Weighted Circle Actions on the Heegaard Quantum Sphere

arXiv:1305.5942v2 [math.QA] 25 Sep 2013 Weighted circle actions on the Heegaard quantum sphere Tomasz Brzeziński and Simon A. Fairfax Abstract. Weighted circle actions on the quantum Heeqaard 3-sphere are considered. The fixed point algebras, termed quantum weighted Heegaard spheres, and their representations are classified and described on algebraic and topological levels. On the algebraic side, coordinate algebras of quantum weighted Heegaard spheres are interpreted as generalised Weyl algebras, quantum principal circle bundles and Fredholm modules over them are constructed, and the associated line bundles are shown to be non-trivial by an explicit calculation of their Chern numbers. On the topological side, the C ∗ -algebras of continuous functions on quantum weighted Heegaard spheres are described and their K-groups are calculated. 1. Introduction This article may be considered as a sequel to [5] in so much as, on the operator algebraic level, it develops the theory of quantum teardrops which include the generic Podleś two-spheres [20]. On the algebraic level, however, though it employs techniques similar to those in [5] it deals with integer gradings on an algebra significantly different from that of the coordiate algebra of the quantum SU(2)-group. 3 The coordinate algebra of the Heegaard quantum 3-sphere O(Spqθ ) [11], [2] is defined for parametres 0 < p, q, θ < 1, with θ irrational, as the complex ∗-algebra generated by a and b satisfying the relations, ab = e2πiθ ba, (1a) (1b) a∗ a − paa∗ = 1 − p, ab∗ = e−2πiθ b∗ a, b∗ b − qbb∗ = 1 − q, (1 − aa∗ )(1 − bb∗ ) = 0. 3 O(Spqθ ) contains two copies of the quantum disc with parametres p and q as ∗-subalgebras and can be interpreted as obtained by glueing of two quantum solid tori [2]. To describe 3 algebraic structure of O(Spqθ ) it is convenient to define (self-adjoint) A := 1 − aa∗ , B := ∗ 1 − bb . In terms of these elements the relations (1b) can be recast as: (2) AB = BA = 0, Aa = paA, Ba = aB, Ab = bA, Bb = qbB. 3 The linear basis for O(Spqθ ) consists of all Ak al bm , Ak a∗l bm , B k al bm , B k a∗l bm and their ∗-conjugates, where k, l, m ∈ N; see [2]. 2010 Mathematics Subject Classification. 58B32; 58B34. Key words and phrases. Heegaard quantum sphere; weighted circle action; principal comodule algebra; Fredholm module. 1 2 TOMASZ BRZEZIŃSKI AND SIMON A. FAIRFAX 3 Standardly, O(Spqθ ) is considered as a Z-graded algebra (compatible with the ∗structure in the sense that the ∗-operation changes the grade to its negative) in two different ways. First a and b are given an equal grade, say, 1. The degree zero subalgebra is generated by polynomials A, B and ab∗ and is known as the coordinate algebra of the mirror quantum sphere [17]. Second, a and b are given opposite grades, say 1 for a and −1 for b. The degree zero subalgebra, generated by A, B and ab, was introduced in [10], where it was shown that its C ∗ -completion is isomorphic to the algebra of continuous functions on the generic (or non-standard) Podleś quantum sphere [20]. We prefer to interpret Z-gradings geometrically as algebraic coactions of the coordinate Hopf algebra O(U(1)) of the circle group. O(U(1)) can be identified with the ∗-algebra C[u, u∗] of polynomials in variables u and u∗ satisying uu∗ = u∗ u = 1. The Hopf algebra structure is given by ∆(u) = u ⊗ u, ǫ(u) = 1, S(u) = u∗ . The two Z-gradings described above correspond to coactions a 7→ a ⊗ u, b 7→ b ⊗ u, and a 7→ a ⊗ u, b 7→ b ⊗ u∗ , respectively. It has been shown in [16], [14] that these coactions 3 are principal, i.e. they make O(Spqθ ) into a principal O(U(1))-comodule algebra [7]. The paper is organised as follows. In Section 2 we equip the coordinate algebra of 3 the Heegaard 3-sphere O(Spqθ ) with Z-gradings determined for a pair of coprime integers k, l, by setting deg(a) = k, deg(b) = l, or, equivalently, with the weighted coaction of O(U(1)), and study the zero degree algebras O(Spq (k, l)). These split into two cases, one in which both k and l are positive, and one in which k is positive and l is negative. We use the notation O(Spq (k, l± )) to distinguish these cases. We list bounded irreducible ∗-representations of these algebras and identify O(Spq (k, l± )) as the generalised Weyl al3 gebras. In Section 3 subalgebras Ak,l of O(Spqθ ) which admit principal O(U(1))-coactions ± 3 that fix O(Spq (k, l )) are identified. In the case k = |l| = 1 these coincide with O(Spqθ ) and in general they can be interpreted as quantum lens spaces; see [19], [23] and [18] for different constructions of lens spaces. Furthermore it is shown that these principal comodule algebras are piecewise trivial but globally non-trivial, and strong connections on them are constructed. They are also given interpretation as endomorphism rings of objects in a specific category. Section 4 deals with noncommutative geometric aspects of O(Spq (k, l± )). More concretely, we construct Fredholm modules over O(Spq (k, l± )) and calculate Chern numbers of line bundles associated to principal comodule algebras constructed in Section 3. Finally, in Section 5 we study algebras of continuous functions on quantum weighted Heegaard spheres, identify them with pullbacks of Toeplitz algebras and calculate their K-groups. 2. The coordinate algebras of quantum weighted Heegaard spheres In this section we gather algebraic properties of quantum weighted Heegaard spheres. 2.1. The definition of quantum weighted Heegaard spheres. A weighted circle 3 coaction on O(Spqθ ) consistent with the algebraic relations (1a) and (1b) is defined for k, l coprime integers, by 3 3 φk,l : O(Spqθ ) → O(Spqθ ) ⊗ O(U(1)), (3) a 7→ a ⊗ uk , b 7→ b ⊗ ul , WEIGHTED CIRCLE ACTIONS ON THE HEEGAARD QUANTUM SPHERE 3 3 3 ) is a right O(U(1))-comodule algeand extend to the whole of O(Spqθ ) so that O(Spqθ 3 bra. The fixed point subalgebras of O(Spqθ ) with respect to coactions φk,l are called the 3 coordinate algebras of quantum weighted Heegaard spheres. Equipping O(Spqθ ) with coaction φk,l is equivalent to making it into a Z-graded algebra with grading determined by deg(a) = k, deg(b) = l. At this stage we need to consider the possible signs for weights k and l. It turns out that the fixed point subalgebra splits into two cases depending on the signs of the weights k and l. Each case can be described by firstly putting k > 0 and l > 0 and secondly k > 0 and l < 0. We write these spaces as, 2 3 3 O(Spq (k, l± )) := O(Spqθ )coO(U (1)) = {x ∈ O(Spqθ ) : φk,l (x) = x ⊗ 1}, where the + sign indicates positive values for l and the negative sign corresponds to negative values of l. Before we describe these algebras in detail, we first need some tools for calculation purposes. The following lemma can be proven by induction. Lemma 2.1. For all m, n ∈ N, ( Q a∗m−n ni=1 (1 − pi A), ∗m n (4a) a a = Qm i n−m , i=1 (1 − p A)a m ∗n (4b) a a = ( am−n Qm Qn i=1 (1 i=1 (1 − p −i+1 −p m≥n m ≤ n. −i+1 A), m ≥ n A)a , m ≤ n, ∗n−m and similarly with a replaced by b (and hence A by B) and p by q. With these at hand we can describe the coordinate algebras of quantum weighted Heegaard spheres by generators and relations. 2 3 Theorem 2.2. The algebra O(Spq (k, l+ )) is the subalgebra of O(Spqθ ) generated by ∗ ∗ l ∗k A = 1 − aa , B = 1 − bb and C+ = a b satifying the relations (5) A∗ = A, (6a) C+∗ C+ B ∗ = B, AB = BA = 0, l Y k Y = (1 − pi A)(1 − q j−k B), AC+ = pl C+ A, C+ C+∗ i=1 j=1 (6b) l k Y Y i = (1−p A)+ (1−q j−k B)−1, i=1 j=1 l Y k Y = (1 − pi−l A)(1 − q j B). i=1 j=1 Alternatively, since AB = 0, we can express (6a) as C+∗ C+ BC+ = q −k C+ B, C+ C+∗ l k Y Y i−l = (1−p A)+ (1−q j B)−1. i=1 j=1 3 2 (k, l− )) is the subalgebra of O(Spqθ ) generated by A = 1 − aa∗ , B = 1 − bb∗ and O(Spq C− = a∗|l| b∗k , satifying the relations (7) A∗ = A, (8a) C−∗ C− B ∗ = B, AB = BA = 0, |l| k Y Y = (1 − pi+l A)(1 − q j−k B), i=1 j=1 AC− = pl C− A, C− C−∗ = BC− = q −k C− B, |l| k Y Y i=1 j=1 (1 − pi A)(1 − q j B). 4 TOMASZ BRZEZIŃSKI AND SIMON A. FAIRFAX Alternatively, since AB = 0, we can express (8a) as (8b) C−∗ C− |l| k Y Y i+l (1−p A)+ (1−q j−k B)−1, = i=1 j=1 C− C−∗ = |l| Y i (1−p A)+ i=1 k Y (1−q j B)−1. j=1 Proof. The first stage is to identify the fixed point subalgebra with respect to the 3 ) is given by Ak a#µ b#ν for k ≥ 0, µ, ν ∈ Z and B k a#µ b#ν given coaction. A basis for O(Spqθ for k > 0, µ, ν ∈ Z, where ( aµ , µ ≥ 0, a#µ := ∗|µ| a , µ < 0, and similarly for the powers of b; see [2]. First note that powers of A and B are automatically fixed by the weighted coaction φk,l . Next, φk,l (aµ bν ) = aµ bν ⊗ ukµ+lν = aµ bν ⊗ 1 =⇒ kµ + lν = 0 (the coinvariance condition). This means that basis elements of the form aµ bν are fixed under the coaction provided kµ = −lν, which in turn means that k|(−lν). On the other hand k and l are coprime so in fact k|(−ν) or αk = −ν for some α ∈ Z. Substituting this back into the coinvariance condition gives kµ = l(αk), i.e. µ = lα and ν = −kα. So, aµ bν = alα b−kα = (al )α (b∗k )α ∼ (al b∗k )α , concluding that al b∗k is a generator from the set of coinvariant elements. This gives 2 3 the full description of O(Spq (k, l+ )) as the subalgebra of O(Spqθ ) generated by A, B l ∗k 2 − and C+ = a b . Similarly, when l is negative O(Spq (k, l )) is generated by A, B and C− = a∗−l b∗k . 2 Next we determine the relations between the generators for both algebras O(Spq (k, l± )), considering the positive case first. Equations (2) immediately imply that A∗ = A, B ∗ = B, AB = 0, AC+ = pl C+ A, and BC+ = q −k C+ B. By Lemma 2.1, C+ C+∗ l Y k Y = (a b )(b a ) = (a a )(b b ) = (1 − pi−l A)(1 − q j B), l ∗k k ∗l l ∗l ∗k k i=1 j=1 and C+∗ C+ = (al b∗k )∗ (al b∗k ) = (a∗l al )(bk b∗k ) = l Y k Y (1 − pi A)(1 − q j−k B). i=1 j=1 The relations for the negative case are proven by similar arguments. ⊔ ⊓ 2 Remark 2.3. The algebras O(Spq (k, l± )) can be understood as polynomial ∗-algebras generated by elements A, B and C± that satisfy relations in Theorem 2.2. By the Diamond Lemma such algebras have linear bases Ar C±s , Ar C±∗s , B r C±s , B r C±∗s . The assignments 2 described in Theorem 2.2 determine surjective ∗-algebra maps onto O(Spq (k, l± )) viewed 3 as subalgebras of O(Spqθ ). These send basis elements into linearly independent elements 2 hence are also injective and thus establish isomorpisms between O(Spq (k, l± )) and algebras generated by elements A, B and C± that satisfy relations in Theorem 2.2. WEIGHTED CIRCLE ACTIONS ON THE HEEGAARD QUANTUM SPHERE 5 2 2.2. Representations of O(Spq (k, l± ). Bounded irreducible ∗-representations of co2 ordinate algebras O(Spq (k, l± ) are derived and classified by standard methods applicable to all algebras of this kind; see in particular the proof of [16, Theorem 2.1] or, for example, the proof of [5, Proposition 2.2]. Proposition 2.4. Up to unitary equivalence, the following is the list of all bounded 2 irreducible ∗-representations of O(Spq (k, l± ). For all m ∈ N, let Vm ∼ = l2 (N) be a separable m Hilbert space with orthonormal basis en for n ∈ N. For s = 0, 1, ..., |l| − 1, t = 0, ..., k − 1, 2 2 the representations πs1 : O(Spq (k, l+ )) → End (Vs ), πt2 : O(Spq (k, l+ )) → End (Vt ) and −2 2 2 πs−1 : O(Spq (k, l− )) → End (Vs ) and πt : O(Spq (k, l− )) → End (Vt ) are given by (9a) πs±1 (A)esn = pn|l|+sesn , πs±1 (B)esn πs1 (C+ )esn = 0, = l Y (1 − pi+nl+s )1/2 esn+1 , i=1 πs−1 (C− )esn (9b) (9c) πt±2 (A)etn = 0, πt±2 (B)etn = |l| Y = (1 − pi+(n−1)|l|+s )1/2 esn−1 , i=1 q nk+tetn , πt±2 (C± )etn k Y = (1 − q j+(n−1)k+t )1/2 etn−1 . j=1 Furthermore, there are one-dimensional representations in each case given by A, B 7→ 0, C± 7→ λ where λ ∈ C such that |λ| = 1, which we denote by πλ± . 2.3. Quantum Heegaard spaces as generalised Weyl algebras. Generalised Weyl algebras are defined in [3] as follows. Definition 2.5. Let D be a ring, σ = (σ1 , ..., σn ) a set of commuting automorphisms of D and ã = (ae1 , ..., aen ) a set of (non-zero) elements of the centre Z(D) of D such that σi (aej ) = aej for all i 6= j. The associated generalised Weyl algebra D(σ, ã) of degree n is a ring generated by D and the 2n indeterminates X1+ , ..., Xn+ , X1− , ..., Xn− subject to the following relations, for all α ∈ D: (10) (11) Xi− Xi+ = aei , Xi+ Xi− = σi (aei ), Xi± α = σi±1 (α)Xi± , [Xi− , Xj− ] = [Xi+ , Xj+ ] = [Xi+ , Xj− ] = 0, ∀i 6= j, where [x, y] = xy − yx. We call ã the defining elements and σ the defining automorphisms of D(σ, ã). Note that in the degree one case the relations (11) are null. 2 Proposition 2.6. The algebras of coordinate functions O(Spq (k, l± )) are degree one generalised Weyl algebras. 2 Proof. Set D = C[A, B]/hAB, BAi. In the positive case O(Spq (k, l+ )), X + = C+∗ and X − = C+ , the automorphism σ+ of D and the defining element ã are l σ+ (A) = p A, σ+ (B) = q −k B, k l Y Y (1 − pi−l A)(1 − q j B). ã = i=1 j=1 6 TOMASZ BRZEZIŃSKI AND SIMON A. FAIRFAX 2 In the negative case O(Spq (k, l− )), X + = C− and X − = C−∗ , the automorphism σ− of D and the defining element ã are −l σ− (A) = p A, k σ− (B) = q B, |l| k Y Y (1 − pi+l A)(1 − q j−k B). ã = i=1 j=1 That these satisfy all the axioms of a degree one generalised Weyl algebra can be checked by routine calculations. ⊔ ⊓ One of the key theorems associated to generalised Weyl algebras [3, Theorem 1.6] provides an insight to the global dimension of such algebras. For example, one can prove that the coordinate algebra O(WPq (k, l)) of the quantum weighted projective line defined in [5] is a degree one generalised Weyl algebra, and then use [3, Theorem 1.6] to conclude that the global dimension of O(WPq (k, l)) is equal to 2 if k = 1 and is infinite otherwise.1 This can be used as an indication that, for the quantum teardrop case k = 1, the classical singularity has been removed (although it is not clear yet, whether O(WPq (1, l)) is a Calabi-Yau algebra). Unfortunately, the hypothesis of [3, Theorem 1.6] fails in the quantum Heegaard case since the basic ring D contains zero divisors. On the 2 other hand, one should not expect the global dimension of O(Spq (k, l± )) to be finite: On the classical level the relation AB = 0 implies that there is a singularity at the origin, which persists in the quantum case. 3. Quantum weighted Heegaard spheres and quantum principal bundles Recall from [7] that, given a Hopf algebra H, a right H-comodule algebra A (with coaction ρ) is said to be a Hopf-Galois extension of its coinvariant subalgebra B if the canonical map can : A ⊗B A → A ⊗ H, a ⊗ a′ 7→ aρ(a′ ), is bijective. If, in addition, the mutliplication map B ⊗ A → A splits as a left Bmodule and right H-comodule map, then A is called a principal comodule algebra (and the coaction ρ is said to be principal). Here we aim to construct O(U(1))-principal 2 comodule algebras with coinvariant subalgebras O(Spq (k, l± )). These can be understood geometrically as coordinate algebras of principal circle bundles over the quantum weighted Heegaard sphere. We follow the general strategy (employed previously in [5] and [6]) of 3 defining a cyclic group algebra coaction on O(Spqθ ) where the space of coinvariant elements with respect to this coaction forms the total space. In Section 3.4 we give a categorical interpretation of this strategy. 2 3 3.1. Circle bundles over Spq (k, l+ ). Since the fixed point algebra of (O(Spqθ ), φk,l), l ∗k for positive l, is generated by C+ = a b , A and B, we need to define a comodule 3 structure on O(Spqθ ) over the cyclic group algebra O(Zm ) that keeps these generators in the invariant part. In terms of the Zm -grading this means deg(al b∗k ) = l deg(a) − k deg(b) = 0 mod m. This equation is satisfied by setting deg(a) = k, deg(b) = l and m = kl. The grading is 3 3 equivalent to a coaction Λk,l : O(Spqθ ) → O(Spqθ )⊗O(Zkl ) given by a 7→ a⊗uk , b 7→ b⊗ul , 1We are grateful to Ulrich Krähmer for pointing this out to us. WEIGHTED CIRCLE ACTIONS ON THE HEEGAARD QUANTUM SPHERE 7 3 which is extended in the usual way to make Λk,l an algebra map and hence (O(Spqθ ), Λk,l ) l k a comodule algebra. The fixed point subalgebra is generated by x = a , y = b , z = A, w = B. This can be seen by taking a basis element and applying Λk,l: Aλ aµ bν is coinvariant if and only if kµ + lν = 0 mod kl, hence µ = lφ and ν = kδ, for some φ, δ ∈ Z. This means that Aλ aµ bν = Aλ (al )φ (bk )δ so al , bk and A are generators. By considering the other basis 3 element the final generator B is identified. The resulting algebra Ak,l = O(Spqθ )coO(Zkl ) is the quotient of C[w, x, y, z] by the relations xy = e2πiθkl yx, (12a) (12b) xx∗ = x∗ y = e−2πiθkl yx∗ , l l k k Y Y Y Y (1−pi−l z), x∗ x = (1−pi z), yy ∗ = (1−q i−k w), y ∗ y = (1−q i w), i=1 i=1 i=1 i=1 (12c) z ∗ = z, w ∗ = w, wz = zw = 0, xw = wx, yz = zy, yw = q −k wy, xz = p−l zx. The circle group algebra coacts on Ak,l by ρk,l : Ak,l → Ak,l ⊗ O(U(1)), w 7→ w ⊗ 1, x 7→ x ⊗ u, y 7→ y ⊗ u, z 7→ z ⊗ 1, making (Ak,l , ρk,l ) a O(U(1))-comodule algebra. The fixed points of this comodule algebra 2 are generated by α = w, β = z, γ = xy ∗ , and thus are isomorphic to O(Spq (k, l+ )) via the map α 7→ B, β 7→ A, γ 7→ C+ . 2 Theorem 3.1. (Ak,l , ρk,l ) is a principal O(U(1))-comodule algebra over O(Spq (k, l+ )). Proof. To prove principality we construct a strong connection for Ak,l , that is a map ω : O(U(1)) −→ Ak,l ⊗ Ak,l , normalised as ω(1) = 1 ⊗ 1, and such that (13a) µ ◦ ω = η ◦ ε, (13b) (ω ⊗ id) ◦ ∆ = (id ⊗ ρk,l ) ◦ ω, (13c) (S ⊗ ω) ◦ ∆ = (flip ⊗ id) ◦ (ρk,l ⊗ id) ◦ ω. Here µ : Ak,l ⊗ Ak,l → Ak,l denotes the multiplication, η : C → Ak,l is the unit map, and flip : Ak,l ⊗ O(U(1)) → O(U(1)) ⊗ Ak,l is the flip. A strong connection for Ak,l is defined by setting ω(1) = 1 ⊗ 1 and then recursively for n ∈ N, (14a) ω(un ) = x∗ ω(un−1)x + f (z)y ∗ ω(un−1)y, (14b) ω(u−n ) = xω(u−n+1)x∗ + f (p−l z)yω(u−n+1)y ∗ , Q where f (z) = 1 − li=1 (1 − pi z). That thus defined ω satisfies (13) is proven by induction. Condition (13a): Note that, since y ∗ y is a polynomial in w with constant term 1, zw = 0 and f (z) has the zero constant term, f (z)y ∗ y = f (z). Hence, in the case n = 1, (µ ◦ ω)(u) = x∗ x + f (z)y ∗ y = x∗ x + 1 − l Y (1 − pi z) = 1, i=1 n by (12b). Now assume that (µ ◦ ω)(u ) = 1 and consider (µ ◦ ω)(un+1) = x∗ (µ ◦ ω(un))x + f (z)y ∗ (µ ◦ ω(un ))y = x∗ x + f (z)y ∗ y = 1, where the second equality is the inductive assumption. The proof for negative powers of u is essentially the same, so the details are omitted. 8 TOMASZ BRZEZIŃSKI AND SIMON A. FAIRFAX Condition (13b): Consider ω for positive powers of u; putting n = 1 gives, ((id ⊗ ρk,l ) ◦ ω)(u) = (id ⊗ ρk,l )(x∗ ⊗ x + f (z)y ∗ ⊗ y) = x∗ ⊗ x ⊗ u + f (z)y ∗ ⊗ y ⊗ u = ω(u) ⊗ u. This is the basis for the inductive proof. Now assume that ((id ⊗ ρk,l ) ◦ ω)(un ) = ((ω ⊗ id) ◦ ∆)(un ) = ω(un ) ⊗ un , and consider ((id ⊗ ρk,l ) ◦ ω)(un+1) = (id ⊗ ρk,l )(x∗ ω(un )x + f (z)y ∗ ω(un )y) = x∗ ω(un )x ⊗ un+1 + f (z)y ∗ ω(un )y ⊗ un+1 = ω(un+1) ⊗ un+1 , where the second equality follows from the induction hypothesis and ρk,l being an algebra map. The proof for ω taking negative powers of u follows a similar argument. Condition (13c): As in the previous conditions, the proofs for positive and negative powers of u are similar, hence only the positive case is displayed. Note that here we need to show that (15) ((flip ⊗ id) ◦ (ρk,l ⊗ id) ◦ ω)(un ) = ((S ⊗ ω) ◦ ∆)(un ) = u−n ⊗ ω(un), for all n. The case n = 1 follows by the same argument as in the preceding proof. Assume that equation (15) is true for an n ∈ N, and consider ((flip ⊗ id) ◦ (ρk,l ⊗ id) ◦ ω)(un+1) = ((flip ⊗ id) ◦ (ρk,l ⊗ id))(x∗ ω(uN )x + f (z)y ∗ ω(un )y) = u−n−1 ⊗ x∗ ω(un )x + u−n−1 ⊗ f (z)y ∗ ω(un)y = u−(n+1) ⊗ ω(un+1), where the second equality follows from the induction hypothesis and ρk,l being an algebra map. This completes the proof of (13c) for positive powers of u. Since (Ak,l, ρk,l ) is a comodule algebra admitting a strong connection it is principal; see [13], [7]. ⊔ ⊓ Recall that an H-comodule algebra A is said to be cleft provided there is a convolution invertible H-comodule map j : H → A. A cleft comodule algebra is automatically principal and corresponds to a principal bundle which only admits trivial associated vector bundles. A comodule algebra is said to be trivial, if there exists an H-comodule algebra map j : H → A. A trivial comodule algebra is automatically cleft. Proposition 3.2. The principal O(U(1))-comodule algebra Ak,l is not cleft. 3 Proof. Since Ak,l ⊆ O(Spqθ ) and, by [18, Theorem 1.10], the only invertible elements 3 in O(Spqθ ) are multiples of 1, the only invertible elements in Ak,l are also the multiples of 1. The convolution invertible map j : O(U(1)) → Ak,l must take the form j(u) = α1 for some α ∈ C∗ . However this violates the right O(U(1))-colinearity of j. ⊔ ⊓ 2 3.2. Circle bundles over Spq (k, l− ). Following the process as in the case of positive 3 3 l, a suitable coaction Φk,l : O(Spqθ ) → O(Spqθ ) ⊗ O(Zk|l| ) for a negative l is arrived k l at as given by a 7→ a ⊗ u , b 7→ b ⊗ u . The fixed point subalgebra is generated by x = a|l| , y = bk , z = A, w = B. This coincides with the algebra Ak,−l , hence positive and negative values of l give rise to the same total space of the quantum principal bundle. The 2 principal coaction of O(U(1)) on Ak,−l that fixes O(Spq (k, l− )) is defined by x 7→ x ⊗ u, y 7→ y ⊗ u∗ , z 7→ z ⊗ 1 and w 7→ w ⊗ 1. WEIGHTED CIRCLE ACTIONS ON THE HEEGAARD QUANTUM SPHERE 9 3.3. Piecewise triviality. A1,1 is a prime example of a locally trivial quantum principal bundle [8], [11], [16], [23] or a piecewise trivial comodule algebra [14, Definition 3.8]. 2 In a similar way Ak,|l| is a piecewise trivial principal comodule algebra over O(Spq (k, l± )). Consider ideals hwi, hzi of Ak,|l| generated by w and z respectively. Clearly hwi ∩ hzi = 0, hence these ideals form a complete covering of Ak,|l| [10]. Each one of them is a subcomodule of Ak,|l| under the O(U(1))-coactions, hence the quotient algebras Ak,|l|/hwi and Ak,|l|/hzi are O(U(1))-comodule algebras. In view of relations (12), the class [y] is a unitary element of Ak,|l|/hwi, while the class [x] is a unitary element of Ak,|l|/hzi. Hence one can define right O(U(1))-colinear ∗-algebra maps jw : O(U(1)) → Ak,|l|/hwi, u 7→ [y]sign(l) ; jz : O(U(1)) → Ak,|l|/hwi, u 7→ [x]. Consequently, Ak,|l|/hwi and Ak,|l|/hzi are trivial principal comodule algebras. The fixed points of O(U(1))-coactions on Ak,|l|/hwi and Ak,|l|/hzi come out as quo2 2 tients O(Spq (k, l± ))/hBi and O(Spq (k, l± ))/hAi, respectively. These are isomorphic to coordinate algebras of quantum real projective planes O(RP2√p (k; +)) and O(RP2√q (|l|; +)), respectively, introduced in [4]. Again, hAi ∩ hBi = 0, so the ideals hAi, hBi constitute a 2 complete covering of O(Spq (k, l± )). Therefore, Ak,|l| are piece trivial principal comodule 2 algebras over O(Spq (k, l± )) in the sense of [14, Definition 3.8]. 3 3.4. Orbifolding. The Z-grading of the algebra O(Spqθ ) that leads to the definition 2 ± of O(Spq (k, l )) gives different weights to generators a and b. There is a way of introducing new, uniform grading, which, following [9] we term orbifolding. The algebras Ak,l introduced in the previous sections gain in this way a natural, categorical interpretation. 3 Consider O(Spqθ ) as a Z-graded algebra with grading compatible with ∗ and deter3 mined from deg(a) = k, deg(b) = l (k positive). The cyclic group Zk|l| acts on O(Spqθ ) by j⊲h = ζ j deg(h) h, 3 where h is a homogeneous element of O(Spqθ ) and ζ is a primitive k|l|-th root of unity. 3 We can form a crossed product O(Spqθ )>⊳ Zk|l| and view it as a category with objects χi , i = 0, . . . , k|l|, the characters of Zk|l| , and morphism sets 3 Hom(χi , χj ) = {h ∈ O(Spqθ ) | deg(h) = (j − i) mod k|l|}. 3 ). Each of the hom-sets The composition is provided by the multiplication in O(Spqθ Hom(χi , χj ) is a Z-graded space with grading deg(h) − j + i b h := . k|l| This grading is compatible with composition, hence each of the End(χi ) is a Z-graded alge3 bra. One easily finds that End(χi ) is a ∗-subalgebra of O(Spqθ ) generated by A, B, a|l| , bk , i.e. it is simply Ak,l of the preceding sections. The hat-gradings come out as: b=B b = 0, A |l| = 1, ac bbk = sign(l), i.e. they coincide with the coactions that make Ak,l a principal comodule algebra over 2 O(Spq (k, l± )). 10 TOMASZ BRZEZIŃSKI AND SIMON A. FAIRFAX 2 4. Fredholm modules and the Chern-Connes pairing for O(Spq (k, l± )) 3 In this section first we associate even Fredholm modules to algebras O(Spqθ ) and 3 use them to construct traces or cyclic cocycles on O(Spqθ ). The latter are then used to calculate the Chern number of a non-commutative line bundle associated to the quantum 2 principal bundle Ak,|l| over the quantum weighted Heegaard spaces O(Spq (k, l± )). First, recall from [12, Chapter IV] that an even Fredholm module over a ∗-algebra A is a quadruple (V, π, F, γ), where V is a Hilbert space of a representation π of A and F and γ are operators on V such that F ∗ = F , F 2 = I, γ 2 = I, γF = −γF , and, for all x ∈ A, the commutator [F, π(x)] is a compact operator. A Fredholm module is said to be 1-summable if [F, π(x)] is a trace class operator for all x ∈ A. In case V is a separable Hilbert space the last condition means X 1 Tr |[F, π(x)]| = h([F, π(x)]∗ [F, π(x)]) 2 ek , ek i < ∞, for all x ∈ A, k∈N where {ek }k∈N is an orthonormal basis for V. 4.1. Fredholm modules. Representations for Fredholm modules are constructed 2 from irreducible representations of O(Spq (k, l± )) listed in Proposition 2.4. In the positive l case we take a cue from [17] and, for every s = 0, 1, 2, . . . , l − 1, t = 0, 1, 2, . . . , k − 1, consider the representation πs,t obtained as a direct sum of representations πs1 and πt2 . The Hilbert space of this representation is denoted by Vs,t and we choose its orthonormal s,t s,t basis fm , m ∈ Z as follows. For m positive the fm correspond to the basis elements esm of 1 s,t the representation space of πs , and for negative m, the fm correspond to the et−m−1 of the Hilbert space of πt2 . In R addition to πs,t we also consider the integral of one-dimensional representations, πc = λ∈S 1 πλ+ dλ. The representation space of πc can be identified with Vs,t , so that (16) s,t s,t s,t s,t πc (A) = πc (B) = 0, πc (C+µ )fm = fm+µ , πc (C+∗µ )fm = fm−µ , for all m ∈ Z, µ ∈ N. Proposition 4.1. For all s = 0, 1, 2, . . . , |l| − 1, t = 0, 1, 2, . . . , k − 1, (Vs,t ⊕ Vs,t , π̄s,t := πs,t ⊕ πc , F, γ), where     0 I I 0 F = , γ= , I 0 0 −I − 2 is a 1-summable Fredholm module over O(Spq (k, l+ )), while (Vs ⊕ Vt , π̄s,t := πs−1 ⊕ 2 πt−2 , F, γ) is a 1-summable Fredholm module over O(Spq (k, l− )). The corresponding Chern characters are (17) ( λt ( λs p q if µ = 0, λ = 6 0, if µ = 0, λ 6= 0, λk λ|l| ± λ #µ ± λ #µ 1−q 1−p τs,t (B C± ) = τs,t (A C± ) = 0 otherwise. 0 otherwise, Here C±#µ ( C±µ , := ∗|µ| C± , µ ≥ 0, µ < 0. WEIGHTED CIRCLE ACTIONS ON THE HEEGAARD QUANTUM SPHERE 11 Proof. It is obvious that F ∗ = F , F 2 = γ 2 = I and F γ + γF = 0. We first deal with 2 the positive l case. By a straightforward calculation, for all x ∈ O(Spq (k, l+ )),   0 πc (x) − πs,t (x) [F, π(x)] = . πs,t (x) − πc (x) 0 Due to the derivation property of a commutator and the fact that the set of trace class operators is a ∗-ideal in the algebra of bounded operators, suffices it to show that πs,t (x)− πc (x) are trace class for x = A, B, C+ . Using the formulae in Proposition 2.4 one easily finds, for all λ, µ ∈ N,  Qlµ i+s+ml 1/2 s,t λ((m+µ)l+s)  ) fm+µ m ≥ 0, p i=1 (1 − p λ µ s,t (18a) πs,t (A C+ )fm = 0 m = −1, ..., −µ,  δ Qkµ (1 − q i−(m+1+µ)k+t )1/2 f s,t m < −µ. λ,0 m+µ i=1 and (18b) s,t πs,t (C+∗µ Aλ )fm   0 Q i+s+(m−µ)l 1/2 s,t = pλ(ml+s) lµ ) fm−µ i=1 (1 − p  Q kµ δ i−(m+1)k+t 1/2 s,t ) fm−µ λ,0 i=1 (1 − q + Set Xλ,µ := πs,t (Aλ C+#µ ) − πc (Aλ C+#µ ). Using (16) we find  Q i+s+ml 1/2 s,t  1 − lµ ) fm  i=1 (1 − p +∗ + 21 s,t (X 0,µ X0,µ ) fm = 0  1 − Qkµ (1 − q i+t−(m+µ)l )1/2 f s,t m i=1 and, for λ 6= 0, (19) + ∗ + 21 s,t (Xλ,µ Xλ,µ ) fm m = 0, 1, ..., µ − 1, m ≥ µ, m < 0. m ≥ 0, m = −µ, ..., −1 m < −µ, ( Q i+s+ml 1/2 s,t ) fm pλ((m+µ)l+s) lµ i=1 (1 − p = 0 m ≥ µ, m < µ. + + We need to estimate the two infinite series for X0,1 and X1,0 . For the first series, note that all factors in the product are less than one, hence, for a non-negative m s,t + ∗ + 21 s,t i h(X0,1 X0,1 ) fm , fm =1− l Y (1 − p i+s+ml i=1 i=1 ≤ 1 − (1 − p ≤p l+s+ml l Y ) ≤1− (1 − pi+s+ml ) 1 2 l+s+ml l ) =p l   X l i=1 i l+s+ml  l X i l p(l+s+ml)i (−1) i i=1 ≤ 2l ps+ml+l . Hence the sum over m ≥ 0 can be estimated by a convergent geometric series and thus it is finite. Similarly for m < 0, using an analogous identity, we find that the sum is finite, + so X0,1 is a trace-class operator. For the second series, ∞ ∞ X X ps + ∗ + 21 s,t s,t h(X1,0 X1,0 ) fm , fm i = pml+s = . l 1 − p m=0 m=0 12 TOMASZ BRZEZIŃSKI AND SIMON A. FAIRFAX + + By defining X̃λ,µ = (πs,t − πc )(B λ C+#µ ), in an analogous way it can be shown that X̃1,0 is also a trace class operator; hence we have shown that (Vs,t ⊕ Vs,t , π̄s,t , F, γ) is a 12 summable Fredholm module over O(Spq (k, l+ )). − − For the negative case, set Xλ,µ := πs−1 (Aλ C−#µ ) − πt−2 (Aλ C−#µ ), X̃λ,µ := πs−1 (B λ C−#µ ) − − πt−2 (B λ C−#µ ), the only non-zero entries of the commutator of F with π̄s,t evaluated at 2 − the basis elements of O(Spq (k, l )). Using the formulae in Proposition 2.4 one finds − that Tr |X0,1 | can be estimated by the difference of two geometric series, while both − − Tr |X1,0 | and Tr |X̃1,0 | are geometric series, very much as in the positive l case. Thus − 2 (Vs ⊕ Vt , π̄s,t :=, F, γ) is a 1-summable Fredholm module over O(Spq (k, l− )) as required. 2 Finally, we can calculate the Chern characters on the basis elements of O(Spq (k, l± )) ± ± using τs,t (x) = Tr (γ π̄s,t (x)). First, for λ 6= 0, ± τs,t (Aλ C±#µ ) = Tr ± (γ π̄s,t (Aλ C±#µ )) = Tr ± (Xλ,µ ) = δµ,0 ∞ X pλ(ml+s) = δµ,0 m=0 pλs , 1 − pλ|l| ± where we noted that if µ 6= 0, then all diagonal entries of Xλ,µ are zero. Similarly, by λt q ± ± considering the traces of X̃λ,µ we find that τs,t (B λ C+#µ ) = δµ,0 1−q ⊓ λk , for λ 6= 0. ⊔ 4.2. The Chern-Connes pairing. Projective modules of sections of line bundles associated to a principal O(U(1))-comodule algebra A with coaction ρ and coinvariant subalgebra B are defined as L[n] := {x ∈ A : ρ(x) = x ⊗ un }, n ∈ Z. In other words, L[n] is the degree n component of A when the latter is viewed as a Zgraded algebra. An idempotent E[n] for L[n] is given in terms of a strong connection ω, E[n]ij = ω(un)[2] i ω(un )[1] j ∈ B, (20) P where ω(un ) = i ω(un)[1] i ⊗ ω(un)[2] i ; see [7, Theorem 3.1]. The traces of powers of each of the E[n] make up a cycle in the cyclic complex of B. In particular Tr E[n] can be paired with the Chern character associated to a Fredholm module over B to give an integer, which identifies isomorphism classes of the E[n]. + Theorem 4.2. For all s = 0, 1, . . . , l−1, t = 0, 1, . . . , k −1, let τs,t be the cyclic cocycle 2 + on O(Spq (k, l )) constructed in Proposition 4.1. Let E[n] be the idempotent determined by + 2 ω(un ) in (14). Then τs,t (k, l+ ))(Tr E[n]) = −n. Consequently, for n 6= 0, the left O(Spq modules L[n] corresponding to E[n] are not free. Proof. We prove the theorem forQ the positive values of n. The negative n case is proven in a similar way. Define f (z) = 1 − li=1 (1 − pi z) and note that, for all s = 0, 1, . . . , l − 1, f (ps−l ) = 1. Lemma 4.3. For positive n, X X (21) ω(un) = ω(un−1)[1] i x∗ ⊗ xω(un−1 )[2] i + ω(un−1)[1] i f (z)y ∗ ⊗ yω(un−1)[2] i i i WEIGHTED CIRCLE ACTIONS ON THE HEEGAARD QUANTUM SPHERE 13 Proof. This is proven by induction. For n = 1, this is simply equation (14a) with n = 1. Assume that equation (21) is true for any r ≤ n. Then, using (14a) and the inductive assumption, we can compute ω(un+1) = x∗ ω(un )x + f (z)y ∗ ω(un )y X ∗ = x∗ ω(un−1)[1] i x ⊗ x} ω(un−1)[2] i x | {z i + X x∗ ω(un−1)[1] i f (z)y ∗ ⊗ yω(un−1)[2] i x i + X i + = X X i ∗ f (z)y ∗ ω(un−1)[1] i x ⊗ x} ω(un−1)[2] i y | {z f (z)y ∗ ω(un−1)[1] i f (z)y ∗ ⊗ yω(un−1)[2] i y ω(un)[1] i x∗ ⊗ xω(un )[2] i + i X ω(un )[1] i f (z)y ∗ ⊗ yω(un)[2] i , i where we indicated grouping of terms over which the definition (14a) of the strong connection ω is applied. ⊔ ⊓ Lemma 4.4. For all positive n, Tr E[n] = gn (z), where gn (z) is a polynomial in z independent of w such that


(22) gn+1 (z) = 1 − f p−l z gn p−l z + f (z)gn (z). Proof. By (21) and the definition of the idempotents E[n], X X Tr E[n] = xω(un−1)[2] i ω(un−1)[1] i x∗ + yω(un−1)[2] i ω(un−1)[1] i f (z)y ∗ i i = xTr E[n − 1]x∗ + yTr E[n − 1]f (z)y ∗. In particular, since Tr E[0] = 1, (23)
Tr E[1] = xx∗ + yf (z)y ∗ = xx∗ + yy ∗f (z) = 1 − f p−l z + f (z), where we used (12) (expressed in terms of the polynomial f ), in particular the fact that z commutes with y, that yy ∗ is a polynomial in w with constant term 1, wz = 0 and f has the zero constant term. Therefore, Tr E[1] is a polynomial in z only (not in w), and it satisfies (22) with g0 = 1. Assume, inductively, that Tr E[n] = gn (z). Then, again extracting the same information from (12) as before and, additionally, using the commutation rule between z and x∗ we obtain,
gn+1 (z) = xgn (z)x∗ + ygn(z)y ∗ f (z) = xx∗ gn p−l z + ygn (z)y ∗ f (z)


= 1 − f p−l z gn p−l z + f (z)gn (z), as required. ⊔ ⊓ With Lemma 4.3 and Lemma 4.4 at hand we can complete the proof of Theorem 4.2. Since z = A, the Chern number + + ξn := τs,t (Tr E[n]) = τs,t (gn (z)), 14 TOMASZ BRZEZIŃSKI AND SIMON A. FAIRFAX is obtained by evaluating the powers of z in polynomial expansion of gn (z) using formulae (17). We will proceed by induction on n, but first write f (z) = l X clm z m , gn (z) = m=1 N X dnr z r . r=0 Note that f (0) = 0, hence in view of (23), g0 (0) = 1, and, consequently gn (0) = 1, by (22). Therefore, dn0 = 1. Apply (23) and (17) to calculate

+ + (g1 (z)) = τs,t ξ1 = τs,t 1 − f p−l z + f (z) ! l l X X l −lm m + cm (p − 1)z = τs,t 1 − =− clm p−lm+sm = −f (ps−l ) = −1. m=1 m=1 The last equality follows from the observation that since s = 0, 1, . . . , l − 1, one of the factors in the product must vanish. Next, assume that ξn = −n, that is N X (24) dnr r=1 psr = −n. 1 − plr Then, using (22) ξn+1 = N X + dnr p−lr τs,t (z r ) r=0 = N X + clm dnr p−l(m+r) τs,t (z m+r ) m=1 r=0 dnr r=1 N X − l X N X l X N X −l(m+r) − 1 (m+r)s p(s−l)r l np − c d p m r lr l(m+r) 1−p 1−p m=1 r=0 + l X N X + clm dnr τs,t (z m+r ) m=1 r=0 N X p(s−l)r s−l = dnr p(r−l)s − f (p ) lr 1 − p r=1 r=0  (s−l)r  N N sr X X p n n p (r−l)s = dr − 1 = d − p − 1 = −n − 1, r lr lr 1 − p 1 − p r=1 r=1 dnr by inductive assumption (24). This completes the proof of the theorem. ⊔ ⊓ 5. Continuous functions on the quantum weighted Heegaard spheres 2 The C ∗ -algebras C(Spq (k, l± )) of continuous functions on the quantum weighted Heegaard spheres are defined as completions of the direct sum of representations classified in Proposition 2.4. In this section we identify these algebras as pullbacks of the Toeplitz algebra and calculate their K-groups, closely following the approach of [21] and[15] (see also [4]). We think about the Toeplitz algebra T concretely as the C ∗ -algebra generated by an unilateral shift U acting on a separable Hilbert space V with orthonormal basis en by Uen = en+1 (V could be any of the spaces Vs in Proposition 2.4). T can be thought of as WEIGHTED CIRCLE ACTIONS ON THE HEEGAARD QUANTUM SPHERE 15 the algebra of continuous functions on the quantum unit disc, and the restriction of these functions to the boundary circle S 1 yields the symbol map, σ : T → C(S 1 ), U 7→ u, where u is the unitary generator of C(S 1 ). Let ι be an automorphism of C(S 1 ) given by u 7→ u∗ . For all k ∈ N and l ∈ Z, define T k,l = {(x1 , . . . , xk+l ) ∈ T ⊕k+l | σ(x1 ) = . . . = σ(xl ) = ι ◦ σ(xl+1 ) = . . . = ι ◦ σ(xk+l )} for positive l and T k,l = {(x1 , . . . , xk−l ) ∈ T ⊕k−l | σ(x1 ) = . . . = σ(xk−l )}, for negative l. In this section we prove Theorem 5.1. For all k ∈ N, l ∈ Z, 2 C(Spq (k, l± )) ∼ = T k,l . (25) Consequently, (26) 2 K 1 (C(Spq (k, l± ))) = 0, 2 K 0 (C(Spq (k, l± ))) = Zk+|l| . Proof. To see that (26) follows from (25), we simply observe that the standard exact sequence (27) 0 // K // σ T // C(S 1 ) // 0, that characterises the Toeplitz algebra in terms of compact operators on V, yields the exact sequence (28) 0 // K⊕k+|l| // T k,l // C(S 1 ) // 0. The sequence (28) gives rise to a six-term exact sequence of K-groups, which can be studied precisely as in [4, Section 4.2] to aid the derivation of the K-groups as stated. 2 Let JA± , JB± denote the closed ∗-ideals of C(Spq (k, l± )) obtained by completing of 2 2 ideals JA± , JB± of O(Spq (k, l± )) generated by A and B, and let ψ : C(Spq (k, l± )) → 2 C(Spq (k, l± ))/(JA± ⊕ JB± ) be the canonical surjection. The image of ψ is generated by ψ(C± ). In view of the relations (6a) and (8a), ψ(C± ) is a unitary operator, hence 2 C(Spq (k, l± ))/(JA± ⊕ JB± ) ∼ = C(S 1 ). Applying πs±1 , πt±2 to JA± ⊕ JB± , one finds that the images contain only compact operators on the corresponding representation spaces Vs , Vt . On the other hand, these images contain all projections onto one-dimensional spaces generated by basis elements, and all step-by-one operators with non-zero weights, hence their completions contain all compact operators. In this way JA± ⊕ JB± can be identified with the direct sum Kk+|l| . Since the direct sum of all irreducible representations of a C ∗ -algebra is faithful, l−1 M s=0 πs±1 ⊕ k−1 M t=0 πt±2 ⊕ M λ∈S 1 πλ± , 16 TOMASZ BRZEZIŃSKI AND SIMON A. FAIRFAX 2 are faithful representations of C(Spq (k, l± )). In the same way as in [21], πλ± factor through πs±1 , πt±2 , hence also l−1 k−1 M M ± ±1 π := πs ⊕ πt±2 s=0 t=0 ± are faithful. It is clear that the images of π are contained in T k+|l| . On the other hand, by inspecting formulae in Proposition 2.4 one easily finds that πs1 (C+ ) − U, πs−1 (C− ) − U ∗ , πt±2 (C± ) − U ∗ are step-by-one operators with coefficients tending to zero. Therefore, they are compact operators and thus are in the kernel of the symbol map. This implies that the image of π ± is contained in T k,l . Summarising the above discussion we obtain commutative diagram with exact rows 0 // K⊕k+|l| // 2 C(Spq (k, l± ))  // C(S 1 ) // 0 _ π±  0 // K ⊕k+|l| // T k,l // C(S 1 ) // 0. This implies the isomorphism (25). ⊔ ⊓ 2 We note in passing that the K 0 -groups of C(Spq (k, l± )) are dependent on or detect the weights. This is in contrast to the classical (commutative) situation in which, irrespective of weights, the K 0 -group of the complex weighted projective line is equal to Z2 [1]. 2 Finally, the description of C(Spq (k, l± )) in terms of pullbacks T k,l of Toeplitz algebras gives one an opportunity to follow [22] and construct Fredholm modules over and compute 2 the index pairing for C(Spq (k, l± )) in a more straightforward way than in the algebraic setup of Section 4. In particular, the very definition of T k,l allows one to combine pairs of projections pri : T k,l → T , (x1 , . . . , xk+|l| ) 7→ xi , into Fredholm modules (pri , prj ), with no restriction on i and j for a negative l, and with i, j ≤ l or i, j > l, for a positive l, since the pri − prj have their images in compact operators. 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