On the Relationship between Classical and Deformed Hopf Fibrations

The $\theta$-deformed Hopf fibration $\mathbb{S}^3_\theta\to \mathbb{S}^2$ over the commutative $2$-sphere is compared with its classical counterpart. It is shown that there exists a natural isomorphism between the corresponding associated module functors and that the affine spaces of classical and deformed connections are isomorphic. The latter isomorphism is equivariant under an appropriate notion of infinitesimal gauge transformations in these contexts. Gauge transformations and connections on associated modules are studied and are shown to be sensitive to the deformation parameter. A homotopy theoretic explanation for the existence of a close relationship between the classical and deformed Hopf fibrations is proposed.


Introduction and summary
The Hopf fibration S 3 → S 2 is the prime example of a non-trivial principal U (1)-bundle over the 2-sphere. From an algebraic perspective, it can be described as a faithfully-flat Hopf-Galois extension, or equivalently as a principal comodule algebra, consisting of the algebra A = O S 3 of functions on S 3 together with the canonically induced coaction δ : A → A ⊗ H of the Hopf algebra H = O(U (1)) of functions on the structure group U (1). Due to its origin in ordinary geometry, this Hopf-Galois extension is special in the sense that the total space algebra A, the structure Hopf algebra H and consequently the base space algebra B := A coH ∼ = O S 2 are commutative.
As Hopf-Galois theory does not require commutative algebras, it provides a natural framework in which to study noncommutative generalizations of principal bundles. In particular, there exists a 1-parameter family of deformations of the Hopf fibration S 3 → S 2 , where the total space algebra is deformed to the Connes-Landi 3-sphere A θ = O S 3 θ and the structure Hopf algebra H and base space algebra B remain undeformed. It is important to emphasize that even This paper is a contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi. The full collection is available at https://www.emis.de/journals/SIGMA/Landi.html though the base space algebra B and structure Hopf algebra H are commutative, these examples are not commutative principal bundles since the total space A θ is a noncommutative algebra. Hence, one should expect certain, potentially subtle, noncommutative geometry features in these examples.
In this work, we shall study in detail the geometric structures on the deformed Hopf fibrations S 3 θ → S 2 that are relevant for gauge theory and compare those with the corresponding structures on the classical Hopf fibration S 3 → S 2 . An interesting observation is that many, however not all, of these geometric structures coincide for these examples even though S 3 θ → S 2 and S 3 → S 2 are not isomorphic as Hopf-Galois extensions. This follows as A θ = O S 3 θ is a noncommutative algebra whilst A = O S 3 is commutative. In more detail, we prove the following results: 1. The associated module functors for the classical and deformed Hopf fibrations are naturally isomorphic, i.e., the theory of associated modules is insensitive to the deformation parameter θ. In physics terminology, this means that we have the same matter fields on the classical and deformed Hopf fibrations.
2. The affine spaces of connections, with respect to suitable Kähler-type differential calculi, for the classical and deformed Hopf fibrations are isomorphic. It is also shown that this isomorphism is compatible with the action of infinitesimal gauge transformations. In physics terminology, this means that we have the same gauge fields on the classical and deformed Hopf fibrations. 3. In contrast to the previous two points, the action of infinitesimal gauge transformations and connections on associated modules does depend on the deformation parameter. Hence, it is different for the classical and deformed Hopf fibrations. In physics terminology, this means that the coupling of gauge fields to matter fields is sensitive to the deformation parameter.
Even though our direct calculations were able to unravel these striking similarities between the classical and deformed Hopf fibrations, they provide no conceptual reason for why these two non-isomorphic Hopf-Galois extensions should behave similarly in certain respects. As a first step towards a more conceptual explanation, we investigate our examples of Hopf-Galois extensions from the homotopy theoretic perspective proposed by Kassel and Schneider in [29]. We shall show that the classical and deformed Hopf fibrations are homotopy equivalent, however in a slightly different way as the one proposed by Kassel and Schneider. In more detail, while the interval object in [29] is modelled by the polynomial algebra C[y] of the affine line, we require an interval object that is modelled by a larger algebra that also contains exponential functions. This is related to the fact that the deformation parameter θ, which we would like to turn to zero by a homotopy equivalence, enters the deformed Hopf fibration in an exponential form q = e 2πiθ . We give some indications, however not a full proof, that this homotopy equivalence could be the reason why the classical and deformed Hopf fibrations have naturally isomorphic associated module functors. A detailed study of these aspects, and in particular of the interplay between homotopy equivalence and connections, is beyond the scope of this paper. However, we hope to come back to this issue in a future work.
The outline of the remainder of this paper is as follows: In Section 2 we provide a brief review of the theory of Hopf-Galois extensions and principal comodule algebras. We also introduce the examples of interest in this work, namely the classical Hopf fibration S 3 → S 2 and its deformation S 3 θ → S 2 by a suitable family of 2-cocycles. In Section 3 we describe the associated module functors for both the classical and deformed Hopf fibrations and prove that they are naturally isomorphic functors. We would like to emphasize that this natural isomorphism is a specific feature of our particular example and not a consequence of the general theory of 2cocycle deformations, see Remark 3.4. Section 4 starts with a brief review of the theory of Atiyah sequences and connections on principal comodule algebras. We shall discuss both the case of universal differential calculi and also the more general case of concordant differential calculi on principal comodule algebras. We then describe in detail the Atiyah sequence for universal and Kähler-type differential calculi on the classical and deformed Hopf fibrations and construct an isomorphism between the affine spaces of classical and deformed connections. Again, we emphasize that this isomorphism results from the specific example under discussion and not from the theory of 2-cocycle deformations, see Remark 4.13. It is also shown that this isomorphism is compatible with the action of infinitesimal gauge transformations. In Section 5 we study gauge transformations and connections on the associated modules of the classical and deformed Hopf fibrations and show that these structures are sensitive to the deformation parameter. In Section 6 we show that the classical and deformed Hopf fibrations are in a suitable sense homotopy equivalent and give some indications why this should imply the properties of associated modules described in Section 3.
Notation and conventions: Throughout the bulk of the paper, by an algebra we mean an associative and unital algebra over a field k. Unless otherwise stated, k = C, the field of complex numbers. The exception is Section 6 where algebras over commutative rings are also admitted. The multiplication map in an algebra A is denoted by µ A : A ⊗ A → A and the unit map by η A : k → A. The unit element 1 A ∈ A, or simply 1 ∈ A, is obtained by evaluating the unit map on 1 ∈ k. We denote the category of left A-modules by A M and that of right A-modules by M A . The full subcategory of finitely generated projective left A-modules is denoted by A P ⊆ A M .
The comultiplication in a Hopf algebra H is denoted by ∆ (or ∆ H if not sufficiently clear from the context), the counit by (or H ), and the antipode by S (or S H ). We always assume that S is a bijective map. For comultiplications ∆ : H → H ⊗ H, right H-coactions δ : V → V ⊗ H and left H-coactions ρ : V → H ⊗ V , we use the following variant of Sweedler's notation (with suppressed summation) The study of connections on noncommutative principal bundles has been initiated in [16] and developed further by the introduction of strong connections in [24,26]. This framework has been extended beyond Hopf algebras in [17,18] and then formalized in terms of principal coalgebra extensions in [14] and principal comodule algebras in [28].
where δ : A → A⊗H is the right H-coaction and δ S : A → H ⊗A is the associated left H-coaction defined by In the context of noncommutative geometry, principal comodule algebras are interpreted as principal bundles. The algebra A is the algebra of functions on the (noncommutative) total space, the Hopf algebra H is the structure (quantum) group and the subalgebra of coinvariants is the algebra of functions on the (noncommutative) base space. Note that since δ is an algebra homomorphism, B is indeed a subalgebra of A. The existence of a strong connection ensures that A is a Hopf-Galois extension of B, i.e., that the canonical Galois map is bijective. Explicitly, the inverse of the canonical Galois map is given by the composite of The Galois property encodes freeness of the action of the structure (quantum) group. The existence of a strong connection also implies that A is an H-equivariantly projective left B-module, i.e., the restriction of the multiplication map to B ⊗ A has a right H-comodule left B-module splitting. Explicitly, In fact, a principal comodule algebra is the same as an H-equivariantly projective Hopf-Galois extension. The projectivity property gives the notion of a principal comodule algebra full geometric meaning, as it implies that A admits a noncommutative connection in the sense of [23]. This is in perfect concord with Cartan's definition of a principal action of a compact Lie group [20]. We return to these differential geometric aspects of principal comodule algebras in Section 4.
Furthermore, a principal comodule algebra is the same as a faithfully-flat Hopf-Galois extension, i.e., a Hopf-Galois extension A of B such that the tensor product functor (−)⊗ B A : M B → M A both preserves and reflects exact sequences. This gives an algebraic geometry flavour to the notion of a principal comodule algebra, as it leads to the faithfully flat descent property. As observed by H.-J. Schneider, in one of the key results of Hopf-Galois theory [33,Theorem I], if H admits an invariant integral (i.e., H is coseparable) such as the Haar measure on the coordinate algebra of a compact quantum group [34], then surjectivity of the canonical Galois map (2.2) implies its injectivity as well as faithful-flatness of A as a left B-module. An explicit construction of a strong connection, in the more general situation of extensions by coalgebras, is given in [7]. As a consequence, Hopf-Galois extensions given by typical Hopf algebras that feature in noncommutative geometry, and in particular those found in the present text, are automatically principal comodule algebras.
The reader interested in studying further the meaning of principal comodule algebras in classical geometry is encouraged to consult [6].
The structure group is described by the * -Hopf algebra H = O(U (1)) of functions on the algebraic circle group U (1). Concretely, H is the commutative * -algebra generated by t, modulo the * -ideal generated by the circle relation The coproduct, counit and antipode read as We endow A with the structure of a right H-comodule * -algebra by defining the right H-coaction δ : A → A ⊗ H on the generators as The compatibility condition δ • * = ( * ⊗ * ) • δ between the coaction and * -involution gives on which the product, unit and involution are defined by The coproduct, counit and antipode are given by We endow A with the structure of a left K-comodule * -algebra by defining the left K-coaction ρ : A → K ⊗ A on the generators as The compatibility condition ρ • * = ( * ⊗ * ) • ρ between the coaction and * -involution implies Recalling the right H-coaction from (2.3), we observe that (A, ρ, δ) is a (K, H)-bicomodule * -algebra, i.e., the diagram commutes and both δ and ρ are algebra maps. Let us recall from, e.g., [32, Section 2.3] that a 2-cocycle on a Hopf algebra K is a convolutioninvertible linear map σ : K ⊗ K → C that is unital, i.e., σ(a ⊗ 1) = (a) = σ(1 ⊗ a) for all a ∈ K, and that satisfies the cocycle condition for all a, b, c ∈ K. Every 2-cocycle σ on K defines a deformation of K into a new Hopf algebra K σ as well as a deformation of the (K, H)-bicomodule algebra (A, ρ, δ) into a deformed (K σ , H)bicomodule algebra (A σ , ρ, δ), see, e.g., [1,Proposition 2.27]. Our focus will be on the family of 2-cocycles defined by (2.8b) (We note in passing that (2.8) may also be interpreted as a U (1)-valued group 2-cocycle σ θ : on the Pontryagin dual T 2 * = Z 2 .) As K = O T 2 is commutative and cocommutative it follows that K θ = K as Hopf algebras. The deformed (K, H)-bicomodule * -algebra (A θ , ρ, δ) is given as follows. As a (K, H)-bicomodule, we have that (A θ , ρ, δ) = (A, ρ, δ), i.e., the left K-coaction and right H-coaction remain undeformed. The product is deformed to the θ -product defined by Definition 3.1. Let (A, δ) be a principal H-comodule algebra with coinvariant subalgebra B = A coH and let (V, ρ) be a left H-comodule. Since δ is a left B-module homomorphism, the cotensor product The corresponding functor is called the associated module functor for the principal comodule algebra A.
As explained in [13], if V is finite-dimensional, then E A (V ) is a finitely generated projective left B-module. In other words, (3.3) restricts to a functor An idempotent for E A (V ) can be explicitly constructed from a strong connection on A and a basis of V , see [13] and [9].

Modules associated to Hopf fibrations
The aim of this section is to compare the associated module functor E A θ for the deformed Hopf fibration (A θ , δ) (cf. Proposition 2.5) with the functor E A for the classical Hopf fibration (A, δ) (cf. Proposition 2.3). As vector spaces, However, the left B-actions on these spaces are different. In view of (3.2), the left B-module structure of E A (V ) comes from the commutative multiplication in A, restricted to B ⊗ A, while the left B-module structure of E A θ (V ) uses the noncommutative multiplication by θ . We shall prove below that the two functors E A θ and E A are naturally isomorphic. This means that the theory of associated modules for the deformed Hopf fibration (A θ , δ) is equivalent to that for the classical Hopf fibration. Loosely speaking, it could be said that associated modules do not depend on the deformation parameter θ.
In order to construct this natural isomorphism we have to analyze the two different left Bmodule structures in more detail. For this, it is convenient to decompose the underlying (K, H)bicomodule (A θ , ρ, δ) = (A, ρ, δ) into irreducible representations. We define the homogeneous (K, H)-bicomodules for all m = (m 1 , m 2 ) ∈ Z 2 and n ∈ Z. (

3.4)
The θ -product of homogeneous elements a ∈ A ((m+n,−m),n) and a ∈ A ((m +n ,−m ),n ) is Proof . Because A is the (K, H)-bicomodule underlying a finitely presented (K, H)-bicomodule algebra, there exists a decomposition A = m∈Z 2 ,n∈Z A (m,n) , cf. [4,Lemma A.3]. As the generators are homogeneous elements we observe that the non-vanishing components are as in (3.4). The formula for the θ -product on homogeneous elements follows directly from (2.9) and (2.8).
It follows from (3.4) that for the * -subalgebra of H-coinvariants.
Given any left H-comodule (V, ρ), the vector space underlying the associated left B-module admits a decomposition . Moreover, the components (3.6) define a natural isomorphism L : E A =⇒ E A θ between the associated module functor for the classical Hopf fibration S 3 → S 2 and the one for the deformed Hopf fibration (A θ , δ).

Proof . For homogeneous elements
These two terms coincide, hence L V is a left B-module isomorphism. Naturality is a straightforward check.
Remark 3.4. We would like to emphasize that Proposition 3.3 is not a consequence of the general theory of 2-cocycle deformations from [10,11] and [1]. In such a setting, it is convenient to observe that the K-coaction on A induces to associated modules, i.e., E A : H M → K B M can be regarded as a functor to the category of left K-comodule left B-modules. Consequently, the deformed associated module functor E A θ : H M → K θ B θ M can be regarded as a functor to the category of left K θ -comodule left B θ -modules, where K θ and B θ are the 2-cocycle deformations of K and B. (In our specific example of interest, we have that K θ = K and B θ = B (cf. Lemma 2.4), but we shall keep these labels to make the discussion below more transparent.) Using the 2-cocycle deformation functor from [11,Proposition 2.4] or [1, Proposition 2.25], which we shall denote by Σ θ , one obtains a commutative diagram relating the associated module functor E A θ for the deformed Hopf fibration to the associated module functor E A for the classical one. In words, each deformed associated module E A θ (V ) can be determined by applying the deformation functor to the classical associated module E A (V ). For general 2-cocycle deformations that is all one can say.
For our special example given by the deformed Hopf fibration A θ , we have that K θ = K and B θ = B are undeformed, hence the deformed and the undeformed associated module functors have the same target category. Proposition 3.3 proves that these two functors are already 'the same' (in the sense of naturally isomorphic) even if we do not use the deformation functor Σ θ . Hence, our natural isomorphism in Proposition 3.3 is more special and stronger than the results from the general theory of 2-cocycle deformations from [10,11] and [1]. 4 The Atiyah sequence and connections 4.1 Differential geometry of principal comodule algebras Let us start with a brief review of some relevant concepts from noncommutative differential calculi, see, e.g., [30] and [8] for more details.
Definition 4.1. A (first-order) differential calculus on an algebra A is a pair Ω 1 (A), d consisting of an A-bimodule Ω 1 (A) and a linear map d : We say that Every differential calculus is a quotient of the universal calculus Γ 1 (A), d u . Recall that the A-bimodule of universal 1-forms Γ 1 (A) := ker µ A ⊆ A ⊗ A is the kernel of the multiplication map and that the universal differential d u : We find that Γ 1 (A), d u is a connected differential calculus. Given any A-subbimodule N ⊆ Γ 1 (A), the A-bimodule Ω 1 (A) = Γ 1 (A)/N and : : : : defines a differential calculus Ω 1 (A), d . Vice versa, every differential calculus is of this form, see, e.g., [30, Proposition 6.1] for a proof.
The case where H is a Hopf algebra was studied in detail in [35]. Any right ideal Q ⊆ H + of the augmentation ideal H + := ker that is invariant under the right adjoint H-coaction ad : plays the role of the dual of the quantum Lie algebra of H relative to Q. By construction, there exists a map κ Q that fits into the commutative diagram for all j a j ⊗ a j ∈ N . The corresponding differential calculus Ω 1 (A), d then satisfies the property that Ω 1 (A) is a right H-comodule A-bimodule and that d : A → Ω 1 (A) is a right H-comodule morphism. Further, given any ad-invariant right ideal Q ⊆ H + , we may require that Q and N are compatible in the sense that ver(N ) ⊆ A ⊗ Q, where the (universal) vertical lift is defined as the linear map ver : Observe that in this case, the (universal) vertical lift descends to a linear map ver : If moreover there is an equality ver(N ) = A⊗Q, we say that the two differential calculi Ω 1 (A), d and Ω 1 (H), d are concordant. This definition is motivated by the result in [27] that the nonuniversal Atiyah sequence (see (4.3) below) associated to a principal H-comodule algebra (A, δ) is short exact if and only if the differential calculi on A and H are concordant. Loosely speaking, this means that in the case of concordant differential calculi one has an identification between the vertical vector fields on A and the quantum Lie algebra of the structure Hopf algebra, which is in analogy to the fundamental vector field construction from classical differential geometry. It is well-known, see, e.g., [15, Part VII, Proposition 6.6], that a right H-comodule algebra (A, δ) is a Hopf-Galois extension of the coinvariant subalgebra B = A coH if and only if the (universal) Atiyah sequence of right H-comodule left A-modules is short exact. In particular, for every principal H-comodule algebra (A, δ), the sequence (4.2) is short exact. The right H-comodule A-bimodule is called the module of universal horizontal 1-forms. Let us also recall that the right H-coactions on both AΓ 1 (B)A and Γ 1 (A) are induced by the tensor product coaction on A ⊗ A. As well, the right H-coaction on A ⊗ H + is the tensor product coaction with H + endowed with the right adjoint H-coaction ad :

Definition 4.2.
A connection with respect to the universal differential calculus on a principal Every connection s is fully determined by its connection form Associated to a connection s : A ⊗ H + → Γ 1 (A), or equivalently to its connection form ω : Strong connections are in one-toone correspondence with maps : H → A ⊗ A satisfying the conditions (2.1) in Definition 2.1, see, e.g., [24] or [14]. A covariant derivative D of a strong connection is a connection on the left B-module A, in the sense of [23]. Every strong connection defines a connection, for the universal calculus, on the associated left It was shown in [27] that for concordant differential calculi Ω 1 (A), d and Ω 1 (H), d , i.e., ver(N ) = A ⊗ Q, exactness of the universal Atiyah sequence (4.2) implies exactness of the induced sequence of right H-comodule left A-modules. The module of horizontal 1-forms is defined by where Ω 1 (B), d is the differential calculus on B that is determined by the B-subbimodule N B := (ker µ B ) ∩ N ⊆ Γ 1 (B). Equivalently, this differential calculus may by obtained from Ω 1 (A), d as a restriction, i.e., Ω 1 (B) = Bd(B) ⊆ Ω 1 (A), see, e.g., [8,Chapter 5]. The short exact sequence (4.3) will be referred to as the Atiyah sequence for the concordant differential calculi Ω 1 (A), d and Ω 1 (H), d .
Similarly to the case of the universal calculus above, associated to a connection s : A connection is said to be strong provided that D(A) ⊆ Ω 1 (B)A. Every strong connection defines a connection for the differential calculus Ω 1 (B), d on the associated left B-module

The universal Atiyah sequence for Hopf fibrations
Consider the (K, H)-bicomodule algebra (A θ , ρ, δ) from Section 2.3 which describes the deformed Hopf fibration. As the underlying right H-comodule algebra (A θ , δ) is a principal comodule algebra (cf. Proposition 2.5), we obtain from (4.2) the corresponding (universal) Atiyah sequence Let us emphasize that B = A coH θ ⊆ A θ is the algebra of functions on the classical 2-sphere (cf. Lemma 2.4) and that the vertical lift (4.1), in the present case, involves the θ -product of A θ , i.e., ver θ : Note that (4.6) is a short exact sequence of (K, H)-bicomodule left A θ -modules. Setting the deformation parameter θ = 0, we obtain the (universal) Atiyah sequence of the classical Hopf fibration S 3 → S 2 This is a short exact sequence of (K, H)-bicomodule left A-modules. Utilizing the 2-cocycle deformation functor from [11,Proposition 2.4] or [1, Proposition 2.25(ii)], we obtain a short exact sequence of (K, H)-bicomodule left A θ -modules As a sequence of (K, H)-bicomodules, is an isomorphism of short exact sequences of (K, H)-bicomodule left A θ -modules.
Proof . Note that (4.9) relates the deformed and undeformed product via µ A θ = µ A • ϕ θ . As a consequence, it restricts to the middle vertical arrow in (4.10). The 2-cocycle property of σ θ implies that the middle vertical arrow is a (K, H)-bicomodule left A θ -module isomorphism. From (2.12) we find that ϕ θ acts as the identity on Together with the previous result, this implies that the left vertical arrow in (4.10) has the claimed domain and codomain and that it is a (K, H)-bicomodule left A θ -module isomorphism. The left square commutes by construction.
The right vertical arrow in (4.10) is a (K, H)-bicomodule left A θ -module isomorphism because the K-coaction on H + is trivial. We see directly that the right square commutes by for all j a j ⊗ a j ∈ Γ 1 (A θ ), where in the second step we used (2.6).

The Atiyah sequence for Kähler forms
For a commutative algebra A, the product map µ A : A ⊗ A → A is an algebra homomorphism when A ⊗ A is endowed with the tensor algebra structure (a ⊗ a )( a ⊗ a ) := (a a) ⊗ (a a ). This implies that ker µ A ⊆ A ⊗ A is an ideal. Recall that the module of Kähler 1-forms on A is defined as the quotient A-bimodule The Kähler differential d : A → Ω 1 (A) is the composition of the universal differential d u : A → Γ 1 (A) and the quotient map Γ 1 (A) Ω 1 (A). In other words, Ω 1 (A), d is the first-order differential calculus presented by the quotient of the universal calculus Γ 1 (A), d u by the Asubbimodule N := (ker µ A ) 2 ⊆ Γ 1 (A).
For a commutative Hopf algebra H, the Kähler differential calculus on H is bicovariant and it corresponds to the ad-invariant right ideal Q = (H + ) 2 , where H + = ker . Given further a commutative principal H-comodule algebra (A, δ), with coinvariants B = A coH , both Γ 1 (A) = ker µ A ⊆ A ⊗ A and A ⊗ H + ⊆ A ⊗ H are ideals with respect to the tensor algebra structures. It is easily checked that the vertical lift ver : Γ 1 (A) → A ⊗ H + is an algebra homomorphism, hence it maps (ker µ A ) 2 to (A ⊗ H + ) 2 = A ⊗ (H + ) 2 . Because the latter map is surjective, the Kähler differential calculi on A and H are concordant. Hence, there is a corresponding short exact Atiyah sequence (4.3) for Kähler forms in which and It might be worth pointing out that, in general, Ω 1 (B) defined above by the restriction of the Kähler 1-forms on A is not necessarily the module of Kähler 1-forms on B. In view of the above comment and example, it is useful to observe the following lemma.
Lemma 4.6. Let A = k[x 1 , . . . , x n ]/J A be a finitely generated algebra and B ⊆ A the subalgebra generated by X 1 , . . . , X m ∈ k[x 1 , . . . , x n ]. If the set {d x (X 1 ), . . . , d x (X m )} is free in the module underlying the Kähler differential calculus Ω 1 (k[x 1 , . . . , x n ]), d x on the algebra k[x 1 , . . . , x n ], then the Kähler differential calculus is injective and hence it defines an isomorphism where the solid vertical arrows arise from (4.11).
Let us consider now the Atiyah sequence (4.3) for the classical Hopf fibration S 3 → S 2 and the Kähler differential calculi on A = O S 3 and H = O(U (1)), i.e., The vector space h ∨ = H + /(H + ) 2 is the algebraic cotangent space of U (1) at the unit element. Its dual is the vector space underlying the Lie algebra of U (1), i.e., the vector space h := Der (H) of derivations relative to : H → C. Recall that X ∈ Der (H) is a linear map X : 2) satisfy the conditions of Lemma 4.6, the horizontal forms in (4.12) are A similar construction applies to the deformed Hopf fibration described by the (K, H)bicomodule algebra (A θ , ρ, δ) from Section 2.3. The key point is that A θ is braided commutative (see, e.g., [4]) in the sense of for all a, a ∈ A θ , with cotriangular structure R θ : K ⊗ K → C given by The product map µ A θ : A θ ⊗ A θ → A θ is an algebra homomorphism when A θ ⊗ A θ is endowed with the braided tensor algebra structure (a ⊗ a )( a ⊗ a ) := R θ ( a −1 ⊗ a −1 )(a θ a 0 ) ⊗ (a 0 θ a ). Consequently, ker µ A θ ⊆ A θ ⊗ A θ is an ideal and the deformed Kähler forms may be defined as The vertical lift in (4.6) is an algebra homomorphism with respect to the braided tensor algebra structures. Hence, it maps (ker µ A θ ) 2 to (A θ ⊗ H + ) 2 = A θ ⊗ (H + ) 2 . In analogy to (4.12), we obtain the quotient short exact sequence Proposition 4.7. The isomorphism ϕ θ : A θ ⊗ A θ → (A ⊗ A) θ given in (4.9) descends to the (K, H)-bicomodule left A θ -module isomorphism This defines an isomorphism of short exact sequences between (4.16) and (4.14).
Proof . Using the 2-cocycle property (2.7) of σ θ , it can be shown that the map ϕ θ : A θ ⊗ A θ → (A⊗A) θ given in (4.9) is an algebra isomorphism with respect to the following algebra structures: As above, the domain A θ ⊗ A θ is endowed with the braided tensor algebra structure, i.e., The codomain (A ⊗ A) θ is endowed with the 2-cocycle deformation of the usual tensor algebra structure on A ⊗ A, i.e., As a consequence, ϕ θ restricts to an isomorphism ϕ θ : (ker µ A θ ) 2 → (ker µ A ) 2 θ , which implies that it descends to the claimed isomorphism ϕ θ between the quotient modules. The explicit expression for ϕ θ given in (4.17) follows from the computation where in the second step we used (4.9) and (2.9), and in the last step we used that d is a Kcomodule map. The statement about short exact sequences follows by using also Proposition 4.4.
Proof . By (4.13) and invertibility of the 2-cocycle σ θ , it follows that where θ denotes the deformed left and right A θ -module structures on Ω 1 (A) θ . Our claim then follows from the isomorphism ϕ θ given in (4.17), because it maps bijectively between

Connections
The aim of this section is to characterize connections with respect to the Kähler differential calculi for both the classical Hopf fibration S 3 → S 2 and the deformed Hopf fibration. It will be shown that they are equivalent in a suitable sense.
We first consider the classical Hopf fibration (A, δ) from Section 2.2. By Definition 4.3, the set of connections is the set of splittings of (4.12) or equivalently the set of connection forms where Hom H denotes the set of right H-comodule morphisms. Let us recall that h = Der (H) ∼ = C is 1-dimensional and choose a basis X ∈ h, e.g., the linear map Let χ ∈ h ∨ be the dual basis defined by χ, X = 1. Composing ver with the evaluation map −, X : h ∨ → C defines the morphism ver X : Ω 1 (A) −→ A , ω −→ ver X (ω) := ver(ω), X .
As the right adjoint H-coaction on h ∨ is trivial, it follows that The bijection is given explicitly by ω(χ) = ω, for the dual basis vector χ ∈ h ∨ . Analogously, the set of connections for the deformed Hopf fibration (A θ , δ) from Section 2.3 is the set of splittings of (4.16) or equivalently the set of connection forms Notice that every connection on (A, δ) and also every connection on (A θ , δ) is strong because of (4.13) and Corollary 4.8.
Remark 4.10. Similar results for connections on modules were proven in [2,3].
Our next aim is to refine the result of Proposition 4.9 by using more explicit features of the example under investigation. Using [25, Section 16.1], the module of Kähler 1-forms on A can be computed as where the right A-action is defined by sa := as, for all a ∈ A and s ∈ Adz 1 ⊕Adz 2 ⊕Adz * 1 ⊕Adz * 2 . The differential d : A → Ω 1 (A) is specified by mapping each generator z 1 , z 2 , z * 1 , z * 2 of A to the corresponding generator dz 1 , dz 2 , dz * 1 , dz * 2 of (4.21) and the Leibniz rule. Moreover, the vertical lift ver X : Ω 1 (A) → A is given by ver X (ada ) = aa 0 X(a 1 ), (4.22) for all a, a ∈ A.
In other words, the affine map ϕ θ preserves the points ω 0 θ and ω 0 , and its linear part given by the identity map id : Ω 1 (B) → Ω 1 (B).
Remark 4.13. Similarly to Remark 3.4, the result in Proposition 4.12 is stronger than the isomorphisms that can be obtained by using the general theory of 2-cocycle deformations from [10,11] and [1]. Concretely, the property that α is unchanged by the isomorphism ω 0 θ +α → ω 0 +α is a particular feature of the example under investigation.

Gauge transformations
We describe a notion of (infinitesimal) gauge transformations for both the deformed and the classical Hopf fibration, together with their actions on connections. We shall show that the identification of connections from Propositions 4.9 and 4.12 intertwines between the deformed and classical gauge transformations. In other words, the theory of connections and their infinitesimal gauge transformations on the deformed Hopf fibration (A θ , δ) is equivalent to that on the classical Hopf fibration S 3 → S 2 .
Let us note that our notion of gauge transformations will be formalized by braided derivations and hence it makes explicitly use of the braided commutativity of A θ . There also exists a more flexible concept of gauge transformations given by right H-comodule left B-module automorphisms f : A → A satisfying f (1) = 1, see, e.g., [12]. Note that such f are not required to be algebra homomorphisms. However, for commutative principal comodule algebras, this definition does not recover the usual concept of gauge transformations in classical geometry, in contrast to our more special approach by braided derivations. As a last remark, let us note that it would also be possible to describe finite gauge transformations by using the noncommutative mapping spaces from [4]. This is technically more involved and will not be discussed here.
We describe presently the case of infinitesimal gauge transformations of the deformed Hopf fibration (A θ , δ), which includes the classical case (A, δ) by setting θ = 0. Consider the associated left B-module E A θ (h) = A θ 2 H h, where the Lie algebra h = Der (H) is endowed with the adjoint left H-coaction, which is trivial as U (1) is Abelian. Hence, E A θ (h) ∼ = B ⊗ h as left B-modules and, using the basis element X ∈ h from (4.19), any element ζ ∈ E A θ (h) can be written as a −→ a θ ζ := a 0 θ bX(a 1 ). (4.25) Remark 4.15. It is easily checked that the action (4.25) of infinitesimal gauge transformations satisfies for all a, a ∈ A θ and ζ ∈ E A θ (h), i.e., E A θ (h) acts on A θ from the right by braided derivations. In particular, this action preserves the left B-module structure on A θ , i.e., (b θ a) θ ζ = b θ (a θ ζ), for all b ∈ B, a ∈ A θ and ζ ∈ E A θ (h).
Remark 4.16. The infinitesimal gauge transformations from Definition 4.14 are a generalization of the analogous concept in ordinary differential geometry. Given a principal G-bundle P → M over a manifold M , infinitesimal gauge transformations are given by the space of sections of the vector bundle V P/G → P/G ∼ = M of vertical tangent vectors modulo G. Using fundamental vector fields, the latter is isomorphic to the space of sections of the adjoint bundle P × ad g → M , which in our noncommutative example is given by E A θ (h). The action in (4.25) is the evident generalization of the usual action of infinitesimal gauge transformations from ordinary differential geometry.
Remark 4.17. Using that the isomorphism Ω 1 (A θ ) ∼ = Ω 1 (A) θ from Proposition 4.7 is an isomorphism of (K, H)-bicomodule A θ -bimodules, we obtain a canonical isomorphism of differential graded algebras where the right hand side is the 2-cocycle deformation of the differential graded algebra of undeformed Kähler forms, see, e.g., [11,Proposition 3.17]. By a convenient abuse of notation, we shall often suppress this isomorphism in what follows, i.e., we simply identify for all a, a ∈ A θ . Because ver X θ is a left B-module morphism, ι θ ζ preserves the left B-module structures too, i.e., ι θ ζ (b θ λ) = b θ ι θ ζ (λ), for all b ∈ B and λ ∈ Ω 1 (A θ ). This map can be extended to the whole of Ω • (A θ ) as a braided anti-derivation, i.e., for all λ ∈ Ω • (A θ ) and all homogeneous λ ∈ Ω m (A θ ).
Then (4.20). Recalling (4.23), it is found As ω θ is H-coinvariant by hypothesis and δ(d θ z i ) = d θ z i ⊗t i with t 1 = t 2 = t and t 3 = t 4 = t * , it follows that δ(a i ) = a i ⊗t * i . Hence, where the third line uses (4.19) and the last step follows by (4.20).

Associated gauge transformations and connections
Let (V, ρ) ∈ H M be a left H-comodule and consider the associated left B-modules E A (V ) = A2 H V and E A θ (V ) = A θ 2 H V from Section 3. By Proposition 3.3, there exists a natural left B-module isomorphism L V : E A (V ) → E A θ (V ). We shall show that this isomorphism does not intertwine between the actions of gauge transformations and connections on associated modules. Physically speaking, this means that the coupling of gauge and matter fields on the deformed Hopf fibration (A θ , δ) is different to that on the classical Hopf fibration S 3 → S 2 .
Let us start with the associated gauge transformations. Using (4.25), there exists an induced action of E A θ (h) on E A θ (V ) which is given by , with the last equality following from the definition of the cotensor product (3.1). Note that each (−) θ ζ : Hence, it defines a notion of infinitesimal gauge transformations that coincides with the analogous concepts from the more standard projective module approach to noncommutative gauge theory, see, e.g., [30]. Setting θ = 0, we obtain a similar expression without the θ -product, i.e., for all a ⊗ v ∈ E A (V ) and ζ = b ⊗ X ∈ E A (h). Let us also recall from Remark 4.20 that E A θ (h) = E A (h) are identified with the identity map L h = id. Proof . Because the H-coaction on V is by hypothesis non-trivial, there exists a non-zero homogeneous element a ⊗ v ∈ A ((m+n,−m),n) ⊗ V n with n = 0. Consider any infinitesimal gauge −m ),0) a non-zero homogeneous element with m = 0. Using (3.6) and (5.2), we compute Using also (5.1) and (3.5), we compute The phase factors differ because n = 0 and m = 0, which proves our claim. Let us now discuss the covariant derivatives (4.4) and their corresponding associated connections (4.5). We fix any connection form ω = ω 0 + α ∈ Con(A, δ) on the classical Hopf fibration (cf. Lemma 4.11) and determine the corresponding connection form on the deformed Hopf fibration (A θ , δ) via the bijection given in Proposition 4.12, i.e., ω θ = ω 0 θ + α ∈ Con(A θ , δ) with the same α.
Let now (V, ρ) ∈ H M be any left H-comodule and consider the corresponding associated connections ∇ : E A (V ) → Ω 1 (B) ⊗ B E A (V ) and ∇ θ : E A θ (V ) → Ω 1 (B) ⊗ B E A θ (V ). The latter are obtained concretely by composing, respectively, D ⊗ id : A2 H V → Ω 1 hor (A)2 H V and D θ ⊗ id : A θ 2 H V → Ω 1 hor (A θ )2 H V with the inverses of the corresponding left B-module isomorphisms  Proof . The proof is analogous to the one for algebras over C, cf. Sections 2.2 and 2.3 and also [19].
We can now show that this principal comodule algebra implements an O(R)-homotopy equivalence between the classical Hopf fibration A 0 and the deformed Hopf fibration A 1 . Proposition 6.3. There exist isomorphisms of H-comodule algebras ev p * (A) ∼ = A p , for p = 0, 1. Hence, the classical Hopf fibration A 0 is O(R)-homotopy equivalent to the deformed Hopf fibration A 1 for any value of the deformation parameter θ ∈ R.
Proof . Using the map C → O(R), c → c that assigns to a complex number the corresponding constant function on R, we can consider A as an algebra over C. The homomorphism of Calgebras A → ev p * (A) = A ⊗ O(R) C, a → a ⊗ O(R) 1 is surjective as a ⊗ O(R) c = ac ⊗ O(R) 1. The kernel of this map is the two-sided ideal of the C-algebra A generated by (f − f (p))1 ∈ A, for all f ∈ O(R). Hence, ev p * (A) is isomorphic to the quotient C-algebra A (f − f (p))1 : f ∈ O(R) , i.e., the evaluation of all coefficient functions in O(R) at p ∈ R. Recalling (6.1), (6.2) and (6.3), we obtain that ev 0 * (A) is isomorphic to the classical 3-sphere due to Q(0) = 1 and that ev 1 * (A) is isomorphic to the deformed 3-sphere with deformation parameter θ as Q(1) = q = e 2πiθ . These isomorphisms are clearly compatible with the H-coactions.
We finish this section by explaining why we believe that Proposition 6.3 is the conceptual reason for the results in Section 3.2 stating that the associated module functors for the classical and deformed Hopf fibration are naturally isomorphic. The following argument is inspired by [29,Remark 2.4(4)]. Let us first note that the underlying vector space of the Hopf algebra H = O(U (1)) admits a decomposition H = n∈Z C n , where C n = C t n is the (1-dimensional) vector space spanned by the n-th power of the generator t. (Recall that t −1 = t * .) As ∆(t n ) = t n ⊗ t n , each C n is a left H-comodule via the coaction ∆ : C n → H ⊗ C n . It follows that A ∼ = n∈Z A H C n is a direct sum of associated modules E A (C n ) = A H C n with C n ∈ H M fin finitedimensional. By [13], each E A (C n ) is a finitely generated projective left B ⊗ O(R)-module and hence it defines an element [E A (C n )] of the zeroth K-theory group K 0 B ⊗ O(R) . Analogously, there exists a decomposition A p ∼ = n∈Z A p H C n into a direct sum of associated modules E Ap (C n ) = A p H C n and we obtain elements [E Ap (C n )] ∈ K 0 (B), for p = 0, 1. Because of Proposition 6.3, we know that [E Ap (C n )] ∈ K 0 (B) is the image under of the element [E A (C n )] ∈ K 0 B ⊗ O(R) , for p = 0, 1. The same holds true for the modules E A (V ), E A 0 (V ) and E A 1 (V ) associated to any finite-dimensional left H-comodule as any such V decomposes as a finite direct sum of C n 's.
If we could prove that the K 0 -groups are invariant under O(R)-homotopy equivalences, the concrete result of Section 3.2, namely that the associated modules E A 0 (V ) and E A 1 (V ) of the classical and deformed Hopf fibrations are isomorphic, would follow from the more conceptual argument in this section. While results in this direction are available for homotopy equivalences described by the polynomial algebra C[y], see, e.g., [5], we are not aware of generalizations to the setup of O(R)-homotopy equivalences. We expect that addressing this question might provide some insights on appropriate choices of the algebra O(R), which we have left unspecified above, besides assuming that it contains exponential functions and polynomials. Developing a theory of O(R)-homotopy equivalences would be useful and interesting also for other typical examples in noncommutative geometry where the deformation parameter appears in an exponentiated form q = e 2πiθ . We hope to come back to this issue in a future work.