The Pullbacks of Principal Coactions

Piotr M. Hajac, Elmar Wagner

Received: December 31, 2013 Revised: August 9, 2014

Communicated by Max Karoubi

Abstract. We prove that the class of principal coactions is closed under one-surjective pullbacks in an appropriate category of algebras equipped with left and right coactions. This allows us to handle cases of C^{∗}-algebras lacking two different non-trivial ideals. It also allows us to go beyond the category of comodule algebras. As an example of the former, we carry out an index computation for noncommutative line bundles over the standard Podleś sphere using the Mayer-Vietoris-type arguments afforded by a one-surjective pullback presentation of the C^{∗}-algebra of this quantum sphere. To instantiate the latter, we define a family of coalgebraic noncommutative deformations of the U(1)-principal bundle S^{7} → ℂP^{3}.

2010 Mathematics Subject Classification: 46L85, 58B32

Keywords and Phrases: coalgebra-Galois extension, entwining structure, equivariant projectivity, strong connection, Hopf algebra, quantum group, index pairing

Contents

1. Introduction and preliminaries 1026 1.1. Pullback diagrams and fibre products 1029 1.2. Odd-to-even connecting homomorphism in K-theory 1029 1.3. Principal coactions and associated projective modules 1031 1.4. Standard Hopf fibration of quantum SU(2) 1036 2. Principality of one-surjective pullbacks 1038 2.1. Principality of images and preimages 1039 2.2. The one-surjective pullbacks of principal coactions are principal 1043 3. The pullback picture of the standard quantum Hopf fibration 1046 3.1. Pullback comodule algebra 1047 3.2. Equivalence of the pullback and standard constructions 1050 3.3. Index pairing 1054 4. Examples of piecewise principal coalgebra coactions 1055 4.1. Piecewise principal coactions from a noncommutative join 1055 4.2. Quantum complex projective spaces ℂP_{q,s}^{3} 1057 Acknowledgements 1058 References 1058

1. Introduction and preliminaries

The idea of decomposing a complicated object into simpler pieces and connecting data is a fundamental computational principle throughout mathematics. In the case of (co)homology theory, it yields the Mayer-Vietoris long exact sequence whose significance and usefulness can hardly be overestimated. The categorical underpinning of all this are pullback diagrams: in a given category they give a rigorous meaning to putting together two objects over a third one.

In the classical setting of spaces, the freeness of a group action is a local property: if it is free on covering pieces, it is free everywhere. Slightly more generally, if we take the pushout of two equivariant maps X_{12} → X_{1} and X_{12} → X_{2} between spaces equipped with free actions, then the induced action on the pushout space is again free provided that at least one of the pushout maps is injective. Our aim is to work out a general noncommutative version of this simple fact. Let us explain our generalization step by step.

First, let us assume that all spaces are compact Hausdorff and all maps are continuous. Then we can easily dualize the aforementioned equivariant pushout to an equivariant pullback of homomorphisms of commutative unital C^{∗}-algebras of all continuous complex-valued functions on compact Hausdorff spaces. Our assumption that at least one of the pushout maps is injective becomes now an assumption that at least one of the pullback maps is surjective. The right action X × G → X of a topological group G translates into a representation of G in the automorphism group of the C^{∗}-algebra C(X) of functions on X:

(1.1)  α: G ∋ g ⟼ α_{g} ∈ Aut(C(X)), (α_{g}f)(x) := f(xg).

Replacing C(X) by an arbitrary unital C^{∗}-algebra still allows us to consider one-surjective equivariant pullback diagrams of G-C^{∗}-algebras.

Next, let G be a compact Hausdorff topological group acting by automorphisms on a unital C^{∗}-algebra A. Then to define the freeness of such an action we need to dualize it to coaction:

(1.2)  δ: A ⟶ C(G, A) = A ¯⊗ C(G), (δ(a))(g) := α_{g}a.

Here C(G, A) is the C^{∗}-algebra of all norm continuous functions from G to A naturally identified with the complete tensor product C^{∗}-algebra (e.g. see [33, Corollary T.6.17]). Now we can replace C(G) by the C^{∗}-algebra of a compact quantum group [34, 36], still consider one-surjective equivariant pullback diagrams of C^{∗}-algebras, and define freeness as a density condition [13].

Furthermore, we can define the Peter-Weyl functor [2] from the category of unital C^{∗}-algebras equipped with an action of a compact quantum group (i.e. a coaction of its C^{∗}-algebra) to the category of comodule algebras over the Hopf algebra of regular functions on the compact quantum group [36]. The main theorem of [2] states that the aforementioned density condition defining the freeness of an action of a compact quantum group on a unital C^{∗}-algebra is, via the Peter-Weyl functor, equivalent to the principality of the coaction of the Hopf algebra of the compact quantum group. Thus the algebraic condition of principality of an appropriate comodule algebra encodes the analytical freeness condition of a compact quantum group action on a unital C^{∗}-algebra. In the commutative case, the latter is equivalent to the freeness of a continuous compact Hausdorff group action on a compact Hausdorff space.

However, the category of principal comodule algebras, despite being very ample and enjoyable, does not encompass all interesting examples coming from quantizations along Poisson structures. A way to obtain a quantum group is by deforming a Poisson-Lie group along its Poisson structure. It is well known that Poisson-Lie groups admit few Poisson-Lie subgroups, so that it is important to consider coisotropic subgroups of Poisson-Lie groups. But the deformation along the Poisson structure of the natural action of a coisotropic subgroup on its Poisson-Lie group leads to a coaction of a coalgebra rather than a Hopf algebra [9, 20]. Such examples motivated the development of general theory of principal coactions of coalgebras on algebras [5–7]. The setting of our paper is based on this theory.

Finally, recall that coactions of discrete groups are defined as coactions of their group-ring Hopf algebras, and it is very easy to understand the principality of such coactions. Indeed, let P be a comodule algebra over a group ring kΓ. This is equivalent to P being graded by Γ:

P = ⨁_{γ∈Γ}P_{γ}, P_{γ} := {p ∈ P | δ(p) = p ⊗ γ} , ∀ γ, γ′ ∈ Γ : P_{γ}P_{γ′} ⊆ P_{γγ′}.

The coaction of kΓ on P is principal if and only if P is strongly graded by Γ [30], i.e.

(1.3)  ∀ γ, γ′ ∈ Γ : P_{γ}P_{γ′} = P_{γγ′}.

The goal of this paper is to prove a general pullback theorem for principal coactions that significantly generalizes the main result of [15] restricted to comodule algebras and pullbacks of surjections. More precisely, our main result is that the pullback of principal coactions over morphisms of which at least one is surjective is again a principal coaction: Theorem 2.2. Let C be a coalgebra, and let P be the pullback of Cequivariant unital algebra homomorphisms π_{1} : P_{1} → P_{12} and π_{2} : P_{2} → P_{12}. Then, granted minor technical requirements, if π_{1} or π_{2} is surjective and both coactions P_{1} → P_{1} ⊗ C and P_{2} → P_{2} ⊗ C are principal, then also the induced coaction P → P ⊗ C is principal.

It may be viewed as a non-linear version of the Milnor construction yielding an odd-to-even connecting homomorphism in algebraic K-theory [23]. Indeed, linearizing our pullback theorem with the help of a corepresentation gives precisely the odd-to-even construction of a projective module defining the connecting homomorphism in K-theory.

We apply this new result in two cases. In the first case, we keep the comodulealgebra setting but take a one-surjective pullback diagram (only one of the defining morphisms is surjective). In the second case, we proceed the other way round, that is we take a pullback diagram given by two surjections but take coactions that are not algebra homomorphisms.

The pullback picture of the standard quantum Hopf fibration gives us our first example. It provides a new way of computing the index pairing for the associated quantum Hopf line bundles (cf. [32]). This index pairing was computed in [14] using a noncommutative index formula, and re-derived in [25]. Here we give yet another method to compute it. This simple example shows the need to generalize from two-surjective to one-surjective pullback diagrams, and the pullback method of index computation seems attractive due to its inherent simplicity.

To obtain our second example, we first show how the piecewise structure [15] of a noncommutative join construction [11] allows one to define a certain class of piecewise principal coactions. Although this class of examples can also be handled by earlier methods, it definitely shows that there are interesting piecewise principal coactions that are not algebra homomorphisms. To obtain a concrete example, we take Pflaum’s instanton bundle S_{q}^{7} → S_{q}^{4} [26] as the noncommutative join of SU_{q}(2), and turn it into the coalgebraic quantum principal bundle S_{q}^{7} → ℂP_{q,s}^{3}. We do it with the help of the canonical surjections π : O(SU_{q}(2)) → O(SU_{q}(2))/J_{q,s} determined by the coideals right ideals J_{q,s} := (O(S_{q,s}^{2}) ∩ kerε)O(SU_{q}(2)), where S_{q,s}^{2} is a generic Podleś quantum sphere [27] and kerε is the kernel of the counit map.

The paper is organized as follows. First, to make our exposition self-contained and to establish notation, we recall fundamental concepts that we use later on. The key Section 2 is devoted to the general pullback theorem for principal coactions of coalgebras on algebras, Section 3 is on deriving the index pairing for quantum Hopf line bundles as a corollary to the pullback presentation of the standard Hopf fibration of SU_{q}(2), and the final Section 4 presents new examples of piecewise principal coactions that go beyond Hopf-Galois theory. Throughout the paper, we work with algebras and coalgebras over a field. The unadorned tensor product stands for the algebraic tensor product over this field. We employ the Heyneman-Sweedler-type notation (with the summation symbol suppressed) for the comultiplication Δ(c) =: c_{(1)} ⊗ c_{(2)} ∈ C ⊗ C and for coactions Δ_{V} (v) =: v_{(0)} ⊗ v_{(1)} ∈ V ⊗ C, _{V} Δ(v) =: v_{(−1)} ⊗ v_{(0)} ∈ C ⊗ V.

The convolution product of two linear maps from a coalgebra to an algebra is denoted by ∗: (f ∗g)(c) := f(c_{(1)})g(c_{(2)}). The set of natural numbers includes 0, that is ℕ := {0, 1, 2, …}.

1.1. Pullback diagrams and fibre products. The purpose of this section is to collect some elementary facts about fibre products. We consider the category of vector spaces as it will be the ambient category for all our pullback diagrams. Let π_{1} : A_{1} → A_{12} and π_{2} : A_{2} → A_{12} be linear maps. The fibre product of these maps is defined by

(1.4)  A_{1} ×_{(π_{1},π_{2})} A_{2} := {(a_{1}, a_{2}) ∈ A_{1} × A_{2} | π_{1}(a_{1}) = π_{2}(a_{2})} .

Together with the canonical projections

(1.5)  pr_{1} : A_{1} ×_{(π_{1},π_{2})} A_{2} ⟶ A_{1},  pr_{2} : A_{1} ×_{(π_{1},π_{2})} A_{2} ⟶ A_{2}

it forms a universal construction completing the initially-given pair of linear maps into the following commutative diagram:

(1.6)  \begin{matrix}A_{1} ×_{(π_{1},π_{2})} A_{2} & ⟶^{pr_{2}} & A_{2} \\ ^{pr}^{1}↓ & & ^{π_{2}}↓ \\ A_{1} & ⟶^{π_{1}} & A_{12} & . \end{matrix}

Such diagrams are called pullback diagrams, and fibre products are often referred to as pullbacks.

Next, if π_{1} : A_{1} → A_{12} and π_{2} : A_{2} → A_{12} are morphisms of *-algebras, then the fibre product A_{1}×_{(π_{1},π_{2})}A_{2} is a *-subalgebra of A_{1} × A_{2}. Furthermore, if we consider the pullback diagram (1.6) in the category of (unital) C^{∗}-algebras, then A_{1}×_{(π_{1},π_{2})}A_{2} with its componentwise multiplication and *-structure is a (unital) C^{∗}-algebra. Much the same, if B is an algebra and π_{1} : A_{1} → A_{12} and π_{2} : A_{2} → A_{12} are morphisms of left B-modules, then the fibre product A_{1}×_{(π_{1},π_{2})}A_{2} is a left B-module via the componentwise left action b.(a_{1}, a_{2}) := (b.a_{1}, b.a_{2}).

1.2. Odd-to-even connecting homomorphism in K-theory. Consider a pullback diagram

(1.7)  \xymatrix{ & A \ar[dl] \ar[dr] \\ A_{1} \ar@{->>}[dr]_{π_{1}} &  & A_{2} \ar[dl]^{π_{2}} \\  & A_{12}}

in the category of unital algebras, and assume that one of the defining morphisms (here we choose π_{1}) is surjective. Then there exists a long exact sequence in algebraic K-theory [23]

(1.8) \xymatrix{⋯ ⟶ K_{1}(A_{12}) \ar[r]^{odd-to-even} & K_{0}(A) ⟶ K_{0}(A_{1} ⊕ A_{2}) ⟶ K_{0}(A_{12}).}

The mapping \xymatrix{K_{1}(A_{12}) \ar[r]^{odd-to-even} & K_{0}(A)} is obtained as follows. First, given left A_{i}-modules E_{i}, i = 1, 2, we obtain left A_{12}-modules π_{i∗}E_{i} defined by A_{12} ⊗_{A_{i}} E_{i}. Since A_{12} is unital, there are canonical morphisms

(1.9)  π_{i∗} : E_{i} ⟶ π_{i∗}E_{i} , π_{i∗}(e) = 1 ⊗_{A_{i}} e.

The modules E_{i} and π_{i∗}E_{i} can also be considered as left modules over the fibre product algebra A via the left actions given by a.e_{i} := pr_{i}(a).e_{i}, for e_{i} ∈ E_{i}, and a.f_{i} := π_{i}(pr_{i}(a)).f_{i}, for f_{i} ∈ π_{i∗}E_{i}. Assume now that h : π_{1∗}E_{1} → π_{2∗}E_{2} is a morphism of left A_{12}-modules. Then h ∘ π_{1∗} : E_{1} → π_{2∗}E_{2} and π_{2∗} : E_{2} → π_{2∗}E_{2} can be lifted to morphisms of left A-modules, and we can consider their pullback diagram in the category of left A-modules:

(1.10)  \xymatrix{ & E_{1} ×_{(h∘π_{1∗},π_{2∗})} E_{2} \ar@{-->}[dl]_{_{pr_{1}}} \ar[dr]^{_{pr_{2}}} \\ E_{1} \ar[d]_{π_{1∗}} &  & E_{2} \ar[d]^{π_{2∗}} \\ π_{1∗}E_{1} \ar[rr]_{h} &  & π_{2∗}E_{2} .}

In [23, Section 2], it is proven in detail that, if E_{i} is a finitely generated projective module over A_{i}, i = 1, 2, and h is an isomorphism, then the fibre product M := E_{1} ×_{(h∘π_{1∗} , π_{2∗})} E_{2} is a finitely generated projective A-module. Furthermore, up to isomorphism, every finitely generated projective module over A has this form, and the A_{i}-modules E_{i} and pr_{i∗}M := A_{i} ⊗_{A} M, i = 1, 2, are naturally isomorphic. In particular, if E_{1} ≅ A_{1}^{n} and E_{2} ≅ A_{2}^{n}, the isomorphism h : π_{1∗}E_{1} → π_{2∗}E_{2} is given by an invertible matrix U ∈ GL_{n}(A_{12}). Using the canonical embedding GL_{n}(A_{12}) ⊆ GL_{∞}(A_{12}), we get a map

(1.11)  GL_{∞}(A_{12}) ∋ U ⟼ M ∈ Proj(A)

given by the pullback diagram

(1.12)  \xymatrix{ & M \ar[dl] \ar[dr] \\ A_{1}^{n} \ar@{->>}[dr]_{π_{1}} &  & A_{2}^{n} \ar[dl]^{π_{2}} \\  & A_{12}^{n} ≅^{U} A_{12}^{n} .}

This map induces an odd-to-even connecting homomorphism on the level of both algebraic [23] and C^{∗}-algebraic [17] K-theory. An explicit description of the module M is as follows. Assume that π_{1} : A_{1} → A_{12} is surjective. Then there exist liftings c, d ∈ Mat_{n}(A_{1}) such that evaluating π_{1} on c and d componentwise yields U^{−1} and U respectively. Applying [12, Theorem 2.1] to our situation yields E_{1} ×_{(h∘π_{1∗} , π_{2∗})} E_{2} ≅ A^{2n}p, where

(1.13)  p := ( \begin{matrix}(c(2 − dc)d, 1) & (c(2 − dc)(1 − dc), 0) \\ ((1 − dc)d, 0) & ((1 − dc)^{2}, 0) \end{matrix} ) ∈ Mat_{2n}(A).

1.3. Principal coactions and associated projective modules. Recall first the general definition of an entwining structure. Let C be a coalgebra with comultiplication Δ and counit ε, and let A be an algebra with multiplication m and unit η. A linear map

(1.14)  ψ : C ⊗ A ⟶ A ⊗ C

is called an entwining structure if and only if it is unital, counital, and distributive with respect to both the multiplication and comultiplication:

ψ ∘ (id ⊗ m) = (m ⊗ id) ∘ (id ⊗ ψ) ∘ (ψ ⊗ id),  ψ ∘ (id ⊗ η) = (η ⊗ id) ∘ flip, (1.15)
(id ⊗ Δ) ∘ ψ = (ψ ⊗ id) ∘ (id ⊗ ψ) ∘ (Δ ⊗ id),  (id ⊗ ε) ∘ ψ = flip ∘ (ε ⊗ id). (1.16)

If ψ is an entwining of a coalgebra C and an algebra A, and M is a right C-comodule and a right A-module, we call M an entwined module [4] when it satisfies the compatibility condition

(1.17)  (ma)_{(0)} ⊗ (ma)_{(1)} = m_{(0)}ψ(m_{(1)} ⊗ a).

Next, let P be an algebra equipped with a coaction Δ_{P} : P → P ⊗ C of a coalgebra C. Define the coaction-invariant subalgebra of P by

(1.18)  B := P^{coC} := {b ∈ P | Δ_{P} (bp) = bΔ_{P} (p) for all p ∈ P }.

We call the inclusion B ⊆ P a C-extension. We call it a coalgebra-Galois C-extension when the canonical left P -module right C-comodule map

(1.19)  can : P ⊗_{B} P ⟶P ⊗ C, p ⊗_{B} p′ ⟼ pΔ_{P} (p′),

is bijective [5]. Note that the bijectivity of can allows us to define the so-called translation map

(1.20)  τ : C ⟶ P ⊗_{B} P, τ(c) := can^{−1}(1 ⊗ c).

Moreover, every coalgebra-Galois C-extension comes naturally equipped with a unique entwining structure that makes P a (P, C)-entwined module in the sense of (1.17). It is called the canonical entwining structure [5], and is very useful in calculations or further constructions. Explicitly, it can be written as:

(1.21)  ψ(c ⊗ p) := can(can^{−1}(1 ⊗ c)p).

An algebra P with a right C-coaction Δ_{P} is said to be e-coaugmented if and only if there exists a group-like element e ∈ C such that Δ_{P} (1) = 1 ⊗ e. We call the C-extension B := P^{coC} ⊆ P e-coaugmented. (Much in the same way, one defines the coaugmentation of left coactions.) For the e-coaugmented coalgebra-Galois C-extensions, one can show that the coaction-invariant subalgebra defined in (1.18) can be expressed as

(1.22)  P^{coC} = {p ∈ P | Δ_{P} (p) = p ⊗ e}.

Indeed, Formula (1.21) allows us to express the right coaction in terms of the entwining

(1.23)  Δ_{P} (p) = ψ(e ⊗ p),

and Equation (1.15) yields the right-in-left inclusion. The opposite inclusion is obvious.

Next, if ψ is invertible, one can use (1.16) to show that the formula

(1.24)  _{P} Δ(p) := ψ^{−1}(p ⊗ e)

defines a left coaction _{P} Δ : P → C ⊗ P. We define the left coaction-invariant subalgebra ^{coC}P as in (1.18), and derive the left-sided version of (1.21). Hence, for any e-coaugmented coalgebra-Galois C-extension with invertible canonical entwining, the right coaction-invariant subalgebra coincides with the left coaction-invariant subalgebra:

(1.25) P^{coC} = {p ∈ P | Δ_{P} (p) = p ⊗ e} = {p ∈ P | _{P} Δ(p) = e ⊗ p} = ^{coC}P.

Finally, we need to assume one more condition on C-extensions to obtain a suitable definition of a principal coaction: equivariant projectivity. It is a pivotal property that guarantees the projectivity of associated modules, and thus leads to index pairings between K-theory and K-homology. Putting together the aforementioned four conditions, we say that a coalgebra C-extension B ⊆ P is principal (or simply that P is principal) [6] iff:

(i) The canonical map can : P ⊗_{B} P →P ⊗ C, p ⊗_{B} p′ ↦ pΔ_{P} (p′), is bijective (Galois condition).

(ii) The right coaction is e-coaugmented for some group-like e ∈ C, i.e.

Δ_{P} (1) = 1 ⊗ e. (iii) The canonical entwining ψ : C ⊗P →P ⊗C, c⊗p ↦ can(can^{−1}(1⊗c)p),

is bijective. (iv) The algebra P is C-equivariantly projective as a left B-module, i.e.

there exists a left-B-linear and right-C-colinear splitting of the multiplication map B ⊗ P → P.

In the framework of coalgebra extensions, the role of connections on principal bundles is played by strong connections [6]. Let P be an algebra and both a left and right e-coaugmented C-comodule. (Note that the left and right coactions need not commute.) A strong connection is a linear map ℓ : C → P ⊗ P satisfying

c̃an ∘ ℓ=1 ⊗ id, (id ⊗ Δ_{P}) ∘ ℓ=(ℓ ⊗ id) ∘ Δ,

(_{P} Δ ⊗ id) ∘ ℓ=(id ⊗ ℓ) ∘ Δ, ℓ(e) = 1 ⊗ 1. (1.26)

Here c̃an : P ⊗ P → P ⊗ C is the lifting of can to P ⊗ P. Assuming that there exists an invertible entwining ψ : C ⊗ P → P ⊗ C making P an entwined module, the first three equations of (1.26) read in the Heyneman-Sweedler-type notation c ↦ ℓ(c)^{⟨1⟩} ⊗ ℓ(c)^{⟨2⟩} as follows:

ℓ(c)^{⟨1⟩}ψ(e ⊗ ℓ(c)^{⟨2⟩}) = ℓ(c)^{⟨1⟩}ℓ(c)^{⟨2⟩}_{(0)} ⊗ ℓ(c)^{⟨2⟩}_{(1)}

= 1 ⊗ c , (1.27)

ℓ(c)^{⟨1⟩} ⊗ ψ(e ⊗ ℓ(c)^{⟨2⟩}) = ℓ(c)^{⟨1⟩} ⊗ ℓ(c)^{⟨2⟩}_{(0)} ⊗ ℓ(c)^{⟨2⟩}_{(1)}

= ℓ(c_{(1)})^{⟨1⟩} ⊗ ℓ(c_{(1)})^{⟨2⟩} ⊗ c_{(2)} , (1.28)

ψ^{−1}(ℓ(c)^{⟨1⟩} ⊗ e) ⊗ ℓ(c)^{⟨2⟩} = ℓ(c)^{⟨1⟩}_{(−1)} ⊗ ℓ(c)^{⟨1⟩}_{(0)} ⊗ ℓ(c)^{⟨2⟩}

= c_{(1)} ⊗ ℓ(c_{(2)})^{⟨1⟩} ⊗ ℓ(c_{(2)})^{⟨2⟩} . (1.29)

Applying id ⊗ ε to (1.27) yields a useful formula

(1.30)  ℓ(c)^{⟨1⟩}ℓ(c)^{⟨2⟩} = ε(c).

It is worthwhile to observe the left-right symmetry of principal extensions. We already noted (see (1.25)) the equality of the left and right coaction-invariant subalgebras. Now let us define the left canonical map as

(1.31)  can_{L} : P ⊗_{B} P ∋ p ⊗ q ⟼ p_{(−1)} ⊗ p_{(0)}q ∈ C ⊗ P.

One can check that it is related to the right canonical map can by the formula [7]

(1.32)  ψ ∘ can_{L} = can.

Also, if ℓ is a strong connection and c̃an_{L} := (id ⊗ m) ∘ (_{P} Δ ⊗ id) is the lifted left canonical map, then c̃an_{L} ∘ ℓ = id ⊗ 1. Hence

(1.33)  c ⊗ p ⟼ ℓ(c)^{⟨1⟩} ⊗ ℓ(c)^{⟨2⟩}p

is a splitting of c̃an_{L} just as

(1.34)  p ⊗ c ⟼ pℓ(c)^{⟨1⟩} ⊗ ℓ(c)^{⟨2⟩}

is a splitting of c̃an.

Lemma 1.1. Let P be an object in the category _{e}^{C}Alg_{e}^{C} of all unital algebras with e-coaugmented left and right C-coactions. Assume that there exists an invertible entwining ψ : C ⊗ P → P ⊗ C making P an entwined module. Then, if P admits a strong connection ℓ, it is principal.

Proof. Following [6], first we argue that

(1.35)  σ : P ∋ p ⟼ p_{(0)}ℓ(p_{(1)})^{⟨1⟩} ⊗ ℓ(p_{(1)})^{⟨2⟩} ∈ B ⊗ P

is a left-B-linear splitting of the multiplication map. Indeed, m∘σ = id because of (1.30), and the calculation

(1.36)  ψ(e ⊗ p_{(0)}ℓ(p_{(1)})^{⟨1⟩}) ⊗ ℓ(p_{(1)})^{⟨2⟩} = p_{(0)}ℓ(p_{(1)})^{⟨1⟩} ⊗ e ⊗ ℓ(p_{(1)})^{⟨2⟩},

obtained using (1.15), proves that σ(P) ⊆ B ⊗ P. This splitting is evidently right C-colinear, so that its existence proves the equivariant projectivity.

Next, let us check that the formula

(1.37)  can^{−1} : P ⊗ C ⟶ P ⊗_{B} P, p ⊗ c ⟼ pℓ(c)^{⟨1⟩} ⊗_{B} ℓ(c)^{⟨2⟩},

defines the inverse of the canonical map can, so that the coaction of C is Galois. It follows from (1.27) that

(1.38)  can(can^{−1}(p ⊗ c)) = pℓ(c)^{⟨1⟩}ℓ(c)^{⟨2⟩}_{(0)} ⊗ ℓ(c)^{⟨2⟩}_{(1)} = p ⊗ c .

On the other hand, taking advantage of (1.30) and (1.35), we see that

can^{−1}(can(p⊗_{B}q)) = pq_{(0)}ℓ(q_{(1)})^{⟨1⟩} ⊗_{B}ℓ(q_{(1)})^{⟨2⟩} = p⊗_{B}q_{(0)}ℓ(q_{(1)})^{⟨1⟩}ℓ(q_{(1)})^{⟨2⟩} = p⊗_{B}q.

Thus the conditions (i) and (iv) of the principality of a C-extension are satisfied. Finally, Condition (ii) is simply assumed, and Condition (iii) follows from the uniqueness of an entwining that makes P an entwined module. □

Note that, if there exists a strong connection ℓ, then (1.37) yields

(1.39)  τ(c) = ℓ(c)^{⟨1⟩} ⊗_{B} ℓ(c)^{⟨2⟩}.

In the Heyneman-Sweedler-type notation, we write τ(c) = τ(c)^{[1]} ⊗_{B} τ(c)^{[2]}. Then the canonical entwining reads

ψ(c ⊗ p) = τ(c)^{[1]}(τ(c)^{[2]} p)_{(0)} ⊗ (τ(c)^{[2]} p)_{(1)}

= ℓ(c)^{⟨1⟩}(ℓ(c)^{⟨2⟩} p)_{(0)} ⊗ (ℓ(c)^{⟨2⟩} p)_{(1)}. (1.40)

Remark 1.2. In [6], there is the converse statement: if P is principal, it admits a strong connection. Thus, among all extensions satisfying the assumptions of Lemma 1.1, principal extensions can be characterized as these that admit a strong connection.

Recall now that classical principal bundles can be viewed as functors transforming finite-dimensional vector spaces into associated vector bundles. Analogously, one can prove that a principal C-extension B ⊆ P defines a functor from the category of finite-dimensional left C-comodules into the category of finitely generated projective left B-modules [6]. Explicitly, if V is a left C-comodule with coaction _{V} Δ, this functor assigns to it the cotensor product

(1.41) P □_{C} V := {_{∑_{i}} p_{i} ⊗ v_{i} ∈ P ⊗ V | _{∑_{i}} Δ_{P} (p_{i}) ⊗ v_{i} = _{∑_{i}} p_{i} ⊗_{V} Δ(v_{i})}.

In particular, if g ∈ C is a group-like element, the formula _{ℂ}Δ(1) := g ⊗ 1 defines a one-dimensional corepresentation, and

(1.42)  P □_{C} ℂ = {p ∈ P | Δ_{P} (p)=p ⊗ g} =: P_{g}

can be viewed as a noncommutative associated complex line bundle. Then the general formula for computing an idempotent E_{g} of the associated module P_{g} out of a corepresentation and a strong connection becomes a very simple special case of [6, Theorem 3.1]: (1.43)

P_{g} ≅ B^{n}E_{g} , (E_{g})_{i,j=1}^{n} := (g_{i}^{R} g_{j}^{L})_{i,j=1}^{n} , ℓ(g) =: ∑_{k=1}^{n} g_{k}^{L} ⊗ g_{k}^{R} ∈ P_{g^{−1}} ⊗ P_{g} .

A fundamental special case of principal extensions is provided by principal comodule algebras. One assumes then that C = H is a Hopf algebra with bijective antipode S, the canonical map is bijective, and P is an H-equivariantly projective left B-module. This brings us in touch with compact quantum groups. Assume that H̄ is the C^{∗}-algebra of a compact quantum group in the sense of Woronowicz [34,36], and H is its dense Hopf *-subalgebra spanned by the matrix coefficients of the irreducible unitary corepresentations. Let P̄ be a unital C^{∗}-algebra and δ : P̄ → P̄ ¯⊗H̄ an injective C^{∗}-algebraic right coaction of H̄ on P̄. (See [1, Definition 0.2] for a general definition and [3, Definition 1] for the special case of compact quantum groups.) Here ¯⊗ denotes the minimal C^{∗}-completion of the algebraic tensor product P̄ ⊗ H̄.

To extend Woronowicz’s Peter-Weyl theory [36] from compact quantum groups to compact quantum principal bundles, one defines [2] the subalgebra P_{δ}(P̄) ⊆ P̄ of elements for which the coaction lands in P̄ ⊗ H, i.e.

(1.44)  P_{δ}(P̄) := {p ∈ P̄ | δ(p) ∈ P̄ ⊗ H}.

One easily checks that it is an H-comodule algebra. We call P_{δ}(P̄) the Peter-Weyl comodule algebra associated to the C^{∗}-coaction δ. It follows from results of [3] and [28] that P_{δ}(P̄) is dense in P̄. Also, it is straightforward to verify [2] that the operation P̄ ↦ P_{δ}(P̄) gives a functor commuting with taking fibre products (pullbacks), and that P_{δ}(P̄)^{coH} coincides with the C^{∗}-algebra P̄^{coH̄}.

Finally, let us recall that, for a compact Hausdorff topological group G and a unital C^{∗}-algebra A, we can use Formula (1.2) to translate a right C(G)-coaction into a G-action. Thus we can use the terminology of right C(G)-comodule C^{∗}-algebras and G-C^{∗}-algebras synonymously. It is important to bear in mind that the Peter-Weyl functor maps G-equivariant *-homomorphisms to colinear homomorphisms of right O(G)-comodule algebras [2].

1.4. Standard Hopf fibration of quantum SU(2). The standard quantum Hopf fibration is given by an action of U(1) on the quantum group SU_{q}(2), q ∈ (0, 1). The coordinate ring of O(SU_{q}(2)) is generated by α, β, γ, δ with relations

αβ = qβα, αγ = qγα, βδ = qδβ, γδ = qδγ, βγ = γβ, (1.45)

αδ − qβγ = 1, δα − q^{−1}βγ = 1, (1.46)

and involution α^{∗} := δ, β^{∗} := −qγ. It is a Hopf *-algebra with comultiplication Δ, counit ε, and antipode S given by

Δ(α) = α ⊗ α + β ⊗ γ, Δ(β) = α ⊗ β + β ⊗ δ, (1.47)

Δ(γ) = γ ⊗ α + δ ⊗ γ, Δ(δ) = γ ⊗ β + δ ⊗ δ, (1.48)

ε(α) = ε(δ) = 1, ε(β) = ε(γ) = 0, (1.49)

S(α) = δ, S(β) = −q^{−1}β, S(γ) = −qγ, S(δ) = α. (1.50)

Let O(U(1)) denote the commutative and cocommutative Peter-Weyl Hopf *-algebra of U(1), and let u stand for its unitary group-like generator. Note that the counit ε and the antipode S satisfy ε(u) = 1 and S(u) = u^{∗}. There is a Hopf *-algebra surjection (1.51) π : O(SU_{q}(2)) ⟶ O(U(1)), π(α) := u, π(δ) := u^{−1}, π(β) := 0 =: π(γ).

Setting Δ_{R} := (id ⊗ π) ∘ Δ, we obtain a right O(U(1))-coaction on O(SU_{q}(2)). On generators, the coaction reads

(1.52) Δ_{R}(α) = α⊗u, Δ_{R}(β) = β ⊗u^{−1}, Δ_{R}(γ) = γ ⊗u, Δ_{R}(δ) = δ⊗u^{−1}.

The *-subalgebra of coaction invariants defines the coordinate ring of the standard Podleś quantum sphere:

(1.53) O(S_{q}^{2}) := O(SU_{q}(2))^{coO(U(1))} = {a ∈ O(SU_{q}(2)) | Δ_{R}(a) = a ⊗ 1} .

One can prove (see [27]) that O(S_{q}^{2}) is isomorphic to the *-algebra generated by B and self-adjoint A satisfying the relations

(1.54)  AB = q^{2}BA, B^{∗}B = A − A^{2}, BB^{∗} = q^{2}A − q^{4}A^{2}.

An isomorphism is explicitly given by the formulas A = −q^{−1}βγ and B = −βα. The irreducible Hilbert space representations of O(S_{q}^{2}) are given by

ρ_{0}(A) = ρ_{0}(B) = 0, ρ_{0}(1) = 1 on ℋ = ℂ, (1.55)

ρ_{+}(A)e_{n} = q^{2n}e_{n}, ρ_{+}(B)e_{n} = q^{n}(1 − q^{2n})^{1/2}e_{n−1} on ℋ = ℓ_{2}(ℕ), (1.56)

where {e_{n} | n = 0, 1, …} is an orthonormal basis of ℓ_{2}(ℕ).

Let C(S_{q}^{2}) denote the universal C^{∗}-algebra generated by A and B satisfying (1.54). From the above representations, it follows that C(S_{q}^{2}) is the minimal unitalization of K(ℓ_{2}(ℕ)), that is,

C(S_{q}^{2}) ≅ K(ℓ_{2}(ℕ)) ⊕ ℂ ⊆ B(ℓ_{2}(ℕ)), (1.57)

(k + α)(k′ + α′) = (kk′ + α′k + αk′) + αα′, k, k′ ∈ K(ℓ_{2}(ℕ)), α, α′ ∈ ℂ.

Here K(ℓ_{2}(ℕ)) and B(ℓ_{2}(ℕ)) denote the C^{∗}-algebras of compact and bounded operators respectively on the Hilbert space ℓ_{2}(ℕ). The isomorphism (1.57) implies that K_{0}(C(S_{q}^{2})) ≅ ℤ ⊕ ℤ, where one generator of K-theory is given by the class of the unit 1 ∈ C(S_{q}^{2}), and the other by the class of the one-dimensional projection onto ℂe_{0} ⊆ ℓ_{2}(ℕ).

Furthermore, K^{0}(C(S_{q}^{2})) ≅ ℤ ⊕ ℤ. We identify one generator of K-homology with the class of the pair of representations [(id, ε)], where id(k + α) = k + α and ε(k + α) = α for all k ∈ K(ℓ_{2}(ℕ)) and α ∈ ℂ. The other generator can be given by the class of the pair of representations [(ε, ε_{0})] with the (non-unital) representation ε_{0} of K(ℓ_{2}(ℕ)) ⊕ ℂ defined by ε_{0}(k + α) = αSS^{∗}, where

(1.58)  S : ℓ_{2}(ℕ) ⟶ ℓ_{2}(ℕ), Se_{n} = e_{n+1},

denotes the unilateral shift on ℓ_{2}(ℕ). (See [22] for a detailed treatment of the K-homology and K-theory of Podleś quantum spheres.)

We shall also consider the coordinate ring of the quantum disc O(D_{q}) generated by z and z^{∗} with the relation

(1.59)  z^{∗}z − q^{2}zz^{∗} = 1 − q^{2}.

Its bounded irreducible Hilbert space representations are given by

μ_{θ}(z) = e^{iθ} on ℋ = ℂ, θ ∈ [0, 2π), (1.60)

μ(z)e_{n} = (1 − q^{2(n+1)})^{1/2}e_{n+1} on ℋ = ℓ_{2}(ℕ). (1.61)

It has been shown in [18] that the universal C^{∗}-algebra of O(D_{q}) is isomorphic to the Toeplitz algebra given as the C^{∗}-algebra generated by the unilateral shift S of Equation (1.58). The representation μ defines then an embedding of O(D_{q}) into T .

Let C(U(1)) denote the C^{∗}-algebra of continuous functions on the unit circle S^{1}, and let u be its unitary generator. The Toeplitz algebra gives rise to the following short exact sequence of C^{∗}-algebras:

(1.62)  0 ⟶ K(ℓ_{2}(ℕ)) ⟶ T ⟶^{σ} C(U(1)) ⟶ 0.

Here the so-called symbol map σ : T → C(U(1)) is given by σ(S) := u. Since S − μ(z) belongs to K(ℓ_{2}(ℕ)), it follows in particular that σ(μ(z)) = u.

Now let us consider the associated quantum line bundles as finitely generated projective modules. They are defined by the one-dimensional corepresentations ℂ ∋ 1 ↦ u^{N} ⊗ 1, N ∈ ℤ, as cotensor products (1.42):

(1.63)  M_{N} := {p ∈ O(SU_{q}(2)) | Δ_{R}(p) = p ⊗ u^{N}}.

Since Δ_{R} is a morphism of algebras, M_{N} is an O(S_{q}^{2})-bimodule. Our next step is to determine explicitly projections describing these projective modules.

For l ∈ \frac{1}{2}ℕ and i, j = −l, −l + 1, …, l, let t_{i,j}^{l} denote the matrix elements of the irreducible unitary corepresentations of O(SU_{q}(2)), so that we have

(1.64)  Δ(t_{i,j}^{l}) = ∑_{k=−l}^{l} t_{i,k}^{l} ⊗ t_{k,j}^{l} ,  ∑_{k=−l}^{l} t_{k,i}^{l∗} t_{k,j}^{l} = ∑_{k=−l}^{l} t_{i,k}^{l} t_{j,k}^{l∗} = δ_{ij} .

By the Peter-Weyl theorem for compact quantum groups [35],

(1.65)  O(SU_{q}(2)) = ⨁_{l∈ \frac{1}{2} ℕ} ⨁_{i,j=−l}^{l} ℂt_{i,j}^{l} .

From the explicit description of t_{i,j}^{l} [19, Section 4.2.4] and the definition of Δ_{R}, it follows that Δ_{R}(t_{i,j}^{l}) = t_{i,j}^{l} ⊗ u^{−2j}, whence t_{i,−j}^{l} ∈ M_{2j}. It can be shown [16, 29] that t_{i,−j}^{|j|}, i = −|j|, …, |j| generate M_{2j} as a left O(S_{q}^{2})-module and

M_{2j} ≅ O(S_{q}^{2})^{2|j|+1}E_{2j} for all j ∈ \frac{1}{2}ℤ, where

(1.66)  E_{2j} := (t_{−|j|,−j}^{|j|} ; ⋮ ; t_{|j|,−j}^{|j|}) (t_{−|j|,−j}^{|j|∗} , ⋯, t_{|j|,−j}^{|j|∗}) ∈ Mat_{2|j|+1}(O(S_{q}^{2})).

It is clear that E_{2j}^{∗} = E_{2j}, and it follows from (1.64) that E_{2j}^{2} = E_{2j}. Hence E_{2j} is a projection.

2. Principality of one-surjective pullbacks

We begin by defining an ambient category for pullback diagrams appearing in the second part of this section. Let P be a unital algebra equipped with both a right coaction Δ_{P} : P → P ⊗ C and a left coaction _{P} Δ : P → C ⊗ P of the same coalgebra C. We do not assume that these coactions commute, but we do assume that they are coaugmented by the same group-like element e ∈ C, i.e. Δ_{P} (1) = 1 ⊗ e and _{P} Δ(1) = e ⊗ 1. For a fixed coalgebra C and a group-like e ∈ C, we consider the category _{e}^{C}Alg_{e}^{C} of all such unital algebras with e-coaugmented left and right C-coactions. Here morphisms are bicolinear algebra homomorphisms.

Since we work over a field, this category is evidently closed under any pullbacks. If π_{1} : P_{1} → P_{12} and π_{2} : P_{2} → P_{12} are morphisms in _{e}^{C}Alg_{e}^{C}, then the fibre product algebra P := P_{1}×_{(π_{1},π_{2})}P_{2} becomes a right C-comodule via

(2.1)  Δ_{P} (p, q) := (p_{(0)}, 0) ⊗ p_{(1)} + (0, q_{(0)}) ⊗ q_{(1)},

and a left C-comodule via

(2.2)  _{P} Δ(p, q) := p_{(−1)} ⊗ (p_{(0)}, 0) + q_{(−1)} ⊗ (0, q_{(0)}).

Also, it is clear that Δ_{P} (1, 1) = (1, 1) ⊗ e and _{P} Δ(1, 1) = e ⊗ (1, 1).

2.1. Principality of images and preimages. In the following lemma, we prove that any surjective morphism in _{e}^{C}Alg_{e}^{C} whose domain is a principal extension can be split by a left-colinear map and by a right-colinear map (not necessarily by a bicolinear map). Note that the first part of the lemma is proved much in the same way as in the Hopf-Galois case [15, Lemma 3.1]:

Lemma 2.1. Let π : P → Q be a surjective morphism in the category _{e}^{C}Alg_{e}^{C} of unital algebras with e-coaugmented left and right C-coactions. If P is principal, then:

(i) The induced map π^{coC} : P^{coC} → Q^{coC} is surjective. (ii) There exists a unital right-C-colinear splitting of π. (iii) There exists a unital left-C-colinear splitting of π. (iv) Q is principal.

Furthermore, if Q′ ∈ _{e}^{C}Alg_{e}^{C}, Q′ ⊆ Q, is principal, then so is π^{−1}(Q′).

Proof. It follows from the right colinearity and surjectivity of π that π(P^{coC}) ⊆ Q^{coC}. To prove the converse inclusion, we take advantage of the left-P^{coC}-linear retraction of the inclusion P^{coC} ⊆ P that was used to prove [6, Theorem 2.5(3)]:

(2.3)  σ_{φ} : P ⟶ P^{coC}, σ_{φ}(p) := p_{(0)}ℓ(p_{(1)})^{⟨1⟩}φ(ℓ(p_{(1)})^{⟨2⟩}) .

Here ℓ is a strong connection on P and φ is any unital linear functional on P. It follows from (1.35) that σ_{φ}(p) ∈ P^{coC}. If π(p) ∈ Q^{coC}, then σ_{φ}(p) is a desired element of P^{coC} that is mapped by π to π(p). Indeed, since

(2.4)  π(p_{(0)}) ⊗ p_{(1)} = π(p)_{(0)} ⊗ π(p)_{(1)} = π(p) ⊗ e,

using the unitality of π, φ, and ℓ(e) = 1 ⊗ 1, we compute

(2.5)  π(σ_{φ}(p)) = π(p_{(0)})π(ℓ(p_{(1)})^{⟨1⟩})φ(ℓ(p_{(1)})^{⟨2⟩}) = π(p).

To show the second assertion, let us choose any unital k-linear splitting of π↾_{P^{coC}} and denote it by α^{coC}. We want to prove that the formula

(2.6)  α_{R}(q) := α^{coC}(q_{(0)}π(ℓ(q_{(1)})^{⟨1⟩}))ℓ(q_{(1)})^{⟨2⟩}

defines a unital right-colinear splitting of π. Since π is surjective, we can write q = π(p). Then, using properties of π, we obtain:

q_{(0)} π(ℓ(q_{(1)})^{⟨1⟩})⊗ ℓ(q_{(1)})^{⟨2⟩} = π(p)_{(0)} π(ℓ(π(p)_{(1)})^{⟨1⟩})⊗ ℓ(π(p)_{(1)})^{⟨2⟩}
= π(p_{(0)})π(ℓ(p_{(1)})^{⟨1⟩})⊗ ℓ(p_{(1)})^{⟨2⟩}

= π(p_{(0)}ℓ(p_{(1)})^{⟨1⟩})⊗ ℓ(p_{(1)})^{⟨2⟩}. (2.7)

Now it follows from (1.35) that the above tensor is in Q^{coC} ⊗ P. Hence α_{R} is well defined. It is straightforward to verify that α_{R} is unital, right colinear, and splits π. (Note that, since q ∈ Q^{coC} implies q_{(0)} ⊗ q_{(1)} = q ⊗ e, we have α^{coC} = α_{R} ↾_{Q^{coC}}.) The third assertion can be proven in an analogous manner.

To prove (iv), we first show that the inverse of the canonical map can_{Q} : Q ⊗_{Q^{coC}} Q → Q ⊗ C (see (1.19)) is given by

(2.8) can_{Q}^{−1} : Q ⊗ C ⟶ Q ⊗_{Q^{coC}} Q, q ⊗ c ⟼ qπ(ℓ(c)^{⟨1⟩}) ⊗_{Q^{coC}} π(ℓ(c)^{⟨2⟩}).

Using the properties of π and ℓ, we get

(can_{Q} ∘ can_{Q}^{−1}) (π(p) ⊗ c) = can_{Q}(π(pℓ(c)^{⟨1⟩}) ⊗_{Q^{coC}} π(ℓ(c)^{⟨2⟩}))
= π (p ℓ(c)^{⟨1⟩} ℓ(c)^{⟨2⟩}_{(0)}) ⊗ ℓ(c)^{⟨2⟩}_{(1)}

= π(p) ⊗ c. (2.9)

Similarly,

(can_{Q}^{−1} ∘ can_{Q})(π(p) ⊗_{Q^{coC}} π(p′)) = can_{Q}^{−1}(π(pp′_{(0)}) ⊗ p′_{(1)})

= π(pp′_{(0)}ℓ(p′_{(1)})^{⟨1⟩}) ⊗_{Q^{coC}} π(ℓ(p′_{(1)})^{⟨2⟩})

= π(p) ⊗_{Q^{coC}} π(p′_{(0)}ℓ(p′_{(1)})^{⟨1⟩}ℓ(p′_{(1)})^{⟨2⟩})

= π(p) ⊗_{Q^{coC}} π(p′). (2.10)

Here we used the fact that π(p′_{(0)}ℓ(p′_{(1)})^{⟨1⟩}) ⊗ ℓ(p′_{(1)})^{⟨2⟩} ∈ Q^{coC} ⊗ P. Hence the extension Q^{coC} ⊆ Q is Galois, and we have the canonical entwining ψ_{Q} : C ⊗ Q → Q ⊗ C.

Our next aim is to prove that ψ_{Q} is bijective. We know by assumption that the canonical entwining ψ_{P} : C ⊗ P → P ⊗ C is invertible. To determine its inverse, recall that the left and right coactions are given by ψ_{P}^{−1}(p ⊗ e) and ψ_{P} (e ⊗ p) respectively. Then apply (1.15) to compute

ψ_{P} ((pℓ(c)^{⟨1⟩})_{(−1)} ⊗ (pℓ(c)^{⟨1⟩})_{(0)} ℓ(c)^{⟨2⟩}) = pℓ(c)^{⟨1⟩}ψ_{P} (e ⊗ ℓ(c)^{⟨2⟩})
= pℓ(c)^{⟨1⟩}ℓ(c)^{⟨2⟩}_{(0)} ⊗ ℓ(c)^{⟨2⟩}_{(1)}

= p ⊗ c. (2.11)

Hence ψ_{P}^{−1}(p ⊗ c) = (pℓ(c)^{⟨1⟩})_{(−1)} ⊗ (pℓ(c)^{⟨1⟩})_{(0)} ℓ(c)^{⟨2⟩}. On the other hand,

ψ_{Q}(c ⊗ π(p)) = π(ℓ(c)^{⟨1⟩})(π(ℓ(c)^{⟨2⟩})π(p))_{(0)} ⊗ (π(ℓ(c)^{⟨2⟩})π(p))_{(1)}
= π(ℓ(c)^{⟨1⟩}) π((ℓ(c)^{⟨2⟩} p)_{(0)}) ⊗ (ℓ(c)^{⟨2⟩} p)_{(1)}

= (π ⊗ id)(ψ_{P} (c ⊗ p)), (2.12)

(id ⊗ π)(ψ_{P}^{−1}(p ⊗ c)) = (p ℓ(c)^{⟨1⟩})_{(−1)} ⊗ π((p ℓ(c)^{⟨1⟩})_{(0)}) π(ℓ(c)^{⟨2⟩})
= (π(p ℓ(c)^{⟨1⟩}))_{(−1)} ⊗ (π(p ℓ(c)^{⟨1⟩}))_{(0)} π(ℓ(c)^{⟨2⟩})

= _{Q}Δ(π(p)π(ℓ(c)^{⟨1⟩}))π(ℓ(c)^{⟨2⟩}). (2.13)

The second part of the above computation implies that the assignment

(2.14)  ψ_{Q}^{−1} : Q ⊗ C ⟶ C ⊗ Q, π(p) ⊗ c ⟼ (id ⊗ π)(ψ_{P}^{−1}(p ⊗ c)),

is well defined. Now it follows from the first part that ψ_{Q}^{−1} is the inverse of ψ_{Q}:

ψ_{Q}(ψ_{Q}^{−1}(π(p) ⊗ c)) = ψ_{Q}((id ⊗ π)(ψ_{P}^{−1}(p ⊗ c)))

= (π ⊗ id)(ψ_{P} (ψ_{P}^{−1}(p ⊗ c))) = π(p) ⊗ c, (2.15)

ψ_{Q}^{−1}(ψ_{Q}(c ⊗ π(p))) = ψ_{Q}^{−1}((π ⊗ id)(ψ_{P} (c ⊗ p)))

= (id ⊗ π)(ψ_{P}^{−1}(ψ_{P} (c ⊗ p))) = c ⊗ π(p). (2.16)

On the other hand, we observe that (π ⊗ π) ∘ ℓ is a strong connection on Q. Combining it with the just-proven existence of a bijective entwining that makes Q an entwined module, we can apply Lemma 1.1 to conclude the proof of (iv).

To prove the final statement of the lemma, note first that π^{−1}(Q′) ∈ _{e}^{C}Alg_{e}^{C}. Next, observe that, if ℓ′ : C → Q′ ⊗ Q′ is a strong connection on Q′, then it is also a strong connection on Q. Now, it follows from (1.40) that for any q ∈ Q′

(2.17)  ψ_{Q}(c ⊗ q) = ℓ′(c)^{⟨1⟩}(ℓ′(c)^{⟨2⟩} q)_{(0)} ⊗ (ℓ′(c)^{⟨2⟩} q)_{(1)} ∈ Q′ ⊗ C.

In much the same way, it follows from the Q-analog of the formula following (2.11) that ψ_{Q}^{−1}(Q′ ⊗ C) ⊆ C ⊗ Q′. Hence to see that ψ_{P} and ψ_{P}^{−1} restrict to π^{−1}(Q′), we can apply (2.12) and (2.14) respectively.

A key step now is to construct a strong connection on π^{−1}(Q′). Let α_{R} and α_{L} be, respectively, right and left colinear unital splittings of π. Their existence is guaranteed by the already proven (ii) and (iii). The map

(2.18)  (α_{L} ⊗ α_{R}) ∘ ℓ′ : C ⟶ π^{−1}(Q) ⊗ π^{−1}(Q)

is bicolinear and satisfies

(2.19)  α_{L}(ℓ′(e)^{⟨1⟩}) ⊗ α_{R}(ℓ′(e)^{⟨2⟩}) = 1 ⊗ 1.

However, it is possible that

1⊗c−(c̃an∘(α_{L} ⊗α_{R})∘ℓ′)(c) = 1⊗c−α_{L}(ℓ′(c_{(1)})^{⟨1⟩})α_{R}(ℓ′(c_{(1)})^{⟨2⟩})⊗c_{(2)} ≠ 0.

To solve this problem, we apply to it the splitting of the lifted canonical map given by a strong connection ℓ (see (1.34)), and add to (α_{L} ⊗ α_{R}) ∘ ℓ′: (2.20) ℓ_{R}(c) := (α_{L} ⊗ α_{R})(ℓ′(c)) + ℓ(c) − α_{L}(ℓ′(c_{(1)})^{⟨1⟩})α_{R}(ℓ′(c_{(1)})^{⟨2⟩})ℓ(c_{(2)}^{⟨1⟩}) ⊗ ℓ(c_{(2)}^{⟨2⟩}).

Now c̃an ∘ ℓ_{R} = 1 ⊗ id, as needed. Also, ℓ_{R}(e) = 1 ⊗ 1 and ((π ⊗ id) ∘ ℓ_{R})(C) ⊆ Q′ ⊗ P. The right colinearity of ℓ_{R} is clear. To check the left colinearity of ℓ_{R}, using the fact that P is a ψ_{P} entwined and e-coaugmented module, we show that (m_{P} ∘ (α_{L} ⊗ α_{R}) ∘ ℓ′) ∗ ℓ is left colinear. (Here m_{P} is the multiplication of P.) First we note that

(2.21) (_{P} Δ⊗id)∘((m_{P} ∘(α_{L} ⊗α_{R})∘ℓ′)∗ℓ) = (id⊗(m_{P} ∘(α_{L} ⊗α_{R})∘ℓ′)∗ℓ)∘Δ

is equivalent to

α_{L}(ℓ′(c_{(1)})^{⟨1⟩})α_{R}(ℓ′(c_{(1)})^{⟨2⟩})ℓ(c_{(2)})^{⟨1⟩} ⊗e⊗ℓ(c_{(2)})^{⟨2⟩}

= ψ_{P} (c_{(1)} ⊗ α_{L}(ℓ′(c_{(2)})^{⟨1⟩})α_{R}(ℓ′(c_{(2)})^{⟨2⟩})ℓ(c_{(3)})^{⟨1⟩}) ⊗ ℓ(c_{(3)})^{⟨2⟩}. (2.22)

Since c_{(1)} ⊗ α_{L}(ℓ′(c_{(2)})^{⟨1⟩}) ⊗ ℓ′(c_{(2)})^{⟨2⟩} = ψ_{P}^{−1}(α_{L}(ℓ′(c)^{⟨1⟩}) ⊗ e) ⊗ ℓ′(c)^{⟨2⟩}, we obtain

ψ_{P} (c_{(1)} ⊗ α_{L}(ℓ′(c_{(2)})^{⟨1⟩})α_{R}(ℓ′(c_{(2)})^{⟨2⟩})ℓ(c_{(3)})^{⟨1⟩}) ⊗ ℓ(c_{(3)})^{⟨2⟩}
= α_{L}(ℓ′(c_{(1)})^{⟨1⟩}) ψ_{P} (e ⊗ α_{R}(ℓ′(c_{(1)})^{⟨2⟩})ℓ(c_{(2)})^{⟨1⟩}) ⊗ ℓ(c_{(2)})^{⟨2⟩}
= α_{L}(ℓ′(c_{(1)})^{⟨1⟩})α_{R}(ℓ′(c_{(1)})^{⟨2⟩}) ψ_{P} (c_{(2)} ⊗ ℓ(c_{(3)})^{⟨1⟩}) ⊗ ℓ(c_{(3)})^{⟨2⟩}
= α_{L}(ℓ′(c_{(1)})^{⟨1⟩})α_{R}(ℓ′(c_{(1)})^{⟨2⟩}) ψ_{P} (ψ_{P}^{−1}(ℓ(c_{(2)})^{⟨1⟩} ⊗ e)) ⊗ ℓ(c_{(2)})^{⟨2⟩}

= α_{L}(ℓ′(c_{(1)})^{⟨1⟩})α_{R}(ℓ′(c_{(1)})^{⟨2⟩}) ℓ(c_{(2)})^{⟨1⟩} ⊗ e ⊗ ℓ(c_{(2)})^{⟨2⟩}. (2.23)

Hence ℓ_{R} is a strong connection with the property ℓ_{R}(C) ⊆ π^{−1}(Q′) ⊗ P. In a similar manner, we construct a strong connection ℓ_{L} with the property ℓ_{L}(C) ⊂ P ⊗ π^{−1}(Q′). Now we need to apply the splitting of the lifted left canonical map given by ℓ (see (1.33)) to derive the formula

(2.24)  ℓ_{L} := (α_{L} ⊗ α_{R}) ∘ ℓ′ + ℓ − ℓ ∗ (m_{P} ∘ (α_{L} ⊗ α_{R}) ∘ ℓ′).

It is clear that ℓ_{L}(e) = 1 ⊗ 1 and ℓ_{L}(C) ⊆ P ⊗ π^{−1}(Q′). A computation similar to (2.23) shows the right colinearity of ℓ_{L}. Since furthermore ψ_{P} (1 ⊗ c) = c ⊗ 1 for any c ∈ C and c̃an = ψ_{P} ∘ c̃an_{L}, we obtain

(2.25)  c̃an(ℓ_{L}(c)) = ψ_{P} (c̃an_{L}(ℓ(c))) = ψ_{P} (c ⊗ 1) = 1 ⊗ c.

Hence ℓ_{L} is a desired strong connection. Plugging it into (2.20) instead of ℓ, we get a strong connection

(2.26)  ℓ_{LR} = (α_{L} ⊗ α_{R}) ∘ ℓ′ + ℓ_{L} − (m_{p} ∘ (α_{L} ⊗ α_{R}) ∘ ℓ′) ∗ ℓ_{L}

with the property ℓ_{LR} ⊆ π^{−1}(Q′) ⊗ π^{−1}(Q′). Applying now Lemma 1.1 ends the proof of this lemma. □

2.2. The one-surjective pullbacks of principal coactions are principal. Our goal now is to show that the subcategory of principal extensions is closed under one-surjective pullbacks. Here the right coaction is the coaction defining a principal extension and the left coaction is the one defined by the inverse of the canonical entwining (see (1.24)). With this structure, principal extensions form a full subcategory of _{e}^{C}Alg_{e}^{C}. The following theorem is the main result of this paper generalizing the theorem of [15] on the pullback of surjections of principal comodule algebras:

Theorem 2.2. Let C be a coalgebra, e ∈ C a group-like element, and P the pullback of π_{1} : P_{1} → P_{12} and π_{2} : P_{2} → P_{12} in the category _{e}^{C}Alg_{e}^{C} of unital algebras with e-coaugmented left and right C-coactions. If π_{1} or π_{2} is surjective and both P_{1} and P_{2} are principal e-coaugmented C-extensions, then also P is a principal e-coaugmented C-extension.

Proof. Without loss of generality, we assume that π_{1} is surjective. We first show that P inherits an entwined structure from P_{1} and P_{2}.

Lemma 2.3. Let ψ_{1} and ψ_{2} denote the entwining structures of P_{1} and P_{2} respectively. Then P is an entwined module with an invertible entwining structure

(2.27)  ψ = ψ_{1} ∘ (id ⊗ pr_{1}) + ψ_{2} ∘ (id ⊗ pr_{2}).

Here pr_{1} and pr_{2} are morphisms of the pullback diagram as in (1.6).

Proof. Our strategy is to construct a bijective map

(2.28)  ψ̃ : C ⊗ (P_{1} × P_{2}) ⟶ (P_{1} × P_{2}) ⊗ C,

and to show that it restricts to a bijective entwining on C ⊗ P. We put

(2.29)  ψ̃ := ψ_{1} ∘ (id ⊗ p̃r_{1}) + ψ_{2} ∘ (id ⊗ p̃r_{2}).

The symbols p̃r_{1} and p̃r_{2} stand for respective componentwise projections. Their restrictions to P yield pr_{1} and pr_{2}. It is easy to check that the inverse of ψ̃ is given by

(2.30)  ψ̃^{−1} = ψ_{1}^{−1} ∘ (p̃r_{1} ⊗ id) + ψ_{2}^{−1} ∘ (p̃r_{2} ⊗ id).

To show that ψ̃(C ⊗ P) ⊆ P ⊗ C and ψ̃^{−1}(P ⊗ C) ⊆ C ⊗ P, we note first that P_{12} and π_{2}(P_{2}) are principal by Lemma 2.1(iv). Consequently, their canonical entwinings ψ_{12} and ψ_{π_{2}(P_{2})} are bijective. Furthermore, arguing as in the proof of Lemma 2.1, we see that ψ_{π_{2}(P_{2})} = ψ_{12}↾_{C⊗π_{2}(P_{2})} and ψ_{π_{2}(P_{2})}^{−1} = ψ_{12}^{−1}↾_{π_{2}(P_{2})⊗C}. An advantage of having both summands in terms of ψ_{12} is that we can apply (2.12) to compute

((π_{1} ∘p̃r_{1} − π_{2} ∘p̃r_{2}) ⊗ id)∘ψ̃
= (π_{1} ∘p̃r_{1} ⊗id)∘ψ_{1} ∘(id⊗p̃r_{1}) − (π_{2} ∘p̃r_{2} ⊗id)∘ψ_{1} ∘ (id⊗p̃r_{1})
+ (π_{1} ∘p̃r_{1} ⊗id)∘ψ_{2} ∘(id⊗p̃r_{2}) − (π_{2} ∘p̃r_{2} ⊗id)∘ψ_{2} ∘ (id⊗p̃r_{2})
= (π_{1} ⊗id)∘ψ_{1} ∘(id ⊗ p̃r_{1}) − (π_{2} ⊗ id)∘ψ_{2} ∘ (id ⊗ p̃r_{2})
= ψ_{12} ∘ (id ⊗ π_{1}) ∘ (id ⊗ p̃r_{1}) − ψ_{π_{2}(P_{2})} ∘ (id ⊗ π_{2}) ∘ (id ⊗ p̃r_{2})

= ψ_{12} ∘ (id ⊗ (π_{1} ∘ p̃r_{1} − π_{2} ∘ p̃r_{2})). (2.31)

Hence ψ̃(C ⊗ P) ⊆ P ⊗ C. In much the same way, using (2.14) instead of (2.12), we show that ψ̃^{−1}(P ⊗ C) ⊆ C ⊗ P.

It remains to verify that the bijection ψ = ψ̃↾_{C⊗P} is an entwining that makes P an entwined module. The former is proven by checking directly (1.15) and (1.16). The latter follows from the fact that P_{1} and P_{2} are, respectively, ψ_{1} and ψ_{2} entwined modules:

Δ_{P} (pq) = Δ_{P_{1}}(pr_{1}(p)pr_{1}(q)) + Δ_{P_{2}}(pr_{2}(p)pr_{2}(q))
= pr_{1}(p_{(0)})ψ_{1}(p_{(1)} ⊗ pr_{1}(q)) + pr_{2}(p_{(0)})ψ_{2}(p_{(1)} ⊗ pr_{2}(q))
= (pr_{1}(p_{(0)}) + pr_{2}(p_{(0)}))(ψ_{1}(p_{(1)} ⊗ pr_{1}(q)) + ψ_{2}(p_{(1)} ⊗ pr_{2}(q)))

= p_{(0)}ψ(p_{(1)} ⊗ q). (2.32)

This proves the lemma. □

Let α_{L}^{1} and α_{R}^{1} be a unital left-colinear splitting and a unital right-colinear splitting of π_{1}, respectively. Also, let α_{R}^{2} be a right-colinear splitting of π_{2} viewed as a map onto π_{2}(P_{2}). Such maps exist by Lemma 2.1. Furthermore, by [6, Lemma 2.2], since P_{1} and P_{2} are principal, they admit strong connections ℓ_{1} and ℓ_{2} respectively. For brevity, let us introduce the notation

\begin{matrix}α_{L}^{12} := α_{L}^{1} ∘ π_{2}, & α_{R}^{12} := α_{R}^{1} ∘ π_{2}, \\ α_{R}^{21} := α_{R}^{2} ∘ π_{1} ↾_{π_{1}^{−1}(π_{2}(P_{2}))} , & L := m_{P_{1}} ∘ (α_{L}^{12} ⊗ α_{R}^{12}) ∘ ℓ_{2} , \end{matrix} ; (2.33)

where m_{P_{1}} is the multiplication of P_{1}. The situation is illustrated in the following diagram:

(2.34) [figure: C; L; P; _{pr_{1}}; _{pr_{2}}; π_{1}^{−1}(π_{2}(P_{2})); α_{R}^{21}; P_{1}; P_{2} .; α_{L}^{12}; α_{R}^{12}; π_{2}; π_{1}; π_{2}(P_{2}); α_{R}^{2}; π_{2}; P_{12}; α_{L}^{1}; α_{R}^{1}]

Our proof hinges on constructing a strong connection on P out of strong connections on P_{1} and P_{2}. Roughly speaking, the basic idea is to take a strong connection on P_{2}, induce a strong connection on the the common part P_{12}, and prolongate it to P_{1}. To this end, we check that (α_{L}^{12} +id)⊗(α_{R}^{12} +id) is a unital bicolinear map from P_{2} ⊗ P_{2} to P ⊗ P. Therefore, as a first approximation for constructing a strong connection on P, we choose the formula

(2.35)  ℓ_{I} := ((α_{L}^{12} + id) ⊗ (α_{R}^{12} + id)) ∘ ℓ_{2}.

It is a bicolinear map from C to P ⊗ P satisfying ℓ_{I}(e) = 1 ⊗ 1 as needed.

However, it does not split the lifted canonical map:

(c̃an ∘ ℓ_{I})(c) − 1 ⊗ c
= α_{L}^{12}(ℓ_{2}(c)^{⟨1⟩})α_{R}^{12}(ℓ_{2}(c)^{⟨2⟩})_{(0)} ⊗ α_{R}^{12}(ℓ_{2}(c)^{⟨2⟩})_{(1)}
+ ℓ_{2}(c)^{⟨1⟩}ℓ_{2}(c)^{⟨2⟩}_{(0)} ⊗ ℓ_{2}(c)^{⟨2⟩}_{(1)} − 1 ⊗ c
= α_{L}^{12}(ℓ_{2}(c_{(1)})^{⟨1⟩})α_{R}^{12}(ℓ_{2}(c_{(1)})^{⟨2⟩}) ⊗ c_{(2)} + (0, 1) ⊗ c − 1 ⊗ c

= L(c_{(1)}) ⊗ c_{(2)} − (1, 0) ⊗ c ∈ P_{1} ⊗ C. (2.36)

We correct it by adding to ℓ_{I}(c) the splitting of the lifted canonical map on P_{1} ⊗ P_{1} afforded by ℓ_{1} and applied to (1, 0) ⊗ c − L(c_{(1)}) ⊗ c_{(2)}:

ℓ_{II}(c) = ℓ_{I}(c) + ℓ_{1}(c)^{⟨1⟩} ⊗ ℓ_{1}(c)^{⟨2⟩} − L(c_{(1)})ℓ_{1}(c_{(2)})^{⟨1⟩} ⊗ ℓ_{1}(c_{(2)})^{⟨2⟩}

= (ℓ_{I} + ℓ_{1} − L ∗ ℓ_{1})(c). (2.37)

The above approximation to a strong connection on P is clearly right colinear. Using the fact that P_{1} is a ψ_{1}-entwined and e-coaugmented module, we follow the lines of (2.21)–(2.23) to show that L ∗ ℓ_{1} is left colinear. Hence ℓ_{II} is bicolinear. It also satisfies ℓ_{II}(e) = 1 ⊗ 1.

The price we pay for having ℓ_{II}(c)^{⟨1⟩} ℓ_{II}(c)^{⟨2⟩}_{(0)} ⊗ ℓ_{II}(c)^{⟨2⟩}_{(1)} = 1 ⊗ c is that the image of ℓ_{II} is no longer in P ⊗ P. The troublesome term ℓ_{1} − L ∗ ℓ_{1} takes values in P ⊗ P_{1}. Now one would like to compose id ⊗ (id + α_{R}^{21}) with ℓ_{1} − L ∗ ℓ_{1} to force it taking values in P ⊗ P. However, since α_{R}^{21} is defined only on π_{1}^{−1}(π_{2}(P_{2})), we need to replace an arbitrary strong connection ℓ_{1} by a strong connection taking values in P_{1} ⊗ π_{1}^{−1}(π_{2}(P_{2})). Such a strong connection is provided for us by (2.24):

(2.38)  ℓ̃_{1} := (α_{L}^{12} ⊗ α_{R}^{12}) ∘ ℓ_{2} + ℓ_{1} − ℓ_{1} ∗ L.

Inserting ℓ̃_{1} in place of ℓ_{1} allows us to apply the correction map id ⊗ (id + α_{R}^{21}) to obtain

(2.39)  ℓ_{III} = ℓ_{I} + (id ⊗ (id + α_{R}^{21})) ∘ (ℓ̃_{1} − L ∗ ℓ̃_{1}).

To end the proof, let us check that ℓ_{III} is indeed a strong connection on P. First, since ℓ_{I}(C) ⊆ P ⊗ P and (id + α_{R}^{21})(π_{1}^{−1}(π_{2}(P_{2}))) ⊆ P, we conclude that ℓ_{III} takes values in P ⊗ P. Next, it is bicolinear because α_{R}^{21} is right colinear. Also, it is clearly unital. To verify that ℓ_{III} splits the canonical map, first we note that c̃an ∘ (id ⊗ α_{R}^{21}) ∘ (ℓ̃_{1} − L ∗ ℓ̃_{1}) = 0 because m_{P_{1}×P_{2}}(p_{1} ⊗ p_{2}) = 0 for all p_{1} ∈ P_{1} and p_{2} ∈ P_{2}. Combining this with the fact that c̃an ∘ (ℓ′_{1} − L ∗ ℓ′_{1}) does not depend on the choice of a strong connection ℓ′_{1}, we infer that c̃an ∘ ℓ_{III} = c̃an∘ℓ_{II} = 1⊗id. Thus ℓ_{III} is a strong connection on P as desired. Combining this fact with Lemma 2.3 and Lemma 1.1 proves the theorem. □

Putting together the formulas in the proof of Theorem 2.2 , we obtain the following strong connection on P:

ℓ = ((α_{L}^{12} + id) ⊗ (α_{R}^{12} + id)) ∘ ℓ_{2}

+ (η_{1} ∘ ε − L) ∗ ((id ⊗ (id + α_{R}^{21})) ∘ (ℓ_{1} − ℓ_{1} ∗ L + (α_{L}^{12} ⊗ α_{R}^{12}) ∘ ℓ_{2})). (2.40)

3. The pullback picture of the standard quantum Hopf fibration

Recall that the classical Hopf fibration is a U(1)-principal bundle given by the maps

π : S^{3} := {(z_{1}, z_{2}) ∈ ℂ^{2} | |z_{1}|^{2} + |z_{2}|^{2} = 1} ⟶ S^{2} ≅ ℂP^{1}, (3.1)

π((z_{1}, z_{2})) := [(z_{1} : z_{2})],

S^{3} × U(1) ⟶ S^{3}, (z_{1}, z_{2}) ◁ u := (z_{1}u, z_{2}u). (3.2)

To unravel the structure of this non-trivial fibration, we split S^{3} into two disjoint parts:

(3.3) S^{3} = {(z_{1}, z_{2}) ∈ ℂ^{2} | |z_{1}|^{2} < 1, |z_{2}|^{2} = 1 − |z_{1}|^{2}} ∪ {(z_{1}, 0) | |z_{1}| = 1}.

Note that both sets are invariant under the U(1)-action. The second set is U(1), and first set is U(1)-equivariantly homeomorphic to the interior of the solid torus D×U(1) equipped with the diagonal action. (Here D = {z ∈ ℂ | |z| ≤ 1}.) By an appropriate U(1)-equivariant gluing of the boundary torus of D × U(1) with U(1), we recover S^{3} with its U(1)-action:

(3.4)

\xymatrix{ &  & S_{3} \\ ϕ_{1}(z, v) := (z, v\sqrt{1−|z|^{2}}) &  &  &  & ϕ_{2}(u) := (u, 0) \\  & D × U(1) \ar[uur]^{ϕ_{1}} &  & U(1) \ar[uul]_{ϕ_{2}} \\ (ι, id)(u, v) := (u, v) &  &  &  & pr_{1}(u, v) := u. \\  &  & U(1) × U(1) \ar[uur]_{_{pr_{1}}} \ar[uul]^{(ι,id)}}

However, to view D × U(1) as a trivial U(1)-principal bundle, we need to gauge the diagonal action to the action on the right slot. This is achieved with the help of the following homeomorphism intertwining these two actions:

Ψ : D × U(1) ⟶ D × U(1), Ψ(x, v) := (xv, v), Ψ(x, vu) = Ψ(x, v) ◁ u.

Let us denote the restriction of Ψ to U(1) × U(1) by the same symbol. Now we can extend the above pushout diagram to the commutative diagram

(3.5)

\xymatrix{ &  & S_{3} \\ D × U(1) \ar[r]^{Ψ} & D × U(1) \ar[ur]^{ϕ_{1}} &  & U(1) \ar[ul]_{ϕ_{2}} & U(1) \ar[l]_{id} \\  &  & U(1) × U(1) \ar[ur]_{_{pr_{1}}} \ar[ul]^{(ι,id)} \\  &  & U(1) × U(1) , \ar[uull]^{(ι,id)} \ar[u]_{Ψ} \ar[uurr]_{m}}

where m is the multiplication map. The outer diagram is again a pushout diagram of U(1)-spaces, but now its defining U(1)-spaces are trivial U(1)-principal bundles. It is the outer pushout diagram that we shall use to analyse a noncommutative deformation of the Hopf fibration.

3.1. Pullback comodule algebra. We consider the tensor products P_{1} := T ⊗ O(U(1)), P_{2} := ℂ ⊗ O(U(1)) = O(U(1)) and P_{12} := C(U(1)) ⊗ O(U(1)). These algebras are right O(U(1))-comodule algebras for the coaction x⊗u^{N} ↦ x ⊗ u^{N} ⊗ u^{N}, N ∈ ℤ. Moreover, P_{1} and P_{2} are trivially principal with strong connections ℓ_{i} : O(U(1)) → P_{i} ⊗ P_{i} given by

(3.6)  ℓ_{i}(u^{N}) := (1 ⊗ u^{N∗}) ⊗ (1 ⊗ u^{N}), i = 1, 2.

To construct a pullback of P_{1} and P_{2}, we define the following morphisms of right O(U(1))-comodule algebras:

π_{1} : T ⊗ O(U(1)) ⟶ C(U(1)) ⊗ O(U(1)), π_{1}(t ⊗ w) := σ(t) ⊗ w, (3.7)

π_{2} : O(U(1)) ⟶ C(U(1)) ⊗ O(U(1)), π_{2}(w) := Δ(w). (3.8)

Then the fibre product P := T ⊗ O(U(1)) ×_{(π_{1},π_{2})} O(U(1)) defined by the pullback diagram

(3.9)  \xymatrix{ & T ⊗ O(U(1)) ×_{(π_{1},π_{2})} O(U(1)) \ar@{-->}[dl]_{_{pr_{1}}} \ar@{-->}[dr]^{_{pr_{2}}} \\ T ⊗ O(U(1)) \ar[dr]_{π_{1}} &  & O(U(1)) \ar[dl]^{π_{2}} \\  & C(U(1)) ⊗ O(U(1))}

is a right O(U(1))-comodule algebra. By Proposition 2.2, it is principal.

Furthermore, define unital, respectively, left-colinear and right-colinear splittings of π_{1} by

(3.10)  α_{L}^{1}(f ⊗ u^{N}) := T_{f} ⊗ u^{N} =: α_{R}^{1}(f ⊗ u^{N}), N ∈ ℤ.

Here f ∈ C(U(1)) and T_{f} denotes the Toeplitz operator with symbol f. In particular, T_{u^{N}} = S^{N} and T_{u^{∗N}} = S^{∗N}. A right-colinear splitting of the map π_{2} : O(U(1)) → π_{2}(O(U(1))) is given by

(3.11)  α_{R}^{2}(u^{N} ⊗ u^{N}) := u^{N}, N ∈ ℤ.

Inserting the definitions of α_{L}^{1}, α_{R}^{i} and ℓ_{i}, i = 1, 2, into (2.33) and (2.40), we obtain the following strong connection on P:

(3.12) ℓ(u^{N}) = (S^{∗N} ⊗ u^{∗N}, u^{∗N}) ⊗ (S^{N} ⊗ u^{N}, u^{N}),
ℓ(u^{∗N}) = (S^{N} ⊗ u^{N}, u^{N}) ⊗ (S^{∗N} ⊗ u^{∗N}, u^{∗N})

+ ((1 − S^{N}S^{∗N}) ⊗ u^{N}, 0) ⊗ ((1 − S^{N}S^{∗N}) ⊗ u^{∗N}, 0), N ∈ ℕ. (3.13)

Note next that, by construction, we have

(3.14) P = {_{∑k}(t_{k} ⊗ u^{k}, α_{k}u^{k}) ∈ (T ⊗ O(U(1))) × O(U(1)) | σ(t_{k}) = α_{k}u^{k}},

where α_{k} ∈ ℂ. For _{ℂ}Δ(1) := u^{N} ⊗ 1, let

(3.15)  L_{N} := P □_{O(U(1))}ℂ = {p ∈ P | Δ_{P} (p) = p ⊗ u^{N}}.

Then L_{0} = P^{coO(U(1))}, each L_{N} is a left P^{coO(U(1))}-module and P = ⨁_{N∈ℤ} L_{N}. From

(3.16)  Δ_{P} ( ∑_{k}(t_{k} ⊗u^{k}, α_{k}u^{k})) = ∑_{k}(t_{k} ⊗u^{k}, α_{k}u^{k}) ⊗ u^{k},

it follows that

(3.17) L_{N} = {(t ⊗ u^{N}, αu^{N}) ∈ (T ⊗ O(U(1))) × O(U(1)) | σ(t) = αu^{N}}.

The next proposition shows that L_{0} ≅ T ×_{(σ,1)}ℂ is isomorphic to the C^{∗}-algebra of the standard Podleś sphere and that

(3.18)  L_{N} ≅ T ×_{(u^{−N} σ,1)} ℂ,

where T ×_{(u^{N} σ,1)} ℂ is given by the pullback diagram

(3.19)  \xymatrix{ & T ×_{(u^{−N} σ,1)} ℂ \ar@{-->}[dl]_{_{pr_{1}}} \ar@{-->}[dr]^{_{pr_{2}}} \\ T \ar[d]_{σ} &  & ℂ \ar[d]^{α↦α1} \\ C(U(1)) \ar[rr]_{f↦u^{−N} f} &  & C(U(1)).}

Proposition 3.1. The fibre product T ×_{(σ,1)}ℂ is isomorphic to the C^{∗}-algebra C(S_{q}^{2}), and L_{N} is isomorphic to T ×_{(u^{−N} σ,1)} ℂ as a left C(S_{q}^{2})-module with respect to the left C(S_{q}^{2})-action on T ×_{(u^{−N} σ,1)} ℂ given by (t, α) · (h, β) := (th, αβ).

Proof. For N = 0, the mappings T ∋ t ↦ σ(t) ∈ C(U(1)) and ℂ ∋ α ↦ α1 ∈ C(U(1)) are morphisms of C^{∗}-algebras, so that T ×_{(σ,1)}ℂ is a C^{∗}-algebra. Next, recall that C(S_{q}^{2}) ≅ K(ℓ_{2}(ℕ)) ⊕ ℂ (see (1.57)), and define

ϕ : T ×_{(σ,1)}ℂ ⟶ K(ℓ_{2}(ℕ)) ⊕ ℂ, ϕ(t, α) := t, (3.20)

ϕ^{−1} : K(ℓ_{2}(ℕ)) ⊕ ℂ ⟶ T ×_{(σ,1)}ℂ, ϕ^{−1}(k + α) := (k + α, α). (3.21)

Clearly, ϕ : T ×_{(σ,1)}ℂ → B(ℓ_{2}(ℕ)) is a morphism of C^{∗}-algebras. Since ϕ(t, α) = (t − α) + α, and σ(t − α) = 0 by the pullback diagram (3.19), it follows from the short exact sequence (1.62) that t − α ∈ K(ℓ_{2}(ℕ)). Hence ϕ(t, α) ∈ K(ℓ_{2}(ℕ)) ⊕ ℂ. One easily sees that ϕ^{−1} is the inverse of ϕ, so that T ×_{(σ,1)}ℂ ≅ K(ℓ_{2}(ℕ)) ⊕ ℂ.

The fact that T ×_{(u^{−N} σ,1)}ℂ with the given C(S_{q}^{2})-action is a left C(S_{q}^{2})-module follows from the discussion preceding the pullback diagram (1.10) with the free rank one modules E_{1} = T , E_{2} = ℂ and π_{1∗}E_{1} = π_{2∗}E_{2} = C(U(1)). Obviously, L_{N} ∋ (t ⊗ u^{N}, αu^{N}) ↦ (t, α) ∈ T ×_{(u^{−N} σ,1)}ℂ defines an isomorphism of left C(S_{q}^{2})-modules. □

3.2. Equivalence of the pullback and standard constructions. Let us view U(1) as a compact quantum group. We consider its C^{∗}-algebra C(U(1)) of all continuous function together with the obvious coproduct, counit and antipode given by Δ(f)(x, y) = f(xy), ε(f) = f(1) and S(f)(x) = f(x^{−1}), respectively. Furthermore, let ¯⊗ stand for the completed tensor product of C^{∗}-algebras. In our case it is unique because of the nuclearity of the involved C^{∗}-algebras.

Now let π_{2} : C(U(1)) → C(U(1)) ¯⊗C(U(1)) be given by the coproduct, i.e. π_{2}(f)(x, y) := (Δf)(x, y) = f(xy), and let σ denote the symbol map T → C(U(1)). Then P̄ := T ¯⊗C(U(1)) ×_{(π_{1},π_{2})} C(U(1)) is defined by the pullback diagram

(3.22)  \xymatrix{ & T ¯⊗C(U(1)) ×_{(π_{1},π_{2})} C(U(1)) \ar@{-->}[dl]_{_{pr_{1}}} \ar@{-->}[dr]^{_{pr_{2}}} \\ T ¯⊗C(U(1)) \ar[dr]_{π_{1}=σ⊗id} &  & C(U(1)) \ar[dl]^{π_{2}=Δ} \\  & C(U(1)) ¯⊗C(U(1)) .}

With the C(U(1))-coaction given by the coproduct Δ on the right tensor factor C(U(1)), π_{1} and π_{2} are morphisms in the category of right C(U(1))-comodule C^{∗}-algebras. Equivalently, we can view this diagram as a diagram in the category of U(1)-C^{∗}-algebras (see Section 1.3). Therefore, P̄ inherits the structure of a right U(1)-C^{∗}-algebra. Using the counit ε : C(U(1)) → ℂ and the fact that the Peter-Weyl functor commutes with taking pullbacks, we easily conclude that the Peter-Weyl comodule algebra P_{Δ}(P̄) = T ⊗ O(U(1)) ×_{(π_{1},π_{2})} O(U(1)), so that P_{Δ}(P̄) is the comodule algebra P of Section 3.1.

Consider next the *-representation of O(SU_{q}(2)) on ℓ_{2}(ℕ) given by [31]

\begin{matrix}ρ(α)e_{n} := (1 − q^{2n})^{1/2}e_{n−1}, ρ(β)e_{n} := −q^{n+1}e_{n}, \\ ρ(γ)e_{n} := q^{n}e_{n}, ρ(δ)e_{n} := (1 − q^{2(n+1)})^{1/2}e_{n+1}. \end{matrix} (3.23)

Note that ρ(β), ρ(γ) ∈ K(ℓ_{2}(ℕ)). Comparing ρ with the representation μ of O(D_{q}) from (1.61), one readily sees that ρ(O(SU_{q}(2))) ⊆ T . Furthermore, the symbol map σ yields σ(ρ(β)) = σ(ρ(γ)) = 0. Using an appropriate diagonal compact operator k, we also obtain

\begin{aligned}σ(ρ(α)) & = σ(ρ(α) − S^{∗}) + σ(S^{∗}) = σ(kS^{∗}) + σ(S^{∗}) = u^{−1}, \\ σ(ρ(δ)) & = σ(ρ(α))^{∗} = u. \end{aligned} (3.24)

Thus we obtain a U(1)-equivariant *-algebra homomorphism O(SU_{q}(2)) →^{ι} P by setting

(3.25)  ι(α) := (ρ(α) ⊗ u, u), ι(γ) := (ρ(γ) ⊗ u, 0).

One easily checks that the image of a Poincaré-Birkhoff-Witt basis of O(SU_{q}(2)) remains linearly independent, so that ι is injective, and we can consider O(SU_{q}(2)) as a subalgebra of P. In particular, we have ι(M_{N}) ⊆ L_{N} as left O(S_{q}^{2})-modules. (See Section 1.4 and Section 3.1 for the definitions of M_{N} and L_{N} respectively.)

The main objective of this section is to establish a U(1)-C^{∗}-algebra isomorphism between C(SU_{q}(2)) and P̄. The universal C^{∗}-algebra C(SU_{q}(2)) of O(SU_{q}(2)) has been studied in [21] and [35]. Here we shall use the fact from [21, Corollary 2.3] that there is a faithful *-representation ρ̂ of C(SU_{q}(2)) on the Hilbert space ℓ_{2}(ℕ) ¯⊗ℓ_{2}(ℤ) given by

(3.26) ρ̂(α)(e_{n}⊗b_{k}) := (1−q^{2n})^{1/2}e_{n−1}⊗b_{k−1}, ρ̂(γ)(e_{n}⊗b_{k}) := q^{n}e_{n}⊗b_{k−1},

where {e_{n}}_{n∈ℕ} and {b_{k}}_{k∈ℤ} denote the standard bases of ℓ_{2}(ℕ) and ℓ_{2}(ℤ) respectively. To compare (3.26) with [21, Corollary 2.3], one has to apply the unitary transformation

(3.27)  T : ℓ_{2}(ℕ) ¯⊗ℓ_{2}(ℤ) ⟶ ℓ_{2}(ℕ) ¯⊗ℓ_{2}(ℤ), T (e_{n} ⊗ b_{k}) := e_{n} ⊗ b_{k−n}.

A right C(U(1))-coaction on C(SU_{q}(2)) is given by (id⊗π̄)∘Δ, where Δ denotes the coproduct of the compact quantum group C(SU_{q}(2)) and π̄ is the extension of the Hopf *-algebra surjection π : O(SU_{q}(2)) → O(U(1)) defined in (1.51) to C(SU_{q}(2)). Using the faithfulness of ρ̂, we can transfer π̄ to ρ̂(C(SU_{q}(2))). In [21], it is shown that π̄ gives rise to the short exact sequence of C^{∗}-algebras:

(3.28) \xymatrix{0 \ar[r] & K(ℓ_{2}(ℕ)) ¯⊗C(U(1)) \ar@{^{(}->}[r] & ρ̂(C(SU_{q}(2))) \ar[r]^{π̄} & C(U(1)) \ar[r] & 0 .}

Here C(U(1)) is naturally identified with the operator algebra on ℓ_{2}(ℤ) generated by the unilateral down-shift.

Theorem 3.2. The U(1)-C^{∗}-algebras C(SU_{q}(2)) and P̄ are isomorphic.

Proof. First note that ker(pr_{1}) = {(0, y) ∈ P̄ | π_{2}(y) = Δ(y) = 0} = 0. Hence we can identify P̄ with the image of pr_{1} in T ¯⊗C(U(1)). We will prove the theorem by applying the Five Lemma to the following commutative diagram of U(1)-C^{∗}-algebras:

(3.29) \xymatrix{0 \ar[r] & K(ℓ_{2}(ℕ)) ¯⊗C(U(1)) \ar[d]^{id} \ar@{^{(}->}[r] & ρ̂(C(SU_{q}(2))) \ar[r]^{π̄} \ar[d]^{τ} & C(U(1)) \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & K(ℓ_{2}(ℕ)) ¯⊗C(U(1)) \ar@{^{(}->}[r] & pr_{1}(P̄) \ar[r]^{ω} & C(U(1)) \ar[r] & 0 .}

To define τ, recall that we realize C(U(1)) as a concrete C^{∗}-algebra of bounded operators on ℓ_{2}(ℤ) by setting u(b_{k}) = b_{k−1}. Then

(3.30)  ρ̂(α) = ρ(α) ⊗ u = pr_{1}(ρ(α) ⊗ u, u) ∈ pr_{1}(P̄),
ρ̂(γ) = ρ(γ) ⊗ u = pr_{1}(ρ(γ) ⊗ u, 0) ∈ pr_{1}(P̄).

Since C(SU_{q}(2)) is generated by α and γ, we take τ to be the inclusion ρ̂(C(SU_{q}(2))) ⊂ pr_{1}(P̄). Next, we define the U(1)-C^{∗}-algebra homomorphism ω by

(3.31)  ω : pr_{1}(P̄) ⟶ C(U(1)), ω := (ε ∘ σ) ⊗ id.

The surjectivity of ω follows from u^{k} = ω(ρ(α^{k})⊗u^{k}) and u^{−k} = ω(ρ(α^{∗k})⊗u^{∗k}) for all k ∈ ℕ.

To prove the exactness of the lower row, note that

(3.32)  K(ℓ_{2}(ℕ)) ¯⊗C(U(1)) = ker(σ) ¯⊗C(U(1)) ⊆ ker(ω).

Now, let f ∈ pr_{1}(P̄) \ ker(σ) ¯⊗C(U(1)). Then (σ ⊗ id)(f) ≠ 0. By the commutativity of Diagram (3.22), there exists a non-zero element g ∈ C(U(1)) such that (σ ⊗ id)(f) = Δ(g). Hence ω(f) = (ε ⊗ id) ∘ Δ(g) = g ≠ 0, which proves that ker(ω) = K(ℓ_{2}(ℕ)) ¯⊗C(U(1)).

It remains to show that Diagram (3.29) is commutative. This is clear for the left part since τ is just the inclusion. The commutativity of the right part follows from

ω(τ(ρ̂(α))) = ε(σ(ρ(α))) ⊗ u = ε(u)u = u = π̄(ρ̂(α)), (3.33)

ω(τ(ρ̂(γ))) = ε(σ(ρ(γ))) ⊗ u = 0 = π̄(ρ̂(γ)), (3.34)

since α and γ generate C(SU_{q}(2)). Therefore, by the Five Lemma, τ is an isomorphism of U(1)-C^{∗}-algebras. □

By the paragraph below Diagram (3.22), we conclude from Theorem 3.2 that the Peter-Weyl comodule algebra P_{Δ}(C(SU_{q}(2))) and the comodule algebra P are isomorphic. We use this isomorphism to identify associated projective modules. For N ∈ ℤ and the left O(U(1))-coaction on ℂ given by _{ℂ}Δ(1) := u^{N} ⊗ 1, we define a “completed” version of M_{N} (see (1.63)):

(3.35)  M̄_{N} :=P_{Δ}(C(SU_{q}(2))) □_{O(U(1))}ℂ
={p ∈ P_{Δ}(C(SU_{q}(2))) | ((id ⊗ π̄) ∘ Δ)(p) = p ⊗ u^{N}}.

Now it follows from Equation (3.15) that M̄_{N} ≅ L_{N}. Applying the same arguments as at the end of Section 1.4, we infer that M̄_{N} ≅ C(S_{q}^{2})^{N+1}E_{N} , with E_{N} being the projection matrix of Equation (1.66). Taking advantage of these isomorphisms of modules, we prove:

Lemma 3.3. Identifying C(S_{q}^{2}) with K(ℓ_{2}(ℕ)) ⊕ ℂ, we obtain the following isomorphisms of left C(S_{q}^{2})-modules:

(3.36) C(S_{q}^{2})^{N+1}E_{N} ≅ C(S_{q}^{2})p_{N}, p_{N} := S^{N}S^{N∗},  N ≥ 0,
(3.37) C(S_{q}^{2})^{|N|+1}E_{N} ≅ C(S_{q}^{2})^{2}p_{N}, p_{N} := ( \begin{matrix}1 & 0 \\ 0 & 1 − S^{|N|}S^{|N|∗} \end{matrix} ) , N < 0.

Proof. We apply (1.43) to construct projections P_{N}, N ∈ ℤ, from the strong connection given in (3.12) and (3.13). For N < 0, we obtain

(3.38) (P_{N})_{11} = (S^{∗|N|} ⊗ u^{N}, u^{N})(S^{|N|} ⊗ u^{|N|}, u^{|N|}) = (1 ⊗ 1, 1),
(3.39) (P_{N})_{12} = (P_{N})_{21}^{∗} = (S^{∗|N|} ⊗ u^{N}, u^{N})((1 − S^{|N|}S^{∗|N|}) ⊗ u^{|N|}, 0) = 0,
(3.40) (P_{N})_{22} = ((1 − S^{|N|}S^{∗|N|}) ⊗ u^{N}, 0)((1 − S^{|N|}S^{∗|N|}) ⊗ u^{|N|}, 0)
= ((1 − S^{|N|}S^{∗|N|}) ⊗ 1, 0).

Analogously, for N ≥ 0, we get

(3.41)  (P_{N})_{11} = (S^{N} ⊗ u^{N}, u^{N})(S^{∗N} ⊗ u^{∗N}, u^{∗N}) = (S^{N}S^{∗N} ⊗ 1, 1).

Finally, applying the isomorphism (3.20) componentwise to P_{N}, N ∈ ℤ, yields the result. □

The projections p_{N} of Lemma 3.3 can also be obtained from the odd-to-even construction in Section 1.2. First, let N < 0. Since L_{N} ≅ T ×_{(u^{−N} σ,1)}ℂ (see (3.18)), we can apply Formula (1.13) by taking E_{1} = T , E_{2} = ℂ, π_{1∗}E_{1} = π_{2∗}E_{2} = C(U(1)), and choosing h in (1.10) to be the isomorphism given by the multiplication with u^{|N|}. As the symbol map σ applied to S is u (see (1.62)), we can lift u^{|N|} and its inverse u^{−|N|} to S^{|N|} and S^{|N|∗} respectively. Inserting c = S^{|N|∗} and d = S^{|N|} into (1.13) gives T ×_{(u^{−N} σ,1)}ℂ ≅ (T ×_{(σ,1)}ℂ)^{2} Q_{N}, where

(3.42)  Q_{N} = ( \begin{matrix}(1, 1) & (0, 0) \\ (0, 0) & (1 − S^{|N|}S^{|N|∗}, 0) \end{matrix} ) .

Finally, applying the isomorphism (3.20) yields the projection in (3.37). Similarly, for N ≥ 0, we insert c = S^{N} and d = S^{N∗} into (1.13). Since S^{N∗}S^{N} = 1, we obtain T ×_{(u^{−N} σ,1)}ℂ ≅ (T ×_{(σ,1)}ℂ)^{2}Q_{N} with

(3.43)  Q_{N} = ( \begin{matrix}(S^{N}S^{N∗}, 1) & (0, 0) \\ (0, 0) & (0, 0) \end{matrix} ) ,

which is equivalent to T ×_{(u^{−N} σ,1)}ℂ ≅ C(S_{q}^{2})S^{N}S^{N∗}.

3.3. Index pairing. Recall that for a C^{∗}-algebra A, a projection p ∈ Mat_{n}(A), and *-representations ρ_{+} and ρ_{−} of A on a Hilbert space ℋ such that [(ρ_{+}, ρ_{−})] ∈ K^{0}(A) (e.g. see [10, Chapter 4]), one has the following: if the operator Tr_{Mat_{n}}(ρ_{+} − ρ_{−})(p) is trace class, then the formula

(3.44)  ⟨[(ρ_{+}, ρ_{−})], [p]⟩ = Tr_{ℋ} (Tr_{Mat_{n}}(ρ_{+} − ρ_{−})(p))

yields a pairing between the K-homology group K^{0}(A) and the K-theory group K_{0}(A).

In this section, we compute the pairing between the K_{0}-classes of the projective C(S_{q}^{2})-modules describing quantum line bundles and two generators of K^{0}(A). By Lemma 3.3, we can take the projections p_{N} as representatives of respective K_{0}-classes. Their simple form makes it very easy to compute the index pairing.

Theorem 3.4. Let M̄_{N} be the associated modules of (3.35), and let [(id, ε)] and [(ε, ε_{0})] denote the generators of K^{0}(C(S_{q}^{2})) given in Section 1.4. Then, for all N ∈ ℤ,

(3.45)  ⟨[(ε, ε_{0})], [M̄_{N}]⟩ = 1,  ⟨[(id, ε)], [M̄_{N}]⟩ = −N.

Proof. Let N ≥ 0. Then p_{N} = S^{N}S^{N∗} = (S^{N}S^{N∗} − 1) + 1, so that ε(p_{N}) = 1 and ε_{0}(p_{N}) = SS^{∗}. Furthermore, since for any N ∈ ℕ\{0}, the image of the projection 1−S^{N}S^{N∗} is span{e_{0}, …, e_{N−1}} ⊂ ℓ_{2}(ℕ), the projection 1−S^{N}S^{N∗} is trace class. Moreover, with the help of Lemma 3.3 and Formula (3.44), it implies that

⟨[(ε, ε_{0})], [M̄_{N}]⟩ = Tr_{ℓ_{2}(ℕ)}(ε − ε_{0})(p_{N}) = Tr_{ℓ_{2}(ℕ)}(1 − SS^{∗}) = 1, (3.46)

(3.47) ⟨[(id, ε)], [M̄_{N}]⟩ = Tr_{ℓ_{2}(ℕ)}(id − ε)(p_{N}) = Tr_{ℓ_{2}(ℕ)}(S^{N}S^{N∗} − 1) = −N.

For N < 0, we have Tr_{Mat_{2}}(p_{N}) = 2 − S^{|N|}S^{|N|∗} = 2 − p_{|N|}. Combining this with the above index pairing for p_{|N|}, the formulas (ε − ε_{0})(2) = 2(1 − SS^{∗}) and (id − ε)(2) = 0, and (3.44), we obtain

(3.48) ⟨[(ε, ε_{0})], [M̄_{N}]⟩ = Tr_{ℓ_{2}(ℕ)}(ε − ε_{0})(2 − p_{|N|}) = Tr_{ℓ_{2}(ℕ)}(1 − SS^{∗})
= 1,
(3.49) ⟨[(id, ε)], [M̄_{N}]⟩ = Tr_{ℓ_{2}(ℕ)}(id − ε)(2 − p_{|N|}) = Tr_{ℓ_{2}(ℕ)}(1 − S^{|N|}S^{|N|∗})
= −N.

This completes the proof. □

The above theorem agrees with the classical situation. Indeed, the pairing ⟨[(ε, ε_{0})], [M̄_{N}]⟩ yields the rank of the line bundles, and ⟨[(id, ε)], [M̄_{N}]⟩ computes the winding number of the map u^{−N} : S^{1} → S^{1}.

4. Examples of piecewise principal coalgebra coactions

We begin by recalling the piecewise structure [15] of a noncommutative join construction proposed in [11]. Then we specify it to SU_{q}(2) to obtain a quantum instanton bundle S_{q}^{7} → S_{q}^{4} [26] as a piecewise trivial principal comodule algebra. A key step is then to replace the Hopf algebra O(SU_{q}(2)) by the quotient of O(SU_{q}(2)) by a coideal right ideal (O(S_{q,s}^{2}) ∩ kerε)O(SU_{q}(2)) provided by a generic Podleś quantum sphere S_{q,s}^{2}, s ≠ 0 [27]. The quotient coalgebra is isomorphic with O(U(1)) [24]. Applying our main theorem, we will prove that the induced right coaction of O(U(1)) is principal.

4.1. Piecewise principal coactions from a noncommutative join. Let H̄ be the C^{∗}-algebra of a compact quantum group and H its Peter-Weyl Hopf algebra [34,36]. We take the algebra of norm continuous functions C([a, b], H̄) from a closed interval [a, b] to the C^{∗}-algebra H̄, and define

P_{1} := \{f ∈ C([0, \frac{1}{2}], H̄) ⊗ H | f(0) ∈ Δ(H)\}, (4.1)

P_{2} := \{f ∈ C([\frac{1}{2}, 1], H̄) ⊗ H | f(1) ∈ ℂ ⊗ H\}. (4.2)

Here we identify elements of C([a, b], H̄)⊗H with functions [a, b] → H̄⊗H. The P_{i}’s are right H-comodule algebras for the coaction Δ_{P_{i}} = id_{C([a_{i},b_{i}],H̄)} ⊗ Δ, where Δ stands for the coproduct of H. The subalgebras of coaction invariants can be identified with

B_{1} := \{f ∈ C([0, \frac{1}{2}], H̄) | f(0) ∈ ℂ\},
B_{2} := \{f ∈ C([\frac{1}{2}, 1], H̄) | f(1) ∈ ℂ\}.

The comodule algebra P_{2} is evidently the same as B_{2} ⊗ H. Unlike P_{2}, the comodule algebra P_{1} does not coincide with B_{1} ⊗ H. However, there is a cleaving map j : H → P_{1} given by j(h)(t) := (t ↦ h_{(1)}) ⊗ h_{(2)}, that is j(h)(t) := Δ(h) for all t ∈ [0, \frac{1}{2}]. Since j is an algebra homomorphism, it identifies the comodule algebra P_{1} with a smash product B_{1}#H.

Now one can define the noncommutative join of H̄ as the pullback right H-comodule algebra

(4.3)  P := {(p, q) ∈ P_{1} ⊕ P_{2} | π_{1}(p) = π_{2}(q)}

given by the evaluation maps

π_{1} := ev_{\frac{1}{2}} ⊗ id : P_{1} → P_{12} := H̄ ⊗ H, π_{2} := ev_{\frac{1}{2}} ⊗ id : P_{2} → P_{12} := H̄ ⊗ H,

where ev_{t} is defined by the evaluation of functions of C([a, b], H̄) at t ∈ [a, b].

Our next goal is to replace H by a quotient coalgebra without losing principality. Using [7, Example 2.29], it is straightforward to verify the following lemma.

Lemma 4.1. Let H be a Hopf algebra with bijective antipode, let Δ_{P} : P → P ⊗ H be a coaction making P a right H-comodule algebra, and let J be a coideal right ideal of H. Then C := H/J is a coalgebra coacting on P via ρ_{R} := (id ⊗ π) ∘ Δ_{P}, where π : H → C the canonical surjection, and the formula

(4.4)  Ψ : C ⊗ P ∋ π̄(h) ⊗ p ⟼ p_{(0)} ⊗ π(hp_{(1)}) ∈ P ⊗ C

defines a bijective entwining making P an entwined module. The inverse of Ψ is given by

(4.5)  Ψ^{−1}(p ⊗ π(h)) = π(hS^{−1}(p_{(1)})) ⊗ p_{(0)},

and defines a left coaction on P via

(4.6)  ρ_{L} : P ∋ p ⟼ Ψ^{−1}(p ⊗ π(1)) = π(S^{−1}(p_{(1)})) ⊗ p_{(0)} ∈ C ⊗ P.

Lemma 4.2. Let P be a principal H-comodule algebra for Δ_{P} : P → P ⊗ H. Also, let J be a coideal right ideal of H defining a coalgebra C := H/J, let ρ_{R} := (id ⊗ π) ∘ Δ_{P}, with π : H → C the canonical surjection, be its right coaction on P, and let i : C → H be a unital (i.e., i(π(1)) = 1) C-bicolinear (for the coactions Δ_{H} := (id ⊗ π) ∘ Δ and _{H}Δ := (π ⊗ id) ∘ Δ) splitting (i.e., π ∘ i = id). Then P is principal for the coaction ρ_{R}.

Proof. Let ℓ : H → P ⊗ P be a strong connection on P. One can easily check that ℓ ∘ i : C → P ⊗ P is a strong connection on P for the right coaction ρ_{R} := (id ⊗ π) ∘ Δ_{P} and the left coaction ρ_{L} := (π ⊗ id) ∘_{P} Δ, where _{P} Δ(p) := S^{−1}(p_{(1)})⊗p_{(0)} gives the left H-coaction on P viewed as a principal H-comodule algebra. Furthermore, it follows from Lemma 4.1 that

(4.7)  Ψ : C ⊗ P ∋ π(h) ⊗ p ⟼ p_{(0)} ⊗ π(hp_{(1)}) ∈ P ⊗ C

is a bijective entwining making P an entwined module. Therefore, since ρ_{R}(1) = 1 ⊗ π(1), ρ_{L}(1) = π(1) ⊗ 1, and ρ_{L}(p) = Ψ^{−1}(p ⊗ π(1)) for all p ∈ P by (4.6), the principality of P for the C-coaction ρ_{R} follows from Lemma 1.1. □

Combining Lemma 4.2 with Theorem 2.2 yields the following result.

Theorem 4.3. Let H̄ be the C^{∗}-algebra of a compact quantum group, H its Peter-Weyl Hopf algebra, J a coideal right ideal of H and π : H → C := H/J the canonical surjection. Also, let

P_{1} := \{f ∈ C([0, \frac{1}{2}], H̄) ⊗ H | (ev_{0} ⊗ id)(f) ∈ Δ(H)\}, (4.8)

P_{2} := \{f ∈ C([\frac{1}{2}, 1], H̄) ⊗ H | (ev_{1} ⊗ id)(f) ∈ ℂ ⊗ H\}, (4.9)

be right and left C-comodules for the right and left coactions

(4.10)  ρ_{R}^{i} := (id ⊗ π) ∘ Δ_{P_{i}}, ρ_{L}^{i} := (π ⊗ id) ∘_{P_{i}}Δ, i = 1, 2,

respectively. Here Δ_{P_{i}} := id ⊗ Δ and _{P_{i}}Δ := (S^{−1} ⊗ id) ∘ flip ∘ Δ_{P_{i}}. Then, if there exists a unital bicolinear splitting i : C → H of π : H → C, the pullback algebra and π(1)-coaugmented C-comodule

(4.11)  P := \{(p_{1}, p_{2}) ∈ P_{1} × P_{2} | (ev_{\frac{1}{2}} ⊗ id)(p_{1}) = (ev_{\frac{1}{2}} ⊗ id)(p_{2})\}

is principal.

First observe that, if both of π_{1} and π_{2} defining the pullback diagram (2.34) are surjective, then (2.40) simplifies to

(4.12)  ℓ = ((α_{L}^{12} + id) ⊗ (α_{R}^{12} + id)) ∘ ℓ_{2}
+ (η_{1} ∘ ε − m_{P_{1}} ∘ (α_{L}^{12} ⊗ α_{R}^{12}) ∘ ℓ_{2}) ∗ ((id ⊗ (id + α_{R}^{21})) ∘ ℓ_{1}).

Indeed, since now α_{R}^{21} is defined on the whole comodule P_{1}, a special connection ℓ̃_{1} constructed in (2.39) can be replaced by any strong connection ℓ_{1} on P_{1}. (Note that specializing (4.12) to comodule algebras coincides with what was obtained in [15].)

Next, specializing to the setting of Theorem 4.3, observe that the formulas

α^{1} : P_{12} ⟶ P_{1}, α^{1}(h̄ ⊗ h) := 2th̄ ⊗ h + (1 − 2t)ε̄(h̄)h_{(1)} ⊗ h_{(2)}, (4.13)

α^{2} : P_{12} ⟶ P_{2}, α^{2}(h̄ ⊗ h) := 2(1 − t)h̄ ⊗ h + (2t − 1)ε̄(h̄) ⊗ h, (4.14)

where ε̄ is any unital linear functional on H̄, define unital C-bicolinear splittings of π_{1} and π_{2} respectively. Hence we can take α_{L}^{12} = α_{R}^{12} = α^{1} ∘ π_{2} and α_{R}^{21} = α^{2} ∘ π_{1}. Combining this with the fact that a cleaving map j defines a strong connection via ℓ := (j^{−1} ⊗ j) ∘ Δ, we obtain very explicit formulae for strong connections on P_{1} and P_{2}:

(4.15)  ℓ_{1} := (j_{1}^{−1} ⊗ j_{1}) ∘ Δ ∘ i, ℓ_{2} := (j_{2}^{−1} ⊗ j_{2}) ∘ Δ ∘ i.

Here j_{1} : H → P_{1}, j_{1}(h) := (t ↦ h_{(1)}) ⊗ h_{(2)}, and j_{2} : H → P_{2}, j_{2}(h) := 1 ⊗ h are cleaving maps for P_{1} and P_{2} respectively.

4.2. Quantum complex projective spaces ℂP_{q,s}^{3}. Finally, we specify H̄ to be C(SU_{q}(2)), H = O(SU_{q}(2)), J = (O(S_{q,s}^{2}) ∩ kerε)O(SU_{q}(2)), and ε̄ : C(SU_{q}(2)) → C to be the counit. Here O(S_{q,s}^{2}) stands for the coordinate algebra of a Podleś quantum sphere S_{q,s}^{2}, s ∈ [0, 1], [8]. (Note that the case s = 0 brings us to the comodule-algebra setting.) The most interesting part of this structure is the unital bicolinear splitting of π : O(SU_{q}(2)) → O(SU_{q}(2))/J given by [8, Proposition 6.3].

All of this defines a family of noncommutative deformations of the U(1)-principal bundle S^{7} → ℂP^{3}. More precisely, we obtain deformations of a U(1)-principal action on

(4.16)  S^{7} := {(z_{1}, z_{2}, z_{3}, z_{4}) ∈ ℂ^{4} | |z_{1}|^{2} + |z_{2}|^{2} + |z_{3}|^{2} + |z_{4}|^{2} = 1}

given by

(4.17)  (z_{1}, z_{2}, z_{3}, z_{4})e^{iφ} = (z_{1}e^{iφ}, z_{2}e^{−iφ}, z_{3}e^{iφ}, z_{4}e^{−iφ}).

However, this is isomorphic to the diagonal action of U(1) on S^{7}, so that the quotient space is again ℂP^{3}. Thus out of Pflaum’s S^{7} we obtain a family of quantum projective spaces ℂP_{q,s}^{3}. A very explicit Mayer-Vietoris-type formula for a strong connection on S_{q}^{7} → ℂP_{q,s}^{3} should allow us to study the K-theoretic aspects of the tautological line bundle over ℂP_{q,s}^{3}, but this is beyond the scope of the present paper.

Acknowledgements

The authors gratefully acknowledge financial support from the following research grants: PIRSES-GA-2008-230836, 1261/7.PR UE/2009/7, N201 1770 33, and 189/6.PR UE/2007/7. It is a pleasure to thank Tomasz Brzeziński for helping us with Section 2, Simon Brain and Ulrich Krähmer for proofreading the manuscript, and Nicola Ciccoli for Poisson-geometric consultations.

References

[1] Baaj, S., G. Skandalis: Unitaires multiplicatifs et dualité pour les produits croisés de C^{∗}-algèbres. Ann. Sci. École Norm. Sup. (4) 26 (1993), 425–488.

[2] Baum, P. F., K. De Commer, P. M. Hajac: Free actions of compact quantum groups on unital C^{∗}-algebras, arXiv:1304.2812.

[3] Boca, F. P.: Ergodic actions of compact matrix pseudogroups on C^{∗}-alge-bras. In Recent advances in operator algebras (Orléans, 1992), Astérisque 232 (1995), 93–109.

[4] Brzeziński, T.: On modules associated to coalgebra Galois extensions. J. Algebra 215 (1999), 290–317.

[5] Brzeziński, T., P. M. Hajac: Coalgebra extensions and algebra coextensions of Galois type. Commun. Algebra 27 (1999), 1347–1367.

[6] Brzeziński, T., P. M. Hajac: The Chern-Galois character. C.R. Acad. Sci. Paris, Ser. I 338 (2004), 113–116.

[7] Brzeziński, T., P. M. Hajac: Galois-type extensions and equivariant projectivity. In Quantum Symmetry in Noncommutative Geometry, P. M. Hajac (ed.), EMS Publishing House, to appear.

[8] Brzeziński, T., S. Majid: Quantum geometry of algebra factorisations and coalgebra bundles. Commun. Math. Phys. 213 (2000), 491–521.

[9] Ciccoli, N.: Quantization of co-isotropic subgroups. Lett. Math. Phys. 42 (1997), 123–138.

[10] Connes, A.: Noncommutative geometry. Academic Press, San Diego, 1994.

[11] D _{}abrowski, L., T. Hadfield, P. M. Hajac: Noncommutative join constructions, arXiv:1402.3024.

[12] D _{}abrowski, L., T. Hadfield, P. M. Hajac, R. Matthes, E. Wagner: Index pairings for pullbacks of C*-algebras. Banach Center Publ. 98 (2012), 67– 84.

[13] Ellwood, D. A.: A new characterisation of principal actions. J. Funct. Anal. 173 (2000), 49–60.

[14] Hajac, P. M.: Bundles over quantum sphere and noncommutative index theorem. K-Theory 21 (2000), 141–150.

[15] Hajac, P. M., U. Krähmer, R. Matthes, B. Zieliński: Piecewise principal comodule algebras. J. Noncommut. Geom. 5 (2011), 591–614.

[16] Hajac, P. M., S. Majid: Projective module description of the q-monopole. Commun. Math. Phys. 206 (1999), 247–264.

[17] Hajac, P. M., A. Rennie, B. Zieliński: The K-theory of Heegaard quantum lens spaces, J. Noncommut. Geom. 7 (2013), 1185–1216.

[18] Klimek, S., A. Lesniewski: A two-parameter quantum deformation of the unit disc. J. Funct. Anal. 115 (1993), 1–23.

[19] Klimyk, K. A., K. Schmüdgen: Quantum Groups and Their Representations. Springer, Berlin, 1997.

[20] Lu, J.-H.: Moment maps at the quantum level. Commun. Math. Phys. 157 (1993), 389–404.

[21] Masuda, T., Y. Nakagami, J. Watanabe: Noncommutative Differential Geometry on the Quantum SU(2), I: An Algebraic Viewpoint. K-Theory 4 (1990), 157–180.

[22] Masuda, T., Y. Nakagami, J. Watanabe: Noncommutative differential geometry on the quantum two sphere of Podleś. I: An Algebraic Viewpoint. K-Theory 5 (1991), 151–175.

[23] Milnor, J.: Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, 1971.

[24] Müller, E. F., H.-J. Schneider: Quantum homogeneous spaces with faithfully flat module structures. Israel J. Math. 111 (1999), 157–190.

[25] Neshveyev, S., L. Tuset: A local index formula for the quantum sphere. Commun. Math. Phys. 254 (2005), 323–341.

[26] Pflaum, M.: Quantum groups on fibre bundles. Commun. Math. Phys. 166 (1994), 279–315.

[27] Podleś, P.: Quantum Spheres. Lett. Math. Phys. 14 (1987), 193–202.

[28] Podleś, P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups. Commun. Math. Phys. 170 (1995), 1–20.

[29] Schmüdgen, K., E. Wagner: Representations of cross product algebras of Podleś’ quantum spheres. J. Lie Theory 17 (2007), 751–790.

[30] Ulbrich, K.-H.: Vollgraduierte Algebren. Abh. Math. Sem. Univ. Hamburg 51 (1981), 136–148.

[31] Vaksman, L. L., Ya. S. Soĭbel’man: An algebra of functions on the quantum group SU(2). (Russian) Funktsional. Anal. i Prilozhen. 22, (1988), 1–14; translation in Funct. Anal. Appl. 22 (1989), 170–181.

[32] Wagner, E.: Fibre product approach to index pairings for the generic Hopf fibration of SU_{q}(2). J. K-Theory 7 (2011), 1–17.

[33] Wegge-Olsen, N. E.: K-theory and C^{∗}-algebras. A friendly approach. Oxford University Press, New York, 1993.

[34] Woronowicz, S. L.: Compact matrix pseudogroups. Commun. Math. Phys. 111 (1987), 613–665.

[35] Woronowicz, S. L.: Twisted SU(2) group. An example of a noncommutative differential calculus. Publ. RIMS, Kyoto Univ. 23 (1987), 117–181.

[36] Woronowicz, S. L.: Compact quantum groups. In Symétries quantiques (Les Houches, 1995), A. Connes, K. Gawedzki, J. Zinn-Justin (eds.), North-Holland 1998, 845–884.

Piotr M. Hajac Instytut Matematyczny Polska Akademia Nauk ul. Śniadeckich 8 Warszawa 00-656 Poland pmh@impan.pl and Katedra Metod Matematycznych Fizyki Uniwersytet Warszawski ul. Hoża 74 Warszawa 00-682 Poland Elmar Wagner Instituto de Física y Matemáticas Universidad Michoacana Ciudad Universitaria Edificio C-3 Morelia C.P. 58040 Mexico elmar@ifm.umich.mx