Quantum Isometry Group of Deformation: A Counterexample

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ANNALES MATHÉMATIQUES

BLAISE PASCAL

Debashish Goswami & Arnab Mandal

Quantum Isometry Group of Deformation: A Counterexample Volume 26, n^o1 (2019), p. 55-65.

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Publication éditée par le laboratoire de mathématiques Blaise Pascal de l’université Clermont Auvergne, UMR 6620 du CNRS

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Quantum Isometry Group of Deformation: A Counterexample

Debashish Goswami Arnab Mandal

Abstract

We give a counterexample to show that the quantum isometry group of a deformed finite dimensional spectral triple may not be isomorphic with a deformation of the quantum isometry group of the undeformed spectral triple.

1. Introduction

Quantum groups play an important role in several areas of mathematics and physics, often as some kind of generalised symmetry objects. Beginning from the pioneering work by Drinfeld, Jimbo, Manin, Woronowicz and others nearly three decades ago ([9, 12, 19, 21]

and references therein) there is now a vast literature on quantum groups both from algebraic and analytic (operator algebraic) viewpoints. Generalizing the concept of group actions on spaces, notions of (co)actions of quantum groups on possibly noncommutative spaces have been formulated and studied by many mathematicians in recent years. In 1998, S. Wang [20] initiated this programme by defining quantum automorphism groups of certain mathematical structures (typically finite sets, matrix algebras etc.). After that, a number of mathematicians including Banica, Bichon and others ([1, 2, 8] and references therein) formulated the notion of quantum symmetries of finite metric spaces and finite graphs. With the motivation of connecting Wang’s quantum automorphism groups with a more geometric framework, one of the authors of the present article [14] defined and proved existence of an analogue of the group of isometries of a Riemannian manifold, in the framework of the so-called compact quantum groups à la Woronowicz. In fact, he considered the more general setting of noncommutative manifold, given by spectral triples defined by Connes [10] and under some mild regularity conditions, he proved the existence of a universal compact quantum group (termed as the quantum isometry group) acting on theC^∗-algebra underlying the noncommutative manifold such that the action also commutes with a natural analogue of Laplacian of the spectral triple. We refer the reader to the original article [14] and the recently published book [7] for the details of the theory of quantum isometry groups. In this context, a very natural and important question is whether the quantum isometry group of deformation of some noncommutative

First author is partially supported by J.C Bose Fellowship and grant from D.S.T(Govt of India).

Keywords:Compact quantum groups, Quantum isometry group, Spectral triple.

2010Mathematics Subject Classification:58B34, 46L87, 46L89.

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manifold is isomorphic with certain deformed version of the quantum isometry group of the original undeformed noncommutative manifold. That is, whether the functor “QISO”

which assigns to a noncommutative manifold its quantum isometry group “commutes”

with the functor of deformation. This question has been answered in the affirmative for Rieffel type cocycle deformation by Goswami, Bhowmick, Joardar [5, 15] and for the more general “monoidal deformation” by Sadeleer [11]. The computation of the quantum isometry groups of Podles spheres [6] indicates that there may be an affirmative answer for a bigger class of deformations. However, it is not true in general as the isometry group of a classical Riemannian manifold may drastically change by a slight perturbation of the Riemannian metric. The aim of the present article is to give an example of flat deformation of finite dimensional spectral triples on theC^∗-algebras of finite groups for which the corresponding quantum isometry groups are not flat deformation. Note that such examples can’t be produced for classical Riemannian geometry since there exists only one Riemannian metric, so that there is no room for deformations.

2. Background Materials

We very briefly discuss the basic definitions and recall some standard facts about noncommutative geometry and quantum isometry groups. We refer [10, 17, 21] for more details. Let us fix some notational convention. We denote the algebraic tensor product and spatial (minimal)C^∗-tensor product by⊗and ˇ⊗respectively. We’ll use the leg-numbering notation. LetHbe a complex Hilbert space,K(H )theC^∗-algebra of compact operators on it, andQa unitalC^∗-algebra. The multiplier algebraM(K(H )⊗ Q)ˇ has two natural embeddings intoM(K(H )⊗ Qˇ ⊗ Q), one obtained by extending the mapˇ x7→x⊗1 and the second one is obtained by composing this map with the flip on the last two factors.

We will writeω¹² andω¹³ for the images of an elementω ∈ M(K(H )⊗ Q)ˇ under these two maps respectively. We’ll denote byH ⊗ Q¯ the HilbertC^∗-module obtained by completingH ⊗ Qwith respect to the norm induced by theQvalued inner product hhξ⊗q, ξ⁰⊗q⁰ii:=hξ, ξ⁰iq^∗q⁰, whereξ, ξ⁰∈ Handq,q⁰∈ Q.

2.1. Spectral triple and compact quantum groups

Definition 2.1. A spectral triple is a triple(A^∞,H,D)whereHis a separable Hilbert space,A^∞ is a∗-subalgebra ofB(H ), (not necessarily norm closed) andDis a self adjoint (typically unbounded) operator such that for alla∈ A^∞, the operator[D,a]has a bounded extension. Such spectral triple is also called an odd spectral triple. If in addition, we haveγ∈ B(H )satisfyingγ=γ^∗=γ⁻¹,Dγ=−γDand[a, γ]=0 for alla∈ A^∞,

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then we say the quadruplet(A^∞,H,D, γ)is an even spectral triple. The operatorDis called the Dirac operator corresponding to the spectral triple.

We say that the spectral triple is of compact type ifA^∞is unital andDhas compact resolvent. In this article, we will consider only odd spectral triple.

Definition 2.2. A compact quantum group (CQG in short) is a pair(Q,∆), whereQ is a unitalC^∗- algebra and∆:Q → Q⊗ Qˇ is a unitalC^∗-homomorphism (called the comultiplication), such that

(1) (∆⊗id)∆=(id⊗∆)∆as homomorphismQ → Q⊗ Qˇ ⊗ Qˇ (coassociativity).

(2) The spaces∆(Q)(1⊗ Q) =Span{∆(b)(1⊗a) |a,b ∈ Q}and∆(Q)(Q ⊗1)= Span{∆(b)(a⊗1) |a,b∈ Q}are dense inQ⊗ Q.ˇ

A CQG morphism from(Q₁,∆₁)to another(Q₂,∆₂)is a unitalC^∗-homomorphism π:Q₁7→ Q₂such that(π⊗π)∆₁=∆₂π.

Definition 2.3. We say that a CQG(Q,∆)acts on a unitalC^∗-algebraBif there is a unital C^∗-homomorphism (called action)α:B → B⊗ Qˇ satisfying the following:

(1) (α⊗id)α=(id⊗∆)α.

(2) The linear span ofα(B)(1⊗ Q)is norm-dense inB⊗ Q.ˇ

Definition 2.4. Let(Q,∆)be a CQG. A unitary representation ofQon a Hilbert space H is aC-linear mapUfromHto the Hilbert moduleH⊗ Q¯ such that

(1) hhU(ξ),U(η)ii=hξ, ηi1Q, whereξ, η∈ H.

(2) (U⊗id)U=(id⊗∆)U.

(3) Span{U(H )Q}is dense inH⊗ Q.¯

Given such a unitary representation we have a unitary element Ue belonging to M(K(H )⊗ Q)ˇ given byU(ξe ⊗b)=U(ξ)b,(ξ ∈ H, b ∈ Q)satisfying(id⊗∆)(eU)= Ue¹²Ue¹³.

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2.2. Quantum isometry groups

In [14] the first author introduced the notion of quantum isometry group of a spectral triple satisfying certain regularity conditions. We refer to [3, 4, 14] for the original formulation of quantum isometry groups and its various avatars including the quantum isometry group for orthogonal filtrations.

Definition 2.5. Let (A^∞,H,D) be a spectral triple of compact type (à la Connes).

Consider the categoryQ(D) ≡Q(A^∞,H,D)whose objects are(Q,U), where(Q,∆)is a CQG having a unitary representation U on the Hilbert spaceH satisfying the following:

(1) Uecommutes with(D ⊗1Q).

(2) (id⊗φ) ◦ad

Ue(a) ∈ (A^∞)⁰⁰for alla ∈ A^∞ and φis any state on Q, where adUe(x):=U(xe ⊗1)eU^∗forx∈ B(H ). Note thatad

Ueis faithful onA^∞. A morphism between two such objects(Q,U)and(Q⁰,U⁰)is a CQG morphismψ:Q → Q⁰such thatU⁰=(id⊗ψ)U. If a universal object exists inQ(D)then we denote it by QI SO⁺(A^∞,H,D)and the corresponding largest Woronowicz subalgebra for which adUf0is faithful, whereU₀is the unitary representation ofQI SO⁺(A^∞,H,D), is called the quantum group of orientation preserving isometries and denoted byQI SO⁺(A^∞,H,D).

Let us state Theorem 2.23 of [4] which gives a sufficient condition for the existence of QI SO⁺(A^∞,H,D).

Theorem 2.6. Let(A^∞,H,D)be a spectral triple of compact type. Assume thatDhas one dimensional kernel spanned by a vectorξ ∈ Hwhich is cyclic and separating for A^∞and each eigenvector ofDbelongs toA^∞ξ. Then QISO⁺(A^∞,H,D)exists.

Here we briefly discuss a specific case of interest for us. For more details see [16, Section 2.2].

Let Γbe a finitely generated discrete group with a symmetric generating setSnot containing the identity ofΓ(symmetric meansg∈Sif and only ifg⁻¹ ∈S) and letlbe the corresponding word length function. We define an operatorDΓbyDΓ(δ_g)=l(g)δ_g, whereδ_gdenotes the vector inl²(Γ)which takes value 1 at the pointgand 0 at all other points. Note thatδ_g,g ∈ Γ forms an orthonormal basis ofl²(Γ). Letτ be the faithful positive functional on the reduced groupC^∗-algebraC_r^∗(Γ)given byτ(Í

gc_gλ_g)=c_e, whereeis the identity element ofΓ. Then QISO⁺(CΓ,l²(Γ),DΓ)exists by Theorem 2.6, takingδ_eas the cyclic separating vector forCΓ.

We also refer to [3] for the related notion of orthogonal filtration and note that the above quantum isometry group is the quantum symmetry group of the orthogonal

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filtration onCΓwith respect to the tracial stateτand the filtrationF_l =(V_1,k)_k_≥0, where V_1,k =Span{λ_g:l(g)=k}. Moreover, we make the following useful observation:

Theorem 2.7. LetP0andP1be the orthogonal projections onto the subspacesV1,0,V1,1

respectively and letP₂=1− (P₀+P₁). DefineD=P₁+2P2. Then for a finite groupΓ, QI SO⁺(CΓ,l²(Γ),D)exists and is isomorphic withQI SO⁺(CΓ,l²(Γ),DΓ).

Proof. LetQ₁=QI SO⁺(CΓ,l²(Γ),DΓ)andQ₂ =QI SO⁺(CΓ,l²(Γ),D). It follows from Theorem 2.10 and Theorem 3.2 of [3] thatQ2is the quantum symmetry group of the orthogonal filtrationF_D =(V_2,k)_k=0,1,2and with respect to the traceτ, whereV_2,k =V_1,k fork=0,1 andV2,2 =∪_k_≥2V1,k =Span{λ_g|l(g) ≥2}. AsF_l is a refinement ofF_D, it is clear thatQ₁ is a subobject ofQ₂(in the category ofF_D preserving quantum groups).

Hence it suffices to show thatQ₂is also a subobject ofQ₁in the category ofF_lpreserving quantum groups. But by definition, the coaction ofQ₂leavesV_1,1invariant and preserves the traceτ, hence it is an object in the categoryC_τintroduced in [16]. By Lemma 2.14

of [16] we conclude thatQ2is a subobject ofQ1.

LetΓ₁andΓ₂be two finite groups with identity elementseande⁰respectively. Assume that F₁ =(V_i)i=0,1,...,k and F₂ =(eV_j)_j_=0,1,...,_l are two orthogonal filtrations ofC^∗(Γ₁) andC^∗(Γ₂)with respect to the canonical tracesτ₁ andτ₂ respectively. Moreover, let QF_i be the quantum isometry groups ofC^∗(Γ_i)fori =1,2. For eachg ∈ Γ₁,g⁰ ∈ Γ₂ consider the subspacesV_(i,g⁰)=Span{b⊗λ_g⁰|b∈V_i}andVe_(g,j)=Span{λ_g⊗a|a∈Ve_j} foralli=0,1, . . . ,kandj =0,1, . . . ,linside the vector spaceC^∗(Γ₁) ⊗C^∗(Γ₂). Clearly, F =(V(i,g⁰))_i₌0,1,...,k,g⁰∈Γ₂andF⁰=(eV(g,j))_j₌0,1,...,l,g∈Γ₁are two orthogonal filtrations for C^∗(Γ₁) ⊗C^∗(Γ₂), i.e.C^∗(Γ₁) ⊗C^∗(Γ₂)=É

i,g⁰V_(i,g⁰)andC^∗(Γ₁) ⊗C^∗(Γ₂)=É

i,gVe_(g,i) with respect to the stateτ₁ ⊗τ₂. LetQF andQF⁰ be the quantum isometry groups of C^∗(Γ₁) ⊗C^∗(Γ₂)corresponding to the filtrationsF andF⁰respectively. Let us assume that {s₁, . . . ,s_p}is a generating set of the groupΓ₁andV₁=Span{λ_s₁, . . . , λ_s_p}. Furthermore, assume that the action ofQF₁ onC^∗(Γ₁)is defined byα₁(λ_s_i)=Íp

j=1λ_s_j ⊗qji, where the underlyingC^∗-algebra ofQF₁is generated byq_{i j}’s fori,j =1, . . . ,p.

Theorem 2.8. QF QF₁ ⊗C^∗(Γ₂).

Proof. First of all, note that any actionαonC^∗(Γ₁) ⊗C^∗(Γ₂)is determined byα(λ_s_i⊗λ_e⁰) and α(λ_e ⊗λ_g⁰) forall i = 1, . . . ,p and g⁰ ∈ Γ₂. Let αF be the action of QF on C^∗(Γ₁) ⊗C^∗(Γ₂). AsV_(0,g⁰)andV_(1,e⁰)are members of the filtrationF,αFmust preserve each of them and hence αF must satisfyαF(λ_e ⊗λ_g⁰) = λ_e⊗λ_g⁰ ⊗q_g⁰ ∀ g⁰ ∈ Γ₂ and αF(λ_s_i ⊗λ_e⁰) = Íp

j=1λ_s_j ⊗λ_e⁰ ⊗q⁰_ji forall i = 1, . . . ,p where {q_g⁰,q⁰_ji,g⁰ ∈ Γ₂,i,j = 1, . . . ,p} ⊆ QF. In fact, as {λ_e⊗λ_g⁰, λ_s_i ⊗λ_e⁰,g⁰ ∈ Γ₂,i = 1, . . . ,p} is a set of generators for the C^∗-algebra C^∗(Γ₁) ⊗C^∗(Γ₂) and αF is faithful, the set

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{q_g⁰,q⁰_ji,g⁰∈Γ₂,i,j=1, . . . ,p}is a set of generators of theC^∗-algebraQF. Now define β : C^∗(Γ₁) ⊗C^∗(Γ₂) 7→ C^∗(Γ₁) ⊗C^∗(Γ₂) ⊗ QF₁ ⊗C^∗(Γ₂) by β = σ₂₃◦ (α₁ ⊗∆_Γ₂), where σ₂₃ denotes the isomorphism between C^∗(Γ₁) ⊗ QF₁ ⊗C^∗(Γ₂) ⊗C^∗(Γ₂) and C^∗(Γ₁) ⊗C^∗(Γ₂) ⊗ QF₁ ⊗C^∗(Γ₂)which interchanges the second and third tensor copies and∆_Γ₂is the usual coproduct onC^∗(Γ₂). Clearly,βis aC^∗-action ofQ_F₁⊗C^∗(Γ₂)on C^∗(Γ₁) ⊗C^∗(Γ₂)and it satisfiesβ(λ_e⊗λ_g⁰) =λ_e⊗λ_g⁰ ⊗1Q_F

1 ⊗λ_g⁰ ∀ g⁰ ∈ Γ₂ and β(λ_s_i ⊗λ_e⁰)=Íp

j=1λ_s_j ⊗λ_e⁰⊗q_ji⊗λ_e⁰foralli =1, . . . ,p, where 1Q_F

1 is the unit of QF₁. Thus,βpreserves the filtrationF which implies the existence of a well defined surjectiveC^∗-homomorphism fromQF toQF₁ ⊗C^∗(Γ₂)sending q_g⁰ toλ_g⁰andq⁰_ji to q_ji,g⁰∈Γ₂,i,j =1, . . . ,p. We claim that there is an inverse of this morphism, namely a C^∗-homomorphism fromQF₁⊗C^∗(Γ₂)toQFwhich sendsλ_g⁰toq_g⁰andq_jitoq⁰_ji. To this end, observe that asαFis aC^∗-homomorphism andαF(λ_e⊗λ_g⁰)=λ_e⊗λ_g⁰⊗q_g⁰, we must have(λ_g⁰

1⊗q_g⁰

1)(λ_g⁰

2 ⊗q_g⁰

2)=λ_g⁰

1g⁰₂ ⊗q_g⁰

1g⁰₂forallg₁⁰,g₂⁰ ∈Γ₂, i.e.q_g⁰

1 ·q_g⁰

2=q_g⁰

1g⁰₂. Similarly,q_e⁰ =1 andq_g⁰⁻¹=(q_g⁰)⁻¹ =q^∗_g0. Thus,g⁰7→q_g⁰is a group homomorphism fromΓ₂to the group of unitaries ofQF, hence by the universality ofC^∗(Γ₂)we get a C^∗-homomorphism (sayρ) fromC^∗(Γ₂)toQ_Fsuch that ρ(λ_g⁰)=q_g⁰ ∀g⁰∈Γ₂. Next, note thatαFpreserves theC^∗-subalgebraC^∗(Γ₁) ⊗λ_e⁰(C^∗(Γ₁)) ⊆C^∗(Γ₁) ⊗C^∗(Γ₂), and clearly the restriction ofαF to this subalgebra gives aF₁- preserving action onC^∗(Γ₁).

Hence we get aC^∗-homomorphism, sayθ, fromQF₁ toQF such thatθ(q_ji)= q⁰_ji ∀ i,j=1, . . . ,p. Moreover, asC^∗(Γ₁) ⊗λ_e⁰andλ_e⊗C^∗(Γ₂)are commutative subalgebras ofC^∗(Γ₁) ⊗C^∗(Γ₂), their images underαFmust commute too. From this, it easily follows thatq_g⁰·q⁰_ji=q⁰_ji·q_g⁰ ∀g⁰∈Γ₂,i,j =1, . . . ,p.Thus, the images ofθandρcommute implying the existence of aC^∗-homomorphismθ⊗ρ:Q_F₁⊗C^∗(Γ₂) 7→ Q_Fwhich sends

q_ji⊗λ_g⁰toq_g⁰·q⁰_ji.

Similarly, it can be shown thatQF⁰ QF₂ ⊗C^∗(Γ₁).

2.3. Deformation

There are different notions of deformation of spaces, algebras and operators found in the literature. Let us specify the notion which we are concerned with. For a more general, abstract setting of deformation theory we refer to the seminal work of Gerstenhaber and others (see, e.g. [13] and references therein).

We will consider families of vector spacesW_h,h∈I, whereIis an open interval inR. Definition 2.9. Let {W_h}_h∈I be a family of vector subspaces of some finite dimensional vector space W. {W_h} is called a continuous deformation of W_h₀ if there areW-valued continuous functions w_i,h,i = 1, . . . ,n (where dim(W) = n) such that W_h=Span{w₁(h), . . . ,w_n(h)} ∀h∈I.

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From now on, we just call it deformation by dropping the word continuous. Moreover, {W_h}_h_∈I is called a flat deformation ofWh0if dim(Wh)does not depend onh.

Definition 2.10. Let{V_h}_h∈Ibe a family of filtered vector spaces, i.e.Vh =∪_i_≥0Fi(V_h) with the conditonF₀(V_h) ⊆. . .F_i(V_h) ⊆ · · · ⊆V_hof finite dimensional subspaces ofV_h. {V_h}is called a deformation ofV_h₀ if for each i, the family of subspaces{F_i(V_h)}is a deformation ofF_i(V_h₀)as in Definition 2.9.

Such deformation is called a flat deformation if dim(Fi(V_h))does not depend onhfor any fixedi.

Definition 2.11. Let{A_h}_h∈Ibe a family of filtered unital algebras, i.e.A_h=∪_i_≥0F_i(A_h) with the conditonsF₀(A_h) ⊆. . .F_i(A_h) ⊆ · · · ⊆A_hof finite dimensional subspaces of A_handF_i(A_h).F_j(A_h) ⊆F_i+j(A_h) ∀i,j. Moreover, we assume thatF₀(A_h)=C.1 for all h. ThenA_his called a deformation ofA_h₀if for each i, there is a finite dimensional vector spaceViandVivalued continuous functionsw_i,j(h),j=1, . . . ,ni (where dim(Vi)=ni) such that F_i(V_h) = Span{wi,1(h), . . . ,w_i,n_i(h)} and the map h 7→ w_i,k(h).w_j,l(h) is continuous foralli,j,k,lwith 1≤k ≤ni, 1≤l ≤nj.

Remark 2.12. Any finitely generated algebra can be considered as a filtered algebra. Given such an algebraAwith a unit 1 and a finite set of generators{a₁, . . . ,a_k}, we consider the filtration given byA₀=C1,A₁=Span{1,a₁, . . . ,a_k},A_n=Span{1,a_i₁. . .a_i_m,1≤m≤ n,il ∈ {1, . . . ,k} ∀l}. Hence the definition of deformation of filtered algebras applies to any arbitrary finitely generated algebra.

Definition 2.13. The family of filtered algebras{A_h}_h_∈I is called a flat deformation of A_h₀if for any fixedi, the dimension ofF_i(A_h)does not depend onh.

Remark 2.14. It is clear that a flat deformation{A_h}of A_h₀ is uniquely determined by a family •_h of associative algebra multiplication from A_h₀ ⊗A_h₀ to A_h₀ such that h7→ω(a•_hb)is continuous for allω∈A_h⁰

0 (dual vector space),a,b∈A_h₀. In fact, one can take this as a definition of flat deformation, for not necessarily filtered algebraAh0. Definition 2.15. A family(A_h,H_h,D_h)of finite dimensional spectral triples of compact type will be called a flat deformation if

(1) There exists continuous functionsλ_i(h), i=1, . . . ,Nsuch thatλ_i(h),λ_j(h) ∀ i, j,{λi(h)}is the set of eigenvalues ofDh.

(2) IfV_i(h)denotes the eigenspaces ofD_hcorresponding toλ_i(h), then{V_i(h),h∈ I}

is a flat deformation of finite dimensional vector spaces∀i.

(3) {A_h}_h∈I is a flat deformation of filtered unital *-algebras too.

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Finally, we need an appropriate notion of flat deformation of Hopf algebras, which are finitely generated as algebras.

Definition 2.16. Let{(Q_h,∆_h)}_h_∈I be a family of Hopf algebras with∆_h denoting the coproduct. Suppose that each{Q_h}_h∈I is finitely generated as an algebra and{Q_h}_h∈I is a flat deformation ofQ_h₀ with respect to the filtered algebra structure corresponding to a finite generating set ofQh0. Thus by Remark 2.14 we can identify eachQh with Q_h₀ for any fixedh₀ ∈ Iso that the multiplication map•_his continuous in the sense discussed in that remark. If furthermore,∆_hviewed as a map fromQ_h₀ toQ_h₀ ⊗Q_h₀is continuous in a similar sense, i.e.h7→ (β⊗id)(∆_h(q))is continuous∀q∈Q_h₀and for all linear functional βonQ_h₀ ⊗Q_h₀, we say that(Q_h,∆_h)}_h∈I is a flat deformation of finitely generated Hopf algebras.

Remark 2.17. By construction of [3] the underlying Hopf algebra of the quantum isometry group for an orthogonal filtration is always finitely generated, hence the above definition applies to such Hopf algebras. A similar remark can be made about the underlying Hopf algebra ofQI SO(A,H,D)of a finite dimensional spectral triple.

3. The Counterexample

It is now natural to ask the following question: does the quantum isometry groups of a flat deformation of spectral triples again form a flat deformation of compact quantum groups?

The answer to this question is negative even if the spectral triples considered are finite dimensional, as shown by the counterexample given by the anonymous referee, which is briefly described below.

ConsiderA^∞=C²acting on the 2 dimensional Hilbert spaceH =C²and two Dirac operators

D₀=1/2 1 1

1 1

, D₁=1/5 1 2

2 4

,

which are rank 1 projections. We take a homotopy Dt connecting D0 and D1 via rank 1 projections, then(A^∞,H,D_t)is a flat deformation of spectral triples, satisfying the assumption of Theorem 2.6. Notice that the only quantum automorphism group of A^∞ isZ2, which preserves D₀ but not D₁. HenceQI SO⁺(A^∞,H,D₀)is Z2 but QI SO⁺(A^∞,H,D₁)is trivial.

However, we would like to give another counterexample, where the initial and final spectral triples of the deformed family arise from a finite group with two different generating sets. From now onwards the two groupsZ9×Z3andZ9o Z3are denoted by Γ₁andΓ₂respectively. Moreover, we denote the identity elements ofΓ₁andΓ₂byeand

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e⁰respectively. Note that

Γ₁ ={a,b|ab=ba,a⁹=b³=e}, Γ₂={x,y|xyx⁻¹=y⁴,x³=y⁹=e⁰}.

ConsiderV₀=Cλ_e,V₁=Span{λ_a, λ_a⁻¹, λ_b, λ_b⁻¹}andV₂=Span{λ_t|t∈Γ₁−{e,a,a⁻¹,b,b⁻¹}}. Observe thatF₁=(V_i)_i=0,1,2gives an orthogonal filtration forC^∗(Γ₁). Similarly, consider Ve0=Cλe⁰,Ve1=Span{λx, λ_x⁻¹, λy, λ_y⁻¹},Ve2 =Span{λs|s∈Γ₂− {e⁰,x,x⁻¹,y,y⁻¹}}and F₂=(eV_i)_i=0,1,2is an orthogonal filtration forC^∗(Γ₂). Moreover, for eachg∈Γ₁,g⁰∈Γ₂ and i = 0,1,2 we can consider the subspaces Ve_(g,i) = Span{λ_g ⊗a|a ∈ Ve_i} and V_(i,g⁰)=Span{b⊗λ_g⁰|b∈V_i}insideC^∗(Γ₁) ⊗C^∗(Γ₂). Clearly,F =(V_(i,g⁰₎)_i=_0,1,2,g⁰∈Γ₂

and F⁰ = (eV_(g,i))_i=0,1,2,g∈Γ₁ are two orthogonal filtrations for C^∗(Γ₁) ⊗C^∗(Γ₂), i.e.

C^∗(Γ₁)⊗C^∗(Γ₂)=É

(i,g⁰)V_(i,g⁰)andC^∗(Γ₁)⊗C^∗(Γ₂)=É

(g,i)Ve_(g,i). Assume thatg⁰7→g is a bijective map fromΓ₂toΓ₁. We can consider a unitary operatorUonl²(Γ₁) ⊗l²(Γ₂) such thatU(V_(i,g⁰₎) =Ve_(g,i). Observe that σ(U)is a finite subset of the unit circle as l²(Γ₁) ⊗l²(Γ₂)is a finite dimensional vector space. Using an appropriate branch of loga- rithm not intersecting the setσ(U)we get a self adjoint matrixSsuch thatU=e^iS. Then we can define a family of unitary operatorsU^h=e^ihS ∀h ∈ [0,1]. Consider the spaces V_(i,g⁰),h =U^h(V_(i,g⁰))forallhand define the operatorsD_h=Í

i=0,1,2,g⁰∈Γ₂n(i,g⁰)P_(i,g⁰),h

on l²(Γ₁) ⊗ l²(Γ₂), where n(i,g⁰) ∈ N∪ {0},n(i₁,g₁⁰) , n(i₂,g₂⁰) if (i₁,g₁⁰) , (i₂,g₂⁰) andP_(i,g⁰_),his the orthogonal projection ontoV_(i,g⁰_),h. Observe thatV_(i,g⁰_),0 =V_(i,g⁰)and V_(i,g⁰_),1 =Ve_(g,i)for eachg⁰ ∈Γ₂ andi =0,1,2. Consider the family of spectral triples (A_h,H_h,D_h), whereA_h =C^∗(Γ₁) ⊗C^∗(Γ₂),H_h =l²(Γ₁) ⊗l²(Γ₂)forall handD_h is defined as above. Note that this family is clearly a flat deformation in the sense of Definition 2.15 andQI SO⁺(C^∗(Γ₁) ⊗C^∗(Γ₂),l²(Γ₁) ⊗l²(Γ₂),D_h)exists by Theorem 2.6, takingξ=δ_e⊗δ_e⁰as a cyclic, separating vector forC^∗(Γ₁) ⊗C^∗(Γ₂). Moreover, using Theorem 2.7, we have

QF₁ QI SO⁺(C^∗(Γ₁),l²(Z9) ⊗l²(Z3),DΓ1), QF₂ QI SO⁺(C^∗(Γ₂),l²(Z9) ⊗l²(Z3),DΓ2).

However, QI SO⁺(C^∗(Γ₁),l²(Z9) ⊗ l²(Z3),DΓ1) and QI SO⁺(C^∗(Γ₂),l²(Z9) ⊗ l²(Z3), DΓ2)have been already computed in [16, 18] respectively and one has the following:

QI SO⁺(C^∗(Γ₁),l²(Z9) ⊗l²(Z3),DΓ1)[C^∗(Z9) ⊕C^∗(Z9)] ⊗ [C^∗(Z3) ⊕C^∗(Z3)], QI SO⁺(C^∗(Γ₂),l²(Z9) ⊗l²(Z3),DΓ2)C^∗(Z9o Z3) ⊕C^∗(Z9o Z3)

with the coproduct structures discussed in [16] and [18] respectively. From this, we can easily conclude the following:

Theorem 3.1. QI SO⁺(C^∗(Γ₁) ⊗C^∗(Γ₂),l²(Γ₁) ⊗l²(Γ₂),D_h)is not a flat deformation.

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Proof. Note that the dimension of the underlying vector space of QI SO⁺(C^∗(Γ₁), l²(Z9) ⊗l²(Z3),DΓ1) is 108, whereas the dimension of the underlying vector space ofQI SO⁺(C^∗(Γ₂),l²(Z9) ⊗l²(Z3),DΓ2)is 54.Thus, by Theorem 2.8, the dimensions of the underlying vector spaces of QI SO⁺(C^∗(Γ₁) ⊗C^∗(Γ₂),l²(Γ₁) ⊗l²(Γ₂),D0)and QI SO⁺(C^∗(Γ₁) ⊗C^∗(Γ₂),l²(Γ₁) ⊗l²(Γ₂),D₁)are 2916 and 1458 respectively. Hence the familyQI SO⁺(C^∗(Γ₁) ⊗C^∗(Γ₂),l²(Γ₁) ⊗l²(Γ₂),D_h)can’t be a flat deformation.

Acknowledgement

We thank the anonymous referee for the first counterexample described in Section 3 and also for pointing out some mistakes in the previous version of this article and valuable suggestions for improvement.

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Debashish Goswami Indian Statistical Institute 203,B.T.Road

Kolkata-700108 INDIA

goswamid@isical.ac.in

Arnab Mandal

School of Mathematical Sciences National Institute of Science Education and Research Bhuvaneswar, HBNI Jatni-752050

INDIA

arnabmaths@gmail.com