Characterization of SU (ℓ + 1)-equivariant spectral triples for q 7 0 the odd dimensional quantum spheres 0 2 n Partha Sarathi Chakraborty and Arupkumar Pal a J 4 February 2, 2008 2 ] A Dedicated to Prof. K. R. Parthasarathy on his seventieth birthday. Q . Abstract h t The quantum group SU (ℓ+1) has a canonical action on the odd dimensional sphere a q m Sq2ℓ+1. All odd spectral triples acting on the L2 space of Sq2ℓ+1 and equivariant under this [ action have been characterized. This characterization then leads to the construction of an optimum family of equivariant spectral triples having nontrivial K-homology class. These 1 v generalize the results of Chakraborty & Pal for SU (2). q 4 9 AMS Subject Classification No.: 58B34, 46L87, 19K33 6 1 Keywords. Spectral triples, noncommutative geometry, quantum group. 0 7 0 1 Introduction / h t a Noncommutative differential geometry, which more commonly just goes by the name noncom- m mutative geometry, is an extension of noncommutative topology and was initially developed in : v order to handle certain spaces like the leaf space of foliations or duals of groups whose topology i X or geometry are difficult to study using machinery available in classical geomtry or topology. r a As the subject developed, more and more examples were found that are further away from classical spaces but can be handled by noncommutative geometric methods. For quite sometime though, it was commonly believed that quantum groups or their ho- mogeneous spaces, which are rather far removed from classical manifolds, are not covered by the formalism of noncommutative geometry. This notion changed with ([2]), wherethe authors treated thecaseofthequantumSU(2)groupandfoundafamilyofspectraltriplesactingonits L -space that are equivariant with respect to its natural (co)action. This family is optimal, in 2 the sense that given any nontrivial equivariant Dirac operator D acting on the L space, there 2 exists a Dirac operator D belonging to this family such that signD is a compact perturbation of signD and there exist reals a and b such that e |D| ≤ a+b|D|. e e 1 Later Dabrowski et al [13] constructed another equivariant spectral triple for SU (2) on two q copies of the L space, which was shown in [4] to be equivalent to a direct sum two spectral 2 triples constructed in [2]. Equivariant triples for the two dimensional Podles spheres were constructed in [11] and [12]. In a more recent paper ([10]), D’Andrea et al gave a construction of an equivariant spectral triple for the quantum four dimensional spheres. Our aim in the present paper is to look for higher dimensional counterparts of the spectral triples found in [2]. We first formulate precisely what one means by an equivariant spectral triplein ageneral set up. We thenuseacombinatorial method, implicitly usedin [2]and[3], to characterize completely all odd spectral triples acting on the L space of the odd dimensional 2 sphereS2ℓ+1 (seesection3forthedescription)andequivariantundertheactionoftheSU (ℓ+1) q q group for all ℓ > 1. This also leads to the construction of an optimum family of equivariant nontrivial (2ℓ +1)-summable odd spectral triples sharing all the properties of the triples for SU (2) in [2]. q One should mention in this context that the construction by Hawkins & Landi ([14]) does notdealwithequivariance, andmoreimportantly, theyproducea(bounded)Fredholmmodule, not a spectral triple, which is essential for determining the smooth structure, giving a metric on the state space and also help in computing the index map through a local Chern character. The paper is organised as follows. In the next section, we will describe the combinatorial method that was earlier used implicitly in [2] and [3]. We also formulate the notion of equivariance. This has been done using the quantum group at the function algebra level rather than passing on to the quantum universal envelopping algebra level. In section 3, we describe the C∗-algebra of continuous functions on the odd dimensional quantum spheres and state some of their relevant properties. In section 4, we briefly recall the quantum group SU (ℓ+1) and its q representation theory. In particular, we describe a nice basis for the L space and study the 2 Clebsch-Gordon coefficients. These are then used to describe the action by left multiplication on the L space explicitly. In section 5, we give a description of the L space of the sphere and 2 2 give a natural covriant representation on it. In the last section, we give a precise characterization of the singular values and of the sign, which helps us to produce an optimal family of equivariant Dirac operators, extending the results of [2] in the present case. 2 Preliminaries 2.1 Equivariance Suppose G is a compact group, quantum or classical, and A is a unital C∗-algebra. Assume that G has an action on A given by τ : A→ A⊗C(G), so that (id⊗∆)τ = (τ ⊗id)τ, ∆ being the coproduct. In other words, we have a C∗-dynamical system (A,G,τ). Definition 2.1 Acovariantrepresentation(π,u)of(A,G,τ)consistsofaunital*-representation 2 π : A → L(H), a unitary representation u of G on H, i.e. a unitary element of the multiplier algebra M(K(H)⊗C(G)) such that they obey the condition (π⊗id)τ(a) = u(π(a)⊗I)u∗ for all a ∈A. Definition 2.2 Suppose (A,G,τ) is a C∗-dynamical system. An odd G-equivariant spectral data for (A,G,τ) is a quadruple (π,u,H,D) where 1. (π,u) is a covariant representation of (A,G,τ) on H, 2. π faithful, 3. u(D⊗I)u∗ = D⊗I, 4. (π,H,D) is an odd spectral triple. 2.2 The general scheme Let G be a graph and (V ,V ) be a partition of the vertex set. We say that (V ,V ) admits an 1 2 1 2 infinite ladder if there exist infinite number of disjoint paths each going from a point in V 1 to a point in V . Here two paths are disjoint means that the set of vertices of one does not 2 intersect the set of vertices of the other. SupposeH is aHilbertspace, andD is a self-adjoint operator on H with compact resolvent. Then D admits a spectral resolution d P , where the d ’s are all distinct and each P γ∈Γ γ γ γ γ is a finite dimensional projection. Assume now onward that all the d ’s are nonzero. Let P γ c be a positive real. Let us define a graph G as follows: take the vertex set V to be Γ. c Connect two vertices γ and γ′ by an edge if |d −d | < c. Let V+ = {γ ∈ V : d > 0} and γ γ′ γ V− = {γ ∈ V : d < 0}. This will give us a partition of V. This partition has the important γ propertythat(V+,V−)doesnotadmitaninfiniteladder. Thisiseasytosee,becauseifthereis a path from γ to δ and d > 0, d < 0, then for some αon thepath, one musthave d ∈[−c,c]. γ δ α Since the paths are disjoint, it would contradict the compact resolvent condition. We will call such a partition a sign-determining partition. We will use this knowledge about the graph. We start with a self-adjoint operator with discretespectrum. FirstchooseabasisthatdiagonalizestheoperatorD. Nextweusetheaction of the algebra elements on the basis elements of H and the boundedness of their commutators with D. This gives certain growth restrictions on the d ’s. These will give us some information γ abouttheedgesinthegraph. Weexploitthisknowledgetocharacterizethosepartitions(V ,V ) 1 2 of the vertex set that are sign-determining, i. e. do not admit any infinite ladder. The sign of the operator D must be of the form P − P where (V ,V ) is a sign-determining γ∈V1 γ γ∈V2 γ 1 2 partition. Of course, for a given c, the graph G may have no edges, or too few edges (if the P cP singular values of D happen to grow too fast), in which case, we will be left with too many 3 sign-determining partitions. Therefore, we would like to characterize those partitions that are sign-determining for all sufficiently large values of c. In general the scheme outlined above will be extremely difficult to carry out, as the action of thealgebra elements withrespecttothebasis thatdiagonalizes D may bequitecomplicated, andthereforeusingboundednessofcommutatorconditionswillingeneralbeverydifficult. This is where equivariance plays an extremely crucial role. It gives us a nice basis that diagonalizes D, so that the boundedness of commutator conditions are simpler and the subsequent steps become much more tractable. 3 The odd dimensional quantum spheres Let q ∈ [0,1]. The C∗-algebra A ≡ C(S2ℓ+1) of the quantum sphere S2ℓ+1 is the universal ℓ q q C∗-algebra generated by elements z ,z ,...,z satisfying the following relations (see [15]): 1 2 ℓ+1 z z = qz z , 1 ≤ j < i≤ ℓ+1, i j j i z z∗ = qz∗z , 1 ≤ i 6=j ≤ ℓ+1, i j j i z z∗−z∗z +(1−q2) z z∗ = 0, 1≤ i ≤ ℓ+1, i i i i k k k>i X ℓ+1 z z∗ = 1. i i i=1 X The K-theory groups for these algebras were computed in [23] and [15]. Proposition 3.1 ([23],[15]) K (A ) =K (A )= Z. 0 ℓ 1 ℓ The group SU (ℓ+1) has an action on S2ℓ+1. Before we describe the action, let us recall q q the definition of the quantum group SU (ℓ+1). The C∗-algebra C(SU (ℓ+1)) is the universal q q C∗-algebra generated by {u :i,j = 1,··· ,ℓ+1} obeying the relations: ij u∗ u = δ I, u u∗ = δ I ki kj ij ik jk ij k k X X (−q)I(k1,k2,···,kℓ+1)u ···u = (−q)I(j1,j2,···,jℓ+1) ji’s distinct j1k1 jℓ+1kℓ+1  0 otherwise ki’sXdistinct  where I(k ,k ,··· ,k ) is the number of inversions in (k ,k ,··· ,k ). The group laws are 1 2 ℓ+1  1 2 ℓ+1 given by the folowing maps: ∆(u ) = u ⊗u (Comultiplication) ij ik kj k X S(u ) = u∗ (Antipode) ij ji ǫ(u ) = δ (Counit) ij ij 4 The map τ(z ) = z ⊗u∗ i k ki k X extendstoa*-homomorphismτ fromA intoA ⊗C(SU (ℓ+1))andobeys(id⊗∆)τ = (τ⊗id)τ. ℓ ℓ q In other words this gives an action of SU (ℓ+1) on A . q ℓ 4 Preliminaries on SU (ℓ + 1) q Ournextjobwillbetogetadescriptionofthecovariantrepresentationofthesystem(A ,SU (ℓ+ ℓ q 1),τ) on L (S2ℓ+1). For this we need a few facts on the representation theory of SU (ℓ+1). In 2 q q the first subsection we describe an important indexing of the basis elements of the representation space of the irreducibles. Then we describe the Clebsch-Gordon coefficients and compute certain estimates. In the last subsection, we write down explicitly the left multiplication operator on L (SU (ℓ+1)). 2 q 4.1 Gelfand-Tsetlin tableaux Irreducible unitary representations of the group SU (ℓ + 1) are indexed by Young tableaux q λ = (λ ,...,λ ), where λ ’s are nonnegative integers, λ ≥ λ ≥ ... ≥ λ (Theorem 1.5, 1 ℓ+1 i 1 2 ℓ+1 [24]). Write H for the Hilbert space where the irreducible λ acts. There are various ways λ of indexing the basis elements of H . The one we will use is due to Gelfand and Tsetlin. λ According to their prescription, basis elements for H are parametrized by arrays of the form λ r r ··· r r 11 12 1,ℓ 1,ℓ+1 r r ··· r  21 22 2,ℓ  r= ··· ,     r r  ℓ,1 ℓ,2    r   ℓ+1,1    where r ’s are integers satisfying r = λ for j = 1,...,ℓ+1, r ≥ r ≥ r ≥ 0 for ij 1j j ij i+1,j i,j+1 all i, j. Such arrays are known as Gelfand-Tsetlin tableaux, to be abreviated as GT tableaux for the rest of this section. For a GT tableaux r, the symbol r will denote its ith row. It i· is well-known that two representations indexed respectively by λ and λ′ are equivalent if and only if λ −λ′ is independent of j ([24]). Thus one gets an equivalence relation on the set of j j Young tableaux {λ = (λ ,...,λ ) : λ ≥ λ ≥ ... ≥ λ ,λ ∈ N}. This, in turn, induces an 1 ℓ+1 1 2 ℓ+1 j equivalence relation on the set of all GT tableaux Γ = {r : r ∈ N,r ≥ r ≥ r }: one ij ij i+1,j i,j+1 says r and s are equivalent if r −s is independent of i and j. By Γ we will mean the above ij ij set modulo this equivalence. We will denote by uλ the irreducible unitary indexed by λ, {e(λ,r) : r = λ} will denote 1· an orthonormal basis for Hλ and uλrs will stand for the matrix entries of uλ in this basis. The 5 symbol 11 will denote the Young tableaux (1,0,...,0). We will often omit the symbol 11 and just write u in order to denote u11. Notice that any GT tableaux r with first row 11 must be, for some i ∈{1,2,...,ℓ+1}, of the form (r ), where ab 1 if 1≤ a ≤ i and b = 1, r = ab  0 otherwise.  Thus such a GT tableaux is uniquelydetermined by the integer i. We will write just i for this GT tableaux r. Thus for example, a typical matrix entry of u11 will be written simply as u . ij Let r = (r ) be a GT tableaux. Let H (r) := r −r and V (r) := r −r . ab ab a+1,b a,b+1 ab ab a+1,b An element r of Γ is completely specified by the following differences V (r) H (r) H (r) ··· H (r) H (r) 11 11 12 1,ℓ−1 1,ℓ V21(r) H21(r) H22(r) ··· H2,ℓ−1(r)  D(r) = . ···     Vℓ,1(r) Hℓ,1(r)      The differences satisfy the following inequalities b b H (r) ≤ V (r)+ H (r), 1 ≤ a≤ ℓ, 0 ≤ b ≤ a−1. (4.1) a−k,k+1 a+1,1 a−k+1,k+1 k=0 k=0 X X Conversely, if one has an array of the form V H H ··· H H 11 11 12 1,ℓ−1 1,ℓ V21 H21 H22 ··· H2,ℓ−1  , ···     Vℓ,1 Hℓ,1      where V ’s and H ’s are in N and obey the inequalities (4.1), then the above array is of the ij ij form D(r) for some GT tableaux r. Thus the quantities V and H give a coordinate system a1 ab for elements in Γ. The following diagram explains this new coordinate system. The hollow circles stand for the r ’s. The entries are decreasing along the direction of the arrows, and the ij V ’s and the H ’s are the difference between the two endpoints of the corresponding arrows. ij ij 6 j // ◦ //◦ // ◦ // ◦ (cid:127)(cid:127)?? (cid:127)(cid:127)?? (cid:127)(cid:127)?? (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) V11 (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) H11 (cid:127)(cid:127) H12 (cid:127)(cid:127) H13 (cid:15)(cid:15) (cid:127) (cid:127) (cid:127) i ◦ (cid:127)(cid:127)//??◦ (cid:127)(cid:127)//??◦ (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) (cid:127) V21 (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) H21 (cid:127)(cid:127) H22 (cid:15)(cid:15) ◦(cid:15)(cid:15) (cid:127) //◦(cid:127) (cid:127)(cid:127)?? (cid:127) (cid:127) (cid:127) (cid:127) V31 (cid:127)(cid:127) (cid:127)(cid:127) H31 (cid:15)(cid:15) (cid:127) ◦ 4.2 Clebsch-Gordon coefficients In this subsection, we recall the Clebsch-Gordon coefficients for the group SU (ℓ+1). This will q be important in writing down the natural representation of C(S2ℓ+1) on L (S2ℓ+1) explicitly. q 2 q Look at the representation u11⊗uλ acting on H ⊗H . Therepresentation decomposes as a 11 λ direct sum ⊕ uµ, i.e. one has a corresponding decomposition ⊕ H of H ⊗H . Thus one has µ µ µ 11 λ two orthonormal bases {eµs} and {e1i1 ⊗eλr}. The Clebsch-Gordon coefficient Cq(11,λ,µ;i,r,s) is defined to be the inner product heµs,e1i1⊗eλri. Since 11, λ and µ are just the first rows of i, r and s respectively, we will often denote the above quantity just by C (i,r,s). q Next, we will compute the quantities C (i,r,s). We will use the calculations given in ([16], q pp. 220), keeping in mind that for our case (i.e. for SU (ℓ+1)), the top right entry of the GT q tableaux is zero. Let M = (m ,m ,...,m ) ∈ Ni be such that 1 ≤ m ≤ ℓ+2−j. Denote by M(r) the 1 2 i j tableaux s defined by r +1 if k = m , 1 ≤ j ≤ i, jk j s = (4.2) jk  r otherwise.  jk With this notation, observe now that C (i,r,s) will be zero unless s is M(r) for some M ∈ Ni.  q (Onehastokeep inmindthoughthatnotalltableaux oftheformM(r)isavalidGTtableaux) From ([16], pp. 220), we have i−1 (1,0) r r +e (1,0) r r +e C (i,r,M(r)) = a· a· ma × i· i· mi , q aY=1* (1,0) ra+1· (cid:12)(cid:12) ra+1· +ema+1 + * (0,0) ri+1· (cid:12)(cid:12) ri+1· + (cid:12) (cid:12) (4.3) (cid:12) (cid:12) where e stands for a vector (in the app(cid:12)ropriate space) whose kth coordinate(cid:12)is 1 and the rest k 7 are all zero, r stands for the jth row of the tableaux r, and j· 2 ℓ+2−a (1,0) r r +e [r −r −i+k] a· a· j = q−raj+ra+1,k−k+j × a,i a+1,k q * (1,0) ra+1· (cid:12)(cid:12)(cid:12) ra+1·+ek + Yii=6=1j [ra,i−ra,j −i+j]q (cid:12) ℓ+1−a (cid:12) [r −r −i+j −1] a+1,i a,j q × , (4.4) [r −r −i+k−1] a+1,i a+1,k q i=1 Yi6=k (1,0) ra· ra·+ej 2 = q 1−j+ ℓi=+11−ara+1,i− ℓii+=6=21j−ara,i! * (0,0) ra+1· (cid:12) ra+1· + P P (cid:12) (cid:12) (cid:12) ℓ+1−a[r −r −i+j −1] (cid:12) × i=1 a+1,i aj q , (4.5)  ℓ+2−a[r −r −i+j]  Q i=1 a,i aj q i6=j   where for an integer n, [n] denotes the q-numberQ(qn −q−n)/(q −q−1). After some lengthy q but straightforward computations, we get the following two relations: (1,0) r r +e a· a· j = A′qA, (4.6) (cid:12)* (1,0) ra+1· (cid:12) ra+1· +ek +(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (1,0) r (cid:12) r +e (cid:12) (cid:12) a· (cid:12) a· j (cid:12)= B′qB, (4.7) (cid:12)* (0,0) ra+1· (cid:12) ra+1· +(cid:12) (cid:12) (cid:12) (cid:12) where (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (r −r )+(r −r ) if j 6= k, a+1,b a,b a+1,j∧k a,j∨k A = j∧k K > 0 such that K < κ(r,M(r)) < K for all r. 2 1 1 2 Proof: Observe that ℓ+1 ℓ+1ℓ+2−i ℓ ψ(r) = (ρ,λ(r)) = − r + r . 1j ij 2 j=1 i=2 j=1 X X X Therefore ℓ ℓ ℓ min{ψ(r) : r = λ}= − λ + (k−1)λ . 1 i k 2 1 k=2 X X This implies that d12 = q−2ℓ ℓ1λi+ ℓk=2(k−1)λk(1+o(q)), λ P P which gives us 1 dλ 2 = q2ℓ−M1+1(1+o(q)). d (cid:18) λ+ek(cid:19) Next, qψ(r)−ψ(M(r)) = q−2ℓ ℓj+=11r1j+ ℓi=+21 ℓj+=21−irij+2ℓ( ℓj+=11r1j+1)−( ℓi=+21 ℓj+=21−irij+i−1) = q2ℓ−i+1. P P P P P P Thus κ(r,M(r)) = qℓ−i−M1+2(1+o(q)). Hence the conclusion follows. 2 5 Covariant representation Let us write G for SU (ℓ+1) and H for SU (ℓ). H is a subgroup of G. This means that there q q is a C∗-epimorphism φ :C(G) → C(H) obeying ∆ φ = (φ⊗φ)∆ . In such a case, one defines H G the quotient space G\H by C(G\H) := {a∈ C(G) : (φ⊗id)∆(a) = I ⊗a}. The group G has a canonical right action C(G\H) → C(G\H) ⊗ C(G) coming from the restriction of the comultiplication ∆ to C(G\H). Let ρ denote the restriction of the Haar state 10

Characterization of $SU_q(\ell+1)$-equivariant spectral triples for the odd dimensional quantum spheres

Similar Characterization of $SU_q(\ell+1)$-equivariant spectral triples for the odd dimensional quantum spheres