Hopf coactions on odd spheres

Journal of Algebra 539 (2019) 305–325

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Journal of Algebra www.elsevier.com/locate/jalgebra

Hopf coactions on odd spheres Suvrajit Bhattacharjee, Debashish Goswami ∗ Stat-Math Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700108, India

a r t i c l e

i n f o

Article history: Received 20 September 2018 Available online 22 August 2019 Communicated by Gunter Malle Keywords: Hopf algebra Co-action Quantum group Odd quantum sphere Quantum isometry group

a b s t r a c t We prove that the q-deformed unitary group, i.e., Uq (N ), is the universal compact quantum group in the category of (compact) quantum groups which coact on the q-deformed odd sphere Sq2N −1 leaving the space spanned by the natural set of generators invariant and preserving the unique SUq (N ) invariant functional on Sq2N −1 . Using this, we identify Uq (N ) as the quantum group of orientation preserving isometries (in the sense of Bhowmick and Goswami [5]) for a natural spectral triple associated with Sq2N −1 constructed by Chakraborty and Pal [9]. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Symmetry is one of the most fundamental and ubiquitous concepts in mathematics and other areas of science. Classically, symmetry of some mathematical structure is understood as the group of automorphisms or isomorphisms (in a suitable sense) of the structure. Generalizing the concept of group, Drinfel d and Jimbo ([11,18,19]) introduced the notion of quantum groups. Later on, Woronowicz ([31,33]) formulated an analogue of this notion in the analytical framework of C ∗ -algebras. * Corresponding author. E-mail addresses: [email protected] (S. Bhattacharjee), [email protected] (D. Goswami). https://doi.org/10.1016/j.jalgebra.2019.08.012 0021-8693/© 2019 Elsevier Inc. All rights reserved.

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It is natural to conceive of quantum groups as ‘symmetry objects’ for certain mathematical structures, e.g., rings or C ∗ -algebras. Indeed, Manin ([23]) pioneered the idea of viewing q-deformation of classical Lie groups (e.g. SLq (N )) as some kind of ‘quantum automorphism group’. A similar approach was taken by Wang ([29]) who defined quantum permutation group of finite sets and quantum automorphism group of finite dimensional matrix algebras. This was followed by flurry of work by several other mathematicians including Banica, Bichon and others ([1,2,6]). The second author of the present article and his collaborators (including Bhowmick, Skalski and others) approached the problem from a geometric perspective and formulated an analogue of the Riemannian isometry groups in the framework of (compact) quantum groups acting on C ∗-algebras. We refer the reader to [16] and the references therein for a comprehensive account of the theory of quantum isometry groups. See also [13,12]. A useful procedure of producing genuine examples of ‘non-commutative spaces’ is to deform the coordinate algebra or some other suitable function algebra underlying a classical space. In this context, it is natural to ask the following: what is the quantum isometry group of a non-commutative space obtained by deforming a classical space? It is expected that under mild assumptions, it should be isomorphic with a deformation of the isometry group of the classical space, at least when the classical space is connected. Indeed, for a quite general class of cocycle deformation (called the Rieffel deformation), such a result has been proved by Bhowmick, Goswami and Joardar ([17]). See also [14,15]. However, no such general result has yet been achieved for the Drinfel d-Jimbo type q-deformation of semisimple Lie groups and the corresponding homogeneous spaces. The goal of the present paper is to make some progress in this direction. We have been able to prove the above result for q-deformed odd spheres, i.e., Sq2N −1 ([28]). Classically (for q = 1), these are nothing but the complex N − 1 dimensional spheres {(z1 , · · · , zN ) |  2 N i |zi | = 1} inside C . The universal group that can act ‘linearly’, i.e., leaves the span of the complex coordinates z1 , · · · , zN invariant and also preserves the canonical inner-product on span{z1 , · · · , zn } coming from the standard inner-product of C N , is the unitary group U (N ). We have proved in Section 6 (Theorem 6.6) a q-analogue of this result. More precisely, we have proved the following theorem. Theorem. Let Q be a Hopf ∗-algebra coacting on O(Sq2N −1 ) by ρ making it a ∗-comodule algebra, where we have viewed O(Sq2N −1 ) as a ∗-coideal subalgebra of O(SUq (N )). Moreover, suppose that  i) ρ leaves the subspace V = span{z1 , · · · , zN } invariant, i.e., ρ(zi ) = j zj ⊗ qij for some qji ∈ Q. We write q = (qji ) for the matrix of Q-valued coefficients; ii) ρ preserves the inner product on V induced by the Haar functional. Then there is a unique ∗-morphism Ψ : O(Uq (N )) → Q such that (id ⊗ Ψ)ρu = ρ.

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Using this, we have also identified (Theorem 7.12) Uq (N ) with the (orientation preserving) quantum isometry group of a natural spectral triple on Sq2N −1 constructed in [9]. Remark. We can compare the above result with [12, Theorem 1.1]. In [12], the algebra A on which Hopf coactions are considered is commutative, which is replaced by a q-deformed quantized function algebra in the present article. Moreover, flexibility of choice of a non-degenerate bilinear form in [12, Theorem 1.1, Condition (i)] is gone in our case; we have the somewhat rigid requirement of preserving a canonical non-degenerate sesquilinear form coming from the Haar functional. In fact, a special advantage of working with a commutative algebra in [12] is that any bilinear form on A admits a natural extension (as a bilinear form) on A ⊗ A which is invariant under the flip map. This is no longer true for quantized function algebra, if we consider the obvious q-analogue of the flip, associated with the natural braiding. 2. Coquasitriangular Hopf algebras In this and the following sections we introduce some well known material on cosemisimple Hopf algebras and coquasitriangular Hopf algebras. The main object of investigation, the Vaksman-Soibelman (also called quantum or odd) sphere, is also introduced. Let H be a Hopf algebra with comultiplication Δ, counit , antipode S, unit 1 and multiplication m. We use Sweedler notation throughout, i.e., for the coproduct we write Δ(a) = a1 ⊗ a2 and for a right coaction ρ, we write ρ(x) = x0 ⊗ x1 . Definition 2.1. [20, p. 331] A coquasitriangular Hopf algebra is a Hopf algebra H equipped with a linear form r : H ⊗ H → C such that the following conditions hold: i) r is invertible with respect to the convolution, that is, there exists another linear form r : H ⊗ H → C such that rr = rr =  ⊗  on H ⊗ H; ii) mH op = r ∗ mH ∗ r on H ⊗ H; iii) r(mH ⊗ id) = r13 r23 and r(id ⊗ mH ) = r13 r12 on H ⊗ H ⊗ H, where r12 (a ⊗ b ⊗ c) = r(a ⊗ b)(c), r23 (a ⊗ b ⊗ c) = (a)r(b ⊗ c) and r13 (a ⊗ b ⊗ c) = (b)r(a ⊗ c), a, b, c in H. Remark 2.2. A linear form r on H ⊗ H with the properties (i)–(iii) is called a universal r-form on H. Since linear forms on H ⊗ H correspond to bilinear forms on H × H, we can consider any linear form r : H ⊗ H → C as a bilinear form on H × H and write r(a, b) := r(a ⊗ b), a, b ∈ H. Then the above conditions (i)–(iii) read as

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r(a1 , b1 )r(a2 , b2 ) = r(a1 , b1 )r(a2 , b2 ) = (a)(b),

(1)

ba = r(a1 , b1 )a2 b2 r(a3 , b3 ),

(2)

r(ab, c) = r(a, c1 )r(b, c2 ),

(3)

r(a, bc) = r(a1 , c)r(a2 , b),

(4)

with a, b, c ∈ H. Remark 2.3. [20, p. 334] It can be shown that r(S(a), S(b)) = r(a, b). Let H be a coquasitriangular Hopf algebra with universal r-form r. For right H-comodules V and W we define a linear mapping rV,W : V ⊗ W → W ⊗ V by rV,W (v ⊗ w) = r(v1 , w1 )w0 ⊗ v0 ,

(5)

v ∈ V , w ∈ W. Remark 2.4. [20, p. 333] It can be shown that rV,W is an isomorphism of the right H-comodules V ⊗ W and W ⊗ V . The compatibility of a universal r-form and a ∗ structure is described in the following definition. Definition 2.5. [20, p. 336] A universal r-form r of a Hopf ∗-algebra H is called real if r(a ⊗ b) = r(b∗ ⊗ a∗ ). 3. The quantum semigroup Mq (N ) Let q be a positive real number. We now introduce some of the well known deformations of classical objects. Definition 3.1. [20, p. 310; 26] The FRT bialgebra, also called the coordinate algebra of the quantum matrix space, denoted O(Mq (N )), is the free unital C-algebra with a set of N 2 generators {uij | i, j = 1, · · · , N } and defining relations uik ujk = qujk uik , uik ujl

−

uki ukj = qukj uki ,

i < j, uil ujk = ujk uil , j i −1 j i ul uk = (q − q )uk ul ,

i < j,

k < l, i < j,

(6) (7)

k < l.

(8)

Proposition 3.2. There is a unique bialgebra structure on the algebra O(Mq (N )) such that Δ(uij ) =

 k

uik ⊗ ukj ,

and

(uij ) = δij ,

i, j = 1, · · · , N.

(9)

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ˆ : CN ⊗ The above construction can be realized more conceptually as follows. Let R C N → C N ⊗ C N be the linear operator whose matrix with respect to the standard basis of C N is given by ij ˆ mn R = q δij δin δjm + (q − q −1 )δim δjn θ(j − i),

(10)

where θ is the Heaviside symbol, that is, θ(k) = 1 if k > 0 and θ(k) = 0 if k ≤ 0. Let ˆ nm and ˆ be R ˆ − . Also, let R ˇ be the “dual” operator defined by R ˇ ij = R the inverse of R mn ji ˇ −ij = R ˆ −ij . R mn mn ˆ satisfies Remark 3.3. [20, p. 309] It is known that R ˆ − qI)(R ˆ + q −1 I) = 0, (R

(11)

where I is the identity operator. The following shows that Mq (N ) is universal in a sense. Proposition 3.4. [20, p. 305] i) There is a linear map φ : C N → C N ⊗O(Mq (N )) such that C N is a right comodule of (2) ˆ is a comodule morphism, i.e., (R⊗id)φ ˆ ˆ O(Mq (N )) with coaction φ and R = φ(2) R, (2) N N (2) where φ is the induced coaction on C ⊗C given by φ (v ⊗w) = v0 ⊗w0 ⊗v1 w1 (φ(v) = v0 ⊗ v1 ); ii) If A is any other bialgebra and ψ : C N → C N ⊗ A is a right coaction of A on C N ˆ is a comodule morphism (in the sense described above) then there exists such that R a unique bialgebra morphism Θ : O(Mq (N )) → A such that (id ⊗ Θ)φ = ψ. Since O(Mq (N )) is only a bialgebra, we want to construct a Hopf algebra out of it. For that we need the following definition. Definition 3.5. [20, p. 312] The quantum determinant, denoted Dq , is the element of O(Mq (N )) defined by 

(−q)(π) u1π(1) · · · uN π(N ) ,

(12)

π∈SN

where SN is the symmetric group on N letters and (π) is the number of inversions in π. Remark 3.6. [20, pp. 312, 313] It is an important fact that Dq is central, nonzero and group-like in O(Mq (N )). We recall that group-like means Δ(Dq ) = Dq ⊗ Dq . Applying ( ⊗ id) on the identity Δ(Dq ) = Dq ⊗ Dq yields Dq = (Dq )Dq , hence (Dq ) = 1 as Dq = 0.

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4. The quantum group SUq (N ) The deformation of the special linear group is realized as follows. Definition 4.1. [20, p. 314] The coordinate algebra of the quantum special linear group is defined to be the quotient O(SLq (N )) = O(Mq (N ))/Dq − 1 of the algebra O(Mq (N )) by the two-sided ideal generated by the element Dq − 1. The following shows that O(SLq (N )) is indeed a Hopf algebra. Proposition 4.2. There is a unique Hopf algebra structure on the algebra O(SLq (N )) with comultiplication Δ and counit  such that Δ(uij ) =



uik ⊗ ukj

and

(uij ) = δij .

(13)

k

The antipode S of the Hopf algebra is given by S(uij ) = (−q)i−j



k

N −1 1 (−q)(π) ukπ(l · · · uπ(l , 1) N −1 )

(14)

π∈SN −1

where {k1 , · · · , kN −1 } := {1, · · · , N } \ {j} and {l1 , · · · , lN −1 } := {1, · · · , N } \ {i} as ordered sets. φ

→ C N ⊗ O(Mq (N )) → C N ⊗ O(SLq (N )) gives the natural The composite C N − coaction of O(SLq (N )) on C N . As quantum groups are understood to be quasitriangular Hopf algebras, quantum function algebras are assumed to be coquasitriangular Hopf algebras. We want to think of O(SLq (N )) as the quantum function algebra of SL(N ). Theorem 4.3. [20, p. 339] O(SLq (N )) is a coquasitriangular Hopf algebra with universal r-form rt uniquely determined by ki ˆ jl rt (uij ⊗ ukl ) = tR ,

(15)

where t is the unique positive real number such that tN = q −1 . Remark 4.4. It can be shown that the morphism rC N ,C N induced by the universal r-form ˆ Let us denote it by σ. rt of O(SLq (N )), equals tR. The following resembles complex conjugation.

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Proposition 4.5. [20, p. 316] There is a unique ∗-structure on the Hopf algebra O(SLq (N )) given by (uij )∗ = S(uji ), making it into a Hopf ∗-algebra. Let us now introduce the quantum version of the real form SU (N ) of SL(N ). Definition 4.6. The coordinate algebra of the quantum special unitary group SUq (N ) is the Hopf ∗-algebra O(SLq (N )) of the above proposition. Theorem 4.7. [20, p. 340] The universal r-form rt of O(SUq (N )) is real, in the sense of Definition 2.5. The algebraic counterpart of classical Peter-Weyl theorem for compact groups is contained in the following: Definition 4.8. [20, p. 403] A Hopf algebra H is said to be cosemisimple if there exists a unique linear form h : A → C, which we call the Haar functional, such that h(1) = 1, and for all a in H (id ⊗ h)Δ(a) = h(a)1,

(h ⊗ id)Δ(a) = h(a)1.

At this point, it is natural to recall the analytic counterpart of compactness. Definition 4.9. [33,24] A compact quantum group (CQG) is given by a pair (S, Δ), where S is a unital C ∗ algebra and Δ is a unital C ∗ homomorphism Δ : S → S ⊗ S (⊗ is C ∗ -algebraic minimal tensor product) satisfying i) (Δ ⊗ id) ◦ Δ = (id ⊗ Δ) ◦ Δ; ii) Each of the linear spans of Δ(S)(S ⊗ 1) and Δ(S)(1 ⊗ S) is norm-dense in S ⊗ S. If the comultiplication is understood, we simply denote the compact quantum group by S. It is well known ([33]) that there is a canonical dense ∗-algebra S of S, consisting of the matrix coefficients of the finite dimensional unitary representation (to be defined in Section 7) of S, which becomes a Hopf ∗-algebra with comultiplication Δ|S . We say that the compact quantum group (S, Δ) acts on a unital C ∗ -algebra B, if there is a unital C ∗ -homomorphism α : B → B ⊗ S satisfying i) (α ⊗ id) ◦ α = (id ⊗ Δ) ◦ α; ii) the linear span of α(B)(1 ⊗ S) is norm-dense in B ⊗ S. Given such an action, there exists ([29,25]) a dense unital ∗-subalgebra B of B such that B is a ∗-comodule algebra over S. Definition 4.10. [20, p. 416] A compact quantum group (CQG) algebra is a cosemisimple Hopf ∗-algebra H with Haar functional h such that h(a∗ a) > 0, for all a = 0.

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It is known ([10]) that H is a CQG algebra if and only if it is isomorphic to the dense Hopf ∗-algebra S of a compact quantum group S. The following captures the compactness of the real form SU (N ). Theorem 4.11. [20, p. 418] O(SUq (N )) is a CQG algebra. In fact, there is a natural C ∗ -norm on O(SUq (N )). Upon completion with respect to this norm, one gets the unital C ∗ -algebra SUq (N ) which is a compact quantum group in the sense of Definition 4.9. Moreover, SUq (N ) is the universal C ∗ -algebra generated by O(SUq (N )). We end this section by recalling a standard fact. Let V be a finite dimensional vector space with inner-product , . Let H be a Hopf ∗-algebra with a coaction ρ : V → V ⊗ H on V . We say that ρ preserves the inner-product if v0 , w0 v1∗ w1 = v, w 1 for all v, w ∈ V , where we use Sweedler notation, i.e., ρ(v) = v0 ⊗ v1 and likewise for ρ(w). Proposition 4.12. [20, p. 402] If ρ preserves the inner-product on V and W is a subcomodule of V then the orthogonal complement W ⊥ (with respect to , ) is also a subcomodule of V . 5. The odd sphere We now introduce the main example to be studied. We remark that these are q-deformations of the complex (N −1)-dimensional (hence odd dimensional real) spheres. One could as well define q-deformations of even dimensional real spheres which are quantum homogeneous spaces of O(SOq (n)) for appropriate n. We do not deal with these ˆ has two because our proof of some of the main results crucially use the fact that R eigenvalues whereas in the case of O(SOq (n)), it has three. Definition 5.1. [30, p. 2; 28] The coordinate algebra O(Sq2N −1 ) of the quantum sphere is the free unital C-algebra with a set of 2N generators {zi , zi∗ | i = 1, · · · , N } and defining relations zi zj = qzj zi ,

zi∗ zj∗ = q −1 zj∗ zi∗ ,

zi zj∗ = qzj∗ zi ,

i
i = j  zi zi∗ − zi∗ zi + q −1 (q − q −1 ) zk zk∗ = 0,

(16) (17) (18)

k>i N 

zi zi∗ = 1,

i=1

together with the ∗-structure (zi )∗ = zi∗ and (zi∗ )∗ = zi .

(19)

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There is a natural C ∗ -norm on O(Sq2N −1 ). Upon completion with respect to this norm, one gets the unital C ∗ -algebra Sq2N −1 which is the universal C ∗ -algebra generated by O(Sq2N −1 ). Proposition 5.2. [9, p. 34] Putting zi = u1i and zi∗ = (u1i )∗ = S(ui1 ) gives an embedding of O(Sq2N −1 ) into O(SUq (N )) making O(Sq2N −1 ) into a quantum homogeneous space for O(SUq (N )) with the coaction ΔR (zi ) =



zj ⊗ uji ,

ΔR (zi∗ ) =

j



zj∗ ⊗ S(uij ).

(20)

j

The compact quantum group SUq (N ) acts on Sq2N −1 in the C ∗ -algebraic sense, lifting the above coaction on O(Sq2N −1 ). Classically, the sphere S 2N −1 can be described as a homogeneous space for SU (N ). The above proposition states the quantum version of it. Moreover, one can view the sphere as a homogeneous space for U (N ) also. As expected, the last statement continues to hold in the quantum world too. We need the following definition. Definition 5.3. [20, p. 313] The coordinate algebra of the quantum general linear group is defined to be the quotient O(GLq (N )) = O(Mq (N ))[t]/tDq − 1 of the polynomial algebra O(Mq (N ))[t] in t over O(Mq (N )) by the two-sided ideal generated by the element tDq − 1. Proposition 5.4. There is a unique Hopf algebra structure on the algebra O(GLq (N )) with comultiplication Δ and counit  such that Δ(uij ) =



uik ⊗ ukj

(uij ) = δij .

and

(21)

k

The antipode S of the Hopf algebra is given by S(uij ) = (−q)i−j Dq −1



k

N −1 1 (−q)(π) ukπ(l · · · uπ(l 1) N −1 )

and

S(Dq ) = Dq −1 ,

(22)

π∈SN −1

where {k1 , · · · , kN −1 } := {1, · · · , N } \ {j} and {l1 , · · · , lN −1 } := {1, · · · , N } \ {i} as ordered sets. We have the following analogue of the real form U (N ). Proposition 5.5. [20, p. 316] There is a unique ∗-structure on the Hopf algebra O(GLq (N )) given by (uij )∗ = S(uji ), making it into a Hopf ∗-algebra.

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Definition 5.6. The coordinate algebra of the quantum unitary group Uq (N ) is the Hopf ∗-algebra O(GLq (N )) of the above proposition. In this ∗-algebra, the quantum determinant Dq becomes a unitary element. In analogy with U (N ), the fact that Uq (N ) is a compact quantum group is reflected in the following theorem. Theorem 5.7. [20, p. 418] O(Uq (N )) is a CQG algebra. Thus we have the analogues of Proposition 4.5 and Theorem 4.10. Moreover, Proposition 5.2 remains true in this case and makes O(Sq2N −1 ) a quantum homogeneous space for O(Uq (N )). There is a natural C ∗ -norm on O(Uq (N )). Upon completion with respect to this norm, one gets the unital C ∗ -algebra Uq (N ) which is a compact quantum group in the sense of Definition 4.9. Uq (N ) is also the universal C ∗ -algebra generated by O(Uq (N )) and it acts on Sq2N −1 in the C ∗ -algebraic sense lifting the algebraic coaction on O(Sq2N −1 ). We end this section with a lemma. Lemma 5.8. Suppose Q is a Hopf ∗-algebra and qji ∈ Q, i, j = 1, · · · , N such that the following hold:  i) Δ(qji ) = k qki ⊗ qjk and (qji ) = δij ; ii) qji satisfy the FRT relations (6), (7) and (8); iii) q satisfies qq∗ = In , where q = (qji ) and q∗ = qt , q = ((qji )∗ ). Then there is a unique ∗-morphism Ψ : O(Uq (N )) → Q between these Hopf ∗-algebras such that Ψ(uij ) = qji . Proof. Define Φ : O(Mq (N )) → Q on the generators uij by Φ(uij ) = qji . Then by hypotheses i), ii), Definition 3.1 and Proposition 3.2, Φ extends to a bialgebra morphism. Using Remark 3.6, we conclude: Δ(Φ(Dq )) = (Φ ⊗ Φ)Δ(Dq ) = Φ(Dq ) ⊗ Φ(Dq )

(23)

(Φ(Dq )) = (Dq ) = 1.

(24)

and

Thus (23), (24) imply that Φ(Dq ) is a nonzero group-like element in Q. Moreover, applying (S ⊗ id) and (id ⊗ S) on either side of (23), we observe S(Φ(Dq ))Φ(Dq ) = Φ(Dq )S(Φ(Dq )) = (Φ(Dq ))1 = 1. Hence Φ(Dq ) is invertible with Φ(Dq )−1 = S(Φ(Dq )) We also have qji Φ(Dq ) = Φ(Dq )qji , by the centrality of Dq , (see Remark 3.6), hence qji commute with Φ(Dq )−1 = S(Φ(Dq )).

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˜ : O(Mq (N ))[t] → Q by Now define Φ ˜ i ) = qi , Φ(u j j

˜ and Φ(t) = S(Φ(Dq )).

˜ sends the ideal This is well-defined because qji commute with S(Φ(Dq )). Clearly, Φ ˜ generated by tDq − 1 to 0. Hence Φ descends to Ψ : O(GLq (N )) → Q which is, by Proposition 5.4, a bialgebra morphism. We are left to show that Ψ is a ∗-map, with respect to the Hopf ∗-algebra structure on O(GLq (N )) defined in Proposition 5.5 i.e., the Hopf ∗-structure of O(Uq (N )). To this  end, applying m(S ⊗ id) and m(id ⊗ S) on either side of the relation Δ(qji ) = k qki ⊗ qjk (hypothesis i)), it follows that the antipode S satisfies S(q)q = qS(q) = IN , i.e., S(q) = q−1 , where S(q) = (S(qji )). This, together with hypothesis iii), implies S(q) = q∗ i.e., S(qji ) = (qij )∗ . Since Ψ is a bialgebra morphism, it preserves the antipode and so Proposition 5.5 implies that Ψ is a ∗-morphism. Finally, the proof of uniqueness is straightforward, hence omitted. 2 6. Main results We start with a basic observation. Lemma 6.1. Let H be any cosemisimple Hopf ∗-algebra with the Haar functional h. Then for a, b ∈ H, h(a∗1 b)a∗2 = h(a∗ b1 )S(b2 ). Proof. By definition, h(x)1 = h(x1 )x2 for x ∈ H. Applying the antipode S, we get h(x)1 = h(x1 )S(x2 ). Now, h(a∗1 b)a∗2 = h(a∗1 b1 )S(a∗2 b2 )a∗3 = h(a∗1 b1 )S(b2 )S(a∗2 )a∗3 = h(a∗1 b1 )S(b2 )(a∗2 ) = h(a∗1 (a∗2 )b1 )S(b2 ) = h(a∗ b1 )S(b2 ).

2

The following lemma exploits the relation between the two apparently different concepts, namely coquasitriangularity and faithfulness of the Haar functional. Lemma 6.2. Let H be a coquasitriangular CQG algebra with Haar functional h and real universal r-form r. Let the induced inner-product be denoted by , , i.e., a, b = h(a∗ b). Let V be any subcomodule of H and rV,V be the induced morphism on V ⊗ V . Then rV,V is hermitian with respect to the restricted inner-product.

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Proof. We have rV,V (v ⊗ w), v  ⊗ w = r(v1 , w1 )w0 ⊗ v0 , v  ⊗ w = r(w1∗ , v1∗ )w0 , v  v0 , w =

(we use reality of r)

r(h(w0∗ v  )w1∗ , h(v0∗ w )v1∗ )

= r(h(w∗ v0 )S(v1 ), h(v ∗ w0 )S(w1 )) =

(using Lemma 6.1)

r(S(v1 ), S(w1 ))w, v0 v, w0

= r(v1 , w1 )v ⊗ w, w0 ⊗ v0 



= v ⊗ w, rV,V (v ⊗ w ) .

(by Remark 2.3) 2

Lemma 6.3. Let A and Q be Hopf ∗-algebras, B ⊂ A a ∗-coideal subalgebra of A and a comodule algebra over Q with coaction ρ : B → B ⊗ Q such that ρ(b∗ ) = ρ(b)∗ for all b ∈ B. Suppose that A is cosemisimple and ρ preserves the restriction of the Haar functional h of A to B i.e., (h ⊗ id)ρ(b) = h(b)1 for all b ∈ B. Then ρ preserves the induced inner-product on B given by a, b = h(a∗ b), in the sense described in the paragraph preceding Proposition 4.12. The proof is straightforward, hence omitted. Proposition 6.4. Let π : O(Uq (N )) → O(SUq (N )) be the quotient homomorphism and ρu , ρsu , respectively, be the corresponding coactions on O(Sq2N −1 ) so that (id ⊗ π)ρu = ρsu . Then ρu preserves the restriction of the Haar functional h on O(Sq2N −1 ). Proof. Let f be any linear functional on O(Sq2N −1 ) such that f (1) = 1. Let f  be defined as (f ⊗ hu )ρu , where hu is the Haar functional of Uq (N ). Then f  is ρu invariant. By (id ⊗ π)ρu = ρsu , we get that f  is ρsu invariant too. It is well known that the restriction of h is the only functional with this property [20]. Hence, the conclusion follows. 2 We think the following is well known. We included it because we couldn’t find it in the literature. kN −1 ∗ lN −1 ∗ lN Proposition 6.5. The set {z1k1 · · · zN (zN · · · (z1∗ )l1 , z1k1 · · · zNN−1 (zN ) · · · (z1∗ )l1 | −1 ) k1 , · · · , kN , lN −1 , · · · , l1 ∈ N ∪ {0}, lN ∈ N} is a vector space basis of O(Sq2N −1 ). k

Proof. It is a simple application of Bergman’s Diamond lemma [3].

2

We state and prove below the main result concerning Hopf coactions on the quantum spheres satisfying suitable conditions. Theorem 6.6. Let Q be a Hopf ∗-algebra coacting on O(Sq2N −1 ) by ρ making it a ∗-comodule algebra, where we have viewed O(Sq2N −1 ) as a ∗-coideal subalgebra of O(SUq (N )). Moreover, suppose that

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317

 i) ρ leaves the subspace V = span{z1 , · · · , zN } invariant, i.e., ρ(zi ) = j zj ⊗ qij for some qji ∈ Q. We write q = (qji ) for the matrix of Q-valued coefficients; ii) ρ preserves the inner product on V induced by the Haar functional. Then there is a unique ∗-morphism Ψ : O(Uq (N )) → Q such that (id ⊗ Ψ)ρu = ρ. Before we go to the proof, we prove a lemma. Lemma 6.7. In the notation of Theorem 6.6, let μ be the multiplication of O(Sq2N −1 ) ˆ − qI). restricted to V ⊗ V . Then ker(μ) = im(R Proof. We first claim that ker(μ) = span{zi ⊗ zj − qzj ⊗ zi ; i < j}. Clearly, zi ⊗ zj −  qzj ⊗ zi ∈ ker(μ). Moreover, v = i,j cij zi ⊗ zj ∈ V ⊗ V (cij ∈ C) is in ker(μ) if and only if 0=

 ij

cij zi zj =

  (cij + q −1 cji )zi zj + cii zi2 i
(by (16)).

i

It follows, by the linear independence of {zi zj , zi2 ; i < j} (Proposition 6.5), that cii = 0  for all i and cij + q −1 cji = 0 for all i < j. Hence, v reduces to the form v = i
i < j, i > j,

(25)

i = j. 2

Proof of Theorem 6.6. We start with the observation that since V is a Q-comodule, we  have that Δ(qji ) = k qki ⊗ qjk and (qji ) = δij . ˆ Now recall the map σ from Remark 4.4. By Lemma 6.2, σ is hermitian. Since σ = tR 2N −1 ˆ and t is real, R is also hermitian. The multiplication of O(Sq ) is a Q-comodule morphism, O(Sq2N −1 ) being a Q-comodule algebra. Since V is assumed to be a subcomodule of O(Sq2N −1 ), the restriction μ of the multiplication to V ⊗ V is also a Q-comodule morphism. We recall that V ⊗V is a Q-comodule with coaction ρV ⊗V (v⊗w) = v0 ⊗w0 ⊗v1 w1 , where ρ(v) = v0 ⊗ v1 . As μ is a Q-comodule morphism, we have ρμ = (μ ⊗ id)ρV ⊗V . ˆ − qI) (from Lemma 6.7 above) is also a Q-comodule whose orthogThus ker(μ) = im(R ˆ + q −1 I, by Remark 3.3. Hence, by Proposition 4.12, onal complement is the image of R −1 ˆ ˆ im(R + q I) is a Q-comodule. R has two eigenvalues, namely, q and −q −1 , the correˆ + q −1 I) and im(R ˆ − qI), respectively. Since ρ preserves sponding eigenspaces being im(R ˆ both the eigenspaces, R becomes a Q-comodule morphism. By Proposition 3.4, q then satisfies the FRT relations (6), (7), (8).

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N By assumption, ρ preserves the relation i=1 zi zi∗ = 1. Applying ρ to both sides, comparing coefficients and using Proposition 6.5 we get that qq∗ = In . Thus, Lemma 5.8 yields a unique ∗-morphism Ψ : O(Uq (N )) → Q such that Ψ(uij ) = i qj . Moreover, (id ⊗ Ψ)ρu and ρ agree on the generators zi , hence they are equal. For any other Ψ satisfying (id ⊗ Ψ )ρu = ρ, we get by evaluating both sides on the generators zi , that Ψ (uij ) = qji . Hence, by the uniqueness in Lemma 5.8, Ψ = Ψ. 2 N Remark 6.8. The equation (19) can also be written as i=1 q −2i zi∗ zi = q −2 . Now applying ρ to both sides, comparing coefficients and using Proposition 6.5, we see that q satisfies EqE −1 qt = qt EqE −1 = In , where E is the matrix 1 diag (1, q 2 , q 4 , · · · , q 2(n−1) ), n−1 q [n]q

[n]q :=

q n − q −n . q − q −1

See [4,27]. We finally have the following theorem. Theorem 6.9. Consider the category C consisting of Hopf ∗-algebras satisfying the hypotheses of Theorem 6.6 as objects and Hopf ∗-algebra morphisms intertwining the coactions as morphisms. Then O(Uq (N )) is a universal object in this category. Proof. By Proposition 6.4 and Lemma 6.3, O(Uq (N )) is an object in this category. Then Theorem 6.6 shows that O(Uq (N )) is universal with that property. 2 7. An application We provide an application of the main result of the previous section, namely, that of determining the quantum isometry group of the odd sphere, in the sense of [5]. We begin by defining unitary representations of a compact quantum group. Definition 7.1. [31] A unitary representation of a compact quantum group (S, Δ) on a Hilbert space H is a map u from H to H ⊗ S such that the element u ˜ ∈ M (K(H) ⊗ S) given by u ˜(ξ⊗b) = u(ξ)(1⊗b) (ξ ∈ H, b ∈ S) is a unitary satisfying (id⊗Δ)˜ u=u ˜(12) u ˜(13) , where for an operator X ∈ B(H1 ⊗H2 ) we have denoted by X(12) and X(13) the operators X ⊗ IH2 ∈ B(H1 ⊗ H2 ⊗ H2 ) and Σ23 X(12) Σ23 respectively (where Σ23 being the unitary on H1 ⊗ H2 ⊗ H2 that switches the two copies of H2 ). Irreducible unitary representations of the quantum group SUq (N ) are indexed by Young tableaux λ = (λ1 , · · · , λN ), where λi ’s are nonnegative integers λ1 ≥ · · · ≥ λN [32, p. 42]. Let Hλ be the carrier Hilbert space corresponding to λ whose basis elements are parametrized by arrays of the form

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⎡ ⎢ ⎢ r=⎢ ⎢ ⎣r

r11 r21 .. . N −1,1

rN 1

r12 r22 .. .

··· ···

r1,N −1 r2,N −1

r1N

rN −1,2

319

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

where rij ’s are integers satisfying r1j = λj for j = 1, · · · , N , rij ≥ ri+1,j ≥ ri,j+1 ≥ 0 for all i, j. Such arrays are known as Gelfand-Tsetlin (GT) tableaux (see [9, p. 29] for details). For a GT tableaux r, ri will denote its ith row. It was shown by Woronowicz ([31]) that any compact quantum group possesses a Haar functional in the sense of Definition 4.8. Let us denote the Haar functional of SUq (N ) again by h. Let L2 (SUq (N )) denote the corresponding GNS space and L2 (Sq2N −1 ) denote the closure of Sq2N −1 in L2 (SUq (N )). Proposition 7.2. [9, p. 34] Assume N > 2. The restriction of the right regular representation of SUq (N ) to L2 (Sq2N −1 ) decomposes as a direct sum of the irreducibles, with each copy occurring exactly once, given by the Young tableau λn,k = (n + k, k, k, · · · , k, 0) with n, k ∈ N0 . Let the irreducible representation corresponding to the Young tableau λn,k = (n + k, k, · · · , k, 0) be denoted by Vn,k . According to our previous notation V = V1,0 , the irreducible with Young tableau λ1,0 = (1, 0, · · · , 0). Moreover, observe that V0,1 , the irreducible with Young tableau λ0,1 = (1, 1, · · · , 1, 0), is the conjugate corepresentation ∗ to V . In our previous notation, this is nothing but the span of z1∗ , · · · , zN , i.e., V0,1 = ∗ ∗ V = {v | v ∈ V }. Consider the space W ⊗(n,k) := V1,0 ⊗ · · · ⊗ V1,0 ⊗ V0,1 ⊗ · · · ⊗ V0,1 .       n times

k times

Let us denote the image of it in the algebra under multiplication by W •(n,k) := V1,0 • · · · • V1,0 • V0,1 • · · · • V0,1 .       n times

k times

Let Λ⊗(n,k) be the set of the Young tableaux for the irreducible representations occurring in the decomposition of W ⊗(n,k) and Λ•(n,k) be the corresponding set for the decomposition of W •(n,k) . The following proposition gives a description of W •(n,k) . Proposition 7.3. i) Vn,k occurs with multiplicity exactly one in the orthogonal decomposition of W •(n,k) into irreducibles;

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ii) If a copy of Vm,l occurs in the irreducible decomposition of the orthogonal complement (Vn,k )⊥ of Vn,k in W •(n,k) then we must have: either, or,

l < k and m ≤ n + k − l; l = k and m < n.

Proof. By Proposition 7.2, it follows that any λ ∈ Λ•(n,k) must be of the form λm,l = (m + l, l, · · · , l, 0) and has multiplicity one. It is known (see e.g., [8, p. 326, Proposition 10.1.16]) that any λ ∈ Λ⊗(n,k) is dominated by λn,k = (n + k, k, · · · , k, 0) which is equivalent to l ≤ k and m + l ≤ n + k. ii) follows immediately from these remarks. For i), let us first remark that O(SUq (N )), hence O(Sq2N −1 ) is an integral domain (see [21, p. 98]). Now, if v and v are the primitive vectors for V1,0 and V0,1 , respectively, then v n v k ∈ W •(n,k) is nonzero and belongs to the weight space corresponding to λn,k , which follows from the definition of weight spaces as in [8, p. 324] and the fact that each Ki is group-like ([8, p. 281], so that Ki (xy) = (Ki x)(Ki y)). 2 The following will be useful in constructing a non-commutative structure on the sphere. Proposition 7.4. [9, p. 35] Let Γ0 be the set of all GT tableaux rnk given by

nk rij

⎧ ⎪ ⎪n + k ⎨ =

0 ⎪ ⎪ ⎩k

if

i=j=1

if

i = 1, j = N

otherwise,

for some n, k ∈ N. Let Γnk 0 be the set of all GT tableaux with top row (n+k, k, k, · · · , k, 0). Then the family of vectors {ernk ,s | n, k ∈ N, s ∈ Γnk 0 } form a complete orthonormal basis for L2 (Sq2N −1 ). We recall the following definition. Definition 7.5. [22, p. 93; 7] A spectral triple (A, H, D) of compact type is given by a unital ∗-algebra with a faithful representation π : A → B(H) on the Hilbert space H together with a self-adjoint operator D = D∗ on H with the following properties: / R, is a compact operator on H; i) The resolvent (D − λ)−1 , λ ∈ ii) [D, a] := Dπ(a) − π(a)D ∈ B(H), for any a ∈ A. In what follows, by a spectral triple, we mean a spectral triple of compact type. Let us now put a non-commutative structure on the quantum sphere.

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321

Theorem 7.6. [9, p. 39] Let A = O(Sq2N −1 ) and H be L2 (Sq2N −1 ). Take π to be the inclusion. Finally, define the operator D : er,s → d(r)er,s on L2 (Sq2N −1 ) where the d(r)’s are given by

nk

d(r ) =

 −k n+k

if

n=0

if

n > 0.

Then (A, H, D), as constructed above, is a spectral triple on O(Sq2N −1 ). Now let us recall the notion of the quantum isometry group of a non-commutative manifold. Definition 7.7. [5, p. 2538] A quantum family of orientation preserving isometries for the spectral triple (A, H, D) is given by a pair (S, u) where S is a unital C ∗ -algebra and u is a linear map from H to H ⊗ S such that u ˜ given by u ˜(ξ ⊗ b) = u(ξ)(1 ⊗ b), extends to a unitary element of M (K(H) ⊗ S) satisfying ˜; i) for every state φ on S, uφ D = Duφ where uφ = (id ⊗ φ) ◦ u ii) (id⊗φ)◦adu (a) ∈ A , for all a ∈ A and for all state φ on S, where adu (x) = u ˜(x⊗1)˜ u∗ for x ∈ B(H). In case the C ∗ -algebra S has a coproduct Δ such that (S, Δ) is a compact quantum group and U is a unitary representation of (S, Δ) on H, it is said that (S, Δ) acts by orientation preserving isometries on the spectral triple. One considers the category Q(A, H, D) whose objects are triples (S, Δ, u), where (S, Δ) is a compact quantum group acting by orientation preserving isometries on the given spectral triple, with u being the corresponding unitary representation. The morphisms are homomorphisms of compact quantum groups that also “intertwines” (see [5]) 0 , Δ0 , u0 ) (say) exists in this category the unitary representations. If a universal object (S ∗ then the C -algebra S0 generated by {(tξ,η ⊗id)(adu0 (a)), ξ, η ∈ H, q ∈ A} is a compact quantum group and it is called the quantum group of orientation preserving isometries. Here, tξ,η : B(H) → C is given by tξ,η (X) =< ξ, Xη >. We record some results from [5] for the reader’s convenience. Theorem 7.8. [5, p. 2547] Let (A, H, D) be a spectral triple and assume that D has a one dimensional eigenspace spanned by a unit vector ξ, which is cyclic and separating for the algebra A. Moreover, assume that each eigenvector of D belongs to the dense subspace 0 , Δ0 , u0 ) in the category Q(A, H, D). Aξ of H. Then there is a universal object (S For the spectral triple in Theorem 7.6, the cyclic separating vector ξ is 1O(Sq2N −1 ) .

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Definition 7.9. [5, p. 2548] Let (A, H, D) be the spectral triple in Theorem 7.6. Let  Q(A, H, D) be the category with objects (S, α) where S is a compact quantum group with an action on A such that i) α is h preserving;  = (D  ⊗ id)α, where D  i.e., αD  is the operator A → A given ii) α commutes with D,  by D(a)ξ = D(aξ), ξ as in Theorem 7.8 which is 1O(Sq2N −1 ) in our case.  and D are in one-one correspondence. In fact, Note that the eigenspaces of D  are of the form {a ∈ A | aξ ∈ Vλ }, where Vλ is the finite dimeneigenspaces of D sional eigenspace of D with respect to eigenvalue λ.  H, D) Proposition 7.10. [5, p. 2549] There exists a universal object S in the category Q(A, ∗  and it is isomorphic to the C -subalgebra S0 of S0 obtained in Theorem 7.8. We conclude by describing the quantum isometry group of the sphere. This generalizes [5, Theorem 4.13, p. 2559]. Lemma 7.11. Given a compact quantum group S with an action α on A, the following are equivalent:  H, D), (A, H, D) as in Theorem 7.6; i) (S, α) is an object of the category Q(A, ii) α is linear (meaning it preserves V as in Theorem 6.6) and preserves h; iii) α preserves each irreducible Vn,k , occurring in Proposition 7.2.  it preserves the eigenspaces of D,  in particProof. i) =⇒ ii): Since α commutes with D,  ular V , and by definition of the category Q(A, H, D), it preserves h. ii) =⇒ iii): Recall the notation W •(n,k) from the discussion below Proposition 7.2. Clearly, α preserves W •(n,0) as it is a homomorphism and preserves V1,0 . It also preserves ∗ W •(0,k) because it is a ∗-homomorphism and preserves V0,1 = V1,0 . Hence α preserves •(n,k) W . We will use this fact to show α preserves Vn,k for all n and k. Let P(k) be the statement “α preserves Vn,k for all n”. We now proceed to prove this statement for all k by induction. We break the proof in several steps. Step 1: We prove that P(0) holds. Thus we need to show α preserves Vn,0 for all n. We use induction on n. Clearly, α preserves V0,0 . Next, we assume that α preserves Vm,0 for all m < n. By Proposition 7.3, each irreducible contained in (Vn,0 )⊥ is of the form Vm,0 with m < n, which is preserved by α. Hence α preserves Vn,0 , by Proposition 4.12. Step 2: Now we assume P(l) holds for all l < k, i.e., α preserves Vn,l for all n and for all l < k. We have to prove that P(k) holds, i.e., α preserves Vn,k for all n. We use induction on n (with fixed k) to prove this.

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Step 2a: We prove that α preserves V0,k . By Proposition 7.3, each irreducible occurring in (V0,k )⊥ is of the form V0,l with l < k. By assumption α preserves V0,l for all l < k. Hence, by Proposition 4.12, α preserves V0,k . Step 2b: Next we assume that α preserves Vm,k for all m < n and prove that it also preserves Vn,k (k fixed). By Proposition 7.3, each irreducible contained in (Vn,k )⊥ is of the form Vm,l with l < k or Vm,k with m < n. α preserves Vm,l with l < k (the main induction hypothesis that P(l) holds for all l < k) and Vm,k with m < n, by the induction hypothesis at the beginning of the present step. Hence, by Proposition 4.12, α preserves Vn,k . This completes the induction on n, proving α preserves Vn,k for all n. So we have proved P(0) holds and P(k) holds assuming P(l) holds for all l < k. This completes the induction on k, thus completing the proof of ii) =⇒ iii).  invariant, hence, iii) =⇒ i): Condition iii) implies that α leaves each eigenspaces of D  Since h(1) = 1 and ker(h) is the span of all Vn,k with n + k = 0, α it commutes with D. preserves ker(h). But then, a −h(a)1 ∈ ker(h), so that α(a −h(a)1) = α(a) −h(a)(1 ⊗1) ∈ ker(h) ⊗ S, implying (h ⊗ id)(α(a)) = h(a)1. Hence, α preserves h. 2 Theorem 7.12. The quantum group of orientation preserving isometries for the spectral triple in Theorem 7.6 is the compact quantum group Uq (N ).  Proof. By Proposition 7.10, there exists a universal object (S, α) in Q(A, H, D). By ii) and iii) of Lemma 7.11, α preserves h and leaves the algebra generated by {Vn,k ; n, k ∈ N ∪ {0}}, i.e., O(Sq2N −1 ) invariant. Moreover, as each Vn,k is finite dimensional, α(O(Sq2N −1 )) ⊂ O(Sq2N −1 ) ⊗ S, where S is the dense Hopf ∗-algebra inside the compact quantum group S mentioned after Definition 4.9. In particular, by ii) of Lemma 7.11, the coaction of S on O(Sq2N −1 ) satisfies the hypotheses of Theorem 6.6, hence is an object of the category C as in Theorem 6.9. Thus we have a unique morphism Ψ : O(Uq (N )) → S in C. As Uq (N ) is the universal C ∗ -algebra corresponding to O(Uq (N )), Ψ extends to a C ∗ -algebra morphism (again denoted by Ψ) Uq (N ) → S. It is clearly a morphism of compact quantum groups and intertwines the two (C ∗ -algebraic) actions, hence a morphism  in Q(A, H, D). On the other hand, it follows from Lemma 7.11 that Uq (N ) is an object in the category  Q(A, H, D) as the canonical action of the compact quantum group Uq (N ) on Sq2N −1 satisfies ii) of Lemma 7.11. Hence we have a unique morphism Θ : S → Uq (N ) in  Q(A, H, D). Clearly, ΨΘ is the identity morphism, since S is the universal object. To show that ΘΨ is the identity morphism, we observe that it is a morphism of compact quantum groups, hence takes O(Uq (N )) to itself. Since O(Uq (N )) is universal in C, ΘΨ is identity, at least on O(Uq (N )). We recall that Uq (N ) is the universal C ∗ -algebra generated by O(Uq (N )), hence ΘΨ lifts uniquely to Uq (N ), implying that ΘΨ is also the identity morphism. Thus S is isomorphic to Uq (N ). 2

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Acknowledgments The first author is grateful to Aritra Bhowmick, Jyotishman Bhowmick and Sugato Mukhopadhyay for helpful discussions that led to many improvements of the paper. The second author is partially supported by J.C. Bose National Fellowship and Research Grant awarded by D.S.T. (Govt. of India). Both the authors thank the anonymous referee for several suggestions and corrections leading to improvement of the paper. References [1] Teodor Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2) (2005) 243–280, https://doi.org/10.1016/j.jfa.2004.11.002, MR2146039. [2] Teodor Banica, Quantum automorphism groups of small metric spaces, Pacific J. Math. 219 (1) (2005) 27–51, https://doi.org/10.2140/pjm.2005.219.27, MR2174219. [3] George M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (2) (1978) 178–218, https://doi.org/10.1016/0001-8708(78)90010-5, MR506890. [4] Jyotishman Bhowmick, Francesco D’Andrea, Biswarup Das, Ludwik Dąbrowski, Quantum gauge symmetries in noncommutative geometry, J. Noncommut. Geom. 8 (2) (2014) 433–471, https:// doi.org/10.4171/JNCG/161, MR3275038. [5] Jyotishman Bhowmick, Debashish Goswami, Quantum group of orientation-preserving Riemannian isometries, J. Funct. Anal. 257 (8) (2009) 2530–2572, https://doi.org/10.1016/j.jfa.2009.07.006, MR2555012. [6] Julien Bichon, Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (3) (2003) 665–673, https://doi.org/10.1090/S0002-9939-02-06798-9, MR1937403. [7] Alain Connes, Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994, MR1303779. [8] Vyjayanthi Chari, Andrew Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1995, corrected reprint of the 1994 original, MR1358358. [9] Partha Sarathi Chakraborty, Arupkumar Pal, Characterization of SUq (l + 1)-equivariant spectral triples for the odd dimensional quantum spheres, J. Reine Angew. Math. 623 (2008) 25–42, https:// doi.org/10.1515/CRELLE.2008.071, MR2458039. [10] Mathijs S. Dijkhuizen, Tom H. Koornwinder, CQG algebras: a direct algebraic approach to compact quantum groups, Lett. Math. Phys. 32 (4) (1994) 315–330, https://doi.org/10.1007/BF00761142, MR1310296. [11] V.G. Drinfel d, Quantum groups, in: Proceedings of the International Congress of Mathematicians, vols. 1, 2, Berkeley, CA, 1986, Amer. Math. Soc., Providence, RI, 1987, pp. 798–820, MR934283. [12] Pavel Etingof, Debashish Goswami, Arnab Mandal, Chelsea Walton, Hopf coactions on commutative algebras generated by a quadratically independent comodule, Comm. Algebra 45 (8) (2017) 3410–3412, https://doi.org/10.1080/00927872.2016.1236934, MR3609348. [13] Pavel Etingof, Chelsea Walton, Semisimple Hopf actions on commutative domains, Adv. Math. 251 (2014) 47–61, https://doi.org/10.1016/j.aim.2013.10.008, MR3130334. [14] Pavel Etingof, Chelsea Walton, Finite dimensional Hopf actions on algebraic quantizations, Algebra Number Theory 10 (10) (2016) 2287–2310, https://doi.org/10.2140/ant.2016.10.2287, MR3582020. [15] Pavel Etingof, Chelsea Walton, Finite dimensional Hopf actions on deformation quantizations, Proc. Amer. Math. Soc. 145 (5) (2017) 1917–1925, https://doi.org/10.1090/proc/13356, MR3611308. [16] Debashish Goswami, Jyotishman Bhowmick, Quantum Isometry Groups, Infosys Science Foundation Series, Springer, New Delhi, 2016, Infosys Science Foundation Series in Mathematical Sciences, MR3559897. [17] Debashish Goswami, Soumalya Joardar, Quantum isometry groups of noncommutative manifolds obtained by deformation using dual unitary 2-cocycles, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014) 076, https://doi.org/10.3842/SIGMA.2014.076, MR3261868. [18] Michio Jimbo, A q-difference analogue of U (g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1) (1985) 63–69, https://doi.org/10.1007/BF00704588, MR797001. [19] Michio Jimbo, A q-analogue of U (gl(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (3) (1986) 247–252, https://doi.org/10.1007/BF00400222, MR841713.

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