The Structure of Line Bundles over Quantum Teardrops

Over the quantum weighted 1-dimensional complex projective spaces, called quantum teardrops, the quantum line bundles associated with the quantum principal U(1)-bundles introduced and studied by Brzezinski and Fairfax are explicitly identified among the finitely generated projective modules which are classified up to isomorphism. The quantum lens space in which these quantum line bundles are embedded is realized as a concrete groupoid C*-algebra.


Introduction
In the theory of noncommutative topology or geometry [6], a generally noncommutative C *algebra A or a dense "core" * -subalgebra A ∞ of it is viewed respectively as the algebra C(X q ) of continuous functions or the algebra O(X q ) of coordinate functions on an imaginary spatial object X q , called a noncommutative space or a quantum space. In many interesting cases, such an imaginary nonexistent space X q is closely related to or actually originated from a classical counterpart, a well-defined topological space or manifold X, and we view X q or its "function algebra" C(X q ) or O(X q ) as a quantization of the classical spatial object X.
There have been very intriguing discoveries that a lot of topological or geometric concepts or properties of a space X are also carried by (the function algebra of) its quantum counterpart X q . For example, the concept of a vector bundle E [12] over a compact space X can be reformulated in the noncommutative context as a finitely generated projective left modules Γ(E q ) over C(X q ), viewed as the space of continuous cross-sections of some imaginary noncommutative or quantum vector bundle E q over X q , as suggested by Swan's work [25]. Beyond the well-known K-theoretic study of such noncommutative vector bundles up to stable isomorphism, the classification of them up to isomorphism for C * -algebras was made popular by Rieffel [18,19] and completed for some interesting quantum spaces by him and others [1,16,19,20,22].
When the spatial object X is actually a topological group G, the quantization encompasses the group structure by requiring C(G q ) or O(G q ) to have an additional Hopf * -algebra structure, and we call G q or its function algebra a quantum group. Generalizing further, we view a surjective Hopf * -algebra homomorphism O(G q ) → O(H q ) as giving a quantum subgroup H q of a quantum group G q , and view the coinvariant * -subalgebra O(G q /H q ) of O(G q ) for the canonical coaction as defining a "quantum homogeneous space" G q /H q . More generally, This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel given a coaction ∆ R : O(X q ) → O(X q ) ⊗ O(H q ) of a compact quantum group H q on a compact quantum space X q , the coinvariant * -subalgebra O(X q /H q ) of O(X q ) defines a "quantum quotient space" X q /H q .
Classically some internal structure of a vector bundle E over a space X is often carried by a principal G-bundle P over X for some structure group G represented on some vector space V such that E = P × G V . The concept of quantum principal bundles has evolved and become well developed through years of study [5,9]. In a recent work of Brzeziński and Fairfax [3], the quantization of weighted 1-dimensional complex projective spaces WP(k, l), called teardrops by Thurston, and of principal bundles over them was studied. In particular, the quantum principal U(1)-bundles and the associated quantum line bundles over the quantum teardrops WP q (k, l) were introduced and analyzed by Brzeziński and Fairfax. More concretely, they found a family of quantum line bundles L[n], n ∈ Z, inside a quantum principal U(1)-bundle C(L q (l; 1, l)) over WP q (k, l) and showed that the continuous function C * -algebra C(WP q (k, l)) is isomorphic to the unitization (K l ) + of l copies of the algebra K of compact operators.
In this paper, we first show that each of C(L q (l; 1, l)) and C(WP q (1, l)) can be realized as a concrete groupoid C * -algebra [17], following the groupoid approach to study C * -algebras as initiated by Renault [17] and popularized by Curto, Muhly, and Renault [7,14]. Then we explicitly identify the completed quantum line bundles L[n] among the well-known classified isomorphism classes of all finitely generated projective left modules over (K l ) + . This identification exhibits an interesting connection between "winding numbers" and "ranks".

Projective modules
From the analysis point of view, since the category of isomorphism classes of unital commutative C * -algebras is equivalent to the category of homeomorphism classes of compact Hausdorff spaces, the category of isomorphism classes of C * -algebras provides a natural context for the development of noncommutative topology or geometry.
In this context, Swan's theorem [25] makes it legitimate to call an (isomorphism class of) finitely generated projective left module E over a unital C * -algebra A an (isomorphism class of) noncommutative vector bundle over A or more precisely the (generally imaginary, nonexistent) underlying quantum space. On the other hand, a projection p in the C * -algebra M n (A) defines a left A-module endomorphism ξ ∈ A n → ξp ∈ A n on the left free A-module A n , and its image is a finitely generated projective left A-module E := A n p. It is well-known that this association establishes a bijective correspondence between the unitary equivalence classes of projections p in M ∞ (A) := ∞ n=1 M n (A), where M n (A) is embedded in M n+1 (A) in the canonical way for each n, and the isomorphism classes of finitely generated projective left modules E over A [2].
Two finitely generated projective left modules E, F over A are called stably isomorphic if they become isomorphic after being augmented by the same finitely generated free A-module, i.e. E ⊕ A k ∼ = F ⊕ A k for some k ∈ N. The K 0 -group of A classifies such finitely generated projective modules up to stable isomorphism. The cancellation problem dealing with whether two stably isomorphic finitely generated projective left modules are actually isomorphic goes beyond K-theory and is in general an interesting but difficult question. It was Rieffel's pioneering work [18,19] that brought the cancellation problem to the attentions and interest of researchers in the theory of C * -algebras. Over some basic geometrically motivated quantum spaces, the finitely generated projective left modules have been successfully classified [1,16,19,20,22].
As a simple example, we now describe the classification of finitely generated projective left modules over a fairly elementary C * -algebra, which is relevant to our main result later.
Let K be the algebra of all compact operators on a separable infinite-dimensional Hilbert space H, say, 2 . Recall that for a C * -algebra A, we use A + to denote its unitization, a unital C * -algebra equal to A ⊕ C as a vector space and endowed with the algebra multiplication (a, s)(b, t) := (ab + sb + ta, st) and involution (a, s) * := (a * , s) for (a, s), (b, t) ∈ A ⊕ C. In particular, (K l ) + ≡ (⊕ l s=1 K) + for l ∈ N denotes the unitization of the direct sum of l copies of K.
The classification of all isomorphism classes of finitely generated projective left modules over (K l ) + , or equivalently, all unitary equivalence classes of projections in M ∞ ((K l ) + ) is fairly well understood as summarized below. In the following, we use I to denote the multiplicative unit of the unital C * -algebra (K l ) + , and I r to denote the identity matrix in M r ((K l ) + ), while denotes the standard n×n identity matrix in M n (C) ⊂ K for any integer n ≥ 0 (with M 0 (C) = 0 and P 0 = 0 understood). In particular, ⊕ l j=1 P k j ∈ K l for integers k j ≥ 0.
with r ∈ N and n j , m j ∈ Z ≥ such that n j m j = 0 for all j represent all unitarily inequivalent classes of projections in M ∞ ((K l ) + ).

Quantum spaces and principal bundles
We recall the definition of a compact quantum group by Woronowicz [28] as a unital separable C * -algebra A with a comultiplication ∆ such that (A ⊗ 1) ∆A and (1 ⊗ A) ∆A are dense in A ⊗ A. It is known [27,28] that a compact quantum group A contains a dense * -subalgebra A ∞ , forming a Hopf * -algebra (A ∞ , ∆, * , S, ε), and has a Haar state h ∈ A * satisfying h(1) = 1 and For a quantum subgroup H q of a compact quantum group G q given by a surjective Hopf * - for the coaction ∆ R defines a "quantum homogeneous space" G q /H q . A fundamental example is the quantum odd-dimensional sphere S 2n+1 q = SU q (n + 1)/SU q (n) [26] with q ∈ (0, 1) generated by z 0 , . . . , z n subject to the relations defines a "quantum quotient space" X q /H q . An interesting example is the quantum weighted complex projective space WP q (l 0 , . . . , l n ) with q ∈ (0, 1) [3], for pairwise coprime numbers l 0 , . . . , l n ∈ N, which is the quantum quotient space for the coaction of for i = 0, . . . , n.
As special cases, this includes the quantum complex projective space CP n q when l 0 = · · · = l n = 1, and the so-called quantum teardrop WP q (k, l) with coprime k, l when n = 1.
Brzeziński and Fairfax [3] determined that S 3 q is a quantum principal U(1)-bundle over WP q (k, l), or more precisely, the algebra O(S 3 q ) is a principal O(U(1))-comodule algebra over O(WP q (k, l)), if and only if k = l = 1. This result is consistent with the classical U(1)-action (z, w) → (u k z, u l w) for u ∈ T on S 3 . Furthermore they found that the quantum lens space L q (l; 1, l) [11] provides the total space of a quantum principal U(1)-bundle over WP q (1, l), where L q (l; 1, l) is the quantum quotient space defined by the coaction ρ : (2)) generated by c := α l and d := β, and a well-defined coaction Corresponding to the irreducible (1-dimensional) representations of U(1) indexed by n ∈ Z, we have the irreducible corepresentations of O(U(1)) on some left comodules denoted as W n . Following the general theory of constructing finitely generated projective modules from quantum principal bundles and finite-dimensional corepresentations [4], Brzeziński and Fairfax took the cotensor product of O(L q (l; 1, l)) with W n over O(U(1)) to get a finitely generated projective module L[n] ⊂ O(L q (l; 1, l)) over O (WP q (1, l)), naturally called a quantum line bundle over WP q (1, l), and they computed an idempotent matrix E[n] over O(WP q (1, l)) implementing the projective module L[n] with complicated entries E[n] ij = ω(u n ) [2] i ω(u n ) [1] j , where ω(u n ) = i ω(u n ) [1] i ⊗ ω(u n ) [2] i comes from a strong connection ω : O(U(1)) → O(L q (l; 1, l)) ⊗ O(L q (l; 1, l)), and showed in particular that the O(WP q (1, l))-module L [1] is not free. Furthermore Brzeziński and Fairfax found the enveloping C * -algebra of O(WP q (k, l)) as C(WP q (k, l)) ∼ = (K l ) + and computed its K-groups from the exact sequence It is then a natural and interesting question to identify explicitly the completed quantum line bundles over C (WP q (1, l)) for all n ∈ Z among the finitely generated projective modules over (K l ) + already well classified.
4 Quantum lens space as groupoid C * -algebra In the past, there have been successful studies of the structure of some interesting C * -algebras [7,14,21,23,24] by realizing them first as a concrete groupoid C * -algebra, following the groupoid approach to C * -algebras initiated by Renault [17] and popularized by the work of Curto, Muhly, and Renault [7,14]. In this section, we first identify the C * -algebra C(L q (l; 1, l)) for q ∈ (0, 1) with a concrete groupoid C * -algebra, and then find an explicit description of the structure of C(L q (l; 1, l)). We construct the groupoid directly from the irreducible representations of C(L q (l; 1, l)) classified by Brzeziński and Fairfax [3]. Our approach should be compared with the machinery developed by Kumjian, Pask, Raeburn, Renault, and Paterson in [13,15] that associates groupoid C * -algebras to graph C * -algebras.
Before proceeding further, we introduce an open subgroupoid F of G defined by Letρ be the representation of the groupoid C * -algebra C * (F) induced off the counting mea- r-discrete groupoid),ρ is faithful. We note that the representation space ofρ is isomorphic to and that where the argument ofρ is understood as an element of C c (F) ⊂ C c (G) with value equal to Now viaρ −1 •π, we can view c, d as elements of C c (F) ⊂ C * (F) and hence view C(L q (l; 1, l)) as embedded in C * (F). Applying functional calculus to d * d, we can get Cδ (0,0,p)s ⊂ C(L q (l; 1, l)) for all p ∈ Z ≥ and 1 ≤ s ≤ l, and then by composing with c * and d * , we get Cδ (0,1,p)s and Cδ (1,0,p)s contained in C(L q (l; 1, l)) for any p ∈ Z ≥ and 1 ≤ s ≤ l, which generate the convolution * -subalgebra On the other hand, for any n ∈ Z, the |n|-th power of c or c * provides an element of C c (F) having a nonvanishing positive value at every point in while vanishing at all other points of F. So the C * -subalgebra C (L q (l; 1, l)) of C * (F) contains all elements of C c (F) and hence equals C * (F).
We summarize: In the general theory of groupoid C * -algebras [17], open invariant subsets and their complements in the unit space of a groupoid give rise respectively to closed ideals and quotients of its groupoid C * -algebra, and under suitable conditions the association is bijective which broadens a result of Gootman and Rosenberg [8] for transformation groups.
Decomposing the base space of C * (F) and the quotient which can be summarized as follows.
In fact, from the above analysis, we actually have the following explicit description C(L q (l; 1, l)) = (a 1 , . . . , a l ) ∈ ⊕ l s=1 C(T, T ) : σ • a 1 = · · · = σ • a l constant on T in terms of the standard Toeplitz C * -algebra T and its symbol map σ : T → C(T).

Line bundles over quantum teardrops
In this section, we identify concretely the quantum line bundles L[n] over C(WP q (1, l)) ∼ = (K l ) + for q ∈ (0, 1). First we recall that the coaction ρ l of O(U(1)) on O(L q (l; 1, l)) gives a Z-grading of O(L q (l; 1, l)) with c of degree 1 and d of degree −1, such that O(WP q (1, l)) generated by b := cd and a := dd * is the degree-0 component of O(L q (l; 1, l)), while L[n] is the degree-n component of O(L q (l; 1, l)) for general n ∈ Z [3]. Now we introduce a compatible Z-grading on the convolution * -algebra C c (F), based on the groupoid structure. We define the homogeneous degree-n component as C c (F) n := C c F n for the open set Note that F = n∈Z F n and C c (F) = ⊕ n∈Z C c (F n ) becomes a Z-graded * -algebra with deg(δ (k,m,p)s ) = k−m. Furthermore c ∈ C c (F 1 ) and d ∈ C c (F −1 ) for the generators c, d ∈ O(L q (l; 1, l)) ⊂ C c (F) of O(L q (l; 1, l)). So this groupoid Z-grading on C c (F) when restricted to the * -subalgebra O(L q (l; 1, l)) ⊂ C c (F) coincides with the original Z-grading on O(L q (l; 1, l)). So when viewed as elements of C c (F), the elements of L[n] ⊂ O(L q (l; 1, l)) are homogeneous of degree n. That is Also note that C c (F) 0 = C c (F 0 ) where F 0 ⊂ F consisting of (0, 0, ∞) and elements of the form (m, m, p) s with p, m + p ∈ Z ≥ is an open subgroupoid of F. It is clear that the * -algebra Z-grading structure on C c (F) makes each C c (F) n a left C c (F) 0 -module.
In particular, C(WP q (1, l)) is realized as the groupoid C * -algebra of the subgroupoid F 0 of F.
Let L[n] be the completion of L[n] in C * (F) = C(L q (l; 1, l)). In the following, we show that L[n] is a finitely generated projective left module over C(WP q (1, l)) ⊂ C * (F), and hence we can make the canonical identification It is easy to see that the O(WP q (1, l))-module structure on L[n] by left multiplication in C(L q (l; 1, l)) is consistent with the C c (F) 0 -module structure on C c (F) n under the embeddings of O(WP q (1, l)) ≡ L[0] ⊂ C c (F) 0 and L[n] ⊂ C c (F) n into C * (F) = C(L q (l; 1, l)).
On the other hand, we have C c (F) n ⊂ L[n] ⊂ C(L q (l; 1, l)) ≡ C * (F), using our knowledge of the |n|-th power of c or c * and that C c (F 0 ) ⊂ L[0]. So Note that for all m ∈ Z, More generally, for all m ∈ Z, Identifying (p + m, p) s ∈ X m with p in the s-th copy of Z ≥ in l s=1 Z ≥ , we get a unitary operator that intertwinesρ(b)| 2 (Xm) andρ(a)| 2 (Xm) with π ⊕ (b) and π ⊕ (a) respectively. More generally, Furthermore, since u m •ρ(χ Cn ) • u −1 m−n = ⊕ l s=1 S n with S the backward unilateral shift on 2 (Z ≥ ) as defined previously, for the characteristic function χ Cn ∈ C c F n of the open and compact set C n := {(0, −n, p) s : n ≤ p ∈ Z ≥ } ∪ {(0, −n, ∞)} ⊂ F n , we have u m •ρ L[n] • u −1 m−n = u m •ρ C c F n • u −1 m−n = ⊕ l s=1 K + C ⊕ l s=1 S n which is isomorphic, as a left (⊕ l s=1 K) + -module, to ⊕ l s=1 K + ⊕ ⊕ l s=1 K + I 1 ⊕ ⊕ l s=1 P n if n ≥ 0, and to ⊕ l s=1 K + I − ⊕ l s=1 P −n if n < 0, where we recall that I 1 denotes the identity matrix in M 1 ((⊕ l s=1 K) + ) while I denotes the identity element of (⊕ l s=1 K) + , and hence I 1 ⊕ (⊕ l s=1 P n ) ∈ M 2 ((⊕ l s=1 K) + ) while I − ⊕ l s=1 P −n ∈ ⊕ l s=1 K + = M 1 ⊕ l s=1 K + .
As summarized below, we have the modules L[n] identified concretely among the finitely generated projective left modules over (K l ) + enumerated earlier in Section 2.
It is interesting to note that this theorem exhibits some kind of an index relation between the "winding number" n of the line bundle L[n] and the "rank" of its representative projection I 1 ⊕ (⊕ l j=1 P n ) or I − (⊕ l j=1 P −n ). Finally, we mention the classification of isomorphism classes of finitely generated projective left modules over the quantum 3-sphere by Bach [1] which shows that the projections 1 ⊗ P k with k ≥ 0 and I r with r ∈ N represent all unitarily inequivalent classes of projections in M ∞ (C(S 3 q )). In view of this classification, we observe that C(S 3 q ) ⊗ C(WPq(1,l)) L[n] for all n ∈ Z is the same rank-1 free module over C(S 3 q ), showing that these non-isomorphic quantum line bundles L[n] over WP q (1, l) pull back to the same quantum line bundles over S 3 q via the quotient map S 3 q → WP q (1, l). This phenomenon resembles the well-known classical result that the pullback, to the total space P , of a vector bundle P × G V → X associated with a principal G-bundle P → X for some G-vector space V is always trivial. In fact, this classical theorem has a general quantum counterpart [10].