maps on H ⊗ Q (see [16]). This algebra 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 w 12 and w 13 for the images of an element w ∈ M(K(H)⊗Q) by these two maps respectively. Note that if H is finite dimensional then M(K(H)⊗Q) is isomorphic to B(H) ⊗ Q (we don't need any topological completion).
Identifying M(K(H)⊗Q) with L(H ⊗ Q), we can also view U as a Q linear map on H ⊗ Q. Clearly U is determined by its restriction on the subspace H ⊗ 1 and by a slight abuse of notation we shall denote the restriction also by U , which is a C-linear map ξ → U (ξ ⊗ q) ∈ H ⊗ Q for ξ ∈ H, q ∈ Q. This gives an equivalent picture of a unitary representation. This is easy to see that the restriction U : H → H ⊗ Q satisfies 1. << U (ξ), U (η) >>=< ξ, η > 1 Q for ξ, η ∈ H.
Equivalently, the orthogonal projection p on H 1 satisfies U (p ⊗ 1) = (p ⊗ 1)U . A unitary representation U of a CQG is said to be irreducible if there is no proper invariant subspace.
It is a well known fact that every irreducible unitary representation is finite dimensional.
We denote by Rep(Q) the set of inequivalent irreducible unitary representations of Q. For π ∈ Rep(Q), let d π and {q π jk : j, k = 1, ..., d π } be the dimension and matrix coefficients of the corresponding finite dimensional representation, say U π respectively. For each π ∈ Rep(Q), we have a unique d π × d π complex matrix F π such that (i) F π is positive and invertible with T r(F π ) = T r(F −1 π ) = M π > 0(say). (ii) h(q π ij q π * kl ) = 1 Mπ δ ik F π (j, l). Corresponding to π ∈ Rep(Q), let ρ π sm be the linear functional on Q given by ρ π sm (x) = h(x π sm x), s, m = 1, ..., d π for x ∈ Q, where x π sm = (M π )q π * km (F π (k, s)). Also let ρ π = dπ s=1 ρ π ss . Given a unitary representation V on a Hilbert space H, we get a decomposition of H as where m π is the multiplicity of π in the representation V and V | H π i is same as the representation U π . The subspace H π = ⊕ i H π i is called the spectral subspace of type π corresponding to the irreducible representation π. It is nothing but the image of the spectral projection given by (id ⊗ ρ π )V .
Recall from [22], the modular operator Φ = S * S, where S is the anti unitary acting on the L 2 (h) (where L 2 (h) is the GNS space of Q corresponding to the Haar state on which Q has left regular representation) given by S(a.1) := a * .1 for a ∈ Q. The one parameter modular automorphism group (see [22]) say Θ t , corresponding to the state h is given by Θ t (a) = Φ it aΦ −it . Note that here we have used the symbol Φ for the modular operator as ∆ has been used for the coproduct. From (ii), we see that In particular Φ maps L 2 (h) π i into L 2 (h) π i for all i.

Action on von Neumann algebras by conjugation of unitary representation
We now discuss an analogue of 'action' (as in [21]) in the context of von Neumann algebra implemented by a unitary representation of the CQG. Given a unitary representation V of a CQG Q on a Hilbert space H, often we consider the * homomorphism ad V on B(H) or on some suitable von Neumann subalgebra M of it. We say ad V leaves M invariant if (id ⊗ φ)ad V (M) ⊂ M for every state φ of Q. Then taking ρ π as above, we define M π = P π (M), where P π = (id ⊗ ρ π ) • ad V : M → M is the spectral projection corresponding to the representation π. Clearly each P π is SOT continuous. We define M 0 := Sp{M π ; π ∈ Rep(Q)}, which is called the spectral subalgebra. Then we have the following:

Proof:
This result must be quite well-known and available in the literature but we could not find it written in this form, so we give a very brief sketch. The proof is basically almost verbatim adaptation of some arguments in [15] and [31]. First, observe that the spectral algebra M 0 remains unchanged if we replace Q by the reduced quantum group Q r which has the same irreducible representations and dense Hopf- * algebra Q 0 . This means we can assume without loss of generality that the Haar state is faithful. The injective normal map β := ad V restricted to M can be thought of as an action of the quantum group (in the von Neumann algebra setting as in [15], [31]) Q r ′′ (where the double commutant is taken in the GNS space of the Haar state) and it follows from the results in [31] about the implementability of locally compact quantum group actions that there is a faithful normal state, say φ, on M such that β is Q r ′′ -invariant, i.e. (φ ⊗ id)(β(a)) = φ(a)1 ∀a ∈ M. We can replace M, originally imbedded in B(H), by its isomorphic image (to be denoted by M again) in B(L 2 (φ)) and as the ultra-weak topology is intrinsic to a von Neumann algebra, it will suffice to argue the ultra-weak density of M 0 in M ⊂ B(L 2 (φ)). To this end, note that Vaes has shown in [31] that β : M → M⊗Q r ′′ extends to a unitary representation on L 2 (φ), which implies in particular that M 0 is dense in the Hilbert space L 2 (φ). From this the ultra-weak density follows by standard arguments very similar to those used in the proof of Proposition 1.5 of [15], applying Takesaki's theorem about existence of conditional expectation. For the sake of completeness let us sketch it briefly. Using the notations of [31] and noting that δ = 1 for a CQG, we get from Proposition 2.4 of [31] that V φ commutes with the positive self adjoint operator ∇ φ ⊗ Q where ∇ φ denotes the modular operator i.e. generator of the modular automorphism group σ φ t of the normal state φ. Clearly, this implies that β := ad V φ satisfies the following: where τ t is the automorphism group generated by Q −1 . Next, as in Proposition 1.5 of [15], consider the ultra-strong * closure M l of the subspace spanned by the elements of the form (id ⊗ ω)(β(x)), x ∈ M , ω is a bounded normal functional on Q ′′ r . It is enough to prove that M l = M, as that will prove the ultra-strong * density (now also the ultra-weak density) of M l in M. This is clearly a von Neumann subalgebra as β is coassociative, and . Then by Takesaki's theorem ( [24], 10.1) there exists a unique normal faithful conditional expectation E from M to M l satisfying E(x)P = P xp where P is the orthogonal projection as in [15]. Clearly, the range of P contains elements of the form (id ⊗ ω)(β(x)). So in particular it contains M 0 , which is dense in L 2 (φ). Thus P = 1 and E(x) = x proving M l = M. ✷ From this, it is easy to conclude the following by verbatim adaptation of the arguments of [21] and [23]. Proof: (1) Recall the linear functionals ρ π sm from Subsection 2.1. Let P π sm := (id ⊗ ρ π sm ) • ∆ : Q → Q and E π sm := (id ⊗ ρ π sm )ad V : M → M for π ∈ Rep(Q) and s, m = 1, ..., d π . Then P π sm (Q) ⊂ Sp{q π is : i = 1, ..., d π }. It follows from the arguments of [21], (2), Let {e πi1 } i∈J be a basis for M π1 for π ∈ Rep(Q), and e πis := E π s1 (e πi1 ), s = 1, ..., d π , i ∈ J . Then by (2), {e πis } i∈J is a basis for M πs . Letting M π i = Sp{e πis : s = 1, ..., d π } and using (3), we get (2) Here we reproduce the arguments of [23] for the sake of completeness. For where S is a finite subset of Rep(Q). Observe that for each π ∈ S, i, j = 1, Applying (id ⊗ id ⊗ ρ π kl ) on both sides, we get (3) follows from (2) and the SOT continuity of each P π . ✷ For x ∈ M 0 , we shall use the natural analogue of Swedler's notation, i.e. write ad V (x) = x (0) ⊗ x (1) .

Dual of a compact quantum group
Let (Q, ∆) be a compact quantum group and Q 0 be its canonical dense Hopf * -algebra as in the Subsection 2.1. We defineQ 0 to be the space of linear functionals on Q defined by q → h(aq), for a ∈ Q 0 , where h is the Haar state on Q. ThenQ 0 is a subspace of the dual Q ′ . For π ∈ Rep(Q), let F π be the positive invertible matrix as in the Subsection 2.1. Recall also the linear functionals ρ π sm 's from Subsection 2.1. Then we have from Lemma 8.1 of [17], ρ π pr (q π pr ) = 1 and ρ π pr is zero on all other matrix elements.
Note that (see [17])Q 0 is a * -subalgebra of Q ′ and as a * -algebra it is nothing but the algebraic direct sum of matrix algebras of the form ⊕ π∈Rep(Q) M dπ where M dπ is the full matrix algebra of size d π × d π and the matrix unit m π pq identified with ρ π pq is given by |e π p >< e π q |, where {e π 1 , ..., e π dπ } is the standard orthonormal basis. Clearly we get a non-degenerate pairing betweenQ 0 = ⊕ π∈Rep(Q) M dπ and Q 0 given by < q π ij , m π ′ ps >= δ ip δ js δ ππ ′ . In fact it can be shown thatQ 0 is a multiplier Hopf * -algebra in the sense of [25]. It has a unique C * norm and by taking the completion we get a C * algebra. It is called the dual discrete quantum group of Q denoted byQ. It has a coassociative * -homomorphism ∆ :Q → M(Q⊗Q) called the dual coproduct satisfying (id⊗∆)∆ = (∆ ⊗ id)∆. Proof: .., m π is equivalent to the irreducible representation of type π. Moreover fix orthonormal basis e π ij , j = 1, ..., d π , i = 1, ..., m π for H π i such that for all π ∈ Rep(Q). Now for a fixed π ∈ Rep(Q), p, r = 1, ..., d π observe that Π U (ρ π pr )(ξ) = 0 for all ξ ∈ H π ′ i and for π = π ′ . Also Π U (ρ π pr )(e π ij ) = δ jr e π ip , i.e. Π U (ρ π pr )| H π i is nothing but the rank one operator |e π ip >< e π ir |. This proves that Π U (ω) is bounded for ω ∈Q 0 , and moreover identifyingQ 0 with the direct sum of matrix algebras ⊕ π∈Rep(Q) M dπ , we see that Π U is nothing but the map which sends X ∈ M dπ to X ⊗ 1 C mπ in B(H). This proves that Π U extends to a non-degenerate * -homomorphism. ✷

Quantum isometry groups
Before defining quantum isometry group, we recall the definition of a spectral triple of compact type. Now let us briefly sketch the formulation of quantum isometry groups of spectral triples given by Goswami and Bhowmick in [7], [12].
where m π,k is the multiplicity of the irreducible representation of type π on H k and I k is some finite subset of Rep(Q). Since R commutes with V , R preserves direct summands of H k . Then we have the following: for some T π,k ∈ B(C m π,k ), where F π 's are as in Subsection 2.1.

Proof:
Only if part: let {e i } dπ i=1 and {f j } m π,k j=1 be orthonormal bases for C dπ and C m π,k respectively. Also let R(e i ⊗ f j ) = s,t R(s, t, i, j)e s ⊗ f t .
We have V * (e i ⊗ f j ⊗ 1 Q ) = k e k ⊗ f j ⊗ q * ik . We denote the restriction of the trace of B(H) on C dπ ⊗ C m π,k again by T r. Let a ∈ B(C dπ ⊗ C m π,k ) and χ(a) := T r(a.R). Then we have On the other hand Thus (χ ⊗ h)ad V (a) = χ(a) implies: Now fix u 0 , t 0 and consider a ∈ B(H) such that a(e u0 ⊗ f t0 ) = e p ⊗ f q and zero on the other basis elements. Then from (5), we get which gives i R(i,t0,i,q) Mπ F π (p, u 0 ) = R(u 0 , t 0 , p, q).
This proves that R| H k = ⊕ π∈I k F π ⊗ T π,k with some T π,k ∈ B(C m π,k ) given by T π,k (t 0 , q) = i R(i,t0,i,q) Mπ . The if part is straightforward and was essentially done in [13].✷ Let us note that in case of quantum group of orientation preserving Riemannian isometry, QISO + R (D) really depends on the von Neumann algebra (A ∞ ) ′′ , not on the algebra A ∞ itself. More precisely, we have the following proposition, which follows from the definition of QISO + R (D) and needs no proof: Proposition 3.9 Let (A ∞ , H, D, R) be as in [7]. If we have a SOT dense subalgebra . In [12], for a spectral triple satisfying some mild regularity conditions, a noncommutative analogue of the Hodge Laplacian was constructed which is denoted by L = L D . We denote the category whose objects are triples (S, ∆, α) where (S, ∆) is a CQG and α is an action of the CQG on the manifold commuting with the Laplacian (as in the sense of definition 2.11 of [12]) by Q ′ L and morphism between two such objects is a CQG morphism intertwining the actions (see [12]).
It is shown in [12] that under certain regularity conditions on the spectral triple, universal object exists in this category and we denote the universal object by QISO L and call it the quantum isometry group of the spectral triple.

Cocycle twisting
In Subsections 4.1 and 4.2, we recollect some well known facts about cocycle twist of CQG by dual unitary cocycles. There is nothing new in these two subsections and we mostly state the results with appropriate references. However, for convenience of the reader, we occasionally sketch a few proofs and give some details.

Cocycle twist of a compact quantum group
Let Q be a compact quantum group. Recall from Subsection 2.1 the dense Hopf * -algebra Q 0 spanned by the matrix coefficients of its inequivalent irreducible representations. Also recall the dual discrete quantum groupQ of Q. With these notations we have the following A dual unitary 2-cocycle σ is said to be normalized if (1⊗p ǫ )σ = 1⊗p ǫ and (p ǫ ⊗ 1)σ = p ǫ ⊗1, where p π 's are the minimal projections of ⊕ π∈Rep(Q) M dπ . Without loss of generality we can always assume a cocycle to be normalized and hence forth in our paper all dual unitary 2-cocycles are normalized. This means in particular σ(a, 1) = σ −1 (a, 1) = ǫ(a) for all a ∈ Q 0 . Recall from Subsection 2.3, the discrete quantum groupQ 0 with the coprod-uct∆ defined by∆(ω)(a ⊗ b) := ω(ab), w ∈Q 0 and a, b ∈ Q 0 . We can deform the discrete quantum groupQ 0 using σ ∈ M (Q⊗Q). The * algebraic structure do not change where the coproduct changes bŷ ∆ σ (.) = σ∆(.)σ −1 .
Using the cocycle condition of σ, it can be easily shown that∆ σ is again coassociative. It can be shown that (Q 0σ ,∆ σ ) is again a discrete quantum group. By Proposition 3.12 and discussions before Proposition 4.5 of [5], we see that we have a CQG dual to the discrete quantum group (Q 0σ ,∆ σ ). Now σ viewed as a linear functional on Q 0 ⊗ alg Q 0 satisfies the cocycle condition (see page 64 of [18]) for a, b, c ∈ Q 0 (Swedler's notation). We can deform Q 0 using σ to obtain a new Hopf * -algebra Q σ 0 . Then Q σ 0 andQ 0σ again form a non degenerate pairing. The product, * and κ are changed by the following formulas whereas the coproduct remains unchanged. For a, b ∈ Q 0 , κ σ (a) := W (a (1) )κ(a (2) )W −1 (a (3) ).
We refer the reader to page 65 of [18] for a proof of co-associativity of the new product as well as other relations of Hopf- * algebra. We note that the unit of the deformed Hopf * -algebra remains unchanged. For this note that (a. σ 1) = σ −1 (a (1)(1) , 1)a (1)(2) σ(a (2) , 1).
As the cocycle is normalized, for all a ∈ Q 0 . So a. σ 1 = ǫ(a (1)(1) )a (1)(2) ǫ(a (2) ) = a. Similarly it can be shown that 1. σ a = a for all Q 0 . We note that Q 0 σ is a dense Hopf * -algebra of the compact quantum group which has the discrete quantum groupQ 0σ as its dual, so we can consider the universal compact quantum group containing Q σ 0 as a dense Hopf * -algebra.
Definition 4.2 Call the universal CQG containing Q σ 0 as a dense Hopf *algebra to be the cocycle twist of the CQG by a dual unitary 2-cocycle σ and denote it by (Q σ , ∆).
Let us now discuss how one gets a dual unitary 2-cocycle on a CQG from such a dual unitary 2-cocycle on its quantum subgroup. Given two CQG's Q 1 , Q 2 and a surjective CQG morphism π : Q 1 → Q 2 which identifies Q 2 as a quantum subgroup of Q 1 , i.e. Q 2 ≤ Q 1 , it can be shown that π maps the Hopf * -algebra (Q 1 ) 0 onto (Q 2 ) 0 . By duality we get a map sayπ from (Q 2 ) ′ 0 to (Q 1 ) ′ 0 and it is easy to check that this indeed maps the dense multiplier Hopf * -algebrâ (Q 2 ) 0 ⊂Q 2 to(Q 1 ) 0 . Indeedπ lifts to a non degenerate * -homomorphism from M(Q 2 ) to M(Q 1 ). So given a dual unitary 2-cocycle σ on Q 2 , we get a dual unitary 2-cocycle σ ′ := (π ⊗π)(σ) ∈ M(Q 1⊗Q1 ). It is easy to check that σ ′ is again a dual unitary 2-cocycle on Q 1 . We shall often use the same notation for both σ ′ and σ i.e. denote σ ′ by σ under slight abuse of notation for convenience.

Proof:
By the universality of Q, it is enough to prove the Hopf- * algebra isomorphism. (Q σ 0 ) σ −1 ∼ = Q 0 Again for that it is enough to check the * -algebra structure. For Similarly we can show that the * -structure is preserved.✷

Unitary representations of a twisted compact quantum group
Let Q be a universal compact quantum group (as in the sense of 2.1) with a dual unitary 2-cocycle σ. In this Subsection we state (with brief sketches of proofs) a few well-known facts about the representation and the Haar state of Q σ .
Proposition 4.5 (i) For π ∈ Rep(Q), there is an irreducible unitary representation π σ of Q σ , which is the same as π as a linear map.

Proof:
(i) This follows since the C * tensor categories of unitary representations of the quantum groups Q and Q σ are the same ( [5]). However we give a brief proof using the fact that bothQ 0 and (Q 0 ) σ are * -isomorphic to ⊕ π∈Rep(Q) M dπ . Moreover recall the notation m π pq and the non-degenerate pairing discussed in the Subsection 2.3. We have < κ σ (q π ij ) * σ , m π pq > = < q π ij , m π * pq > = < q π ij , m π qp >(since the * structure does not change) = δ iq δ jp But we know < q π ji , m π pq >= δ iqjp . Hence by non-degeneracy of the pairing κ σ (q ij ) = q * σ ji , i.e. π σ is again a unitary representation of Q σ .✷ Thus, given a unitary representation U (possibly infinite dimensional) of Q on a Hilbert space H with the spectral decomposition H = ⊕ π∈Rep(Q),i H π i , we get a unitary representation U σ of Q σ on H defined uniquely as U σ | H π i = π σ for all i. It is clear that U → U σ is a bijective correspondence between unitary representations of Q and Q σ . Proposition 4.6 (i)The Haar state for the deformed compact quantum group is the same as that of the undeformed compact quantum group. (ii) The operator F σ π corresponding to the twisted CQG Q σ given by δ ik F σ π (j, l) = M dπ h(q π ij . σ q π * σ kl ) is related to F π by the following: where c π is some positive constant and A π ξ := (id ⊗ v)π σ ξ, v is defined by equation (10).
Proof: (i) This is a straightforward consequence of the isomorphism of the coalgebra structures of Q and Q σ . To be more precise, denote the Haar state of Q and Q σ by h and h σ respectively and observing from Proposition 4.5 that the matrix coefficients of π σ (π ∈ Rep(Q)) are {q π ij : i, j = 1, ..., d π }, we have h σ (q π ij ) = h(q π ij ) = 0 if π is non trivial and 1 for trivial representation. (ii) By equation (1), we see that the modular operator Φ| L 2 (h) π i = F π for all π and i. Let Φ σ be the modular operator for the CQG Q σ and S σ be the corresponding anti unitary operator. Then recalling the definition of the deformed * of Q σ (equation (7)), we see that Φ σ = SC, where C = AB = BA, A is the operator given by A(a) = (id ⊗ v)∆(a) and B is given by B(a) = (v −1 ⊗ id)∆(a) for a ∈ Q 0 . Then the modular operator Φ σ = S σ * S σ = A * B * ΦBA. We know Φ σ | L 2 (h) π i = F σ π for all π and i. As both A and A * map L 2 (h) π i into itself for all π and i, B * ΦB also does so. Fix some i and let P π i be the projection onto Sp {q π ij : j = 1, ..., d π }. It is clear from the definition of B that B(q π kl ) = m b π km q π ml and B * (q π kl ) = m d π km q π ml , for some constants b km and d km 's. Thus In particular, taking c π = c π,1 and denoting the restrictions of A and A * on Sp {q π 1j : 1 ≤ j ≤ d π } by A π and A * π respectively, we write F σ π = c π A * π F π A π and as F σ π is positive invertible, c π must be a positive constant.✷  (1) and V (ξ) = ξ (0) ⊗ ξ (1) with Swedler's notation. (2) ρ σ P π is SOT continuous, where P π is the spectral projection corresponding to π ∈ Rep(Q).

Deformation of a von Neumann algebra by dual unitary 2-cocycles
Proof: Using the facts that Π V (σ i ) and b i (0) 's are bounded operators we can conclude that ρ σ (b) ∈ B(H) for all b ∈ M 0 , which proves (1). The proof of (2) is very similar and hence omitted. ✷ Definition 4. 8 We call (ρ σ (M 0 )) ′′ the deformation of M by σ and denote it by M σ . Remark 4.9 1. If there is a weakly dense C * -subalgebra C of M such that the restriction of ad V leaves C invariant and also gives a C * -action on it, then the C * -algebraic deformation C σ in the sense of [20] can be defined. One can prove (which we plan to discuss in a forthcoming article) that Given such a dual unitary 2-cocycle we get corresponding induced dual unitary 2-cocycle on QISO + R (D), which will again be denoted by σ with a slight abuse of notation. Let U be a unitary representation of QISO + R (D) on H. Then U commutes with D. From now onwards we shall denote the von Neumann algebra (A ∞ ) ′′ inside B(H) by M. As in section 3 (right after Remark 3.6), decompose H into the eigenspaces H k , k ≥ 1 and also decompose each H k into spectral subspaces, say:  Hence (id ⊗ φ)ad U (b) ∈ G for all bounded linear functionals φ of QISO + R (D) i.e. G is ad U invariant SOT-dense * subalgebra of M. Now (1),(2) follow from part (3) of Lemma 2.4, taking A 0 to be the span of the subspaces P π (G), π ∈ Rep(Q). We also get (3) by 2 of Remark 4.9. To prove (4), observe that ∀a ∈ A 0 , ρ σ (a) ∈ B(H) by the proof of (1) of the Lemma 4.7. So we only need to check that [D, ρ σ (a)] ∈ B(H) for all a ∈ A 0 . With the notations used before, consider a ∈ A 0 and ξ in the linear span of H k 's, writing ad

Deformation of a spectral triple by dual unitary 2cocycles
[D, a i (0) ]Π U (σ i )(ξ) (using the commutativity of D and U ).
As [D, a i (0) ] is bounded for all i = 1, ..., k, we conclude that [D, ρ σ (a)] is bounded for all a ∈ A 0 .✷ Remark 4. 11 We can re-cast the deformed spectral triple (A σ 0 , H, D) in the framework of Neshveyev-Tuset ( [20]). Consider the image K of the representation V viewed as an isometric Hilbert space operator from H to H ⊗ L 2 (Q, h) (where h is the Haar state of Q). It can be shown that (we plan to discuss this in a forthcoming article) is the realisation of the deformed spectral triple in the Neshveyev-Tuset framework. We have the following: and v is as in equation (10).

Proof:
Recall the decomposition (Section 3) of the Hilbert space where m π,k is the multiplicity of the irreducible representation π on H k and I k is some finite subset of Rep(Q). By Theorem 4.5, V σ is again a unitary representation of Q σ on the Hilbert space H. Also note that ad V is algebraic over A 0 . Let a ∈ A 0 and ξ ∈ N , where N is the subspace of H given by span of H k 's. Then we have On the other hand, for all a ∈ A 0 . Also for φ ∈ (Q σ ) * , So in particular (id ⊗ φ)ad Vσ (A 0 ) ⊂ (A 0 ) ′′ . V σ commutes with D since as a linear map V σ is same as V . By Theorem 3.8, R has the form ⊕ π∈I k ,k≥1 F π ⊗ T π,k , for some T π,k ∈ B(C m π,k ) so that Π V (v) * RΠ V (v) is of the form ⊕ π,k A * π F π A π ⊗ T π,k , for some T π,k where A π , F π 's are as in (ii) of Proposition 4.6. Now by (ii) of Proposition 4.6, we see that F σ π = c π A * π F π A π for some positive constant c π . Then R σ is of the form ⊕ π∈I k ,k≥1 F σ π ⊗ c −1 π T π,k , which implies by the 'if part' of Theorem 3.8 that ad Vσ preserves the R σ -twisted volume. ✷ Remark 4.13 By looking at the proof we can easily conclude that if (Q, V ) is an object in the category Q ′ (D), then (Q σ , V σ ) is an object in the category Q ′ (D σ ). Now replacing Q by QISO + R (D) and using the dual unitary 2-cocycle on QISO + R (D) induced from σ on its quantum subgroup, we get: Corollary 4.14 QISO + R (D) σ ≤ QISO + R σ (D σ ).

Remark 4.18
We can also obtain similar result for the Laplacian based approach i.e. in the category Q ′ L , whenever it makes sense. Combining with Proposition 3.11, we conclude that the quantum isometry groups (QISO + I and QISO L ) for the noncommutative manifolds obtained by Rieffel deformation of an arbitrary compact, connected Riemannian manifold are similar deformation of the classical isometry groups.

Remark 4.19
Viewing the Rieffel deformation as a special case of cocycle twist, the above theorem in fact improves the result (Theorem 5.13 of [7]) obtained by Bhowmick and Goswami by removing the assumption about a nice dense subalgebra on which the adjoint action of the quantum isometry group is algebraic. In fact, techniques of this paper have enabled us to prove existence of such nice algebra in general.
Remark 4.20 It should be mentioned that our result does not cover the examples constructed by Neshveyev-Tuset in [19]. In that paper they considered the problem of deforming a spectral triple coming from classical compact Lie groups using a Drinfeld twist. It should be noted that although 2-cocycle is an example of a Drinfeld twist, our scheme fails for a general Drinfeld twist. The main problem lies in transporting a unitary Drinfeld twist to a compact quantum group from its quantum subgroup. We are grateful to Sergey Neshveyev for pointing out this difficulty which was overlooked mistakenly in the previous version of this paper. However,it seems that techniques developed in [11] can be adapted to get the main result of this paper for the spectral triples of [19]. We are working in that direction and hope to report it in another article.