Noncommutative principal bundles through twist deformation

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.


Introduction
Given the relevance of Lie groups and principal bundles, the noncommutative analogues of these structures have been studied since the early days of noncommutative (NC) geometry, first examples being quantum groups and their coset spaces. The algebraic structure underlying NC principal bundles is that of Hopf-Galois extension, the structure group G being replaced by a Hopf algebra H (e.g. that of functions on G, or more in general a neither commutative nor cocommutative Hopf algebra). Presently, in the literature, there are many examples of NC principal bundles, most of them can be understood as deformation quantization of classical principal bundles, see e.g. [9,8]. Our NC geometry study of principal bundles is in this deformation quantization context, and specifically Drinfeld twist (or 2-cocycle) deformations [17,16]. We provide a general theory where both the base space and the fibers are deformed, this allows to recover previously studied examples as particular cases, including a wide class of NC principal bundles on quantum coset spaces, as well as the NC instanton bundle on the θ-sphere S 4 θ [14,23,24].
Drinfeld twist deformation is indeed a powerful method. It applies to any algebra A that carries an action of a group (more in general a coaction of a Hopf algebra K). Given a twist on the group one first deforms the group in a quantum group and then canonically induces via its action a deformation of the algebra. Similarly, modules over algebras are twisted into modules over deformed algebras, in particular into NC vector bundles. This program has been successfully extended in [4] to the differential geometry of NC vector bundles. It has led to a theory of arbitrary (i.e., not necessary equivariant) connections on bimodules and on their tensor products that generalizes the notion of bimodule connection introduced in [28,19]. The construction is categorical, and in particular commutative connections can be canonically quantized to NC connections. As sharpened in [5,6] the categorical setting is that of closed monoidal categories.
In the next level of complexity after algebras and (bi)modules we find (A, H)-relative Hopf modules, i.e. (bi)modules with respect to an algebra A and comodules with respect to an Hopf algebra H (in particular we will be concerned with the example A⊗H, that in the commutative case corresponds to the algebra of functions on the total space of a principal bundle tensored that on the structure group). They are the first objects of our interest because principality of NC bundles is bijectivity of a map (the so-called canonical map) between relative Hopf modules.
In this paper we thus first deform the category of (A, H)-relative Hopf modules by considering a twist associated with H itself (this is the special degenerate case where K = H).
Next we consider the case where there is a different Hopf algebra K that coacts on A and on (A, H)-relative Hopf modules, and study how to twist deform this category using twists on K, and then using twists both on K and on H. Studying this category we are canonically led to twist deform classical principal bundles into NC principal bundles and more generally to prove that NC principal bundles are twisted into new NC principal bundles. The key point is to relate the canonical map between the twisted modules to the initial canonical map, so to deduce bijectivity of the first from bijectivity of the second. This is achieved via a set of isomorphisms that are explicitly constructed and have a categorical interpretation as components of natural isomorphisms.
Considering a twist on the "structure group" H leads to a deformation of the fibers of the principal bundle; this result was also obtained in [27] with a different proof that disregards the natural categorical setting we are advocating. Considering a twist on "the external symmetry Hopf algebra" K (classically associated with a subgroup of the automorphism group of the bundle) leads to a deformation of the base space. Combining twists on H and on K we obtain deformations of both the fibers and the base space.
The categorical context of relative Hopf modules and of their twist deformations we set up is furthermore used in order to prove that principal H-comodule algebras (i.e. Hopf-Galois extensions that admit the construction of associated vector bundles) are deformed into principal H-comodule algebras (Corollary 3.19), here principality is captured by a linear map that is not in general K-equivariant and that has to be properly deformed. This deformation is explicitly given and shown to be related to the natural isomorphism proving the equivalence of the categories of Hopf algebra modules and of twisted Hopf algebra modules as closed monoidal categories. This same categorical context is relevant for planned further investigations in the geometry of NC principal bundles, in particular in the notion of gauge group and of principal connection. Indeed both gauge transformations and connections, as is the case for connections on NC vector bundles [4], will not in general be K-equivariant maps.
Complementing the global description of principal G-bundles as G-manifolds with extra properties, there is the important local description based on trivial principal bundles and on transition functions. We therefore also present a local theory of twists deformations of NC principal bundles, based on a sheaf theoretic approach that complements the initial global approach. The explicit example of the θ-sphere S 4 θ is detailed. Finally we observe that the present study is mainly algebraic so that the examples treated are either in the context of formal deformation quantization, using Fréchet Hopf-Galois extensions on the ring C[[ ]] (cf. the main Example 3.24), or obtained via abelian Drinfeld twists associated with tori actions on algebraic varieties. However these latter NC algebras can be completed to C * -algebras by the general deformation construction of Rieffel [32]; furthermore, also deformations of smooth manifolds based on nonabelian Drinfeld twists can be constructed nonformally [7]. It is then promising to combine these nonformal deformation techniques with the algebraic and categorical ones here developed in order to consider nonformal deformations of principal bundles. This is even more so because, contrary to the well established theory of NC vector bundles (consider for example finite projective C *modules over C * -algebras), a general characterization of NC principal bundles beyond the algebraic level and in terms of NC topology is still missing. In particular we are interested in the wide class of nonformal NC principal bundles that could be obtained via twists based on an external symmetry Hopf algebra K. The present paper is also motivated by this program and can be seen as the first step toward its accomplishment.
The outline of the paper is the following: in §2.1 we recall the basic definitions and results about Hopf-Galois extensions, while in §2.2 we review some results from the theory of deformations of Hopf algebras and comodule algebras by 2-cocycles and extend them to the category of relative Hopf-modules, relevant to our study. The main results of the present paper are contained in §3: in three successive subsections we study deformations of H-Hopf-Galois extensions by 2-cocycles on the structure group H ( §3.1), on an external Hopf algebra K of symmetries ( §3.2), and the combination of these deformations ( §3.3). In §4 we apply the theory developed to deformations of quantum homogeneous spaces ( §4.1) and to encompass sheaves of Hopf-Galois extensions ( §4.2), providing also examples. Appendix A reviews the close relationship between the theory of 2-cocycles and that of Drinfeld twists, and Appendix B clarifies the relationship between one of our deformation maps (the Gmap) and the natural transformation which establishes that twisting may be regarded as an equivalence of closed monoidal categories. Appendix C presents a complementary study of the twisted sheaf describing the instanton bundle on S 4 θ .

Background
Notation: We work in the category of K-modules, for K a fixed commutative ring with unit 1 K . We denote the tensor product over K just by ⊗. Morphisms of K-modules are simply called K-linear maps. In the following, all algebras are over K and assumed to be unital and associative. The product in an algebra A is denoted by m A : A ⊗ A → A, a ⊗ b → ab and the unit map by η A : K → A, with 1 A := η A (1 K ) the unit element. Analogously all coalgebras are assumed to be over K, counital and coassociative. We denote the coproduct and counit of a coalgebra C by ∆ C : C → C ⊗ C and ε C : C → K respectively. We use the standard Sweedler notation for the coproduct: ∆ C (c) = c (1) ⊗ c (2) (sum understood), for all c ∈ C, and for iterations of it ∆ n C = (id ⊗ ∆ C ) • ∆ n−1 C : c → c (1) ⊗ c (2) ⊗ · · · ⊗ c (n+1) , n > 1. We denote by * the convolution product in the dual K-module C ′ := Hom(C, K), ( f * f ′ )(c) := f (c (1) ) f ′ (c (2) ), for all c ∈ C, f, f ′ ∈ C ′ . Finally, for a Hopf algebra H, we denote by S H : H → H its antipode. For all maps mentioned above we will omit the subscripts referring to the co/algebras involved when no risk of confusion can occur. Many of the examples presented will concern co/algebras equipped with an antilinear involution ( * -structure); we will assume all maps therein to be compatible with the * -structure. To indicate an object V in a category C we frequently simply write V ∈ C. Finally, all monoidal categories appearing in this paper will have a trivial associator, hence we can unambiguously write V 1 ⊗ V 2 ⊗ · · · ⊗ V n for the tensor product of n objects.
M H (with A ⊗ A ∈ M H via δ A⊗A as above). Explicitly, a right H-comodule algebra A is an algebra which is a right H-comodule and such that This is equivalent to require the coaction δ A : A → A ⊗ H to be a morphism of unital algebras (where A ⊗ H has the tensor product algebra structure), for all a, a ′ ∈ A , A morphism between two right H-comodule algebras is a morphism in M H which preserves products and units. We shall denote by A H the category of right H-comodule algebras.
If V is a right H-comodule and also a left A-module, where A ∈ A H , it is natural to require the A-module structure (or A-action) ⊲ V : A ⊗ V → V, a ⊗ v → a ⊲ V v to be a morphism in the category M H , i.e. δ V • ⊲ V = (⊲ V ⊗ id) • δ A⊗V (with A ⊗ V ∈ M H via δ A⊗V as above). We thus define the category of relative Hopf modules: Definition 2.1. Let H be a bialgebra and A ∈ A H . An (A, H)-relative Hopf module V is a right H-comodule with a compatible left A-module structure, i.e. the left A-action ⊲ V is a morphism of H-comodules according to the following commutative diagram Explicitly, for all a ∈ A and v ∈ V, (1) v (1) . (2.7) A morphism of (A, H)-relative Hopf modules is a morphism of right H-comodules which is also a morphism of left A-modules. We denote by A M H the category of (A, H)-relative Hopf modules.

Remark 2.2. If V is a left A-module then V ⊗H is a left (A⊗H)-module via the left (A⊗H)-action
The commutativity of the diagram (2.6) is equivalent to Explicitly, for all a ∈ A and v ∈ V, Given a left A-module V and a K-module W, the K-module V ⊗ W is a left A-module with left action defined by ⊲ V⊗W := ⊲ V ⊗ id, i.e. Proof. We prove that the compatibility condition (2.7) between the left A-action ⊲ V⊗W (see (2.11)) and the right H-coaction δ V⊗W (see (2.3)) is satisfied: where in the third passage we have used that (2.7) holds for the left A-action ⊲ V and the right H-coaction δ V .
Remark 2.5. More abstractly, Lemma 2.4 states that A M H is a (right) module category over the monoidal category M H , see e.g. [29]. Indeed, we have a bifunctor (denoted with abuse of notation also by ⊗) ⊗ : Lemma 2.4 and to a morphism ( f : Analogously, given a right A-module V and a K-module W, the K-module W ⊗ V is a right A-module with right action defined by ⊳ W⊗V := id ⊗ ⊳ V . We omit the proof of the corresponding The map is called the canonical map. The extension B ⊆ A is called an H-Hopf-Galois extension provided the canonical map is bijective. The notion of Hopf-Galois extensions in this general context of (not necessarily commutative) algebras appeared in [22]. It generalizes the classical notion of Galois field extensions and with a noncommutative flavor it can be viewed as encoding the data of a principal bundle. We refer the reader to the references [26], [10,Part VII] and examples therein. See also Example 2.13 below.
In the special case when A is commutative (and hence also B ⊆ A is commutative), then A ⊗ B A is an algebra and the canonical map χ is an algebra morphism. In general however A is noncommutative and also B is not contained in the center of A, so A ⊗ B A does not even inherit an algebra structure. As we shall now show, in the general case the canonical map χ is a morphism in the category of relative Hopf modules A M A H .
The tensor product A ⊗ A is an object in A M A H because of Lemma 2.4 (take V = A with left A-action given by the product in A and W = A) and of Lemma 2.6 (take V = A with right A-action given by the product in A and W = A); the compatibility between the left and the right A-actions is immediate: The right H-coaction δ A⊗A : (2.14) (The notation H is in order to distinguish this structure from the Hopf algebra structure  (1) .
This right A-action is easily seen to be a morphism in M H , indeed the diagram Proof. We show that the canonical map is a morphism of right H-comodules, for all a, a ′ ∈ A, It is immediate to see that χ is a morphism of left and right A-modules. 1 Similarly on the tensor product A ⊗ H we also have the H-comodule structure δ A⊗H induced by the right regular coaction (coproduct) of H. Notice that if A is isomorphic to the H-comodule B ⊗ H with right coaction id B ⊗ ∆ (hence in particular if A is cleft, see page 11), then the H-comodules A ⊗ H, δ A⊗H and A ⊗ H, δ A⊗H are isomorphic. The isomorphism is given by Let, as in the previous example, H be the right H-comodule algebra with the coaction given by the coproduct ∆. Then A := B ⊗ H is a right H-comodule algebra (with the usual tensor product algebra and right H-comodule structure). We have A coH ≃ B and χ : is easily seen to be invertible; hence B ⊆ A is an H-Hopf-Galois extension.
Example 2.13. Let G be a Lie group, M a manifold and π : P → M a principal G-bundle over M with right G-action denoted by r P : P × G → P , (p, g) → p g. (All manifolds here are assumed to be finite-dimensional and second countable). We assign to the total space P its space of smooth functions C ∞ (P) and recall that it is a (nuclear) Fréchet space with respect to the usual C ∞ -topology. Even more, the Fréchet space A = C ∞ (P) is a unital Fréchet algebra with (continuous) product m := diag * P : A ⊗ A → A and unit η := t * P : K → A. Here A ⊗ A ≃ C ∞ (P × P) denotes the completed tensor product and the product and unit are defined as the pull-back on functions of the diagonal map diag P : P → P × P and the terminal map t P : P → pt to a point. Similarly, B = C ∞ (M) is a Fréchet algebra and H = C ∞ (G) is a Fréchet Hopf algebra with co-structures and antipode defined by the pull-backs of the Lie group structures on G. (In a Fréchet Hopf algebra also the antipode, counit and coproduct ∆ : H → H ⊗H are continuous maps). The right G-action r P : P × G → P induces the structure of a Fréchet right H-comodule algebra on A and we denote the (continuous) right H-coaction by δ A := r * P : A → A ⊗ H. The H-coinvariant subalgebra is A coH = C ∞ (P/G) and A coH ≃ B = C ∞ (M) is the pull-back of the isomorphism M ≃ P/G of the principal G-bundle P → M. The canonical map in the present case may be obtained by considering the pull-back of the smooth map where P× M P := {(p, q) ∈ P×P | π(p) = π(q)} is the fibered product. This map is an isomorphism of right G-spaces, because the G-action on the fibers of P is free and transitive. It follows that the canonical map 2 χ :  2 The topological tensor product over B is defined as follows: Consider the two parallel continuous linear maps m ⊗ id and id ⊗ m from A ⊗ B ⊗ A to A ⊗ A, which describe the right and respectively left B-action on A.
This condition captures the algebraic aspect of triviality of a principal bundle. We recall that the normal basis property is equivalent to the existence of a convolution invertible map j : H → A (called cleaving map) that is a right H-comodule morphism, i.e.
(In order to prove that j * j = η A • ε use A-linearity of χ −1 , then that θ is an H-comodule map and then recall the definition of χ. In order to prove thatj * j = η A • ε it is convenient to set To conclude this subsection, let us recall the definition of principal comodule algebra which, as it is the case for principal bundles, allows for the construction of associated vector bundles (i.e. associated finitely generated and projective B-modules). Among the equivalent formulations we consider the one here below [15] (see also [10, Part VII, §6.3 and §6.4]) because it will be easily seen to be preserved by twist deformations. In particular if H is a Hopf algebra with bijective antipode over a field, the condition of equivariant projectivity of A is equivalent to that of faithful flatness of A [33]. Moreover, by the Theorem of characterization of faithfully flat extensions [34], if H is cosemisimple and has a bijective antipode, then surjectivity of the canonical map is sufficient to prove the principality of A.

Deformations by 2-cocycles
We review some results from the theory of deformations of Hopf algebras H and their comodule (co)algebras by 2-cocycles γ : H ⊗ H → K. The notion of 2-cocycle is dual to that of Drinfeld twist. In Appendix A we detail this duality for the reader more familiar with the Drinfeld twist notation. We omit some of the proofs of standard results, see e.g. [16] and also [21, §10.2], or, in the dual Drinfeld twist picture, [25, §2.3]. We also extend results from the category of H-comodules to those of relative Hopf (co)modules and bicomodules (cf. Proposition 2.21, Proposition 2.25, and Proposition 2.27) which will be relevant to our construction in §3.

Hopf algebra 2-cocycles
Let H be a Hopf algebra and recall that H ⊗ H is canonically a coalgebra with coproduct (2) and counit ε H⊗H (h ⊗ k) = ε(h)ε(k), for all h, k ∈ H. In particular, we can consider the convolution product of K-linear maps H ⊗ H → K. Definition 2.15. A K-linear map γ : H ⊗ H → K is called a convolution invertible, unital 2-cocycle on H, or simply a 2-cocycle, provided γ is convolution invertible, unital, i.e. γ (h ⊗ 1) = ε(h) = γ (1 ⊗ h), for all h ∈ H, and satisfies the 2-cocycle condition for all g, h, k ∈ H.
The following lemma is easily proven. The stated equalities will be used for computations in the next sections.
Given a 2-cocycle γ, with the help of (iii) and (iv), it is possible to prove that the maps are the convolution inverse of each other.

Twisting of right H-comodules
The deformation of a Hopf algebra H by a 2-cocycle γ : H ⊗ H → K affects also the category then V with the same coaction, but now thought of as a map with values in V ⊗ H γ , is a right H γ -comodule. This is evident from the fact that the comodule condition (cf. (2.1)) only involves the coalgebra structure of H, and H γ coincides with H as a coalgebra. When thinking of V as an object in M H γ we will denote it by V γ and the coaction by The equivalence between the categories M H and M H γ extends to their monoidal structure. We denote by (M H γ , ⊗ γ ) the monoidal category corresponding to the Hopf algebra H γ . Explicitly, for all objects V γ , Proof. The invertibility of ϕ V,W follows immediately from the invertibility of the cocycle γ. The fact that it is a morphism in the category M H γ is easily shown as follows: where the coaction δ (V⊗W) γ is given by The monoidal categories are equivalent (actually they are isomorphic) becauseγ twists back H γ to H and V γ to V so that the monoidal functor (Γ, ϕ) has an inverse (Γ, ϕ), where Γ : M H γ → M H is the inverse of the functor Γ and ϕ V γ ,W γ : , then these additional structures are also deformed by the 2-cocycle γ : H ⊗H → K. Let us illustrate this for the category A H of right H-comodule algebras: Recall that an object A ∈ A H is an object A ∈ M H together with two M H -morphisms, m : A ⊗ A → A and η : K → A, which satisfy the axioms of an algebra product and unit. Using the functor Γ of Theorem 2.19, we can assign to this data the object Γ(A) = A γ ∈ M H γ and the two M H γ -morphisms Γ(m) : Γ(A ⊗ A) → Γ(A) and Γ(η) : Γ(K) → Γ(A). The deformed algebra structure m γ , η γ on A γ ∈ A H γ is now defined by using the components ϕ -,-(cf. (2.26)) of the natural isomorphism ϕ, and the commutative diagrams in the category M H γ . The deformed product m γ is associative due to the 2-cocycle condition of γ, and η γ is the unit for m γ since γ is unital. Explicitly we have that η γ = η and the deformed product reads as This construction provides us with a functor Γ : A H → A H γ ; indeed it can be easily checked that Γ(ψ) := ψ : A γ → A ′ γ is a morphism in A H γ for any A H -morphism ψ : A → A ′ . Using again the convolution inverseγ of γ we can twist back A γ to A. In summary we have obtained By a similar construction one obtains the functors (all denoted by the same symbol) Explicitly, the deformed left A γ -actions are given by while the deformed right A γ -actions read as (1) .
The A γ -module and A γ -bimodule properties follow again from the 2-cocycle condition and unitality of γ. Moreover the bifunctors ⊗ described in the Remarks 2.5, 2.7 and 2.8 on module categories are preserved, i.e., the M H γ -isomorphisms (2.26), that in the context of Theorem 2.19 define the natural isomorphism ϕ : where here Γ is any of the three functors described above, and where ⊗ γ is the tensor product corresponding to the Hopf algebra H γ .
For example we here prove that, for all V ∈ A M H and W ∈ M H , the M H -isomorphism (2) ⊗ v (2) where in the third line we used (2.7). On the other hand, we have These two expressions coincide because of the 2-cocycle condition (cf. (ii) in Lemma 2.16).
In summary, we have obtained Proposition 2.21. Given a 2-cocycle γ : H ⊗ H → K the following functors induce equivalences of categories: We finish this subsection by studying the twisting of the category C H of right H-comodule coalgebras. An object C ∈ C H is an object C ∈ M H together with two M H -morphisms which satisfy the axioms of a coalgebra. Morphisms in C H are H-comodule maps which are also coalgebra maps (i.e., preserve coproducts and counits). Given now a 2-cocycle γ : H ⊗ H → K, we can use the functor Γ : M H → M H γ in order to assign to an object C ∈ C H (with coproduct ∆ C and counit ε C ) the object C γ ∈ C H γ with coproduct ∆ Cγ and counit ε Cγ defined by the commutative diagrams ' ' P P P P P P P P P P P P P P in the category M H γ . Notice that ε Cγ = ε C and that the deformed coproduct explicitly reads as It is easy to check that C γ is an object in C H γ and that the assignment Γ : C H → C H γ is a functor (as before, Γ acts as the identity on morphisms). In summary, we have obtained Example 2.23. The right H-comodule H is a comodule coalgebra with coproduct and counit canonically inherited from the Hopf algebra H, i.e., ∆ H = ∆ and ε H = ε. For ease of notation we will omit the indices and denote by δ H , ∆, ε the comodule coalgebra structure of H. Cocycle deformations of H will be relevant in §3.1.

Twisting of left K-comodules
Of course, similar twist deformation constructions as in §2.2.2 are available for left Hopf algebra comodules rather than right ones. We briefly collect the corresponding formulae as they will be needed in §3. As we later consider also the case where two (in general different) Hopf algebras coact from respectively the left and the right, we denote the Hopf algebra which coacts from the left by K. (2.36) The Sweedler notation for the left K-coaction is We denote by K M the category of left K-comodules; the morphisms in K M are K-linear maps that preserve the left K-coactions, i.e. a K-linear map ψ : Notice that K M is a monoidal category with bifunctor ⊗ : K M × K M → K M defined by equipping the tensor product (of K-modules) V ⊗ W with the tensor product coaction

The tensor product of morphisms is again
and the unit object in K M is K together with the left K-coaction ρ K := η K : K → K ⊗ K ≃ K given by the unit in K.
Let σ : K ⊗ K → K be a 2-cocycle on K (we use the symbol σ in order to distinguish it from 2-cocycles on the Hopf algebra H). Proposition 2.17 provides us with a deformed Hopf algebra K σ . We can further construct a functor Σ : K M → K σ M by assigning to an object Analogously to Theorem 2.19 we have Theorem 2.24. The functor Σ : K M → K σ M induces an equivalence between the monoidal categories The category K A of left K-comodule algebras and the categories K A M, K M A and K A M A of relative Hopf modules are defined analogously to the case where the Hopf algebra coacts from the right. As in Remark 2.5 and in Remark 2.  In particular (ii) and (iii) induce the following equivalences of K Mand K σ M-module categories:

Twisting of (K, H)-bicomodules
Let K and H be two (in general different) Hopf algebras. As our last scenario we consider the situation where we have K-modules V together with a left K-coaction Evaluated on an element v ∈ V this condition reads We denote by K M H the category of (K, H)-bicomodules, where K M H -morphisms are K-linear maps that are both M H -comodule and K M-comodule morphisms. It is a monoidal category; the tensor product of V, W ∈ K M H is the object V ⊗ W ∈ K M H with left K-comodule structure ρ V⊗W given in (2.39) and right H-comodule structure δ V⊗W given in (2.3). Notice that δ V⊗W and ρ V⊗W are compatible in the sense of (2.44), for all v ∈ V and w ∈ W, where in the second and third passage we have used that both V and W are objects in K M H and so their coactions satisfy the compatibility condition (2.45).
Given a 2-cocycle σ : K⊗K → K and a 2-cocycle γ : H⊗H → K, we have by §2.2.3 and §2.2.2 the monoidal functors (Σ, ϕ ℓ ) : We therefore can construct two monoidal functors Proof. As functors, Σ • Γ is equal to Γ • Σ as both functors act as the identity on objects and on morphisms. Thus, we just have to prove that the diagram In the third equality we have used the bicomodule property (2.45) for V and W.
In short the above proposition states that it does not matter if we first deform by σ and then by γ or if we first deform by γ and then by σ.
Let us now consider the category K A H of (K, H)-bicomodule algebras, where objects and leads to the following functors, which induce equivalences of categories. (1) . In the cases (ii), (iii) and (iv), Γ(ϕ ℓ V,W ) • ϕ σ V, σ W are isomorphisms in the corresponding categories respectively); they are the components of the natural iso-

Twisting of Hopf-Galois extensions
Suppose B = A coH ⊆ A is an H-Hopf-Galois extension with total space A, base space B and structure Hopf algebra H (see Definition 2.9). We are interested in studying how the invertibility of the canonical map χ behaves under deformations via 2-cocycles. We are in particular interested in deforming classical principal bundles (cf. Example 2.13) in order to obtain noncommutative principal bundles or quantum principal bundles, i.e., principal comodule algebras (cf. Definition 2.14) obtained via deformation of principal bundles. Let us observe that if we consider a 2-cocycle on the structure Hopf algebra H and use it to twist the data (A, B, H), the deformation of the base space B turns out to be trivial as a direct consequence of the triviality of the right H-coaction on coinvariants. In the language of noncommutative principal bundles this means that by twisting a classical principal bundle with a 2-cocycle on H, we only have the possibility to obtain a noncommutative principal bundle with a classical (i.e. not deformed) base space. In order to obtain a more general theory, which also allows for deformations of the base space, we shall also consider the case of A carrying an external symmetry (described by a second Hopf algebra K) that is compatible with the right H-comodule structure. Indeed, by assuming the total space A to be a (K, H)-bicomodule algebra, we can use a 2-cocycle on K to induce a deformation of A, which in general also deforms the subalgebra B of H-coinvariants.
Notice that it would be also possible to develop this theory by assuming the existence of an action of a Hopf algebra U (dual to K) and use a twist F ∈ U ⊗ U, rather than a 2-cocycle on K, to implement the deformation (see the discussion in Appendix A). Nevertheless, we shall use here the language of coactions as usual in the literature on Hopf-Galois extensions.
Example 3.1. In the setting of Example 2.13, a natural choice for the Hopf algebra U is the universal enveloping algebra of the Lie algebra of G-equivariant vector fields on P, i.e. the Lie algebra of derivations of A = C ∞ (P) which commute with the right G-action. The Hopf algebra U describes the infinitesimal automorphisms of the principal G-bundle π : P → M. A natural choice for the Hopf algebra K would be the Hopf algebra of functions on a finitedimensional Lie subgroup of the group of automorphisms φ : P → P of the bundle.
We therefore consider the following three scenarios: §3. In all these cases we shall show that Hopf-Galois extensions and principal comodule algebras are respectively deformed into Hopf-Galois extensions and principal comodule algebras. In particular principal bundles are deformed into noncommutative principal bundles. Our proof relies on relating the canonical map of the twisted bundle with the canonical map of the original bundle via a commutative diagram in the appropriate category.

Deformation via a 2-cocycle on the structure Hopf algebra H
Let γ : H ⊗ H → K be a 2-cocycle on H which we use to deform H into a new Hopf algebra H γ with the same co-structures and unit, but different product and antipode given in Proposition 2.17. Using the techniques from §2.2.2, we can deform A ∈ A H into A γ ∈ A H γ by introducing the twisted product (2.28). As we have already observed above, the algebra structure of the subalgebra of H-coinvariants B ⊆ A does not change under our present class of 2-cocycle deformations, since the coaction of H on the elements of B is trivial. In other words, the subalgebra of coinvariants B γ = A coH γ γ of A γ is isomorphic (via the identity map) to B = A coH as an algebra (see (2.28)).
We shall relate the twisted canonical map χ γ : Our strategy is as follows: First, we notice that applying the functor Γ : we relate the two morphisms Γ(χ) and χ γ in A γ M A γ H γ via the natural transformation ϕ -,-(cf. (2.26)) and an isomorphism G introduced in Theorem 3.4 below after a few technical lemmas. The role of the isomorphism G is to relate the two twist deformations of H into right H γ -comodule coalgebras: H γ and H γ ; recall Example 2.23. While H γ is the deformation of the H-comodule coalgebra H, in H γ we first deform the Hopf algebra H to H γ and then regard it as an H γ -comodule coalgebra. The isomorphism G is related to the natural isomorphism proving the equivalence of the categories of Hopf algebra modules and twisted Hopf algebra modules as closed categories, cf. Appendix B.
Let us now discuss the construction in detail. By Proposition 2.10 we have that the canonical map χ : Applying the functor Γ : The right H γ -coaction on (A ⊗ B A) γ is given by the right H-coaction on A ⊗ B A, and the right A γ -action reads Analogously, on (A ⊗ H) γ the left A γ -action reads the right H γ -coaction on (A ⊗ H) γ is given by the right H-coaction on A ⊗ H , and the right A γ -action reads (2) .
We proceed with the second step and introduce the isomorphism G mentioned above relating the two deformations of H when thought of as a Hopf algebra or as a right Hcomodule coalgebra H. We first need the following technical lemmas. Recall the definition of u γ and its convolution inverse from (2.22).
On the other hand, the twisted Hopf algebra H γ can be considered as a right H γ -comodule coalgebra, denoted by H γ , via the H γ -adjoint coaction and the coproduct ∆ : (2) .
is an isomorphism of right H γ -comodule coalgebras, with inverse Proof. It is easy to prove by a direct calculation that where in the fourth passage we used u γ (h (6) )ū γ (h (5) ) = ε(h (5) ), and in the fifth h (6) (6) ). Next, we prove that G is a coalgebra morphism, i.e.
for all h ∈ H γ . The last equality follows from comparison with (3.8).
Remark 3.5. If we dualize this picture by considering a dually paired Hopf algebra H ′ (and dual modules) then the right H-adjoint coaction dualizes into the right H ′ -adjoint action, ζ ◭ ξ = S(ξ (1) )ζξ (2) for all ζ, ξ ∈ H ′ . If we further consider a mirror construction by using left adjoint actions rather than right ones, then the analogue of the isomorphism G is the isomorphism D studied in [2] and more in general in [4]. Explicitly, as explained in Appendix B, the isomorphism G is dual to the isomorphism D relative to the Hopf algebra H ′op cop with opposite product and coproduct; this latter is a component of a natural transformation determining the equivalence of the closed monoidal categories of left H ′op cop -modules and left (H ′ γ ) op cop -modules. After these preliminaries we can now relate the twisted canonical map χ γ with the original one χ.

Theorem 3.6. Let H be a Hopf algebra and A an H-comodule algebra. Consider the algebra extension B = A coH ⊆ A and the associated canonical map
the left vertical arrow is well defined. We prove that the diagram (3.12) commutes. We obtain for the composition (id ⊗ γ G) • χ γ the following expression On the other hand, from (2.26) and (2.14) we have , where we have used (3.7). The last two terms simplify, giving the desired identity. From the properties of the canonical map (Proposition 2.10) and from Proposition 2.21 it immediately follows that all arrows in the diagram are H γ , and show that all arrows in the diagram are also morphisms in A γ M A γ H γ .
Notice that since on B = A coH ⊆ A the H-coaction is trivial, it follows that ϕ B,H = id and as K-linear maps θ γ = θ. Recalling from Definition 2.14 the notion of principal H-comodule algebra it is easy to show that deformations by 2-cocycles γ : H ⊗ H → K preserve this structure.

Corollary 3.9. A is a principal H-comodule algebra if and only if
in the category B γ M H γ , from which it follows that s γ is a section of m γ . The reverse implication follows using the convolution inverseγ of γ that twists back A γ to A and (B ⊗ A) γ to B ⊗ A, so that, given the section s γ of m γ , the section of m is Γ

Completion of the proof of Theorem 3.6 (the right
We here complete the proof of Theorem 3.6, i.e., we show that the diagram (3.12) is a diagram in the category . This is the case if all morphisms in (3.12) are We are therefore left to introduce a right A γ -module structure on A γ ⊗ γ H γ and prove that id ⊗ γ G and ϕ A,H are right A γ -modules morphisms (Lemma 3.11). To this aim let us recall that the right A-action on A ⊗ H is given by (a ⊗ h) ⊳ A⊗H c = ac (0) ⊗ hc (1) , for all a, c ∈ A and h ∈ H (cf. (2.16)). Applying Proposition 2.21 (iii), we observe that the right A γ -module structure on (A ⊗ H) γ is given by for all a ∈ A, h ∈ H and c ∈ A γ . Again by (2.16), the right A γ -module structure on By construction, H γ and H γ are isomorphic Hopf algebras via the isomorphism G.
As a simple consequence of this corollary, every right H γ -comodule is also a right H γcomodule; just use the isomorphism G between the Hopf algebras H γ and H γ in order to turn a right H γ -comodule structure into a right H γ -comodule structure. In particular, we have that the right H γ -comodule algebra A γ is a right H γ -comodule algebra with coaction given by G(a (1) ) .
Using this right H γ -comodule structure on A γ we canonically define the right A γ -module structure on A γ ⊗ γ H γ by (cf. (2.16)), Lemma 3.11. The vertical arrows id⊗ γ G and ϕ A,H in diagram (3.12) are right A γ -module morphisms.

Deformation via a 2-cocycle based on an external symmetry K
In this section we first define the notion of external symmetry (with Hopf algebra K) of an H-Hopf-Galois extension, and study the corresponding category. Then we deform this extension with a 2-cocycle on K. Consider a Lie group L acting via diffeomorphisms on both the total manifold and the base manifold of a bundle P → M, these actions being compatible with the bundle projection (hence L acts via automorphisms of P → M). We say that L is an external symmetry of P → M. Considering algebras rather than manifolds (cf. Example 2.13), we term a Hopf algebra K an external symmetry of the extension B ⊂ A, if A is a (left) K-comodule algebra with B a K-subcomodule algebra. If we consider principal G-bundles P → M then we also require G-equivariance of the L-action on the total manifold leading to algebras A that are (K, H)-bicomodules algebras, whose category is denoted K A H and defined in §2.2.4 before Proposition 2.27.
We are thus led to term a Hopf algebra K an external symmetry of an H-Hopf Galois The requirement that B = A coH is a K-subcomodule of A holds automatically true in particular if K is a flat module. We recall that K is a flat K-module if any short exact sequence In particular all modules are flat if K is a field or the ring of formal power series with coefficients in a field. Proposition 3.12. Let H and K be Hopf algebras, let K be flat as K-module, and let A ∈ K A H ; then B = A coH is a K-subcomodule algebra.
If K is flat we have the associated short exact sequence Now the compatibility between the H-and K-coactions δ A and ρ A (cf. (2.44)) implies that, for where the last equality is due to the exact sequence. This proves that B = A coH is a K-subcomodule of A, and hence a K-subcomodule algebra. (1)

and left K-coaction by ρ
and of the corresponding ones for A ⊗ A (the proof that A ⊗ H and A ⊗ A ∈ K A M and that A⊗A ∈ K M A can be also seen to follow from the property that K A M and K M A are respectively right and left module categories over the monoidal category K M).
Furthermore, since B is a K-subcomodule then it is easy to see that the K-comodule structure of A ⊗ A is induced on the quotient A ⊗ B A, that is therefore an object in the relative Hopf module category K A M A H . We have thus proven the following

Explicitly the K-coactions on A ⊗ B A and on
The canonical map preserves this additional structure: Proposition 3.14. If A ∈ K A H and B = A coH is a K-subcomodule, then the canonical map χ : Proof. Recalling from Proposition 2.10 that the canonical map χ is a morphisms in A M A H , we just have to show that it preserves the left K-coactions, i.e. ρ A⊗H • χ = (id ⊗ χ) • ρ A⊗ B A . This indeed holds true: where in the third and fourth equality we have used the equivariance condition (2.45).
Let us now consider a 2-cocycle σ : K ⊗ K → K on K. We deform according to Proposition 2.17 the Hopf algebra K into the Hopf algebra K σ . Using the machinery of §2.2.3 and §2.2.4 we can also deform the (K, H)-bicomodule algebra A into the (K σ , H)-bicomodule algebra

coH is a K-comodule then it is a (K, H)-bicomodule algebra and is as well deformed into the (K σ , H)-bicomodule algebra
The following theorem relates the twisted canonical map σ χ with the original canonical map χ.

Theorem 3.15. Let A ∈ K A H and B = A coH a K-subcomodule. Given a 2-cocycle
Proof. First we notice that the left vertical arrow is the induction to the quotient of the isomorphism ϕ ℓ A,A : σ A σ ⊗ σ A −→ σ (A ⊗ A) defined in (2.40); it is well defined thanks to the cocycle condition (2.21) for σ. Next let us observe that ϕ ℓ A,H is the identity; indeed, since H is equipped with the trivial left K-coaction h → 1 K ⊗ h and σ is unital, we have for all a, a ′ ∈ σ A.

Corollary 3.16. B ⊆ A is an H-Hopf-Galois extension if and only if σ B ⊆ σ A is an H-Hopf-Galois extension.
Proof. The statement follows from the invertibility of the morphisms ϕ A,H and ϕ A,A in diagram (3.24).
In order to prove that twist deformations of principal H-comodule algebras are principal H-comodule algebras we need the following where in the second equality we used left B-linearity of s, and in the fourth Lemma 3.2.
Finally if s is a K-comodule map then we immediately see that S(s) = Σ(s).
Remark 3.18. Consider a Hopf algebra V dually paired to K. If V, W are left K-comodules, then they are right V-modules and left V op -modules; let ⊲ V and ⊲ W be the corresponding V opactions, cf. Appendix A.2 (recall that V op is the Hopf algebra with opposite product, inverse antipode and same coproduct and counit as V). The set of K-linear maps This map is the same as that in Proposition 3.17 (with B and H trivial). Explicitly where G = g α ⊗ g α , and the product, coproduct and antipode are those of V. We refer to [4] for further properties of this left deformation map.
The categorical viewpoint is also instructive. We first define the functor hom :

Corollary 3.19. A is a principal H-comodule algebra if and only if σ A is a principal H-comodule algebra.
Proof. The proof is similar to that of Corollary 3.9 with the caveat that since s is not a Kcomodule map we have to consider its deformation via the map S. Hence we consider the commutative diagram σ A S(s) ( ( P P P P P P P P P P P P P P σ s where in the third equality we used that m is a K-comodule map, and in the fourth that   (2)) be the Hopf algebra of coordinate functions on SU(2) realized as the * -algebra generated by commuting elements {w i , w * i , i = 1, 2} with w * i w i = 1 and standard Hopf algebra structure induced from the group structure of SU (2).
The classical principal action of SU(2) on S 7 can be described at the algebraic level by the data of the following right coaction of O(SU(2)) on O(S 7 ): (and their * -conjugated α * , β * , with x * = x), form a set of generators for B and from the 7-sphere relation z * i z i = 1 it follows that they satisfy We now apply the theory developed above and deform this extension of commutative algebras by using a symmetry of the classical Hopf bundle. Let K := O(T 2 ) be the commutative * -Hopf algebra of functions on the 2-torus T 2 with generators t j , t * j = t −1 j , j = 1, 2 and co-structures Let σ be the exponential 2-cocycle on K which is determined by its value on the generators: and extended to the whole algebra by requiring σ (ab ⊗ c) = σ a ⊗ c (1) σ b ⊗ c (2) and σ (a ⊗ bc) = σ a (1) ⊗ c σ a (2) ⊗ b , for all a, b, c ∈ O(T n ). There is a left coaction of O(T 2 ) on the algebra O(S 7 ): it is given on the generators as where (τ i ) := (t 1 , t * 1 , t 2 , t * 2 ), and it is extended to the whole of O(S 7 ) as a * -algebra homomorphism. It is easy to prove that the two coactions δ O(S 7 ) and ρ O(S 7 ) satisfy the compatibility condition (2.44), hence they structure O(S 7 ) as a (O(T 2 ), O(SU(2)))-bicomodule algebra; furthermore O(S 4 ) is a (O(T 2 ), O(SU(2)))-subbicomodule algebra as can be easily checked on its generators, or indirectly inferred from Proposition 3.12 (since vector spaces are flat). Explicitly the O(T 2 )-coaction reads  (2))-comodule algebra.
The noncommutative bundle so obtained is the quantum Hopf bundle on the Connes-Landi sphere O(S 4 θ ) that was originally constructed in [23], and further studied in the context of 2-cocycles deformation in [8]. The principality of the algebra inclusion O(S 4 θ ) ⊆ O(S 7 θ ) was first proven in [23, §5] by explicit construction of the inverse of the canonical map. Proposition 3.22 follows instead as a straightforward result of the general theory developed in the present section (out of the principality of the underlying classical bundle).

Combination of deformations
We now consider the combination of the previous two constructions. This leads to Hopf-Galois extensions in which the structure Hopf algebra, total space and base space are all deformed.
As before, we let H and K be Hopf algebras and A ∈ K A H a (K, H)-bicomodule algebra, with B = A coH a K-subcomodule. Let σ : K ⊗ K → K and γ : H ⊗ H → K be 2-cocycles and denote by K σ and H γ the twisted Hopf algebras and by σ A γ :

and have the same external square diagram. Moreover: (i) B ⊆ A is an H-Hopf-Galois extension if and only if σ B ⊆ σ A γ is an H γ -Hopf-Galois extension. (ii) A is a principal H-comodule algebra if and only if σ A γ is a principal H γ -comodule algebra.
Proof. Commutativity follows from commutativity of the internal diagrams, statements (i) and (ii) also immediately follow combining the analogue statements for each of the 2-cocycles γ and σ. The equality of the external square diagrams follows from diagram (2.48) of Proposition 2.26 applied to the left vertical arrows, and from ϕ ℓ   Notice that for G a Lie group we have a (in general degenerate) pairing between the universal enveloping algebra U(g) of its Lie algebra g and C ∞ (G); it is determined by evaluating at the unit element e ∈ G left invariant vector fields on functions. Explicitly, · , · : U(g) × C ∞ (G) → C is defined by extending for all h ∈ C ∞ (G) and v ∈ g, to all U(g) via linearity and requiring ξξ ′ , h = ξ, h (1) ξ ′ , h (2) for all ξ, ξ ′ ∈ U(g). The K-linear maps v, · : C ∞ (G) → K are continuous and since the coproduct ∆ : is continuous also the K-linear maps ξ, · : C ∞ (G) → K are continuous for all ξ ∈ U(g). (These maps are actually the Lie derivative along ξ, L ξ : C ∞ (G) → C ∞ (G), composed with the counit in C ∞ (G); where L v is the Lie derivative along the left invariant vector field defined by v ∈ g, and L is extended to all U(g) by L ξξ ′ = L ξ • L ξ ′ ). Because of this pairing we can assign to a twist F = f α ⊗ f α ∈ U(g) [

Applications
We apply the theory so far developed first to the study of deformations of quantum homogeneous spaces in §4.1, including the explicit example of the even θ-spheres S 2n θ in §4.1.1, and then to the study of deformations of sheaves of Hopf Galois extensions in §4.2, providing the example of the Hopf bundle over S 4 θ as a twisted sheaf in §4.2.1.

Twisting quantum homogeneous spaces associated with quantum subgroups
The theory of twists, in particular the combination of deformations developed in §3.3, can be used to study deformations of bundles over quantum homogeneous spaces arising from Hopf algebra projections. This is the subject of the present subsection. Recall that given a Hopf algebra G, a quantum subgroup of G is a Hopf algebra H together with a surjective bialgebra (and thus Hopf algebra) homomorphism π : G → H. The restriction via π of the coproduct of G induces on G the structure of a right H-comodule algebra. The subalgebra B := G coH ⊆ G of coinvariants is called a quantum homogeneous G-space. When the associated canonical map Given a quantum principal bundle B = G coH ⊆ G over a quantum homogeneous space B and a 2-cocycle γ on H we can consider two different constructions: -On the one hand we can lift the 2-cocycle γ on H to a 2-cocycleγ on G (see Lemma 4.1 below) and thus apply the theory of 2-cocycle deformations for Hopf algebras ( §2.2.1) to deform both G and H into new Hopf algebras G γ and H γ . It turns out that the condition for H to be a quantum subgroup of G is preserved under deformation, i.e. H γ is a quantum subgroup of G γ , and thus there is an associated twisted quantum homogeneous space B γ .
-On the other hand, we can direct the attention to the algebra inclusion B = G coH ⊆ G as a Hopf-Galois extension, and twist it. In this case, we forget the Hopf algebra structure of G and use γ to deform G just as an object in A H , as in §3.1. Denote by G γ the resulting comodule algebra.
These two deformations G γ and G γ of G do not coincide. In particular, G γ is not in general a Hopf algebra and thus the base space of the twisted bundle is no longer a quantum homogeneous space of the total space. Nevertheless the second construction can be reconciled with the first one by applying a further twist deformation and thus considering a combination of deformations as in §3.3. As a corollary of this second approach we obtain that B γ ⊆ G γ is a Hopf-Galois extension. Indeed we show below that, given a 2-cocycle γ on H, quantum principal bundles B = G coH ⊆ G over quantum homogeneous spaces B deform into new quantum principal bundles over new quantum homogeneous spaces.
We proceed by first showing that given a 2-cocycle γ on H we can twist both the Hopf algebras H and G is such a way to still have a quantum homogeneous space.
is a 2-cocycle on G.
We can deform the algebra product and antipode in the Hopf algebra G, and H, by using the 2-cocycles γ and γ respectively. By Proposition 2.17 we obtain two new Hopf algebras which we denote by G γ and H γ . Their algebra products are given respectively by for all g, g ′ ∈ G γ , and for all h, h ′ ∈ H γ . The map π : G → H remains a Hopf algebra homomorphism with respect to the deformed Hopf algebra structures on G γ and H γ : The map is a surjective bialgebra homomorphism.
It follows that the twisting procedure deforms the quantum homogeneous space B = G coH into another quantum homogeneous space B γ = G coH γ γ , which is isomorphic to B only as a K-module but not in general as an algebra.
On the other hand, given a 2-cocycle γ on H, we can deform H into the Hopf algebra H γ as above, but consider G simply as a right H-comodule algebra with coaction given in (4.1) and twist its algebra product accordingly to (2.28). In this way we get an H γ -comodule algebra, G γ , with product for all g, g ′ ∈ G γ . By Corollary 3.7, the extension B = G coH γ γ ⊆ G γ is an H γ -Hopf-Galois extension if and only if the original extension B ⊆ G was H-Hopf-Galois. However, as already remarked above, this twisted bundle has a total space which is just an algebra and the condition for H γ to be a quantum subgroup is lost, and so that of B to be a quantum homogeneous space. To resolve this problem let us consider K = H as an external Hopf algebra of symmetries coacting from the left on G. The Hopf algebra G is also a left Hcomodule algebra via (1) ) ⊗ g (2) . (4.8) Clearly, the left and right H-coactions ρ G and δ G satisfy the compatibility condition (2.44), hence they structure G as an (H, H)-bicomodule. Assume B is a subcomodule for the left H-coaction. We can therefore twist the product in G accordingly to Proposition 2.27 (i) (with the special choice σ = γ : H ⊗ H → K) in order to get an (H γ , H γ )-bicomodule algebra γ G γ with product for all g, g ′ ∈ γ G γ . Theorem 3.23 then implies that γ B := γ G coH γ γ ⊆ γ G γ is an H γ -Hopf-Galois extension if and only if B = G coH ⊆ G is a H-Hopf-Galois extension. Proposition 4.3. The algebra γ G γ is isomorphic to the algebra underlying the Hopf algebra G γ and hence inherits from it a Hopf algebra structure. The subalgebra of coinvariants γ B is isomorphic to the quantum homogeneous space B γ .
Proof. By comparing (4.9) with (4.4) we have that the algebras γ G γ and G γ are isomorphic via the identity map. 2) , because B = B γ as K-modules and hence b, b ′ are right H-coinvariant. Hence the result As a direct consequence of Theorem 3.23 we then obtain that quantum principal bundles over quantum homogeneous spaces deform into quantum principal bundles over quantum homogeneous spaces:

The quantum homogeneous spaces S 2n θ and their associated quantum principal bundles
The θ-spheres S 2n θ were introduced in [14] as noncommutative manifolds with the property that the Hochschild dimension equals the commutative dimension. They were shown to be homogeneous spaces of twisted deformations of SO(2n + 1, R) in [35]. Their geometry was further studied in [13], see also [1]. We here revisit their explicit construction and as a corollary of the previous section conclude that the Hopf algebra of noncommutative coordinate functions O(SO θ (2n + 1, R)) is a quantum principal bundle over the quantum homogeneous space O(S 2n θ ) of noncommutative coordinate functions on the sphere. We then immediately conclude that the Hopf-Galois extension O(S 2n θ ) ⊂ O(SO θ (2n + 1, R)) is a principal comodule algebra.
We begin by introducing the algebra of coordinate functions on SO(2n, R), on SO(2n+1, R) and on their quotient S 2n . Let O(M(2n, R)), n ∈ N be the commutative * -algebra over C with generators a ij , b ij , a * ij = * (a ij ), b * ij = * (b ij ), i, j = 1, . . . n. It is a bialgebra with coproduct and counit given in matrix notation as where .
⊗ denotes the combination of tensor product and matrix multiplication, ½ is the identity matrix and capital indices I, J run from 1 to 2n. The Hopf algebra of coordinate functions on SO(2n, R) is the quotient O(SO(2n, R)) = O(M(2n, R))/I Q where I Q is the bialgebra ideal defined by (4.11) In matrix notation the * -structure in O(M(2n, R)) is given by * (M) = QMQ so that I Q is easily seen to be a * -ideal. The * -bialgebra O (SO(2n, R)) is a * -Hopf algebra with antipode Similarly, for the odd case let O(M(2n + 1, R)), n ∈ N, be the commutative * -bialgebra with generators . . n, and x = * (x). The coproduct and counit are given as The algebra of coordinate functions on SO(2n + 1, R) is the quotient O(SO(2n + 1, R)) = O(M(2n + 1, R))/J Q where J Q is the bialgebra * -ideal The * -structure can be written in terms of Q as * (N) = QNQ −1 . The * -bialgebra O (SO(2n+1, R)) is a * -Hopf algebra with antipode S(N) = QN t Q = N † . The (commutative) Hopf algebra O (SO(2n, R)) is a quantum subgroup of O(SO(2n + 1, R)) with surjective Hopf algebra morphism (4.14) Hence there is a natural right coaction of O(SO(2n, R)) on O(SO(2n + 1, R)), given by (cf. (4.1)) Next we consider a 2-cocycle γ on the quantum subgroup O(SO(2n, R)), or rather on its maximal torus T n , and use it to deform the quantum homogeneous space O(S 2n ) and the principal fibration on it. Let O(T n ) be the commutative * -algebra of functions on the n-torus with generators t j , t j * = * (t j ) satisfying t j t j * = 1 = t j * t j (no sum on j) for j = 1, . . . n. It is a Hopf algebra with We consider the exponential 2-cocycle γ on O(T n ) defined on the generators t i by γ t j ⊗ t k = exp iπθ jk ; θ jk = −θ kj ∈ R (4. 15) and extended to the whole algebra by requiring γ (ab ⊗ c) = γ a ⊗ c (1) γ b ⊗ c (2) and γ (a ⊗ bc) = γ a (1) ⊗ c γ a (2) and hence the 2-cocycle γ lifts by pullback to a 2-cocycle on O(SO(2n, R)) (see Lemma 4.1), that we still denote by γ. Now to deform with γ the Hopf algebra O(SO(2n, R)) into the noncommutative Hopf algebra O (SO θ (2n, R)). The twisted algebra product is given by (cf. Since γ (T I ⊗ T K ) = (γ (T K ⊗ T I )) −1 and similarly forγ, it follows that the generators in O(SO θ (2n, R)) satisfy the commutation relations Explicitly, setting λ IJ := (γ(T I ⊗T J )) 2 , so that λ ij = exp(2iπθ ij ), and sinceγ(T J ⊗T L ) = γ(T L ⊗T J ), they read together with their * -conjugated. It is also not difficult to show the equivalence of the quotient conditions (4.11) with the relations where the quantum determinant is defined by show that for each permutation σ we have the equality Next expand the twisted products in (4.19) in terms of the commutative products using γ (ab ⊗ c) = γ a ⊗ c (1) γ b ⊗ c (2) as well as the equivalent relationγ(ab ⊗ c) =γ(a ⊗ c (2) )γ(b ⊗ c (1) ) for all a, b, c ∈ O(SO(2n, R)) and notice that (4.19) becomes the usual determinant of M.
From (4.17) and (4.18) we see that the twisted Hopf algebra O(SO θ (2n, R)) can be described algebraically as the algebra over C freely generated by the matrix entries M IJ modulo the ideal implementing the relations (4.18). The twisted Hopf algebras O(SO θ (n, R)) were studied in [3] (see also [31]) and in [13] as symmetries of θ-planes and spheres.
We can lift the 2-cocycle from the quantum subgroup O(SO(2n, R)) to the Hopf algebra O (SO(2n + 1, R)) by using the projection π in (4.14) (or equivalently we can consider the torus T n embedded in SO(2n + 1)). The resulting Hopf algebra is denoted by O(SO θ (2n + 1, R)). It is the Hopf algebra over C freely generated by the matrix entries N IJ modulo the relations where now T := diag(t 1 , . . . t n , t * 1 , . . . t * n , 1), and where det θ (N) is defined as in (4.19), just consider the permutation group P 2n+1 . As from Lemma 4.2, the quantum homogeneous space B = O(S 2n ) is deformed into the quantum homogeneous space of coinvariants of O(SO θ (2n + 1, R)) under the O(SO θ (2n, R))coaction. This is the subalgebra B θ =: O(S 2n θ ) ⊂ O(SO θ (2n + 1, R)) which is generated by the elements u i , u * i and x entering the last column of the matrix N. Their commutation relations follow from (4.21) while the orthogonality conditions (4.22) imply the sphere relation n i=1 u * i · γ u i + x 2 = 1. By Corollary 4.4 we conclude

Twisting sheaves of Hopf-Galois extensions
In classical geometry a principal bundle over a topological space X can be given in terms of the local data of trivial product bundles over the open sets of a covering of X and a set of transition functions which specify how to glue the local trivial pieces into a (possibly non trivial) global one. A local-type approach to noncommutative principal bundles was given in [30] by using sheaf theoretical methods. A quantum principle bundle consists in the data of two sheaves of C-algebras over a (classical) topological space together with a quantum group, playing the role of the structure group, and a family of sheaf morphisms, satisfying some suitable conditions, as local trivializations. The two sheaves of algebras have to be regarded as the quantum analogues of the sheaves of functions over the base and total space of a classical fibration. The basic idea behind is that of considering a quantum space as a 'quantum ringed space' (M, O M ), i.e. a topological space M whose structure sheaf O M is a sheaf of (not necessarily commutative) algebras rather than of commutative rings.
A refinement of this sheaf theoretical approach to noncommutative bundles was proposed in [12] in terms of sheaves of Hopf-Galois extensions. For simplicity let us here assume all algebras are over a field.
The sheaf A and its subsheaf A coH : U → A (U) coH play the role of noncommutative analogues of the sheaf of functions on the total space, respectively base space, of the bundle. Notice that condition (ii) is equivalent to requiring just the algebra A (X) to be an H-Hopf-Galois extension, indeed it was observed in [12] that the property of being Hopf-Galois restricts locally: if on an open set U, the algebra extension A (U) coH ⊆ A (U) is Hopf-Galois, then A (W) coH ⊆ A (W) is a Hopf-Galois extension for any W ⊆ U. (This is the algebraic counterpart of the well-known classical fact that the restriction of a principal action is still principal).
The notions of quantum principal bundle introduced in [30] and that of locally cleft sheaf of Hopf-Galois extensions are closely related: every locally cleft sheaf of Hopf-Galois extensions is a quantum principal bundle in the sense of [30]. On the other hand, a sufficient condition for a quantum principal bundle in the sense of [30] to be a sheaf of Hopf-Galois extensions (in fact, locally cleft) is that the restriction maps are surjective (see [12, §4]).
Since a (locally cleft) sheaf A of H-Hopf-Galois extensions is in particular a sheaf of H-comodule algebras, given a 2-cocycle in H we can apply the functor Γ in (2.24) and obtain a new sheaf A γ over the same topological space X. The sheaf A γ is a sheaf of H γ -comodule algebras and is defined by A γ (U) := Γ(A (U)) = (A (U)) γ , with restriction maps given by morphisms of H γ -comodule algebras Γ(r UW ) = r UW : A γ (U) → A γ (W), for all W ⊂ U open sets.
By Corollary 3.7, we can conclude that A γ is a (locally cleft) sheaf of H γ -Hopf-Galois extensions if and only if A is a sheaf of (locally cleft) H-Hopf-Galois extensions. The subsheaves A coH γ γ and A coH over X coincide (i.e. they are isomorphic via the identity maps).
Let now K be another Hopf algebra; we may assume the additional (restrictive) condition for the sheaf A to be valued in the category of (K, H)-bicomodule algebras, i.e. A (U) ∈ K A H for each open U and the restriction maps are morphisms of (K, H)-bicomodule algebras. In this case we can deform A also by using a 2-cocycle σ on the external Hopf algebra K, or even by using both σ on K and γ on H. With the same reasoning as above, by using the results obtained in §3.2 and §3.3, the two sheaves σ A and σ A γ obtained in this way are sheaves of Hopf-Galois extensions if and only if the original sheaf A is. In general, the subsheaves σ A coH and σ A coH γ γ of coinvariants will not coincide with A coH . In the following subsection we provide an example. As a first step we define the trivial Hopf-Galois extensions

The Hopf bundle over
where O(α, β, x, c ±1 N ) denotes the * -algebra generated by the S 4 coordinates α, β, x and by c ±1 Next we introduce the restriction maps defining the sheaf A of locally trivial Hopf-Galois extensions, and precisely the trivial restriction map (where i N denotes the canonical injection) and the nontrivial one (defined on the generators and extended as * -algebra map), 3 It is straightforward to check that these restriction maps are morphisms of H-comodule algebras. Since {∅, U N , U S , U NS } is a basis of the topology {∅, U N , U S , U NS , S 4 } the Hopf-Galois extensions in (4.23) and the restriction maps (4.24), (4.25) uniquely define the locally cleft sheaf A on the topology {∅, U N , U S , U NS , S 4 } (to ∅ we assign the one element algebra, terminal object in the category of algebras).
In particular the Hopf-Galois extension on the sphere S 4 is obtained as the pull-back (in the category of * -algebras) (4.26) 3 This restriction map encodes the information on the transition function g NS characterizing the two charts U N , U S description of the Hopf bundle S 7 → S 4 . Indeed we have g NS : (we use the same notation for the coordinate functions and the point coordinates).
From Lemma C.1 in Appendix C (for the notation used see (C.1) and (C.6)) and the Hcomodule algebra isomorphism (C.2) we immediately conclude that the pull-back A (S 4 ) is isomorphic to O(S 7 ) as an H-comodule algebra. Then the subalgebra of coinvariants is O(S 4 ) and the Hopf-Galois extension A (S 4 ) coH ⊆ A (S 4 ) describes the instanton bundle S 7 → S 4 .
Finally the sheaf A is a sheaf of (K, H)-bicomodule algebras, where the K-coactions are given by We can now consider the 2-cocycle σ in (3.31) on K and use it to deform A (S 4 ) coH ⊆ A (S 4 ) to σ A (S 4 ) coH ⊆ σ A (S 4 ), and the commutative and trivial Hopf-Galois extensions in (4.23) into the noncommutative and trivial Hopf-Galois extensions The corresponding restriction maps are

A Twists, 2-cocycles and untwisting
We briefly outline the duality between the notions of Drinfeld twists [17,18] and 2-cocycles that was mentioned in Section 2.2 and illustrate the 'untwisting procedure'.
Given a twist F ∈ U ⊗ U we can deform the bialgebra (or Hopf algebra) U according to the following Proposition A.2. Let F = f α ⊗ f α be a twist on a bialgebra U. Then the algebra U with coproduct for all ξ ∈ U, and counit unchanged is a bialgebra, denoted U F . If moreover U is a Hopf algebra, then the twisted bialgebra U F is a Hopf algebra with antipode S F (ξ) : for all a, a ′ ∈ A, is a left U F -module algebra with respect to the same action. We denote the twisted algebra by A F , with U F -module structure given by ⊲, now thought of as a map U F ⊗ A F → A F .

A.2 Duality between twists and 2-cocycles
We here clarify how the two constructions of deforming by 2-cocycles and twists are dual to each other. Suppose H and U are dually paired bialgebras (or Hopf algebras) with pairing , : U × H → K, i.e., for all ξ, ζ ∈ U and h, k ∈ H we have ξζ, . Then to each invertible and counital twist F = f α ⊗ f α ∈ U ⊗ U there corresponds a convolution invertible and unital 2-cocycle The 2-cocycle condition for γ F follows from the twist condition for F ; indeed condition (A.1) in the F = f α ⊗ f α notation reads as (2) ) . (A.6) If we use F to twist the coproduct in U according to Proposition A.2 and γ F to deform the product in H as in Proposition 2.17, then the deformed bialgebras (or Hopf algebras) U F and H γ F are dually paired via the same pairing , ; indeed, it is easy to prove that (1) . Hence, we can twist the product in A by using F as in (A.3) or by using γ F as in (2.28). The two constructions give the same algebra for all a, a ′ ∈ A. Finally we observe that for a 2-cocycle γ F associated with a twist F = f α ⊗ f α ∈ U ⊗ U, the map ϕ A,A introduced in Theorem 2.19 reads for all a, a ′ ∈ A γ F .
It is straightforward to show that the twisted product · τ * γ in H τ * γ equals the twisted product Since the antipode if it exists is unique we immediately have the statement for Hopf algebras.

B Equivalence of closed monoidal categories and the G-map
In this section we show how the G-map of Theorem 3.4 is related (by duality) to the natural transformation which establishes that twisting may be regarded as an equivalence of closed monoidal categories.
Referring to [4,Section 3.2], it follows that G ′ is precisely the isomorphism D F op cop for the Hopf algebra H ′op cop twisted by the twist F op cop . It has been shown in [5] that such D-maps have a categorical interpretation in terms of the natural isomorphism which establishes that twisting is an equivalence of closed monoidal categories. Hence, in conclusion, the dual of our G-map can be given a categorical interpretation.

C The twisted sheaf of the Hopf bundle over S 4 θ : top down approach
We complement the example of the twisted sheaf in §4.2.1 by presenting a top down approach: we first describe S 7 as a ringed space, then on these algebras (rings) of coordinate functions on opens of S 7 we induce the H-coaction leading to a sheaf A of H-comodule algebras (with H = O(SU(2))). Next we show that this is a locally cleft sheaf of H-Hopf Galois extensions, and as a corollary that it is naturally isomorphic to the sheaf A of §4.2.1, A ≃ A . Finally, in the last of the paragraphs titled in italics, the torus action on π : S 7 → S 4 is pulled back to this sheaf description and the corresponding twist deformation is obtained. In Section C.1 we study the subsheaf of H-coinvariants, it is generated by two copies (of the exponential version) of the Moyal-Weyl algebra on R 4 θ that describe S 4 θ as a ringed space.
For each open set U ⊂ S 4 , A(U) is the algebra of coordinate functions on π −1 (U) ⊂ S 7 . Indeed extending the algebra O(S 7 ) by the generator c −1 N corresponds, geometrically, to restricting to the subspace of S 7 of those points with z 3 z * 3 + z 4 z * 4 never vanishing. Now recalling relation (3.30) between coordinates on S 7 and on S 4 , we see that these are the points with x 1, i.e., they are the points of π −1 (U N ) (we use the same notation for the coordinate functions and the point coordinates). Conversely, enlarging the algebra with c N does not have a geometrical significance, but it is a pure algebraic operation designed to add the square root of the positive real element z 3 z * 3 + z 4 z * 4 = c 2 N already belonging to the algebra O(S 7 ). The same discussion is valid for the elements c ±1 S , so that A(U S ) and A(U NS ) are coordinate algebras on π −1 (U S ) and π −1 (U NS ) respectively. The subalgebras of H-coinvariants are given by N ) denotes the * -subalgebra of A(U N ) generated by the elements α, β, x, c ±1 N , and similarly for the other basic open sets. Notice that A coH (U) = A coH (U) as defined in (4.23). The algebra of global coinvariant sections is A(S 4 ) coH ≃ O(S 4 ), and, similarly to (C.2), it is isomorphic to the pull-back (C.7) In §C.1 we explicitly show that the subsheaf A coH of coinvariant elements (complemented by A(∅) coH = A(∅)) is that of coordinate functions on the opens ∅, U N , U S , U NS , S 4 .

The sheaf A is a locally cleft sheaf of H-Hopf-Galois extensions
The H-comodule algebra isomorphism A(S 4 ) ≃ O(S 7 ) shows that the global sections A(S 7 ) are an H = O(SU(2))-Hopf-Galois extensions of the global coinvariant sections A(S 4 ) coH ≃ O(S 4 ). Recalling the general theory, cf. §4.2, this shows that the sheaf A of H-comodule algebras is a sheaf of H-Hopf-Galois extensions.
In order to prove that A is locally cleft we consider the open covering {U N , U S } of S 4 and show that A(U N ) coH ⊆ A(U N ) and A(U S ) coH ⊆ A(U S ) are cleft extensions.
We first observe that the matrix elements By using the matrix u in (C.3) and the matrix projector P = uu * (whose entries are the generators of S 4 , see (3.29)) we introduce "local trivialization maps": 5
Proof. The inverse maps are given by By using (1 − x) = 2c 2 N , valid in A (U N ) coH , it is easy to show that the map φ N is an algebra map, i.e. preserves the identity ρ −2 N (1 + x 1 x * 1 + x 2 x * 2 ) = 1. An analogous computation, using again (C.5), shows that φ S is an algebra map.
Because of this lemma the algebras A (U N ) coH and A (U S ) coH are interpreted as two (isomorphic) copies of the algebra of coordinate functions on R 4 . (Adding to the algebra generated by x 1 , x 2 , x * 1 , x * 2 the generators ρ ±1 N is geometrically ineffective, similarly for ρ ±1 S ). Specifically, they describe the algebras of coordinate functions on the open sets U N , U S , obtained via stereographic projections from the North and South poles of the 4-sphere. 6 Similarly, the following lemma shows that the algebra A (U NS ) coH is that of coordinate functions on R 4 minus the origin. in B(U S ), has no geometrical significance since 1 + x 1 x * 1 + x 2 x * 2 has always a well defined and invertible square root (being 1 + x 1 x * 1 + x 2 x * 2 ≥ 1). Conversely, from αα * + ββ * + x 2 = 1, it follows that r 2 = x 1 x * 1 + x 2 x * 2 = (1 + x)(1 − x) −1 is defined and is non-zero when the point (α, α * , β, β * , x) we started from belongs to S 4 \{N, S}. The algebra extension of B(U N ) by r −2 considered in Lemma C.3 geometrically corresponds indeed to the restriction to the points in the intersection of the charts U N and U S of S 4 .
Proof. Observe that φ N (r 2 ) = 1 2 (1 + x)c −2 N . Then, since 2c 2 S = (1 + x) (see (C.5)), we immediately conclude that φ N (r 2 ) = φ NS (r 2 ). For the inverse map set φ −1 The restriction maps characterizing the subsheaf of coinvariants B ≃ A coH are the composition of the canonical inclusions i N : A (U N ) coH ֒→ A (U NS ) coH and i S : A (U S ) coH ֒→ A (U NS ) coH with the isomorphisms φ N , φ S and φ −1 NS . These restriction maps are * -algebra homomorphisms and their explicit expression on the generators reads The algebra of global sections B(S 4 ) is the pull-back The algebra B(S 4 ) is generated by the elements (ρ −2 N x i , ρ −2 S y i ), (ρ −2 N x * i , ρ −2 S y * i ), i = 1, 2 and (1 − 2ρ −2 N , 2ρ −2 S − 1) and is a copy of the coordinate algebra O(S 4 ) = O(S 7 ) coH .
Using the isomorphisms of Lemma C.2 and Lemma C.3 it is immediate to induce from A (U) coH the K = O(T 2 )-comodule structure on B(U), (U = U N , U S , U NS ) and to see that the restriction maps (C.16) are K-comodule maps. Considering the twist (3.31) on K we then obtain the noncommutative algebra σ B(U N ) that is the (geometrically trivial central extension via the real elements ρ ± N of the) coordinate algebra on R 4 θ ; i.e. the (exponential version of the Moyal-Weyl) algebra defined by the commutation relations x 1 σ • x 2 = e −2πiθ x 2 σ • x 1 . Similarly for σ B(U S ), and for σ B(U NS ) that is the geometrically nontrivial central extension of σ B(U N ) via the real elements r ±1 . These algebras and the restriction maps Σ(r B N,NS ) = r B N,NS , Σ(r B S,NS ) = r B S,NS , define the sheaf σ B of noncommutative coordinates algebras over S 4 , i.e. define S 4 θ as a ringed space.