K-theory and equivariant spectral triple for the quantum group U_q(2) for complex deformation parameters
aa r X i v : . [ m a t h . OA ] F e b K -theory and equivariant spectral triple for the quantum group U q (2) for complex deformation parameters Satyajit Guin, Bipul Saurabh
February 24, 2021
Abstract
Let q = | q | e iπθ , θ ∈ ( − , , be a nonzero complex number such that | q | 6 = 1 and considerthe compact quantum group U q (2). For θ / ∈ Q \ { , } , we obtain the K -theory of the C ∗ -algebra C ( U q (2)). Then, we produce a spectral triple on U q (2) which is equivariant underits own comultiplication action. The spectral triple obtained here is even, 4 + -summable,non-degenerate, and the Dirac operator acts on two copies of the L -space of U q (2). TheChern character of the associated Fredholm module is shown to be nontrivial. At the end,we compute the spectral dimension of U q (2). AMS Subject Classification No.: B , B , L . Keywords.
Compact quantum group, spectral triple, K -theory, quantum unitary group,equivariance, spectral dimension. In a recent work [22], Woronowicz et al. defined a family of q -deformations of SU (2) for q ∈ C \ { } . This agrees with the compact quantum group SU q (2) ([29],[30]) when q is real.But for q ∈ C \ R , SU q (2) is not a compact quantum group, rather a braided quantum groupin a suitable tensor category. In [24], Woronowicz et al. showed that for a braided quantumgroup A and a compact quantum group B , the semidirect product A ⊠ B becomes a compactquantum group. This semidirect product construction is a C ∗ -algebraic analogue of “bosoni-sation” defined by Majid [23]. Thus, taking A = SU q (2) for q ∈ C \ { } and B = C ( T ), weobtain a genuine compact quantum group. In (Sec. 6, [22]), it is proved that this compactquantum group is the coopposite of the compact quantum group U q (2) defined in [31]. In [20],authors have analyzed the compact quantum group structure of U q (2) by describing its finitedimensional irreducible unitary representations up to equivalence. The matrix coefficients areexpressed in terms of the generators of C ( U q (2)) using the q -Jacobi polynomials, their norms1ave been computed, and the explicit form of the Peter-Weyl decomposition is obtained. Therepresentation ring structure is also determined by showing how the tensor product of twoirreducible representations decomposes into irreducible components. These works lay the foun-dation for investigating geometrical aspects of U q (2) in the sense of Connes ([11],[14],[13]), andpurpose of the present article is precisely that.Central notion in Connes’ formulation of Noncommutative geometry is the notion of “Spec-tral triple”. Let A be a unital C ∗ -algebra and A ⊆ A be a unital dense ⋆ -subalgebra. An odd spectral triple is a tuple ( A , H , D ), where H is a separable Hilbert space on which A actsas bounded operators, and D is an unbounded self-adjoint operator with compact resolventsuch that the commutator [ D , a ] extends to a bounded operator on H for all a ∈ A . If thereis a Z / γ acting on H such that it commutes with a for all a ∈ A andanticommutes with D , then ( A , H , D , γ ) is called an even spectral triple. The operator D iswidely referred as the Dirac operator. Since the kernel of D is finite dimensional, withoutloss of generality, one can assume that D has trivial kernel (P. 316 in [11], P. 446 in [19]).If for p > , |D| − p lies in the Dixmier ideal L (1 , ∞ ) ⊆ B ( H ) then we say that the spectraltriple is p + -summable. Note that given A ⊆ A , there is no general recipe to construct aDirac operator on A satisfying certain desirable properties. If A is countably generated in A , then existence of spectral triple ( A , H , D ) is known ([1], Chap. 4 in [11]), although onehas very little control here over the Hilbert space or representation as it starts with a Fred-holm module. However, if we demand further properties like finite summability, then givena dense subalgebra of a C ∗ -algebra it may not admit any finitely summable spectral triple[10]. Given an even spectral triple ( A , H , D , γ ), taking F = D|D| − it induces a K -homologyclass [( A , H , F, γ )] in K ( A ) consisting of even Fredholm modules. To check nontriviality ofthis class, one pairs it with K ( A ) through the Kasparov product, which we describe briefly.Given a projection P ∈ M n ( A ) define H n = H ⊗ C n , γ n = γ ⊗ I n , F n = F ⊗ I n , P + = γ n P ,and P − = − γ n P . The operator P − F n P + : P + H + n −→ P − H − n is a Fredholm operator, where H n = H + n ⊕ H − n under the grading operator γ n . Index of this Fredholm operator is the value ofthe K − K pairing h [ P ] , [( A , H , F, γ )] i . An even spectral triple ( A , H , D , γ ) is called nontrivialif h [ P ] , [( A , H , F, γ )] i is nonzero for some [ P ] ∈ K ( A ). The requirement of nontrivial pairingis very crucial and reason is discussed in detail in [5].Now, if a compact quantum group G acts on A via the action τ : A −→ A ⊗ C ( G ), so thatwe have a C ∗ -dynamical system ( A , G , τ ), one demands the condition of “equivariance” in orderto capture the geometry of the dynamical system. Let ( π, U ) be a covariant representation ofthe C ∗ -dynamical system ( A , G , τ ) on H [2]. A Dirac operator D is called equivariant underthe action τ if D ⊗ U . If γ ⊗ U , then the even spectraltriple ( A , H , D , γ ) is called G -equivariant. See [28] for detail discussion on equivariant spectraltriples. Several authors have investigated these aspects of noncommutative geometry on variouscompact quantum groups and homogeneous spaces (e.g. [5],[12],[15],[16],[17],[18],[6]). A com-2act quantum group G acts on its underlying C ∗ -algebra A = C ( G ) via the comultiplication∆, and we have the C ∗ -dynamical system ( C ( G ) , G , ∆). A natural choice for A is the denseHopf ⋆ -subalgebra O ( G ) generated by the matrix coefficients of irreducible corepresentationsof ( C ( G ) , ∆). Now, one can desire to produce an explicit G -equivariant Dirac operator on A .Our focus here lies in this direction for compact quantum groups arising from the semidirectproduct construction of Woronowicz et al., for which U q (2) is a nontrivial concrete example.Let us recall the definition of U q (2) from [31]. For any nonzero complex number q , the C ∗ -algebra C ( U q (2)) is the universal C ∗ -algebra generated by a, b, D satisfying the followingrelations : ba = qab, a ∗ b = qba ∗ , bb ∗ = b ∗ b, aa ∗ + bb ∗ = 1 ,aD = Da, bD = q | q | − Db, DD ∗ = D ∗ D = 1 , a ∗ a + | q | b ∗ b = 1 . (1.1)The compact quantum group structure is given by the following comultiplication,∆( a ) = a ⊗ a − ¯ qb ⊗ Db ∗ , ∆( b ) = a ⊗ b + b ⊗ Da ∗ , ∆( D ) = D ⊗ D. (1.2)Let A q := O ( U q (2)) be the ⋆ -subalgebra of the C ∗ -algebra C ( U q (2)) generated by a, b and D .The Hopf ⋆ -algebra structure on it is given by the following :antipode: S ( a ) = a ∗ , S ( b ) = − qbD ∗ , S ( D ) = D ∗ , S ( a ∗ ) = a, S ( b ∗ ) = − (¯ q ) − b ∗ D , counit: ǫ ( a ) = 1 , ǫ ( b ) = 0 , ǫ ( D ) = 1 . For | q | 6 = 1, it is a compact quantum group of non-Kac type, whereas for | q | = 1 it is of Kac type. In this article, we restrict ourselves to the case of | q | 6 = 1. The faithful C ∗ -representationsfor the cases of | q | 6 = 1 and | q | = 1 are different and they lie on different Hilbert spaces (Propn.2 . . | q | = 1 will require different treatment. Weplan to discuss this case in detail elsewhere. The classification obtained in (Thm. 7 .
1, [20])justifies that for the case of | q | 6 = 1, it is enough to assume that | q | < . h on U q (2)is faithful. Combining this with Thm. (2 . , .
8) in [3], it follows that U q (2) is a coamenablecompact quantum group (in fact, further combining with Thm. 6 . U q (2)is coamenable for all q ∈ C \{ } such that q not a root of unity). Since the Haar state is faithful,one gets a faithful representation π h of C ( U q (2)) on L ( h ) given by the left multiplication. Onecan now work with the representation π h ⊕ π h on L ( h ) ⊕ L ( h ) and hope to define a Diracoperator of the form (cid:2) T ∗ T (cid:3) with the grading operator (cid:2) − (cid:3) . In such a case, the conditionof equivariance of the Dirac operator under comultiplication action reduces to the fact thateach matrix element in the Peter-Weyl decomposition is an eigenvector of T , and all matrixelements in a row must be an eigenvectors of T corresponding to the same eigenvalue.Brief description of our work is the following. For q = | q | e √− πθ , we assume that θ / ∈ Q \ { , } . Note that for θ = 0 ,
1, the deformation parameter q is real. In Sec. 2, we first show3hat the C ∗ -algebras C ( U q (2)) for different q ∈ C with | q | 6 = 0 , . K -theory of C ( U q (2)) along with the generatorsof the K -groups. This is achieved by showing that certain closed ideal in C ( U q (2)) can beidentified with K ( ℓ ( N )) ⊗ A θ , where A θ is the noncommutative torus and K ( ℓ ( N )) denotesthe space of compact operators acting on ℓ ( N ). It turns out that one of the generators, say P θ , of K ( C ( U q (2))) is same as the Powers-Rieffel projection in A θ if one replaces P b and
P D with standard unitary generators of A θ , where P denotes the element { bb ∗ =1 } in C ( U q (2)). InSec. 3, we compute the action of the generators of U q (2) on the orthonormal basis elementsof L ( h ). In Sec. 4, we get an orthonormal basis of the fixed point subspace, say E , for theaction of bb ∗ . Observe that π h ( P ) L ( h ) is the orthogonal projection onto E . Sec. 5 is theheart of the article where we produce a 4 + -summable even spectral triple on L ( h ) ⊗ C whichis equivariant under the comultiplication action of U q (2). The Dirac operator is shown to benon-degenerate, which means that our Dirac operator is really a Dirac operator for the fulltangent bundle rather than that of some lower dimensional subbundle. In Sec. 6, the Cherncharacter of the associated Fredholm module is shown to be nontrivial. This is achieved bypairing it with the class of projection P θ . We first show that the C ∗ -algebra B generated by π h ( P b ) and π h ( P D ) is isomorphic to the noncommutative torus A θ . Now, if one looks at theaction of B and the Dirac operator on the truncated Hilbert space π h ( P ) L ( h ), the Fredholmmodule associated with the spectral triple decomposes into a direct sum of infinite Fredholmmodules, one being a compact perturbation of the standard Fredholm module on A θ , and allthe other copies are compact perturbation of the trivial Fredholm module on A θ . Thus, weget the required Fredholm index to be nonzero. Finally in Sec. 7, we compute the spectraldimension [7] of U q (2) and it turns out to be 4. This says that one can not find any equivariantDirac operator acting on L ( h ) which is p -summable for p < K -theory of C ( U q (2)) Let q be a nonzero complex number with the polar decomposition q = | q | e √− πθ , θ ∈ ( − , | q | 6 = 1. Note that for θ = 0 , q is non-real. The followingproposition is along the line of Thm. 2 . SU q (2). Proposition 2.1.
The C ∗ -algebras C ( U q (2)) for different q ∈ C with | q | 6 = 0 , are isomorphic.Proof : Firstly, it is enough to show that for q, q ′ ∈ C ∗ with | q | , | q ′ | <
1, the C ∗ -algebras C ( U q (2)) and C ( U q ′ (2)) are isomorphic. This is because for the cases when | q | , | q ′ | > | q | < , | q ′ | >
1, it follows from the classification of the compact quantum group U q (2) obtainedin (Thm. 7 .
1, [20]), which in particular says that the C ∗ -algebras C ( U q (2)) and C ( U q (2)) areisomorphic. 4ow, if both q, q ′ ∈ ( − , \ { } , then the C ∗ -algebras C ( U q (2)) and C ( U q ′ (2)) are isomor-phic due to the fact that C ( U q (2)) ∼ = C ( SU q (2)) ⊗ C ( T ) in this case (Thm. 2 . C ∗ -algebras C ( SU q (2)) with 0 < | q | < . < | q | < , C ( U q (2)) ∼ = C ( U | q | (2)). The idea of the proof inThm. 2 . C ( D ), as done in [22], according to the defining relations in 1.1, and thenshow that this subalgebra is actually C ( D ) by the Stone-Weirstrass theorem. Then, workingwith the same homeomorphism g on D defined in [22], it follows that a, g ( b ) , D satisfy 1.1 with q replaced by | q | . This will show that C ( U q (2)) ∼ = C ( U | q | (2)) by our argument as in the firststep. Finally, using the second step the proof is completed. ✷ The classification obtained in (Thm. 7 .
1, [20]) justifies that it is enough to assume | q | < θ / ∈ Q \ { , } . Let H be the Hilbert space ℓ ( N ) ⊗ ℓ ( Z ) ⊗ ℓ ( Z ). Consider the right shift operator V : e n e n +1 acting on ℓ ( N ) and the unitary shift operator U : e k e k +1 acting on ℓ ( Z ). It is knownthat for | q | < C ( U q (2)) on H : π ( a ) = q − | q | N V ⊗ ⊗ , π ( b ) = q N ⊗ U ⊗ , π ( D ) = 1 ⊗ e − √− πθN ⊗ U. (2.1)is faithful (Propn. 2 . θ is irrational and mention at the end for θ ∈ { , } . Fororthonomal basis { e n : n = 0 , , . . . } of ℓ ( N ) recall the bra-ket notation | e m ih e n | to denotethe rank one projection e k e m h e n , e k i . Consider the following operators acting on ℓ ( N ) ⊗ ℓ ( Z ) ⊗ ℓ ( Z ), a = V ⊗ ⊗ , b = p ⊗ U ⊗ , D θ = 1 ⊗ e − √− πθN ⊗ U .
Let C ( U ,θ ) be the C ∗ -subalgebra of B (cid:0) ℓ ( N ) ⊗ ℓ ( Z ) ⊗ ℓ ( Z ) (cid:1) generated by a , b and D θ .Since the representation π in 2.1 is faithful, the C ∗ -algebra C ( U q (2)) is concretely realized asthe C ∗ -subalgebra of B (cid:0) ℓ ( N × Z × Z ) (cid:1) generated by a , b and D , where a = q − | q | N V ⊗ ⊗ , b = q N ⊗ U ⊗ , D = D θ . Proposition 2.2.
One has C ( U q (2)) = C ( U ,θ ) as C ∗ -algebras.Proof : Observe that b b ∗ = p ⊗ × a j b b ∗ ( a ∗ ) i = p ij ⊗ ⊗ p ij is the rankone projection | e j ih e i | on ℓ ( N ). This shows that K ⊗ ⊗ ⊆ C ( U ,θ ) where K is the space ofall compact operators on ℓ ( N ). Since a − a = (cid:16)q − | q | N − (cid:17) V ⊗ ⊗ ∈ K ⊗ ⊗ , (2.2)5e have a ∈ C ( U ,θ ). Now, observe that b = ∞ X n =0 ( q n | e n ih e n | ) ⊗ U ⊗ ∞ X n =0 q n a n b ( a ∗ ) n , and this is an element in C ( U ,θ ). Thus, we have C ( U q (2)) ⊆ C ( U ,θ ). To see the reverseinequality, first observe that { } ( b ∗ b ) = p ⊗ ⊗ ∈ C ( U q (2)) . Now, a j ( p ⊗ ⊗ a ∗ ) i = i − Y l =0 q − | q | i − l ) p ij ⊗ ⊗ C ( U q (2)) shows that K ⊗ ⊗ ⊆ C ( U q (2)). Since a − a ∈ K ⊗ ⊗ a ∈ C ( U q (2)). Finally, b = ( p ⊗ ⊗ q N ⊗ U ⊗
1) = ( p ⊗ ⊗ b ∈ C ( U q (2)) . Thus, C ( U ,θ ) ⊆ C ( U q (2)) and this completes the proof. ✷ Let T := C ∗ ( V ) be the Toeplitz algebra. We have the well-known short exact sequence0 −→ K ι −→ T σ −→ C ( T ) −→ σ : V z (here z denotes the standard unitary generator for C ( T )). Consider thehomomorphism τ : C ( U q (2)) −→ C ( T ) ⊗ B ( ℓ ( Z ) ⊗ ℓ ( Z )) given by τ = σ ⊗ ⊗
1, and let I θ = the closed two-sided ideal of C ( U q (2)) generated by b and b ∗ , B θ = C ∗ (cid:0) { τ ( a ) , τ ( D θ ) } (cid:1) = C ∗ (cid:0) { z ⊗ ⊗ , ⊗ e − √− πθN ⊗ U } (cid:1) . Proposition 2.3.
The following chain of C ∗ -algebras −→ I θ ι −→ C ( U q (2)) τ −→ B θ −→ is an exact sequence, where ‘ ι ’ denotes the inclusion map.Proof : Since τ ( b ) = τ ( b ∗ ) = 0, we have I θ ⊆ ker( τ ). Consider any irreducible representation π of C ( U q (2)) such that I θ ⊆ ker ( π ), i,e. π ( b ) = 0. By Thm. 3 . π isone dimensional. Define e π : B θ −→ C by e π ( τ ( a )) = π ( a ) and e π ( τ ( D θ )) = π ( D θ ). Then, π factors through the map τ : C ( U q (2)) −→ B θ . This shows that I θ = ker ( τ ). ✷ Let C θ be the C ∗ -subalgebra of B ( ℓ ( Z ) ⊗ ℓ ( Z )) generated by U ⊗ e − √− πθN ⊗ U .Since θ is irrational, by the universality and simpleness of the noncommutative torus A θ , weget that C θ ∼ = A θ as C ∗ -algebras. 6 emma 2.4. I θ = K ( ℓ ( N )) ⊗ C θ .Proof : We first claim that K ( ℓ ( N )) ⊗ C θ ⊆ C ( U q (2)), and it is a closed two sided ideal. Forthis, observe that p ⊗ C θ ⊆ C ( U q (2)) and hence, a j ( p ⊗ C θ ) a i ⊆ C ( U q (2)). This is same as p ij ⊗ C θ and hence, K ( ℓ ( N )) ⊗ C θ ⊆ C ( U q (2)). It is now a straightforward verification that K ( ℓ ( N )) ⊗ C θ becomes a two sided ideal in C ( U q (2)).To complete the proof, observe that b ∈ K ( ℓ ( N )) ⊗ C θ and hence, I ⊆ K ( ℓ ( N )) ⊗ C θ .Moreover, the image of K ( ℓ ( N )) ⊗ C θ under the map τ is zero and hence, it follows that K ( ℓ ( N )) ⊗ C θ ⊆ ker( τ ) = I θ by Propn. 2.3. ✷ Lemma 2.5.
For j = 0 , , K j ( I θ ) and K j ( B θ ) all are isomorphic to Z . The generators of K ( I θ ) are [ p ⊗ ⊗ and [ p ⊗ p θ ] , where p θ is the Powers-Rieffel projection in A θ with trace θ ,and that of K ( I θ ) are [ p ⊗ U ⊗ − p ) ⊗ ⊗ and [ p ⊗ e − √− πθN ⊗ U + (1 − p ) ⊗ ⊗ .Proof : Since by the definition B θ = C ∗ ( { τ ( a ) , τ ( D θ ) } ), it is immediate that B θ ∼ = C ( T ).Thus, K ( B θ ) ∼ = K ( B θ ) ∼ = Z and K ( B θ ) is generated by [ τ ( a )] and [ τ ( D θ )]. Moreover, K ( B θ ) is generated by [1] and [ P ( τ ( a ) , τ ( D θ ))]. We refer the reader to [9] (or Sec. 4 in [8])for detail exposition of the operation P .By the previous Propn. 2.4 and the fact that C θ is isomorphic to the noncommutativetorus A θ , we get that K ( I θ ) ∼ = K ( I θ ) ∼ = Z . Let p θ be the Powers-Rieffel projection inthe noncommutative torus A θ which generates K ( A θ ) along with [1]. Then, the generators of K ( I θ ) are [ p ⊗ ⊗
1] and [ p ⊗ p θ ]. Moreover, the generators of K ( I θ ) are [ p ⊗ U ⊗ − p ) ⊗ ⊗ p ⊗ e − √− πθN ⊗ U + (1 − p ) ⊗ ⊗ ✷ Theorem 2.6.
For q = | q | e √− πθ with θ irrational, both the K -groups K ( C ( U q (2))) and K ( C ( U q (2))) are isomorphic to Z . The equivalence classes of unitaries [ D ] and [ p ⊗ U ⊗ − p ) ⊗ ⊗ form a Z -basis for K ( C ( U q (2))) . The equivalence classes of projections [1] and [ p ⊗ p θ ] form a Z -basis for K ( C ( U q (2))) , where p θ denotes the Powers-Rieffel projection withtrace θ in the noncommutative torus A θ .Proof : In the six-term exact sequence K ( I θ ) K ( C ( U q (2))) K ( B θ ) K ( B θ ) K ( C ( U q (2))) K ( I θ ) K ( ι ) K ( τ ) δ∂ K ( τ ) K ( ι ) we first consider the index map ∂ : K ( B θ ) −→ K ( I θ ). Since we have the inclusion map inPropn. 2.3, I θ is an ideal in C ( U q (2)) , a is an isometry in C ( U q (2)), and D θ is unitary in C ( U q (2)), we immediately observe that ∂ ([ τ ( a )]) = − [ p ⊗ ⊗
1] and ∂ ([ τ ( D θ )]) = 0 (Propn.9 . . ker ( K ( ι )) = Im ( ∂ ) = h [ p ⊗ ⊗ i ∼ = Z and Im ( K ( τ )) =7 er ( ∂ ) = h [ τ ( D θ )] i ∼ = Z . Hence, Im ( K ( ι )) = ker ( K ( τ )) ∼ = Z by Lemma 2.5. Now, considerthe exponential map δ : K ( B θ ) −→ K ( I θ ). Observe that δ ([1]) = 0 as τ is unital (Propn.12 . . δ ([ P ( τ ( a ) , τ ( D θ ))]) is the subgroupin K ( I θ ) generated by [ D θ (1 − a a ∗ ) + a a ∗ ] by Cor. 4 . D θ (1 − a a ∗ ) + a a ∗ = p ⊗ e − √− πθN ⊗ U + (1 − p ) ⊗ ⊗
1. Hence, ker ( δ ) = Im ( K ( τ )) = h [1] i ∼ = Z and Im ( δ ) = ker ( K ( ι )) = h [ p ⊗ e − √− πθN ⊗ U + (1 − p ) ⊗ ⊗ i ∼ = Z . Thus, Im ( K ( ι )) = ker ( K ( τ )) ∼ = Z by Lemma 2.5. These facts show that K ( C ( U q (2))) ∼ = K ( C ( U q (2))) ∼ = Z . The generators ofthe K -groups are obtained using Lemma 2.5. Observe that K ( C ( U q (2))) will be generated bythe generator of Im ( K ( ι )) ∼ = K ( I θ ) /Im ( ∂ ) and Ker ( δ ), which are [ p ⊗ p θ ] and [1] respectively.Similarly, K ( C ( U q (2))) will be generated by the generator of Im ( K ( ι )) ∼ = K ( I θ ) /Im ( δ ) and Ker ( ∂ ), which are [ p ⊗ U ⊗ − p ) ⊗ ⊗
1] and [ D ] respectively. ✷ Finally, we mention what happens when θ = 0 ,
1. The deformation parameter q is realin these two situations and q ∈ ( − , C ∗ -algebra C ( U ,θ ) and Propns. (2.2,2.3) makesense and hold in these two cases. But the C ∗ -algebra C θ in Lemma 2.4 becomes C ( T ) when θ ∈ { , } . Thus, the generators of K ( I θ ) becomes [1] and [ p ⊗ Bott ] where, “
Bott ” denotesthe Bott projection in M ( C ( T )). The proof of Thm. 2.6 holds along the same line, and weget that the generators of K ( C ( U q (2))) are [1] and [ p ⊗ Bott ]. L ( h ) In this section we describe the action of the genrators a, a ∗ , b, b ∗ , D, D ∗ of U q (2) on the or-thonormal basis of L ( h ) consisting of matrix coefiicients. Let π h : C ( U q (2)) −→ B ( L ( h ))be the GNS representation associated to the Haar state h . Since the Haar state on U q (2)is faithful (Thm. 2 . π h is faithful. We omit the representationsymbol π h and write x instead of π h ( x ) for the action of x ∈ C ( U q (2)). Recall that A q is the ⋆ -subalgebra of C ( U q (2)) generated by the matrix coefficients of all finite dimensional irreduciblerepresentations of U q (2).The Peter-Weyl decomposition obtained in Thm. 4 .
17 in [20] says that the following set (cid:8) | q | − i q | ℓ + 1 | | q | t ℓi,j D k : ℓ ∈ N , k ∈ Z (cid:9) is an orthonormal basis of L ( h ) where, t ℓi,j = X m + n = ℓ − i ≤ m ≤ ℓ − j ≤ n ≤ ℓ + j q n ( ℓ − j − m ) (cid:0) ℓℓ + j (cid:1) / | q | (cid:0) ℓℓ + i (cid:1) / | q | (cid:18) ℓ − jm (cid:19) | q | (cid:18) ℓ + jn (cid:19) | q | a m c ℓ − j − m b n d ℓ + j − n c = − ¯ qDb ∗ and d = Da ∗ . Throughout the article we reserve the following notation, e ℓi,j,k := | q | − i q | ℓ + 1 | | q | t ℓi,j ( D ∗ ) k = | q | − i q | ℓ + 1 | | q | t ℓi,j D − k for the orthonormal basis of L ( h ). For reader’s convenience, we recall the following q -Jacobipolynomial expressions of t ℓi,j D k from § . Theorem 3.1 ([20]) . The matrix coefficients t ℓi,j D − k are expressed in terms of the little q-Jacobi polynomials in the following way :(i) for the case of i + j ≤ , i ≥ j , a − ( i + j ) c i − j (¯ q ) ( j − i )( ℓ + j ) ( ℓℓ + j ) / | q | ( ℓℓ + i ) / | q | (cid:0) ℓ − ji − j (cid:1) | q | P ( i − j, − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k ; (ii) for the case of i + j ≤ , i ≤ j , a − ( i + j ) b j − i q ( i − j )( ℓ + i ) ( ℓℓ + j ) / | q | ( ℓℓ + i ) / | q | (cid:0) ℓ + jj − i (cid:1) | q | P ( j − i, − i − j ) ℓ + i ( bb ∗ ; | q | ) D ℓ + i − k ; (iii) for the case of i + j ≥ , i ≤ j , q ( i − j )( ℓ + i ) ( ℓℓ + j ) / | q | ( ℓℓ + i ) / | q | (cid:0) ℓ + jj − i (cid:1) | q | P ( j − i,i + j ) ℓ − j ( bb ∗ ; | q | )( a ∗ ) i + j b j − i D ℓ + i − k ; (iv) for the case of i + j ≥ , i ≥ j , (¯ q ) ( j − i )( ℓ + j ) ( ℓℓ + j ) / | q | ( ℓℓ + i ) / | q | (cid:0) ℓ − ji − j (cid:1) | q | P ( i − j,i + j ) ℓ − i ( bb ∗ ; | q | )( a ∗ ) i + j c i − j D ℓ + j − k . For n, k ∈ Z and m, r ∈ N , define a n b m ( b ∗ ) r D k = a n b m ( b ∗ ) r D k if n ≥ , ( a ∗ ) − n b m ( b ∗ ) r D k if n < . Theorem 3.2 ([31]) . The set { a n b m ( b ∗ ) r D k : n, k ∈ Z , m, r ∈ N } forms a linear basis of A q for all q ∈ C ∗ . Thm. 3.1 and 3.2 lead us to the following very important theorem, which is the backboneto our search for Dirac operator in Sec. 5.
Theorem 3.3.
The action of the generators of U q (2) on the orthonormal basis element e ℓi,j,k is described by the following : 9 i) D ⊲ e ℓi,j,k = (cid:16) qq (cid:17) ( i − j ) e ℓi,j,k − (ii) D ∗ ⊲ e ℓi,j,k = (cid:16) qq (cid:17) ( i − j ) e ℓi,j,k +1 (iii) b ⊲ e ℓi,j,k = β + ( ℓ, i, j ) e ℓ +1 / i − / , j +1 / , k + β − ( ℓ, i, j ) e ℓ − / i − / , j +1 / , k − (iv) b ∗ ⊲ e ℓi,j,k = β ++ ( ℓ, i, j ) e ℓ +1 / i +1 / , j − / , k +1 + β + − ( ℓ, i, j ) e ℓ − / i +1 / , j − / , k (v) a ⊲ e ℓi,j,k = α + ( ℓ, i, j ) e ℓ +1 / i − / , j − / , k + α − ( ℓ, i, j ) e ℓ − / i − / , j − / , k − (vi) a ∗ ⊲ e ℓi,j,k = α ++ ( ℓ, i, j ) e ℓ +1 / i +1 / , j +1 / , k +1 + α + − ( ℓ, i, j ) e ℓ − / i +1 / , j +1 / , k where, β + ( ℓ, i, j ) = q ℓ − j s (1 − | q | ℓ + j +1) )(1 − | q | ℓ − i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) .β − ( ℓ, i, j ) = − q ℓ − j − ( q ) j − i +1 r q ¯ q s (1 − | q | ℓ − j ) )(1 − | q | ℓ + i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) .β ++ ( ℓ, i, j ) = − q j − i − ( q ) ℓ − j +1 r q ¯ q s (1 − | q | ℓ − j +1) )(1 − | q | ℓ + i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) .β + − ( ℓ, i, j ) = ( q ) ℓ − j s (1 − | q | ℓ + j ) )(1 − | q | ℓ − i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) .α + ( ℓ, i, j ) = s (1 − | q | ℓ − j +1) )(1 − | q | ℓ − i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) .α − ( ℓ, i, j ) = q ℓ − i ( q ) ℓ − j +1 r q ¯ q s (1 − | q | ℓ + j ) )(1 − | q | ℓ + i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) .α ++ ( ℓ, i, j ) = q ℓ − j ( q ) ℓ − i +1 r q ¯ q s (1 − | q | ℓ + j +1) )(1 − | q | ℓ + i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) .α + − ( ℓ, i, j ) = s (1 − | q | ℓ − j ) )(1 − | q | ℓ − i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) . roof : For part ( i ), observe that Da m c ℓ − j − m = (cid:16) | q | q (cid:17) ℓ − j − m ) a m c ℓ − j − m D , Db n = (cid:16) | q | q (cid:17) n b n D , and Dd ℓ + j − n = d ℓ + j − n D . Hence, De ℓi,j,k = (cid:16) qq (cid:17) n (cid:16) qq (cid:17) ( ℓ − j − m ) e ℓi,j,k − = (cid:16) qq (cid:17) ( i − j ) e ℓi,j,k − . Part( ii ) follows easily from this.For part ( iii ), we only show the computations for the case i + j ≤ i ≥ j . For all theother cases, computations go in a similar fashion. We separate two cases. First let’s assumethat i + j ≤ i ≥ j + 1. Take the expression in case ( i ) in Thm. 3.1. That is, t ℓi,j D − k = a − ( i + j ) c i − j (¯ q ) ( j − i )( ℓ + j ) (cid:0) ℓℓ + j (cid:1) / | q | (cid:0) ℓℓ + i (cid:1) / | q | (cid:18) ℓ − ji − j (cid:19) | q | P ( i − j , − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k = a − ( i + j ) c i − j − ( − ¯ q ) (cid:16) | q | ¯ q (cid:17) (¯ q ) ( j − i )( ℓ + j ) (cid:0) ℓℓ + j (cid:1) / | q | (cid:0) ℓℓ + i (cid:1) / | q | (cid:18) ℓ − ji − j (cid:19) | q | b ∗ P ( i − j , − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k +1 . Thus, b ⊲ t ℓi,j D − k = a − ( i + j ) c i − j − q − i − j (cid:16) q | q | (cid:17) i − j − ( − ¯ q ) (cid:16) | q | ¯ q (cid:17) (¯ q ) ( j − i )( ℓ + j ) (cid:0) ℓℓ + j (cid:1) / | q | (cid:0) ℓℓ + i (cid:1) / | q | (cid:18) ℓ − ji − j (cid:19) | q | bb ∗ P ( i − j , − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k +1 = α li,j a − ( i + j ) c i − j − bb ∗ P ( i − j , − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k +1 where, α ℓi,j = q − i − j (cid:16) q | q | (cid:17) i − j − ( − ¯ q ) (cid:16) | q | ¯ q (cid:17) (¯ q ) ( j − i )( ℓ + j ) ( ℓℓ + j ) / | q | ( ℓℓ + i ) / | q | (cid:0) ℓ − ji − j (cid:1) | q | . We find α and α suchthat || t ℓi,j || − α ℓi,j a − ( i + j ) c i − j − bb ∗ P ( i − j , − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k +1 = || t ℓ +1 / i − / , j +1 / || − α a − ( i + j ) c i − j − (¯ q ) ( j − i +1)( ℓ + j +1) (cid:0) ℓ +1 ℓ + j +1 (cid:1) / | q | (cid:0) ℓ +1 ℓ + i (cid:1) / | q | (cid:18) ℓ − ji − j − (cid:19) | q | P ( i − j − , − i − j ) ℓ + j +1 ( bb ∗ ; | q | ) D ℓ + j − k +1 + || t ℓ − / i − / , j +1 / || − α a − ( i + j ) c i − j − (¯ q ) ( j − i +1)( ℓ + j ) (cid:0) ℓ − ℓ + j (cid:1) / | q | (cid:0) ℓℓ + i − (cid:1) / | q | (cid:18) ℓ − j − i − j − (cid:19) | q | P ( i − j − , − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k +1 . Equating coefficients of the element in C ( U q (2)) associated with the constant term of thepolynomial bb ∗ P ( i − j , − i − j ) ℓ + j ( bb ∗ ; | q | ) from the left with that of P ( i − j − , − i − j ) ℓ + j +1 ( bb ∗ ; | q | ) and11 ( i − j − , − i − j ) ℓ + j ( bb ∗ ; | q | ) from the right we get the following equation,0 = || t ℓ +1 / i − / , j +1 / || − α (¯ q ) ( j − i +1)( ℓ + j +1) (cid:0) ℓ +1 ℓ + j +1 (cid:1) / | q | (cid:0) ℓ +1 ℓ + i (cid:1) / | q | (cid:18) ℓ − ji − j − (cid:19) | q | + || t ℓ − / i − / , j +1 / || − α (¯ q ) ( j − i +1)( ℓ + j ) (cid:0) ℓ − ℓ + j (cid:1) / | q | (cid:0) ℓℓ + i − (cid:1) / | q | (cid:18) ℓ − j − i − j − (cid:19) | q | . (3.1)Similarly, equating coefficients of the elements in C ( U q (2)) associated with the highest degreeterm ( | q | bb ∗ ) ℓ + j +1 of the polynomials from both sides we get the following,1 | q | || t ℓi,j || − α ℓi,j ( | q | − ℓ + j ) ; | q | ) ℓ + j ( | q | ℓ − j +1) ; | q | ) ℓ + j ( | q | ; | q | ) ℓ + j ( | q | i − j +1) ; | q | ) ℓ + j = || t ℓ +1 / i − / , j +1 / || − α (¯ q ) ( j − i +1)( ℓ + j +1) (cid:0) ℓ +1 ℓ + j +1 (cid:1) / | q | (cid:0) ℓ +1 ℓ + i (cid:1) / | q | (cid:18) ℓ − ji − j − (cid:19) | q | ( | q | − ℓ + j +1) ; | q | ) ℓ + j +1 ( | q | ℓ − j +1) ; | q | ) ℓ + j +1 ( | q | ; | q | ) ℓ + j +1 ( | q | i − j ) ; | q | ) ℓ + j +1 . From here we get that α = || t ℓ +1 / i − / , j +1 / |||| t ℓi,j || q ℓ − j vuuut (cid:0) ℓ +1 ℓ + i (cid:1) | q | (cid:0) ℓℓ + i (cid:1) | q | vuuut (cid:0) ℓℓ + j (cid:1) | q | (cid:0) ℓ +1 ℓ + j +1 (cid:1) | q | (cid:0) ℓ − ji − j (cid:1) | q | (cid:0) ℓ − ji − j − (cid:1) | q | (cid:16) − | q | i − j ) − | q | ℓ +1) (cid:17) = q ℓ − j s − | q | ℓ +1) − | q | ℓ +2) s − | q | ℓ +1) − | q | ℓ − i +1) s − | q | ℓ + j +1) − | q | ℓ +1) (cid:16) − | q | ℓ − i +1) − | q | i − j ) (cid:17)(cid:16) − | q | i − j ) − | q | ℓ +1) (cid:17) = q ℓ − j s (1 − | q | ℓ + j +1) )(1 − | q | ℓ − i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) . Therefore, from the eqn. 3.1 it follows that α = − q ℓ − j (¯ q ) j − i r ¯ qq s (1 − | q | ℓ − j ) )(1 − | q | ℓ + i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) . Now, for the second case, i,e. when i + j ≤ i = j we need to take the expression in case( ii ) in Thm. 3.1. One can verify the rests as computations are similar.For part ( vi ), again we only show the computations for the case i + j ≤ i ≥ j . Readerscan verify the other cases as the computations are similar. Here also we separate two cases.First let’s assume that i + j ≤ − i ≥ j . Take the expression in case ( i ) in Thm. 3.1.Observe that a ∗ ⊲ t ℓi,j D − k = α ℓij a − ( i + j ) − c i − j P ( i − j , − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k −| q | − i + j ) α ℓij a − ( i + j ) − c i − j bb ∗ P ( i − j , − i − j ) ℓ + j ( bb ∗ ; | q | ) D ℓ + j − k α ℓij = (¯ q ) ( j − i )( ℓ + j ) ( ℓℓ + j ) / | q | ( ℓℓ + i ) / | q | (cid:0) ℓ − ji − j (cid:1) | q | . In order to prove that the right hand side is of thefollowing form C || t ℓi,j D − k |||| t ℓ − / i +1 / , j +1 / D − k || t ℓ − / i +1 / , j +1 / D − k + C || t ℓi,j D − k |||| t ℓ +1 / i +1 / , j +1 / D − k +1 || t ℓ +1 / i +1 / , j +1 / D − k +1 for complex numbers C , C , we first equate coefficients of the ellements in C ( U q (2)) associatedwith the constant terms of the polynomials of both sides to get the following equation, α ℓij || t ℓi,j D − k || − = C || t ℓ − / i +1 / , j +1 / D − k || − (¯ q ) ( j − i )( ℓ + j ) (cid:0) ℓ − ℓ + j (cid:1) / | q | (cid:0) ℓ − ℓ + i (cid:1) / | q | (cid:18) ℓ − j − i − j (cid:19) | q | + C || t ℓ +1 / i +1 / , j +1 / D − k +1 || − (¯ q ) ( j − i )( ℓ + j +1) (cid:0) ℓ +1 ℓ + j +1 (cid:1) / | q | (cid:0) ℓ +1 ℓ + i +1 (cid:1) / | q | (cid:18) ℓ − ji − j (cid:19) | q | . (3.2)Now, equating coefficients of the elements in C ( U q (2)) associated with the highest degree termsof the polynomials of both sides we get the following, − α ℓij || t ℓi,j D k || | q | − i + j +1) ( | q | − ℓ + j ) ; | q | ) ℓ + j ( | q | ℓ − j +1) ; | q | ) ℓ + j ( | q | ; | q | ) ℓ + j ( | q | i − j +1) ; | q | ) ℓ + j = C || t ℓ +1 / i +1 / , j +1 / D − k +1 || − (¯ q ) ( j − i )( ℓ + j +1) (cid:0) ℓ +1 ℓ + j +1 (cid:1) / | q | (cid:0) ℓ +1 ℓ + i +1 (cid:1) / | q | (cid:18) ℓ − ji − j (cid:19) | q | ( | q | − ℓ + j +1) ; | q | ) ℓ + j +1 ( | q | ℓ − j +1) ; | q | ) ℓ + j +1 ( | q | ; | q | ) ℓ + j +1 ( | q | i − j +1) ; | q | ) ℓ + j +1 . This gives us the following, C = q ℓ − i (¯ q ) ℓ − j +1 r q ¯ q s (1 − | q | ℓ + j +1) )(1 − | q | ℓ + i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) . Therefore, from eqn. 3.2 it follows that C = s (1 − | q | ℓ − j ) )(1 − | q | ℓ − i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) . Now, for the second case, i,e. when i + j = 0 and i ≥ j we need to take the expression in case( iv ) in Thm. 3.1. One can verify the rests as computations are similar.For part ( iv ), it is now easy to observe that b ∗ ⊲ e ℓi,j,k = β − ( ℓ + 1 / , i + 1 / , j − / e ℓ +1 / i +1 / , j − / , k +1 + β + ( ℓ − / , i + 1 / , j − / e ℓ − / i +1 / , j − / , k = β ++ ( ℓ, i, j ) e ℓ +1 / i +1 / , j − / , k +1 + β + − ( ℓ, i, j ) e ℓ − / i +1 / , j − / , k β ++ ( ℓ, i, j ) = − (¯ q ) ℓ − j q j − i r ¯ qq s (1 − | q | ℓ − j +1) )(1 − | q | ℓ + i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) ,β + − ( ℓ, i, j ) = (¯ q ) ℓ − j s (1 − | q | ℓ + j ) )(1 − | q | ℓ − i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) . Similarly for part ( v ), observe that a ⊲ e ℓi,j,k = α + − ( ℓ + 1 / , i − / , j − / e ℓ +1 / i − / , j − / , k + α ++ ( ℓ − / , i − / , j − / e ℓ − / i − / , j − / , k − = α + ( ℓ, i, j ) e ℓ +1 / i − / , j − / , k + α − ( ℓ, i, j ) e ℓ − / i − / , j − / , k − where, α + ( ℓ, i, j ) = s (1 − | q | ℓ − j +1) )(1 − | q | ℓ − i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) ,α − ( ℓ, i, j ) = q ℓ − i (¯ q ) ℓ − j +1 r q ¯ q s (1 − | q | ℓ + j ) )(1 − | q | ℓ + i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) . This completes the proof. ✷ These coefficients will play a pivotal role in finding equivariant Dirac operator in Sec. 5.
Notation : − t r ± s := (1 − t r + s )(1 − t r − s ) for t ∈ R + . Corollary 3.4.
We have, bb ∗ ⊲ e ℓi,j,k = γ + ( ℓ, i, j ) e ℓ +1 i,j , k +1 + γ ( ℓ, i, j ) e ℓi,j,k + γ − ( ℓ, i, j ) e ℓ − i,j , k − where, γ + ( ℓ, i, j ) = − q ℓ − i (¯ q ) ℓ − j +1 r q ¯ q (cid:0) − | q | ℓ +2) (cid:1) s (cid:0) − | q | ℓ +1 ± j ) (cid:1)(cid:0) − | q | ℓ +1 ± i ) (cid:1)(cid:0) − | q | ℓ +1) (cid:1)(cid:0) − | q | ℓ +3) (cid:1) γ ( ℓ, i, j ) = | q | ℓ − i ) (cid:0) − | q | ℓ − j +1) (cid:1)(cid:0) − | q | ℓ + i +1) (cid:1)(cid:0) − | q | ℓ +1) (cid:1)(cid:0) − | q | ℓ +2) (cid:1) + | q | ℓ − j ) (cid:0) − | q | ℓ + j ) (cid:1)(cid:0) − | q | ℓ − i ) (cid:1)(cid:0) − | q | . ℓ (cid:1)(cid:0) − | q | ℓ +1) (cid:1) γ − ( ℓ, i, j ) = − q ℓ − j − (¯ q ) ℓ − i r q ¯ q − | q | . ℓ ) s (cid:0) − | q | ℓ ± j ) (cid:1)(cid:0) − | q | ℓ ± i ) (cid:1)(cid:0) − | q | ℓ − (cid:1)(cid:0) − | q | ℓ +1) (cid:1) . Fixed points for the action of bb ∗ It is easy to observe that the spectrum of bb ∗ , denoted by σ ( bb ∗ ), is { , | q | , | q | , · · · } ∪ { } . Fix i, j ∈ Z and k ∈ Z . We abbreviate “closed linear span” by c.l.s. For any λ ∈ C ∗ , define thefollowings, w ij = max {| i | , | j |} ,A ( i, j, k ) = c.l.s. (cid:8) e w ij + mi,j,k + m : m ∈ N (cid:9) E λ ( i, j, k ) = { v ∈ A ( i, j, k ) : bb ∗ v = λv } E λ = { v ∈ L ( h ) : bb ∗ v = λv } . It follows from Cor. 3.4 that A ( i, j, k ) is an invariant subspace of bb ∗ . Using Thm 4 .
17 in [20],it follows that L ( h ) = M i,j ∈ Z , k ∈ Z A ( i, j, k ) . Note that either both i and j are integers or both are half integers. To get solutions ofthe equation bb ∗ v = λv , it is enough to look at solutions in each A ( i, j, k ). Suppose that v = P ∞ m =0 c m e w ij + mi,j,k + m ∈ A ( i, j, k ) is a nonzero solution of bb ∗ v = λv . By Cor. 3.4, we get thefollowing recurrence relations, λc = c γ ( w ij , i, j ) + c γ − ( w ij + 1 , i, j ) , (4.1)and for m ≥ λc m = c m +1 γ − ( w ij + m + 1 , i, j ) + c m γ ( w ij + m, i, j ) + c m − γ + ( w ij + m − , i, j ) . (4.2)Observe that in Cor. 3.4, γ + ( ℓ, i, j ) and γ ( ℓ, i, j ) are always nonzero, and γ − ( ℓ, i, j ) = 0 for w ij < ℓ . Using this, from the relations 4.1 and 4.2 it follows that c = 0 as otherwise v wouldbe zero. We call e w ij i,j,k as the leading term and the corresponding coefficient as the leadingcoefficient. Proposition 4.1.
Let λ ∈ σ ( bb ∗ ) . Then one has the followings, ( i ) dim E λ ( i, j, k ) ≤ . ( ii ) If v is a nonzero vector in E λ , then Dv and D ∗ v are nonzero vectors in E λ . ( iii ) If v ∈ E λ , then av and a ∗ v are in E | q | λ and E λ | q | respectively. ( iv ) If v is a nonzero vector in E λ ( i, j, k ) , then bv is a nonzero vector in E λ ( i − , j + , k ) for i ≤ j , and in E λ ( i − , j + , k − for i > j . v ) If v is a nonzero vector in E λ ( i, j, k ) , then b ∗ v is a nonzero vector in E λ ( i + , j − , k + 1) for i ≥ j , and in E λ ( i + , j − , k ) for i < j . ( vi ) For any nonzero vector v in E λ ( i, j, k ) , av is a nonzero vector in E | q | λ ( i − , j − , k − for i ≥ j , and in E | q | λ ( i − , j − , k ) for i < j . ( vii ) If λ = 1 , then for any nonzero vector v in E λ ( i, j, k ) , a ∗ v is a nonzero vector in E λ | q | ( i + , j + , k + 1) for i ≥ j , and in E λ | q | ( i + , j + , k ) for i < j .Proof :( i ) The eqn. (4.1) and (4 .
2) have a unique solution ( c m ) m ≥ if one fix the leading coefficient c . Now if P ∞ m =0 | c m | < ∞ , then dim E λ ( i, j, k ) = 1 otherwise dim E λ ( i, j, k ) = 0.( ii ) Since D is a unitary, Dv = 0 if v = 0. Moreover, Dv ∈ E λ for v ∈ E λ follows from thefact that D commutes with bb ∗ . Similar reason for D ∗ .( iii ) Follows from the defining relations ba = qab and a ∗ b = qba ∗ .( iv ) That v = 0 in E λ ( i, j, k ) implies bv = 0 follows immediately from the normality of b . Therests follow by looking at the leading coefficient of v and the action of b given by Thm.3.3.( v ) Use the same argument as in part ( iv ).( vi ) Let av = 0. Using the relation a ∗ a + | q | bb ∗ = 1, one has bb ∗ v = | q | v . Since | q | / ∈ σ ( bb ∗ ),one arrives at a contradiction. The rest will follow by analyzing the leading coefficient of v and the action of a ∗ given by Thm. 3.3.( vii ) Use the same argument as in part ( vi ). ✷ The following proposition provides a nontrivial solution of the equation bb ∗ v = v . Moreprecisely, it says that dim E (0 , ,
0) = 1.
Proposition 4.2.
Let v = P ∞ m =0 c m e m , ,m ∈ L ( h ) . The equation bb ∗ v = v has a uniquenonzero solution up to constant scalar multiple.Proof : By Cor. 3.4, we see that bb ∗ ( e m , ,m ) = X ξ = − Υ ξ ( m ) e m + ξ , ,m + ξ − ( m ) = − | q | m − (1 − | q | m ) (1 − | q | m ) p (1 − | q | m − )(1 − | q | m +2 ) , Υ ( m ) = | q | m (1 − | q | m +2 ) (1 − | q | m +2 )(1 − | q | m +4 ) + | q | m (1 − | q | m ) (1 − | q | m )(1 − | q | m +2 ) , Υ ( m ) = − | q | m +1 (1 − | q | m +2 ) (1 − | q | m +4 ) p (1 − | q | m +2 )(1 − | q | m +6 ) . We want to solve bb ∗ v = v i,e. P ∞ m =0 c m e m , ,m = P ∞ m =0 P ξ = − c m Υ ξ ( m ) e m + ξ , ,m + ξ . That is, ∞ X m =0 c m e m , ,m = ∞ X m =0 c m Υ − ( m ) e m − , ,m − + c m Υ ( m ) e m , ,m + c m Υ ( m ) e m +10 , ,m +1 . Equating coefficients of the basis elements of both sides, and observing that Υ ( m −
1) =Υ − ( m ), we get the following, c = c Υ (0) + c Υ − (1) ,c = c Υ − (2) + c Υ (1) + c Υ (0)= c Υ (1) + c Υ (1) + c Υ (0) ,c m = c m +1 Υ ( m ) + c m Υ ( m ) + c m − Υ ( m − ∀ m ≥ . Observe that if c = 0 then c m = 0 for all m >
0, and consequently v = 0. Assume that c = 1.Then, c = − Υ (0)Υ (0) and we have the following recurrence relation, c m +1 = c m (1 − Υ ( m )) − c m − Υ ( m − ( m ) ∀ m ≥ . (4.3)Plugging the value of Υ (0) and Υ (0) we see that c = −| q | q −| q | −| q | . Now, one can verify thatfor m ≥ c m = ( − m +1 c | q | m − s − | q | m +2 − | q | satisfies the recurrence relation 4.3. That is, for all m ≥ c m = ( − m | q | m s − | q | m +2 − | q | . Now, observe that | c m || q | m = | q | m − m s − | q | m +2 − | q | −→ m → ∞ < | q | <
1. Thus, c m = o ( | q | m ) as m → ∞ , i,e. ( c m ) m ≥ ∈ ℓ ( N ). This completes theproof. ✷ Let l ∈ N and consider the operator bb ∗ restricted to the invariant subspace A (0 , , C ∗ -algebra C ∗ (1 , bb ∗ ) ∼ = C ( σ ( bb ∗ )). By the spectral decom-position, there exists a subset Ω = { n l : l ∈ N , n l < n l +1 } of N such that E | q | nl (0 , , = { } ,and we can write A (0 , ,
0) = E (0 , , M l ∈ N E | q | nl (0 , , . By Propn 4.2, we have n = 0. It follows from part ( i ) of the Propn. 4.1 that the cardinalityof Ω is infinite. Proposition 4.3.
One has the followings. ( i ) If dim E ( i, j, k ) = 1 for some k ∈ Z , then for each k ∈ Z we have dim E ( i, j, k ) = 1 . ( ii ) If dim E ( i, j, k ) = 1 , then dim E ( i ′ , j ′ , k ′ ) = 1 for k ′ ∈ Z , and i ′ , j ′ ∈ Z with i ′ + j ′ = i + j . ( iii ) For n l ∈ Ω , dim E ( i, j, k ) = 1 for all k ∈ Z and i, j ∈ Z such that i + j = n l . ( iv ) E = L ∞ l =0 L i + j = n l , k ∈ Z E ( i, j, k ) .Proof :( i ) It is a direct consequence of part ( ii ) of Propn. 4.1.( ii ) This follows by repeatedly applying part ( ii ) , ( iv ) and ( v ) of Propn. 4.1, together withpart ( i ).( iii ) We have dim E | q | nl (0 , ,
0) = 1. Using part ( vii ) of Propn. 4.1 repeatedly, we get thatdim E ( n l , n l , n l ) = 1. By part ( ii ), the claim now follows.( iv ) From part ( iii ), we have E ⊇ L ∞ l =0 L i + j = n l , k ∈ Z E ( i, j, k ). To prove the claim, it isenough to show that dim E ( i, j,
0) = 0 for i, j ∈ Z such that i + j <
0. Let i , j ∈ Z with i + j <
0, and let v = P ∞ m =0 c m e w i j + mi ,j ,m ∈ A ( i , j ,
0) be a nonzero solution of bb ∗ v = v . By Propn. 3.3, we get the following, a ∗ v = c α + − ( w i j , i , j ) e w i j − i + ,j + , + ∞ X m =1 f c m e w i j − + mi + ,j + ,m where f c m ’s are complex numbers. Since i + j <
0, we have i , j = w i j . Hence, α + − ( w i j , i , j ) = 0, which implies that a ∗ v is a nonzero vector as c = 0. This alongwith part ( iii ) of the Propn. 4.1 gives that | q | ∈ σ ( bb ∗ ), which is a contradiction. ✷ H l = M i + j = n l , k ∈ Z E ( i, j, k ) . (4.4)Take a unit vector in E ( n l , n l , n l ), and denote it by | n l , n l , i . For i, j ∈ Z such that i + j = n l ,we define the following | i, j, k i = ( b ∗ ) i − n l ( D ∗ ) ( n l − i + k ) | n l , n l , i if i ≥ n l , ( b ) n l − i ( D ∗ ) k | n l , n l , i if i < n l . (4.5)Observe that b | i, j, k i = (cid:12)(cid:12) i − , j + , k − (cid:11) if i ≥ n l +12 , (cid:12)(cid:12) i − , j + , k (cid:11) if i ≤ n l ; (4.6) b ∗ | i, j, k i = (cid:12)(cid:12) i + , j − , k + 1 (cid:11) if i ≥ n l , (cid:12)(cid:12) i + , j − , k (cid:11) if i ≤ n l − ; (4.7)and D | i, j, k i = e π √− i − n l ) θ | i, j, k − i , D ∗ | i, j, k i = e π √− n l − i ) θ | i, j, k + 1 i . (4.8)Moreover, since b, b ∗ , D and D ∗ act as unitary operators when restricted to the invariant space E , it follows that | i, j, k i is a unit vector in E ( i, j, n l + k ) ⊆ H l . Hence, we can write | i, j, k i = ∞ X m =0 c ( i,j,k ) m e w ij + mi,j,i + j + k + m . (4.9)Note that P ∞ m =0 | c ( i,j,k ) m | = 1 as | i, j, k i is a unit vector. Proposition 4.4.
Define C i,j,k = P ∞ m =1 (cid:12)(cid:12) c ( i,j,k ) m (cid:12)(cid:12) for k ∈ Z and i, j ∈ Z with i + j = n l ∈ Ω for some l ∈ N . Then, one has the following. ( i ) C i,j,k → as i → ±∞ , provided i + j = n l for some fixed l ∈ N . ( ii ) C i,j,k → as i + j → ∞ .Proof :( i ) Using Propn. 3.3 and eqns. (4.8,4.9), we have D | i, j, k i = D ∞ X m =0 c ( i,j,k ) m e w ij + mi,j,i + j + k + m ⇒ e √− π (2 i − n l ) θ | i, j, k − i = ∞ X m =0 c ( i,j,k ) m De w ij + mi,j,i + j + k + m ⇒ e √− π (2 i − n l ) θ ∞ X m =0 c ( i,j,k − m e w ij + mi,j,i + j + k + m − = ∞ X m =0 c ( i,j,k ) m e √− π ( i − j ) θ e w ij + mi,j,i + j + k + m − | c ( i,j,k ) m | = | c ( i,j,k − m | . This shows that forany k , k ∈ Z , C i,j,k = C i,j,k . Hence, without loss of generality, we can take k = 0. Fix l ∈ N . To get lim i →−∞ C i,j, , we assume that i ≤ n l /
2, so that w ij = j . b | i, j, i = b ∞ X m =0 c ( i,j, m e j + mi,j,i + j + m ⇒ | i − , j + 12 , i = ∞ X m =0 c ( i,j, m (cid:16) β ++ ( j + m, i, j ) e j + m + i + , j − , i + j + m +1 + β + − ( j + m, i, j ) e j + m − i + , j − , i + j + m (cid:17) ⇒ ∞ X m =0 c ( i − ,j + , m e j + m + i − ,j + ,i + j + m = ∞ X m =0 (cid:16) c ( i,j, m β ++ ( j + m, i, j )+ c ( i,j, m +1 β − + ( j + m + 1 , i, j ) (cid:17) e j + m + i + , j − , i + j + m +1 . Comparaing co-efficients on both sides, one gets c ( i − ,j + , m = c ( i,j, m β ++ ( j + m, i, j ) + c ( i,j, m +1 β − + ( j + m + 1 , i, j ) . (4.10)By Thm. 3.3, we have the following estimates. | β ++ ( ℓ, i, j ) | ≤ | q | l − i , | β − + ( ℓ, i, j ) | ≤ | q | l − j . (4.11)From eqns. (4.10,4.11), we get | c ( i − ,j + , m | ≤ | q | m | c ( i,j, m | + | q | m +1 | c ( i,j, m +1 | ≤ | q | m − | q | (cid:0) | c ( i,j, m | + | q || c ( i,j, m +1 | (cid:1) . By iterating the process starting with | n l , n l , i , using eqns (4.10,4.11), and the fact that | c ( i,j,m ) m | ≤
1, we have | c ( i,j, m | ≤ | q | m ( n l − i ) − | q | ( | c ( nl , nl , m | + | q || c ( nl , nl , m +1 | + · · · | q | n l − i | c ( nl , nl , m + n l − i | ) ≤ | q | m ( n l − i ) (1 − | q | ) . This proves that C i,j, → i → −∞ . By replacing b with b ∗ in this computations, onecan show that C i,j, → i → ∞ .( ii ) By part ( i ), it is enough to show that C nl , nl , → l → ∞ . Take v ∈ E | q | nl (0 , , k v k = 1. By part (vii) of Propn. 4.1 and using k a k ≤
1, we have( a ∗ ) n l v = M | n l , n l , i (4.12)for some constant M with | M | ≤
1. Now, proceeding as above using eqn. (4.12) and theestimates of | α ++ ( ℓ, i, j ) | and α ++ ( ℓ, i, j ) obtained from Theorem 3.3 we get the claim. ✷ roposition 4.5. One has the followings. ( i ) The set {| i, j, k i : k ∈ Z ; i, j ∈ Z , i + j = n l } is an orthonormal basis of H l . ( ii ) The set {| i, j, k i : k ∈ Z ; i, j ∈ Z , i + j = n l , l ∈ N } is an orthonormal basis of E .Proof : Observe that A ( i, j, k ) is orthogonal to A ( i ′ , j ′ , k ′ ) if ( i, j, k ) = ( i ′ , j ′ , k ′ ). This alongwith the decomposition in 4.4 proves the first part. By Propn. 4.3 part ( iv ), the second partfollows. ✷ In this section, we construct a finitely summable nontrivial Dirac operator on U q (2) that isequivariant under its own comultiplication action. Let T be the following unbounded operatoron L ( h ) with dense domain A q , defined by T ( e ℓi,j,k ) = d ( ℓ, i, k ) e ℓi,j,k where, d ( ℓ, i, k ) = (2 ℓ + 1) + √− k − ℓ − i ) if i = − ℓ, − (2 ℓ + 1) + √− k if i = − ℓ. (5.1) Lemma 5.1.
For each x ∈ A q , the operators [ T, x ] and [ T ∗ , x ] initially defined on A q extendsto bounded operators acting on L ( h ) .Proof : It is enough to show that [ T, x ] where x ∈ { a, a ∗ , b, b ∗ , D, D ∗ } ⊆ A q extends to abounded operator on L ( h ). From Thm. 3.3 we have the followings,[ T, D ]( e ℓi,j,k ) = (cid:0) d ( ℓ, i, k − − d ( ℓ, i, k ) (cid:1)(cid:16) qq (cid:17) i − j e ℓi,j,k − , [ T, D ∗ ]( e ℓi,j,k ) = (cid:0) d ( ℓ, i, k + 1) − d ( ℓ, i, k ) (cid:1)(cid:16) qq (cid:17) i − j e ℓi,j,k +1 , [ T, a ]( e ℓi,j,k ) = (cid:0) d ( ℓ + 12 , i − , k ) − d ( ℓ, i, k ) (cid:1) α + ( ℓ, i, j ) e ℓ + i − ,j − ,k + (cid:0) d ( ℓ − , i − , k − − d ( ℓ, i, k ) (cid:1) α − ( ℓ, i, j ) e ℓ − i − ,j − ,k − , [ T, a ∗ ]( e ℓi,j,k ) = (cid:0) d ( ℓ + 12 , i + 12 , k + 1) − d ( ℓ, i, k ) (cid:1) α ++ ( ℓ, i, j ) e ℓ + i + ,j + ,k +1 + (cid:0) d ( ℓ − , i + 12 , k ) − d ( ℓ, i, k ) (cid:1) α + − ( ℓ, i, j ) e ℓ − i + ,j + ,k , T, b ]( e ℓi,j,k ) = (cid:0) d ( ℓ + 12 , i − , k ) − d ( ℓ, i, k ) (cid:1) β + ( ℓ, i, j ) e ℓ + i − ,j + ,k + (cid:0) d ( ℓ − , i − , k − − d ( ℓ, i, k ) (cid:1) β − ( ℓ, i, j ) e ℓ − i − ,j + ,k − , [ T, b ∗ ]( e ℓi,j,k ) = (cid:0) d ( ℓ + 12 , i + 12 , k + 1) − d ( ℓ, i, k ) (cid:1) β ++ ( ℓ, i, j ) e ℓ + i + ,j − ,k +1 + (cid:0) d ( ℓ − , i + 12 , k ) − d ( ℓ, i, k ) (cid:1) β + − ( ℓ, i, j ) e ℓ − i + ,j − ,k . As | d ( ℓ, i, k ± − d ( ℓ, i, k ) | = 1, it is easy to see that [ T, D ] and [
T, D ∗ ] extends to boundedoperators on L ( h ). Now, consider the right hand side of [ T, a ]( e ℓi,j,k ). We have from 5.1 (cid:12)(cid:12) d (cid:0) ℓ + 12 , i − , k (cid:1) − d ( ℓ, i, k ) (cid:12)(cid:12) = 1 , (cid:12)(cid:12) d (cid:0) ℓ − , i − , k − (cid:1) − d ( ℓ, i, k ) (cid:12)(cid:12) = ℓ + 1 for ℓ + i = 0 , ℓ + i = 0 , . We also have from Thm. 3.31 − | q | ≤ α + ( ℓ, i, j ) = s (1 − | q | ℓ − j +1) )(1 − | q | ℓ − i +1) )(1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) ≤ (1 − | q | ℓ +1) ) p (1 − | q | ℓ +1) )(1 − | q | ℓ +2) ) ≤ max j | α − ( ℓ, i, j ) | = | q | ℓ − i +1 s (1 − | q | ℓ + i ) )(1 − | q | ℓ +1) ) . Hence, (cid:12)(cid:12)(cid:0) d (cid:0) ℓ − , i − , k − (cid:1) − d ( ℓ, i, k ) (cid:1) α − ( ℓ, i, j ) (cid:12)(cid:12) ≤ s (1 − | q | ℓ + i ) )(1 − | q | ℓ +1) ) | q | ℓ − i (cid:12)(cid:12) d (cid:0) ℓ − , i − , k − (cid:1) − d ( ℓ, i, k ) (cid:12)(cid:12) ≤ | q | ℓ − (4 ℓ + 1) for ℓ + i = 1;1 otherwise . This gives the boundedness of the commutator [
T, a ]. The boundedness of [
T, b ] follows simi-larly.Now, consider the right hand side of [
T, b ∗ ]( e ℓi,j,k ). We have from 5.1 (cid:12)(cid:12) d ( ℓ − , i + 12 , k ) − d ( ℓ, i, k ) (cid:12)(cid:12) = 1 , (cid:12) d (cid:0) ℓ + 12 , i + 12 , k + 1 (cid:1) − d ( ℓ, i, k ) (cid:12)(cid:12) = ℓ + 3 for ℓ + i = 0 , − ℓ + i = 0 , − . We also have from Thm. 3.3 β + − ( ℓ, i, j ) = | q | ℓ − j s (1 − | q | ℓ + j ) )(1 − | q | ℓ − i ) )(1 − | q | ℓ ) )(1 − | q | ℓ +1) ) ≤ s − | q | ℓ − i ) − | q | ℓ +1) ≤ max j | β ++ ( ℓ, i, j ) | = | q | ℓ − i s (1 − | q | ℓ + i +1) )(1 − | q | ℓ +2) ) . Hence, (cid:12)(cid:12)(cid:0) d ( ℓ + 12 , i + 12 , k + 1) − d ( ℓ, i, k ) (cid:1) β ++ ( ℓ, i, j ) (cid:12)(cid:12) ≤ s (1 − | q | ℓ + i +1) )(1 − | q | ℓ +2) ) | q | ℓ − i (cid:12)(cid:12) d (cid:0) ℓ + 12 , i + 12 , k + 1 (cid:1) − d ( ℓ, i, k ) (cid:12)(cid:12) ≤ | q | ℓ (4 ℓ + 3) for ℓ + i = 0;1 otherwise . This gives the boundedness of the commutator [
T, b ∗ ]. The boundedness of [ T, a ∗ ] followssimilarly. ✷ Define the faithful representation π eq of C ( U q (2)) on H := L ( h ) ⊗ C by π eq ( x ) = " π h ( x ) 00 π h ( x ) . Let D = " T ∗ T and γ = " − . Immediately from Lemma 5.1, it follows that [ D , π eq ( x )] extends to a bounded operator on H for each x ∈ A q . Lemma 5.2.
The unbounded self-adjoint operator D has compact resolvent. roof : Observe that | T ∗ | − ( e ℓi,j,k ) = | T | − ( e ℓi,j,k ) = | d ( ℓ, i, k ) | − ( e ℓi,j,k ). Now,0 ≤ ℓ + i ≤ ℓ ⇒ k ≥ k − ( ℓ + i ) ≥ k − ℓ ⇒ k ≥ ( k − ( ℓ + i )) ≥ ( k − ℓ ) ⇒ (2 ℓ + 1) + k ≥ (2 ℓ + 1) + ( k − ( ℓ + i )) ≥ (2 ℓ + 1) + ( k − ℓ ) ⇒ ℓ + 1) + k ≤ | d ( ℓ, i, k ) | ≤ ℓ + 1) + ( k − ℓ ) . This shows that | D | − is a compact operator. ✷ Theorem 5.3.
The tuple ( A q , H , π eq , D , γ ) is a + -summable, non-degenerate, even spectraltriple on U q (2) that is equivariant under its own comultiplication action.Proof : Combining Lemmas (5.1, 5.2) it follows that D is a Dirac operator. Equivariance of D follows immediately from the facts that all eigenvectors of | D | are matrix co-efficients, andmatrix co-efficients in a row are eigenvectors corresponding to the same eigenvalue. To prove4 + -summability, fix n ∈ N and let S n = (cid:8) e ℓi,j,k : − ℓ ≤ i, j ≤ ℓ, ℓ ∈ N , k ∈ Z , | d ( ℓ, i, k ) | ≤ n (cid:9) . From (5.1), it follows that if | d ( ℓ, i, k ) | ≤ n , then ℓ ≤ n , − n ≤ i ≤ n , and hence − n ≤ k ≤ n .If one counts the number of such matrix co-efficients, we get the following, S n ≤ n (cid:0) + · · · + (2 n + 1) (cid:1) . This proves that the given spectral triple is 4 + -summable.Now, we show the non-degeneracy of this spectral triple. Recall that a spectral triple suchthat [ D , x ] = 0 implies x ∈ C is called non-degenerate. In view of Thm. 3.2, consider anyfinite subset F ⊆ Z × N × N × Z and scalars C n,m,r,k = 0, and take x = X ( n,m,r,k ) ∈ F C n,m,r,k a n b m ( b ∗ ) r D k in A q such that x / ∈ C . We claim that [ T, x ] = 0. Let η = max {| n | + m + r + | k | : ( n, m, r, k ) ∈ F } .Then, η > x / ∈ C . Since F is a finite set, there exists a tuple ( n , m , r , k ) ∈ F such that η = | n | + m + r + | k | . Fix this tuple and we rename it as ( n, m, r, k ) for notational brevity. Case 1 : n ≥ ℓ ∈ N and take i = r − m − n , j = m − r − n . It is easy to check that − (cid:0) ℓ + η − | k | (cid:1) ≤ i , j ≤ ℓ + η − | k | . D e ℓ + η −| k | i,j,r − k , [ T, x ] (cid:0) e ℓ , , (cid:1)E = D e ℓ + η −| k | i,j,r − k , (cid:2) T, X ( n ′ ,m ′ ,r ′ ,k ′ ) ∈ F C n ′ ,m ′ ,r ′ ,k ′ a n ′ b m ′ ( b ∗ ) r ′ D k ′ (cid:3)(cid:0) e ℓ , , (cid:1)E = C n,m,r,k D e ℓ + η −| k | i,j,r − k , [ T, a n b m ( b ∗ ) r D k ] (cid:0) e ℓ , , (cid:1)E = C n,m,r,k r Y t =1 β ++ (cid:16) ℓ + t − , t − , − t − (cid:17) r + m Y t = r +1 β + (cid:16) ℓ + t − , r − t − , − r + t − (cid:17) η −| k | Y t = r + m +1 α + (cid:16) ℓ + t − , r + 1 − t , − t − m − (cid:17)(cid:16) d (cid:16) ℓ + η − | k | , i, r − k (cid:17) − d ( ℓ, , (cid:17) . We claim that this number is nonzero. For this, in view of Thm. 3.3, we only have to checkwhether the last term ξ := d (cid:16) ℓ + η − | k | , i, r − k (cid:17) − d ( ℓ, , ℓ > i = − ℓ − η −| k | . Thus, from 5.1 weget that ξ = η − | k | + √− (cid:16) r − k − i − η − | k | (cid:17) . If η = | k | , then ℜ ( ξ ) = 0 and we are done. If η = | k | then we have n = m = r = 0 and thus, i = 0. This gives ℑ ( ξ ) = − k = 0, as x / ∈ C . So, in both the situations we have ξ = 0 andconsequently, [ T, x ] = 0. Case 2 : n < ℓ, i, j chosen in the previous case, observe that D e ℓ + η −| k | i,j,r − k − n , [ T, x ] (cid:0) e ℓ , , (cid:1)E = D e ℓ + η −| k | i,j,r − k − n , (cid:2) T, X ( n ′ ,m ′ ,r ′ ,k ′ ) ∈ F C n ′ ,m ′ ,r ′ ,k ′ a n ′ b m ′ ( b ∗ ) r ′ D k ′ (cid:3)(cid:0) e ℓ , , (cid:1)E = C n,m,r,k D e ℓ + η −| k | i,j,r − k − n , [ T, a n b m ( b ∗ ) r D k ] (cid:0) e ℓ , , (cid:1)E = C n,m,r,k r Y t =1 β ++ (cid:16) ℓ + t − , t − , − t − (cid:17) r + m Y t = r +1 β + (cid:16) ℓ + t − , r − t − , − r + t − (cid:17) η −| k | Y t = r + m +1 α ++ (cid:16) ℓ + t − , − m + 1 − t , t − r − (cid:17)(cid:16) d (cid:16) ℓ + η − | k | , i, r − k − n (cid:17) − d ( ℓ, , (cid:17) . Now, consider ξ := d (cid:16) ℓ + η − | k | , i, r − k − n (cid:17) − d ( ℓ, , . ξ = η − | k | + √− (cid:16) r − k − n − i − η − | k | (cid:17) . Again, if η = | k | , then we are done. If η = | k | , by similar argument as in the previous case weget that ℑ ( ξ ) = − k = 0. So, [ T, x ] = 0. ✷ Aim of this section is to prove that the spectral triple constructed in Thm. 5.3 is K -homologicallynontrivial. We first deal the case for θ irrational. More precisely, we show that when we pairthe associated Fredholm module with the equivalence class of projection p ⊗ p θ , an element in K ( C ( U q (2))) obtained in §
2, through the K − K pairing, we get the Fredholm index nonzero.We begin by briefly recalling the standard spectral triple on the noncommutative 2-torus.We denote by u and v the generating unitaries of A θ satisfying the relation uv = e π √− θ vu . Let A be the unital ⋆ -subalgebra of A θ generated by u and v . Consider the following representation, τ : A θ −→ B ( ℓ ( Z )) u U ⊗ , v e − π √− θN ⊗ U .
For a complex valued function f on Z , define the operator T f acting on ℓ ( Z ) as T f ( e m ⊗ e n ) := f ( m, n )( e m ⊗ e n ) . Then, the following tuple (cid:0) A , ℓ ( Z ) ⊗ C , h T m −√− n T m + √− n i , (cid:2) − (cid:3) (cid:1) is a 2 + -summable even spectral triple on the noncommutative torus A θ . To check nontriviality,one pairs it with the K -class of the Powers-Rieffel projection p θ . The projection p θ has a powerseries expression in u and v , which we denote by P ( u, v ). Now, the index ̺ of the followingFredholm operator p θ T m + in √ m n p θ : p θ ℓ ( Z ) −→ p θ ℓ ( Z )is nonzero [21], which proves nontriviality of this spectral triple.Now, let P = { bb ∗ =1 } ∈ C ( U q (2)). Under the GNS representatin π h , P is the orthogonalprojection onto the subspace E . Note that P bP = P b = bP and P DP = P D = DP , as both b and D commute with bb ∗ . Consider the C ∗ -algebra B generated by P b and
P D . Then, B is aunital C ∗ -algebra with the multiplicative unit P . Using the defining relations in 1.1, it followsthat P bP D = e √− πθ P DP b , and
P D is a unitary in B . Now, the relation aa ∗ + bb ∗ = 126mplies that P aa ∗ P + P bb ∗ P = P . It is easy to check that for any vector v in the fixed pointsubspace under the action of bb ∗ , one has a ∗ v = 0 as σ ( bb ∗ ) = {| q | m : m ∈ N } ∪ { } . Thus,it follows that P bb ∗ P = P and consequently, P b is a unitary in B . Hence, by the universalityof the noncommutative torus A θ and its simpleness, there exists an isomorphism ψ : A θ −→ B sending u P b and v P D . So, one can conclude that K ( B ) is generated by [ P ]and [ P ( P b, P D )]. Moreover, Lemma (2.4,2.5) and Thm. 2.6 says that one can view [ p ⊗ p θ ],one of the generators of K ( C ( U q (2))), as [ P ( P b, P D )]. We denote P ( P b, P D ) by P θ . Since P ( P b, P D ) ∈ B , we get that P P θ = P θ P = P θ . (6.1)Let ρ : B −→ B ( P L ( h )) be a representation of B given by ρ ( x ) = π h ( x ) for x ∈ B . Byeqns. (4.6,4.7,4.8) and Propn. 4.5, the closed subspace P L ( h ) is invariant under b, b ∗ , D and D ∗ , and hence the representation ρ is well-defined. Define an operator F : P L ( h ) −→ P L ( h )by F | i, j, k i = f ( i, j, k ) | i, j, k i , where f ( i, j, k ) = (2 w ij +1)+ √− k + j − w ij ) √ (2 w ij +1) +( k + j − w ij ) if i = − w ij , − (2 w ij +1)+ √− k + j − w ij ) √ (2 w ij +1) +( k + j − w ij ) if i = − w ij . (6.2) Proposition 6.1.
The operator P θ F P θ : P θ L ( h ) −→ P θ L ( h ) is a Fredholm operator with ind ( P θ F P θ ) = ̺ . To prove this Proposition, we need to decompose the operator P θ F P θ in the following manner.For l ≥
0, let P l be the projection onto the subspace H l of P L ( h ). It follows from eqns. (4.6,4.8) that H l is invariant under the action of B i.e P l x = xP l for all x ∈ B . This induces arepresentation ρ l : B −→ B ( H l ) defined by ρ l ( x ) = ρ ( x ) | H l for x ∈ B . For l ≥
0, let F l = F | H l and P lθ = ( P θ ) | H l . Hence, we have the following, ρ = ⊕ ∞ l =0 ρ l , P = ⊕ ∞ l =0 P l , F = ⊕ ∞ l =0 F l and P θ F P θ = ⊕ ∞ l =0 P lθ F l P lθ . (6.3) Lemma 6.2.
For l > , the tuple ζ l = (cid:0) B , ρ l ⊕ ρ l , h F ∗ l F l i , (cid:2) − (cid:3) (cid:1) is a finitely summable evenFredholm module. Moreover, for l , l > one has the following, ind ( P l θ F l P l θ ) = ind ( P l θ F l P l θ ) . Proof : For i, j ∈ Z , i + j = n l , one has[ F l , P b ] | i, j, k i = F l P b | i, j, k i − P bF | i, j, k i = F l | i − / , j + 1 / , k i − f ( i, j, k ) P b | i, j, k i = (cid:0) f ( i − / , j + 1 / , k ) − f ( i, j, k ) (cid:1) | i − / , j + 1 / , k i F l , P b ] is a compact operator on H l .Similarly one can show that [ F l , P D ] is compact, which proves that ζ l is a finitely summableeven Fredholm module.Fix l >
0. To prove the last part, we will show that the Fredholm module ζ l is unitarilyequivalent to a Fredholm module which is homotopic to ζ . Define a unitary operator W l : H l −→ H by the following, W l | i, j, k i = | i + n / − n l / , j + n / − n l / , k i , for k ∈ Z , i, j ∈ Z , i + j = n l . (6.4)It follows from eqns. (4.6,4.8) that W l ρ l ( b ) W ∗ l = ρ ( b ), and W l ρ l ( D ) W ∗ l = ρ ( D ), which impliesthat W l ρ l W ∗ l = ρ . Therefore, we have W l P lθ W ∗ l = P θ . (6.5)For 0 ≤ t ≤
1, define F tl | i, j, k i = 2 w ij + 1 + t ( n l − n ) + √− k + j − w ij ) p (2 w ij + 1 + t ( n l − n )) + ( k + j − w ij ) | i, j, k i . Consider the tuple ζ tl = (cid:0) B, ρ ⊕ ρ , h F tl ) ∗ F tl i , (cid:2) − (cid:3) (cid:1) . Similar to the case of ζ l , one can showthat ζ tl is a finitely summable even Fredholm module. Moreover, ζ l = ζ and ζ l = W l ζ l W ∗ l .Thus, ζ is homotopic to W l ζ l W ∗ l and hence, they represent the same element in K -group.Using the K − K pairing, one has the following, ind ( P θ F P θ ) = ind ( P θ W l F W ∗ l P θ )= ind ( W l P lθ F P lθ W ∗ l ) ( by eqn. 6 . ind ( P lθ F P lθ ) , which completes the proof. ✷ Lemma 6.3.
The tuple (cid:0) B , ρ ⊕ ρ , h F ∗ F i , (cid:2) − (cid:3) (cid:1) is a finitely summable even Fredholmmodule, and ind ( P θ F P θ ) = ̺ .Proof : Define a unitary operator W : H −→ ℓ ( Z ) as follows. W | i, j, k i = e i ⊗ e k , for k ∈ Z , i, j ∈ Z , i + j = 0 . (6.6)The tuple (cid:0) B , ρ ⊕ ρ , h F ∗ F i , (cid:2) − (cid:3) (cid:1) is unitarily equivalent to the finitely summable Fredholmmodule (cid:16) B , τ ◦ ψ − ⊕ τ ◦ ψ − , " T r − −√− s ( r − s T r − √− s ( r − s , (cid:2) − (cid:3) (cid:17) W , which is homotopic to the Fredholm module (cid:16) B , τ ◦ ψ − ⊕ τ ◦ ψ − , " T r −√− sr s T r + √− sr s , (cid:2) − (cid:3) (cid:17) , and this completes the proof. ✷ Proof of Thm. . B , ρ ⊕ ρ, (cid:2) F ∗ F (cid:3) , (cid:2) − (cid:3) ) is a finitely summable evenFredholm module as( B , ρ ⊕ ρ, (cid:2) F ∗ F (cid:3) , (cid:2) − (cid:3) ) = ⊕ ∞ k =0 ( B , ρ l ⊕ ρ l , h F ∗ l F l i , (cid:2) − (cid:3) ) . This shows that the operator P θ F P θ : P θ L ( h ) −→ P θ L ( h ) is a Fredholm operator. Therefore,we have ind ( P θ F P θ ) < ∞ . By eqn (6.3), one has ind ( P θ F P θ ) = ∞ X l =0 ind ( P lθ F l P lθ ) . It follows from Lemmas 6.2 that ind ( P lθ F l P lθ ) = 0 for l ≥
1. Now by Lemma 6.3, one canconclude that ind ( P θ F P θ ) = ̺ . ✷ Lemma 6.4.
The truncated operator
P T | T | − P : P L ( h ) −→ P L ( h ) is a compact perturba-tion of F .Proof : Let f ( ℓ, i, m ) = d ( ℓ,i,m ) | d ( ℓ,i,m ) | . Then one has T | T | − | i, j, k i = T | T | − (cid:16) ∞ X m =0 c ( i,j,k ) m e w ij + mi,j,i + j + k + m (cid:17) , = ∞ X m =0 c ( i,j,k ) m f ( w ij + m, i, m + k ) e w ij + mi,j,i + j + m + k . Hence, we get the following,
P T | T | − P | i, j, k i = X D T | T | − | i, j, k i , | i ′ , j ′ , k ′ i E | i ′ , j ′ , k ′ i = D T | T | − | i, j, k i , | i, j, k i E | i, j, k i = (cid:16) ∞ X m =0 (cid:12)(cid:12) c ( i,j,k ) m (cid:12)(cid:12) f ( w ij + m, i, i + j + k + m ) (cid:17) | i, j, k i . Using the facts that P ∞ m =0 (cid:12)(cid:12) c ( i,j,k ) m (cid:12)(cid:12) = 1 and f ( w ij + m, i, i + j + k + m ) = f ( i, j, k ), one has( P T | T | − P − F ) | i, j, k i = (cid:16) ∞ X m =1 (cid:12)(cid:12) c ( i,j,k ) m (cid:12)(cid:12) (cid:0) f ( w ij + m, i, i + j + k + m ) − f ( i, j, k ) (cid:1)(cid:17) | i, j, k i . | f ( w ij + m, i, i + j + k + m ) − f ( i, j, k ) | ≤ , and | f ( w ij + m, i, i + j + k + m ) − f ( i, j, k ) | → m → . ✷ Theorem 6.5.
The Chern character of the spectral triple ( A q , H , π eq , D , γ ) is nontrivial.Proof : Observe that P θ T | T | − P θ = P θ P T | T | − P P θ as P θ P = P P θ = P θ . Therefore by Lemma6.4, the operator P θ T | T | − P θ is a compact perturbation of P θ F P θ . From Propn. 6.1, we getthat ind ( P θ T | T | − P θ ) = ind ( P θ F P θ ) = ̺ . Thus, the value of the K − K pairing coming from the Kasparov product is given by thefollowing, h [ p ⊗ p θ ] , ( A q , π eq , D , γ ) i = ind ( P θ T | T | − P θ ) = ̺ which completes the proof. ✷ We finally conclude the case for θ = 0 ,
1. In these two situations, q is real. Recall from Sec.2 that in these cases K ( C ( U q (2))) is generated by [1] and [ p ⊗ Bott ], where “
Bott ” denotesthe Bott projection is M ( C ( T )). It is well known that the following spectral triple (cid:0) C ∞ ( T ) , ℓ ( Z ) ⊗ C , (cid:20) ∂∂x −√− ∂∂y∂∂x + √− ∂∂y (cid:21) , (cid:2) − (cid:3) (cid:1) is nontrivial. The C ∗ -algebra B becomes C ( T ) and all the proofs in this section remain valid,and hence we have that Thm. 6.5 also holds for θ = 0 , Spectral dimension of a compact quantum group has been introduced in [7]. In this sectionwe compute it for the compact quantum group U q (2). We will mainly follow [27] for thecomputation. To put into the appropriate framework we define the followings,Γ = { γ = ( γ , γ , γ ) : γ ∈ N , γ ∈ Z , γ ∈ {− γ , − γ + 1 , · · · γ }} e γ = a γ − γ b γ + γ D γ = t γ γ , − γ D γ ,e ( γ,j ) = t γ γ ,j D γ ; − γ ≤ j ≤ γ ,ǫ = (1 / , , , ǫ = (0 , , , ǫ = (0 , , / . emma 7.1. With the above set up, one has the followings. ( i ) sup { γ ∈ Γ } k e γ kk e γ + ǫ k < ∞ . ( ii ) sup { γ ∈ Γ } k e γ kk e γ + ǫ − ǫ k < ∞ . ( iii ) sup { γ ∈ Γ } k e γ kk e γ + ǫ ǫ k < ∞ .Proof : Note that for γ = ( γ , γ , γ ) ∈ Γ, e γ = t γ γ , − γ D γ . Hence by Thm 4 .
17 in [20], we have k e γ k = | q | γ √ | γ +1 | | q | . Applying this result, we get k e γ kk e γ + ǫ k = 1. Moreover, k e γ kk e γ + ǫ − ǫ k = | q | γ | q | γ − s | γ + 2 | | q | | γ + 1 | | q | = | q | s | q | γ (1 + | q | + · · · + | q | γ +2 ) | q | γ +1 (1 + | q | + · · · + | q | γ ) ≤ p | q | . Similarly, k e γ kk e γ + ǫ + ǫ k = | q | γ | q | γ +1 s | γ + 2 | | q | | γ + 1 | | q | = 1 | q | s | q | γ (1 + | q | + · · · + | q | γ +2 ) | q | γ +1 (1 + | q | + · · · + | q | γ ) ≤ √ | q | . ✷ Let c be an upper bound of all the suprema mentioned in Lemma 7.1. Take R = { a, b, D, D ∗ } .From [27], recall the growth graph G cR . One takes the vertex set of G cR to be Γ. For r ∈ R , wewrite γ r γ ′ if e ( γ ′ ,j ′ ) = re ( γ,j ) and k e ( γ,j ) kk e ( γ ′ ,j ′ ) k < c (7.1)for some 1 ≤ j ≤ N γ and 1 ≤ j ′ ≤ N γ ′ . Define the edge set of G cR to be the following, E := (cid:8) ( γ, γ ′ ) : γ r γ ′ for some r ∈ R (cid:9) . We write γ → γ ′ if ( γ, γ ′ ) ∈ E . We say that the graph G cR has a root γ ∈ Γ if for any γ ∈ Γ,there is a directed path from γ to γ . The following Lemma says that G cR has a root (0 , , Lemma 7.2.
For γ = ( γ , γ , γ ) ∈ Γ , there is a directed path in G cR from (0 , , to γ , and itis of length less than or equal to γ + | γ | .Proof : By Lemma 7.1, we have γ D γ + ǫ , γ D ∗ γ − ǫ , γ a γ + ǫ − ǫ , γ ∈ Γ. For γ ∈ Γ such that γ = γ , we have γ b γ + ǫ + ǫ . Take γ = ( γ , γ , γ ) ∈ Γ.Then, one has (0 , , b (1 / , , / b · · · · · · b ( γ + γ , , γ + γ . Moreover, we have( γ + γ , , γ + γ a ( γ + γ / , , γ + γ − / a · · · · · · a ( γ , , γ ) . If γ ≥
0, then we have( γ , , γ ) D ( γ , , γ ) D · · · · · · D ( γ , γ , γ ) , and if γ ≤
0, then we have( γ , , γ ) D ∗ ( γ , − , γ ) D ∗ · · · · · · D ∗ ( γ , γ , γ ) . This proves the claim, as the length of this path is 2 γ + | γ | . ✷ Proposition 7.3.
Let L : L ( h ) −→ L ( h ) be an unbounded self-adjoint operator with densedomain A q given by the following, L ( e ( γ,j ) ) = (2 γ + | γ | ) e ( γ,j ) , for γ = ( γ , γ , γ ) ∈ Γ and − γ ≤ j ≤ γ . Then, the tuple (cid:0) A q , L ( h ) , L (cid:1) is a + -summable spectral triple equivariant under the comulti-plication action of U q (2) .Proof : Clearly, the operator L is an unbounded self-adjoint operator with compact resolventequivariant under the U q (2) action. Take any x ∈ { a, b, D } . By Thm. in [20], one can see that xe ( γ,j ) = X β ∈ Γ γ − / ≤ β ≤ γ +1 / γ − ≤ β ≤ γ +1 γ − / ≤ β ≤ γ +1 / c ( β,j ) e ( β,j ) . Using this, we get that k [ L, x ] e ( γ,j ) k ≤ k xe ( γ,j ) k ≤ k x kk e ( γ,j ) k ≤ k e ( γ,j ) k . Appropriate summability of the spectral triple ( A q , L ( h ) , L ) immediately follows from thedefinition of L . ✷ Theorem 7.4.
The spectral dimension of U q (2) is . roof : Let γ = (0 , , ∈ Γ. Define the length function ℓ γ : Γ −→ N as follows, ℓ γ ( γ ) = γ = γ , length of a shortest path from γ to γ ; otherwise.Let L γ be the unbounded positive operator acting on L ( h ) with dense domain A q sending e ( γ,j ) to ℓ γ ( γ ) e ( γ,j ) for all − γ ≤ j ≤ γ and γ ∈ Γ. It follows from Lemma 7.2 that ℓ γ ( γ ) ≤ γ + | γ | .Therefore, we get that inf (cid:8) p : Tr( L − pγ ) < ∞ (cid:9) ≥ . By Propn. 2 . U q (2) is greater than or equal to 4.This, together with Propn. 7.3, proves the claim. ✷ Acknowledgements
Satyajit Guin would like to acknowledge the support of DST INSPIRE Faculty award grantDST/INSPIRE/04/2015/000901, and Bipul Saurabh acknowledges the support of SERB grantSRG/2020/000252.
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Satyajit Guin ( [email protected] ) Department of Mathematics and Statistics,Indian Institute of Technology, Kanpur,Uttar Pradesh 208016, India