Gauge groups and bialgebroids

We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples illustrating these constructions include: Galois objects of Taft algebras, a monopole bundle over a quantum spheres and a not faithfully flat Hopf--Galois extension of commutative algebras. The latter two examples have in fact a structure of Hopf algebroid for a suitable invertible antipode.


Introduction
The study of groupoids on the one hand and gauge theories on the other hand is important in different areas of mathematics and physics. In particular these subjects meet in the notion of the gauge groupoid of a principal bundle. In view of the considerable amount of recent work on noncommutative principal bundles it is desirable to come up with a noncommutative version of groupoids and study their relations to noncommutative principal bundles. For all of this there is a need for a better understanding of bialgebroids.
In the present paper, having in mind applications to noncommutative gauge theories, we consider the Ehresmann-Schauenburg bialgebroid associated with a noncommutative principal bundle as a quantization of the classical gauge groupoid. Classically, bisections of the gauge groupoid are closely related to gauge transformations. In parallel with this result we show that in a rather general context the gauge group of a noncommutative principal bundle is group isomorphic to the group of bisections of the corresponding Ehresmann-Schauenburg bialgebroid. To illustrate the theory we work out all the details of the gauge group of the principal bundle and of the bialgebroid with corresponding group of bisections, for the noncommutative U(1) bundle over the quantum standard sphere, and for a commutative not faithfully flat Hopf-Galois extension obtained in [3] from a particular coaction on the algebra O(SL (2)). In fact, in each of these two cases there is also an invertible antipode which satisfies the conditions for a Hopf algebroid. In general, for a bialgebroid there is a coproduct and a counit but not an antipode. Here we wish to emphasise one important property that the O(SL(2)) example shows, that is that at least for a commutative algebra (of coinvariants) the Hopf-Galois extension needs not be faithfully flat for our constructions to be well defined and our results to be valid.
Part of the paper deals with Galois objects. A Galois object of a Hopf algebra H is a noncommutative principal bundle over a point in a sense: a Hopf-Galois extension of the ground field C. In contrast to the classical case where a bundle over a point is trivial, for the isomorphism classes of noncommutative principal bundles over a point this needs not be the case. An antipode can always be defined for the Ehresmann-Schauenburg bialgebroid of a Galois object which (the bialgebroid that is) is then a Hopf algebra. Notable examples are group Hopf algebras C[G], whose corresponding principal bundle are C[G]-graded algebras and are classified by the cohomology group H 2 (G, C × ), and Taft algebras T N . The equivalence classes of T N -Galois objects are in bijective correspondence with the abelian group C. Thus, part of the paper concerns the Ehresmann-Schauenburg bialgebroid of a Galois object and corresponding groups of bisections, been they algebra maps from the bialgebroid to the ground field (and thus characters) or more general transformations. For these bialgebroids some of the results we report could be and have been obtained in an abstract and categorical way. Here we re-obtained them in an explicit and more workable fashion, for potential applications to noncommutative gauge theory.
Automorphisms of a (usual) groupoid with natural transformations form a strict 2group or, equivalently, a crossed module. The crossed module involves the product of bisections and the composition of automorphisms, together with the action of automorphisms on bisections by conjugation. Bisections are the 2-arrows from the identity morphisms to automorphisms, and the composition of bisections can be viewed as the horizontal composition of 2-arrows. In the present paper this construction is extended to the Ehresmann-Schauenburg bialgebroid of a Hopf-Galois extension by constructing a crossed module for the bisections and the automorphisms of the bialgebroid.
The paper is organised as follows. After a recap in §2 of algebraic preliminaries and notation, in §3 we give the relevant concepts for noncommutative principal bundles (Hopf-Galois extensions), gauge groups and bialgebroids that we need. We then work out in §3.3 the gauge group for the noncommutative U(1) principal bundle over the quantum sphere and in §3.4 for a commutative not faithfully flat Hopf-Galois extension associated to O(SL (2)). In §4, we first have Ehresmann-Schauenburg bialgebroids and the group of their bisections. Then we show that the group of gauge transformations of a noncommutative principal bundle is group isomorphic to the group of bisections of the corresponding Ehresmann-Schauenburg bialgebroid. In §5 we describe the Hopf algebroid structure for the U(1) principal bundle over the quantum sphere in §3.3 and for the commutative not faithfully flat Hopf-Galois extension out of O(SL(2)) considered in §3. 4. In §6 we consider Galois objects with several examples, such as Galois objects for a cocommutative Hopf algebra, in particular group algebras, regular Galois objects (Hopf algebras as self-Galois objects) and Galois objects of Taft algebras. Finally, in §7, we study the crossed module (or 2-group) structure coming from the bisections and the automorphism group of a Ehresmann-Schauenburg bialgebroid. When restricting to Hopf algebras one is lead to the representation theory of crossed modules on them. In the present paper we work out this construction for the Taft algebras; more general results will be reported elsewhere.

Algebraic preliminaries
We recall here some known facts from algebras and coalgebras and corresponding modules and comodules. We also recall the more general notions of rings and corings over an algebra as well as the associated notion of bialgebroid. We move then to Hopf-Galois extensions, as noncommutative principal bundles, and to the definitions of gauge groups.
2.1. Algebras, coalgebras and all that. We work over the field C of complex numbers but this could be substituted by any commutative field k. Algebras (coalgebras) are assumed to be unital and associative (counital and coassociative) with morphisms of algebras taken to be unital (of coalgebras taken counital). For the coproduct of a coalgebra ∆ : H → H ⊗H we use the Sweedler notation ∆(h) = h (1) ⊗h (2) (sum understood) and its iterations: . We denote by * the convolution product in the dual vector space H ′ := Hom(H, C), (f * g)(h) := f (h (1) )g(h (2) ). The antipode of a Hopf algebra H is denoted S.
Given an algebra A, a left A-module is a vector space V carrying a left A-action, that is with a C-linear map Dually, with a coalgebra (H, ∆), a right H-comodule is a vector space V carrying a right H-coaction, that is with a C-linear map δ V : (1) , and the right H-comodule properties read and v (0) ε(v (1) ) = v, for all v ∈ V . The C-vector space tensor product V ⊗ W of two H-comodules is a H-comodule with the right tensor product H-coaction In particular, a right H-comodule algebra is an algebra A which is a right H-comodule such that the multiplication and unit of A are morphisms of H-comodules. This is equivalent to requiring the coaction δ A : A → A ⊗ H to be a morphism of unital algebras (where A⊗H has the usual tensor product algebra structure). Corresponding morphisms are H-comodule maps which are also algebra maps.
In the same way, a right H-comodule coalgebra is a coalgebra C which is a right H-comodule and such that the coproduct and the counit of C are morphisms of Hcomodules. Explicitly, this means that, for each c ∈ C, and ε(c (0) )c (1) = ε(c)1 H . Corresponding morphisms are H-comodule maps which are also coalgebra maps. There are right A-modules and left H-comodule versions of the above.
Next, let H be a coalgebra and let A be a right H-comodule algebra. An (A, H)-relative Hopf module V is a right H-comodule with a compatible left A-module structure. That is the left action A morphism of (A, H)-relative Hopf modules is a morphism of right H-comodules which is also a morphism of left A-modules. In a similar way one can consider the case for the algebra A to be acting on the right, or with a left and a right (bimodule) A-actions.
For an algebra B a B-ring is a triple (A, µ, η). Here A is a B-bimodule with B-bimodule maps µ : A ⊗ B A → A and η : B → A, satisfying the associativity and unit conditions: From [5,Lemma 2.2] there is a bijective correspondence between B-rings (A, µ, η) and algebra automorphisms η : B → A. Starting with a B-ring (A, µ, η), one obtains a multiplication map A ⊗ A → A by composing the canonical surjection A ⊗ A → A ⊗ B A with the map µ. Conversely, starting with an algebra map η : B → A, a B-bilinear associative multiplication µ : A ⊗ B A → A is obtained from the universality of the coequaliser A ⊗ A → A ⊗ B A which identifies an element ab ⊗ a ′ with a ⊗ ba ′ .
Dually, for an algebra B a B-coring is a triple (C, ∆, ε). Here C is a B-bimodule with B-bimodule maps ∆ : C → C ⊗ B C and ε : C → B, satisfying the coassociativity and counit conditions: Let B be an algebra. A left B-bialgebroid L consists of a (B ⊗B op )-ring together with a B-coring structures on the same vector space L with mutual compatibility conditions [26]. From what said above, a (B ⊗B op )-ring L is the same as an algebra map η : B ⊗B op → L. Equivalently, one may consider the restrictions which are algebra maps with commuting ranges in L, called the source and the target map of the (B ⊗ B op )-ring L. Thus a (B ⊗ B op )-ring is the same as a triple (L, s, t) with L an algebra and s : B → L and t : B op → L both algebra maps with commuting range.
For a left B-bialgebroid L the compatibility conditions are required to be (i) The bimodule structures in the B-coring (L, ∆, ε) are related to those of the (ii) Considering L as a B-bimodule as in (2.4), the coproduct ∆ corestricts to an algebra map from L to where L × B L is an algebra via component-wise multiplication.
An automorphism of the left bialgebroid (L, ∆, ε, s, t) over the algebra B is a pair (Φ, ϕ) of algebra automorphisms, Φ : L → L, ϕ : B → B such that: In fact, the map ϕ is uniquely determined by Φ via ϕ = ε • Φ • s and one can just say that Φ is a bialgebroid automorphism. Automorphisms of a bialgebroid L form a group Aut(L) by map composition. A vertical automorphism is one of the type (Φ, ϕ = id B ). The pair of algebra maps (Φ, ϕ) can be viewed as a bialgebroid map (see [24, §4.1]) between two copies of L with different source and target maps (and so B-bimodule structures). If s, t are the source and target maps on L, one defines on L new source and target maps by s ′ := s • ϕ and t ′ := t • ϕ with the new bimodule structure given by b ⊲ ϕ c ⊳ ϕb := s ′ (b)t ′ (b)a, for any b,b ∈ B and a ∈ L (see (2.4)). Therefore one gets a new left bialgebroid with product, unit, coproduct and counit not changed.
From the conditions (2.6), Φ is a B-bimodule map: Φ(b ⊲ c ⊳b) = b ⊲ ϕ Φ(c) ⊳ ϕb . The first condition (2.7) is well defined once the conditions (2.6) are satisfied (the balanced tensor product is induced by s ′ and t ′ ). Conditions (2.6) imply that Φ is a coring map; therefore (Φ, ϕ) is an isomorphism between the starting bialgebroid and the new one.
Finally, we recall from [6, Def. 4.1] the conditions for a Hopf algebroid with invertible antipode. Given a left bialgebroid (L, ∆, ε, s, t) over the algebra B, an invertible antipode S : L → L in an algebra anti-homomorphism with inverse S −1 : L → L such that and satisfying compatibility conditions with the coproduct: for any h ∈ L. These then imply S(h (1) ) h (2) = t • ε • Sh. 5

Noncommutative principal bundles
We start with a brief recall of Hopf-Galois extensions as noncommutative principal bundles. Then we consider gauge transformations as equivariant automorphisms of the total space algebra which are vertical so that they leave invariant the base space algebra.
3.1. Hopf-Galois extensions. These extensions are H-comodule algebras A with a canonically defined map χ : A ⊗ B A → A ⊗ H which is required to be invertible [22].
is an isomorphism. (a ⊗ B a ′ )a ′′ := a ⊗ B a ′ a ′′ and (a ⊗ h)a ′ := aa ′ (0) ⊗ ha ′ (1) . For the H-comodule structure, the right tensor product H-coaction as in (2.1): for all a, a ′ ∈ A, descends to the quotient A ⊗ B A because B ⊆ A is the subalgebra of H-coinvariants. Similarly, A ⊗ H is endowed with the tensor product coaction, where one regards the Hopf algebra H as a right H-comodule with the right adjoint H-coaction The right H-coaction on A ⊗ H is then given, for all a ∈ A, h ∈ H, by Since the canonical Galois map χ is left A-linear, its inverse is determined by the restriction τ : 2) The translation map enjoys a number of properties [23, 3.4] that we list here for later use. For any h, k ∈ H and a ∈ A, b ∈ B: Two Hopf-Galois extensions A, A ′ for the Hopf algebra H of the algebra B are isomorphic provided there exists an isomorphism of H-comodule algebras A → A ′ . This is the algebraic counterpart for noncommutative principal bundles of the geometric notion of isomorphism of principal G-bundles over the same base space.
3.2. The group of gauge transformations. In [7] gauge transformations for a noncommutative principal bundles were defined to be invertible and unital comodule maps, with no additional requirement. In particular they were not asked to be algebra maps. However, the resulting gauge group might be very big, even in the classical case. For example the gauge group of a G-bundle over a point would be much bigger than the structure group G. In contrast, in [2] gauge transformations were taken to be algebra homomorphisms. This property implies in particular that they are invertible. Moreover, recall the notation τ (h) = h <1> ⊗ B h <2> for the translation map. Then for F ∈ Aut H (A) its inverse F −1 ∈ Aut H (A) is given, for all a ∈ A, by Proof. That vertical H-comodule algebra maps are invertible is in [22,Rem. 3.11]. We check the expression of the inverse in (3.10). The B-linearity and the algebra map property assure that the inverse is well defined: For any a ∈ A, using the H-equivariance of F , Now, the action of the canonical map χ yields the equality Indeed, using (3.3), for the right hands side: Next, using equivariance, (3.3) and (3.4), for the left hand side: (2) .
Then bijectivity of the canonical map χ yields the identity (3.12). With B-linearity of F , using id A ⊗ B F on both sides of equality (3.12), the right faithful flatness leads to Thus F −1 is the inverse map of F ∈ Aut H (A). That F −1 is a H-comodule algebra map follows directly from such properties of F .
Elements F ∈ Aut H (A) preserve the (co)-action of the structure quantum group since they are such that (1) ). And they also preserve the base space algebra B. This group will be called the gauge group.
Remark 3.4. A similar proposition was given in [2], for H a coquasitriangular Hopf algebra, and A a quasi-commutative H-comodule algebra. As a consequence, B is in the centre of A. In the present paper, there is no restriction on the coinvariant subalgebra B.

3.3.
Noncommutative U(1)-bundles. Let G be a group and C[G] be its group algebra. Its elements are finite sums λ g g with λ g ∈ C. The algebra product follows from the group product in G, with unit 1 C[G] = e, the neutral element of G. The coproduct, counit and antipode, making C[G] a Hopf algebra, are ∆(g) = g ⊗ g, ε(g) = 1, S(g) = g −1 .
It is known that C[G]-Hopf-Galois extensions are the same as strongly graded algebras over G. Now, an algebra A is G-graded, given by a → a g ⊗ g for a = a g , a g ∈ A g . Then the algebra A is strongly G-graded, that is [15,Thm.8.1.7]).
Let us concentrate on taking is the ideal generated by 1 − zz −1 in the polynomial algebra C[z, z −1 ] in two variables. The Hopf algebra structure of H is now, for all n ∈ Z, the coproduct ∆ : z n → z n ⊗ z n , the antipode S : z n → z −n and the counit ε : z n → 1.
A strongly graded Z-algebra A = ⊕ n∈Z A n with coaction determined by results into a right comodule algebra for H = O(U(1)) which will be referred to as a noncommutative U(1) principal bundle over the algebra B := A 0 . From [16, Cor. I.3.3], for A a strongly Z-graded algebra, the right-modules A 1 and A −1 are finitely generated and projective over A 0 . In fact, the total space algebra can be recovered out of the 'line bundles' A 1 and A −1 as a Pimsner algebra [1]. (1) implies that F respects the grading and in fact F is completely determined by its restrictions to A 1 and A −1 as B-module maps, given that F is required to be the identity on B and that it is then extended as an algebra map.
Let us consider an explicit example of the above construction, that is the U(1) principal bundle over the standard Podleś sphere S 2 q of [17]. With q ∈ R a deformation parameter, the coordinate algebra O(SL q (2)) of the quantum group SL q (2) is the algebra generated by elements a, c and d, b with relations a c = q c a and b d = q d b, a b = q b a and c d = q d c, Then the Hopf algebra H = O(U(1)) coacts on the algebra O(SL q (2)) via 14) The subalgebra of coinvariant elements in O(SL q (2)) for this coaction is the coordinate . As a set of generators for O(S 2 q ) one may take for which one finds the relations ), a noncommutative principal bundle [8], is a faithfully flat Hopf-Galois extension. The translation map on generators of O(U(1)) is The total space algebra decomposes as O(SL q (2)) = n∈Z A n where In particular as B-modules, A −1 is generated by a, c while A 1 is generated by d, b. Any gauge transformation will be then determined by the images with coefficients which are elements in the algebra B, and extended as an algebra map. Let us first consider the classical case, q = 1 of commutative algebras, to clarify the structures. Asking for the coinvariant generators in (3.15) to be left unchanged by F in (3.19) reduces the coefficient to a single one X, any non vanishing function from S 2 → C: 20) and the sphere relation is automatically satisfied. We get Aut H SL(2) = Map(S 2 → C * ). In contrast, when q = 1, requiring that F be an algebra map and so to respect the commutation relations in (3.13), one gets X ∈ C * since the centre of O(S 2 q ) is just the algebra C. Thus Aut H SL q (2) = C * , the non vanishing complex numbers.

3.4.
A gauge group without faithfully flatness. We give an example of the above construction of the gauge group for a Hopf-Galois extension over a commutative algebra of coinvariants which is flat but not faithfully flat.
This example was studied in [3,Ex. 2.4]. Consider the Hopf algebra Then, the algebra of coinvariants is B = C[c, d] and the inclusion B = A coH ⊂ A is a Hopf-Galois extension (that is the corresponding canonical map is bijective). It is shown in [3,Ex. 2.4], that the extension is flat but not faithfully flat. It is easy to see that the corresponding translation map τ : H → A ⊗ B A is given by Consider then the group Aut H (A) of gauge transformations. Any such a map is determined by its values on the generators, being F (c) = c and F (d) = d by the B-linearity. We claim that given F (a) and F (b) their inverse is as in the formula (3.10): Proof. A direct computation using the equivariance: and also using the 'commutation relations' of the algebra (just commutativity in the present case) to remove the terms in x and x 2 .
Next one checks that (3.6) is satisfied for all generators. Then, F −1 • F = id A goes as in (3.11) given the expressions in (3.22): using equivariance and B-linearity, with the relation ad − bc = 1. Conversely, from Lemma 3.5, using B-linearity and the determinant condition: A similar computation shows that F −1 (F (b)) = b = F (F −1 (b)). The group Aut H (A) is not trivial. Besides the identity map it contains for instance unital maps of the kind for h an arbitrary element in B, and F (c) = c, F (d) = d, extended as an algebra map. This F is equivariant and preserves the determinant condition: F (ad − bc) = F (1) = 1.

Ehresmann-Schauenburg bialgebroids
To any Hopf-Galois extension B = A co H ⊆ A one associates a B-coring and a bialgebroid [19] (see [9, §34.13 and 34.14]). These can be viewed as a quantization of the gauge or Ehresmann groupoid that is associated to a principal fibre bundle (see [12]).
4.1. The Ehresmann coring. The coring can be given in a few equivalent ways. Let B = A co H ⊆ A be a Hopf-Galois extension with right coaction δ A : A → A ⊗ H. Recall the diagonal coaction (3.1), given for all a, a ′ ∈ A, by . Let τ be the translation map of the Hopf-Galois extension. We have the following: is the same as the B-bimodule Proof. The B-bimodule structure of C is left and right multiplication by elements of B.
Applying the map m A ⊗ id H to elements of (4.1) one gets aã ∈ B. The above B-coring is called the Ehresmann or gauge coring; we denote it C(A, H). Using the known relation between the coinvariants of a tensor product of comodules and their cotensor product [23, Lemma 3.1], the coring C(A, H) can also be given as a cotensor product A H A.

Definition 4.3. Let C(A, H) be the coring associated with a faithfully flat Hopf-Galois
for all x ⊗x, y ⊗ỹ ∈ C(A, H) (and unit 1 A ⊗ 1 A ). The target and the source maps are We refer to [9, 34.14] for the check that all defining properties are satisfied. When there is no risk of confusion we drop the decoration • C(A,H) in the product.

4.2.
Bisections and gauge groups. The bialgebroid of a Hopf-Galois extension can be viewed as a quantization (of the dualization) of the classical gauge groupoid, recalled in Appendix A, of a (classical) principal bundle. Dually to the notion of a bisection on the classical gauge groupoid there is the notion of a bisection on the Ehresmann-Schauenburg bialgebroid. These bisections correspond to gauge transformations.
The notion of a bisection as in the following definition could be given for any bialgebroid, not only for the Ehresmann-Schauenburg bialgebroid. However, for the general case one would need some additional requirements so to get a proper composition of bisections extending (4.6) below. We shall address this general definition elsewhere.
There is also a middle B-linearity, that is for any bisection σ: for any x ⊗x ∈ C(A, H) and b ∈ B. Indeed, using the associativity in the 2nd step and B-bilinearity and unitality in the 3rd step.  The collection B(C(A, H)) of bisections of the bialgebroid C(A, H) is made a group by the convolution product of any two σ 1 bisections σ 2 : for any element x ⊗x ∈ C(A, H), recalling the B-coring coproduct (4.3).
The product is well defined over the B-balanced tensor product since the bisections are B-bilinear; by the same reason σ 1 * σ 2 is B-bilinear. We are left to show the associativity property (3) in the Definition 4.4. For simplicity write X = x ⊗x and Y = y ⊗ỹ for elements of C(A, H) and use a Sweedler-like notation for the B-coring coproduct (4.3), (2) . Then, where the 2nd and 4th steps use (4.5), and the 3rd step uses (2.5). The product is associative since C(A, H) is coassociative as a B-coring.
The counit of the bialgebroid is a bisection by definition and one checks that ε * σ = σ = σ * ε for any bisection σ and ε is the unit element. The inverse of the bisection σ is  Proof. Firstly, given a bisection σ ∈ B(C(A, H)) we define a map F σ : for any a ∈ A. This is well defined since σ is right B-linear and a (0) ⊗ a (1) and F σ is an algebra map: using (3.7) for the 1st step, (4.5) for the 2nd, (3.9) for the third. Also, F σ is H-equivariant: where the 2nd step uses (3.3). Thus F σ ∈ Aut H (A). 13 Conversely, let F ∈ Aut H (A) a gauge transformation and define σ F ∈ B(C(A, H)) by σ F (a ⊗ã) := F (a)ã, (4.9) for any a ⊗ã ∈ C(A, H). This is well defined since using (4.1). Clearly, σ F is unital and B-bilinear. Also, for any a ⊗ã, a ′ ⊗ã ′ ∈ C(A, H), where the 3rd step uses the fact that F (a ′ )ã ′ ∈ B. Thus σ F is a bisection. The correspondence is bijective: one easily checks that σ Fσ = σ and F σ F = F . Also σ id A = ε and for any a ⊗ã ∈ C (A, H) we have, where the 3th step uses that G(a (0) )a (1) <2> similarly to (3.12), and the 4th step uses (3.5).
Thus the correspondence is a group isomorphism. Via this we can get the inverse (4.7) directly from (3.10): , where the 4th step uses (4.2).
We have already mentioned that gauge transformations for a noncommutative principal bundles could be defined without them being algebra maps [7]. Mainly for the sake of completeness we record here a version of these via bialgebroids and bisections. To distinguish them from the analogous concepts introduced previously, and for lack of a better name, we call them extended gauge transformation and extended bisections.
Thus following [7], the extended gauge group Aut ext H (A) of a Hopf-Galois extension B = A coH ⊆ A consists of invertible H-comodule unital maps F : A → A which are such that F (ba) = bF (a) for any b ∈ B and a ∈ A. The group structure is map composition.
In parallel with this we have then the following. Since extended bisections are B-bilinear, the product (4.6) is indeed well defined, and the counit ε is still the unit element for the product. Thus the collection B ext (C(A, H)) of extended bisections form a group, of which B (C(A, H)) is a subgroup. Notice that now (4.7) is not the inverse in B ext (C(A, H)) for the product in (4.6) since for an extended bisection we are not asking it to be a left character for the B-ring (C(A, H), s), and thus the expression in (4.7) is not right B-linear.
Finally, in analogy with Proposition 4.6 we have the following.  (C(A, H)).
Proof. This uses the same methods as Proposition 4.6. Given F ∈ Aut ext H (A), define its image as in (4.9): σ F (a ⊗ã) = F (a)ã. For all b ∈ B we have, Conversely, given σ ∈ B ext (C(A, H)) define its image in Aut ext H (A) as in (4.8): . The rest of the proof goes as that of Proposition 4.6.

Hopf algebroids
As examples we construct the Ehresmann-Schauenburg bialgebroid for the U(1) principal bundle over the quantum sphere in nonzeroncu1b and for the commutative not faithfully flat Hopf-Galois extension out of O(SL(2)) considered in Sect. 3.4. In both cases there is a suitable invertible antipode satisfying conditions 2.8 and 2.9 for a Hopf algebroid. It is worth stressing that the results for O(SL(2)) in Sect. 3.4 and Sect. 5.2 below show that at least for an algebra of coinvariants which is commutative, the Hopf-Galois extension needs not be faithfully flat for all constructions to be well defined. An analysis of the role of the faithful flatness in a general context will be reported elsewhere.
and their 'conjugated': By using the expression (3.17) for the translation map and the relations (3.13), one checks that the above generators h ⊗ k satisfy the condition h (0) ⊗ τ (h (1) )k = h ⊗ k ⊗ B 1 A as it should be from the alternative description (4.2) of the bialgebroid C(A, H). For instance: The eight generators in (5.1) and (5.2) are not independent. Indeed, define

Similar computations work for the other generators. A direct computation leads to
Then, a direct computation shows that Also, the sphere relations in (3.16) translate into An alternative way to show the relations among the generators is to observe by a direct computation that there are four 'circle' relations: These in turn imply δα + βγ + β α + δ γ = 1 ⊗ 1. (5.7) Notice that the above relations which survive the classical limit q = 1, are constraints among the generators and not commutation relations. For the latter one has the following.
as well as (There are also 'conjugated' relations directly derived from the previous ones.) As a consequence BC = CB, for the generators in (5.3).
Proof. This is the result of a direct computation given the relations in (3.13).

Lemma 5.2. The bialgebroid C(A, H) has a structure of a Hopf algebroid with coproduct (4.3) which results into:
counit (4.4) which results into: and antipode S = S −1 : Proof. The expressions for the counit are clear. For the coproduct, from Definition 4.3, using the expression (3.17) for the translation map. Similarly, Similar computations work for the other generators. Finally, the antipode in (5.10) clearly satisfies (2.8). Then, on the one hand, for condition (2.9) for the generator α: using the relations da − q −1 bc = 1 and ad − qcb = 1. On the other hand, being S −1 = S, is well defined. Using the expression (3.17) for the translation map one gets on generators: Given the generic gauge transformation in (3.20), formula (4.9) determines a generic bisection. This is then given on generators by As before in Sect. 3.3, X is any map from the sphere S 2 → C * when q = 1, while is any non vanishing element X ∈ C * when q = 1.

5.2.
easily seen to be invariant for the diagonal coaction. The following is easily established. Also in the present case the bialgebroid C(A, H) can be given as in (4.2) since the generators h ⊗ k satisfy the condition h (0) ⊗ τ (h (1) )k = h ⊗ k ⊗ B 1 A . Using the expression (3.22) for the translation map, the coinvariance of c and d to pass them over the balanced tensor product, and the relation ad − bc = 1, one computes for instance, and similarly for the other generators. 18 Lemma 5.4. The bialgebroid C(A, H) has a structure of a Hopf algebroid with coproduct (4.3) which results into: counit (4.4) which results into: and antipode: Proof. The form of the counit is clear. For the coproduct, from Definition (4.3) and translation map (3.22), one computes on the generators (5.11), while, crossing d and c over the balanced tensor product, which coincides with the previous expression when using the relation ad − bc = 1. Also, which is the same as Similar computations work for the remaining generators. Finally, the antipode in (5.16) clearly satisfies (2.8). Then, on the one hand, for condition (2.9) for the generator α: using the relation α δ − βγ = 1 ⊗ 1. On the other hand, being S −1 = S, Similar computations go for the other generators. This concludes the proof.
Finally, once again both equations (4.8) and (4.9) are well defined since the map is well defined. Indeed, using once more the expression (3.22) for the translation map one gets on the generators a and b (the not coinvariant ones): having inserted ad − bc = 1 and using the coinvariance of c and d to cross them over the balanced tensor product. Similarly for b one finds: A similar computation goes with the generator b, the statement being trivial for the coinvariant elements c and d. Conversely, using the first relation in (5.13). Similar computations go with the remaining generators.
As an example, one gets for the gauge transformations in (3.23) the bisection: for h an arbitrary element in B.

Galois objects
We shall now consider Galois objects of a Hopf algebra H. Such an object could be thought of as a noncommutative principal bundle over a point. In contrast to the classical result that any fibre bundle over a point is trivial, the set Gal H (C) of isomorphic classes of H-Galois objects needs not be trivial (see [4], [11]). We shall illustrate later on this non-triviality with examples coming from group algebras and Taft algebras.
As already mentioned, the results of this section could be and have been obtained in an abstract and categorical way. Here we re-obtained them in a more explicit and more workable way, having in mind potential application to noncommutative gauge theory. 20 6.1. The bialgebroid of a Galois object.
Recall that an (A, H)-relative Hopf module M is a H-comodule with a compatible Amodule structure. That is the action is a morphism of H-comodules such that δ M (ma) = m (0) a (0) ⊗ m (1) a (1) for all a ∈ A, m ∈ M. The multiplication induces an isomorphism [22], [19, eq. 2.7]). Then, given that C(A, H) = (A ⊗ A) coH , this yields an isomorphism We finally collect some results of [18] Being algebra maps, now bisections are characters of the Hopf algebra C(A, H) with product in (4.6) and inverse in (4.7) that, with the antipode in (6.1) is written σ −1 = σ•S C , as it is the case for characters. From Proposition 4.6 we have the isomorphism Aut H (A) ≃ B(C(A, H)) = Char (C(A, H)). with Char ext (C(A, H)) the group of convolution invertible unital maps φ : C(A, H) → C. Example 6.3. Any Hopf algebra H is a H-Galois object with its coproduct as coaction.
With A = H, the corresponding left bialgebroid becomes We have a linear map φ : (2) ). This is well defined since The map φ is an algebra map: It is also a coalgebra map: ⊗ h)) .

Group Hopf algebras.
With a cocommutative Hopf algebra H and A a Galois object for H, the bialgebroid C(A, H) is isomorphic to H as a Hopf algebra [18, Rem. 3.8; Thm. 3.5]. We work out some of the details for the case of a group algebra whose Galois objects are classified by group cohomology [15,Chap. 8].
Let H = C[G] be a group algebra and let A = ⊕ g∈G A g be a strongly G-graded algebra. If A is a C[G]-Galois object, that is A e = C, each component A g is one-dimensional. If we pick a element u g in each A g , the multiplication of A is determined by the products u g u h for each pair g, h of elements of G: for a non vanishing λ(g, h) ∈ C. Associativity of the product requires that λ satisfies a 2-cocycle condition, that is for any g, h ∈ G, λ(g, h)λ(gh, k) = λ(h, k)λ(g, hk).
With a different nonzero element v g ∈ A g , we have v g = µ(g)u g , for some nonzero µ(g) ∈ C. The multiplication (6.4) will become v g v h = λ ′ (g, h)v gh with that is the two 2-cocycles λ ′ and λ are cohomologous. Thus the multiplication in A depends only on the cohomology class of λ ∈ H 2 (G, C × ), the second cohomology group of G with values in C × . Thus, equivalence classes of C[G]-Galois objects are in bijective correspondence with the cohomology group H 2 (G, C × ).
Example 6.4. For any cyclic group G one has H 2 (G, C × ) = 0 and any corresponding C[G]-Galois object is trivial. On the other hand, H 2 (Z r , C × ) = (C × ) r(r−1)/2 for the free abelian group of rank r ≥ 2. Hence, there are infinitely many isomorphism classes of C[Z r ]-Galois objects (see [11,Ex. 7.13]).
Being H = C[G] cocommutative, as mentioned the bialgebroids C(A, H) are all isomorphic to H as Hopf algebra. It is instructive to see this directly. For any u g ⊗u h ∈ C(A, H) the coinvariance condition u g ⊗ u h ⊗ gh = u g ⊗ u h ⊗ 1 H , requires h = g −1 so that C(A, H) is generated as vector space by elements u g ⊗ u g −1 , g ∈ G, with multiplication with µ(g) = λ(g, g −1 ). Consequently, by rescaling the generators u g → v g = λ(g, g −1 ) − 1 2 u g the multiplication rule (6.4) becomes v g v h = λ ′ (g, h) v gh , with λ ′ (g, h) = Λ(g, h) − 1 2 λ(g, h) that we rename back to λ(g, h). As for the bialgebroid product in (6.5) one has, The group of bisections B(C(A, H)) of C(A, H), and the gauge group Aut H (A) of the Galois object A coincide with the group of characters on C[G], which is the same as Hom(G, C × ) the group (for point-wise multiplication) of group morphisms from G to C × . Explicitly, since F ∈ Aut H (A) is linear on A, on a basis {v g } g∈G of A, it is of the form for complex numbers, f h (g) ∈ C. Then, the H-equivariance of F , requires F (v g ) to be contained in to A g and we get f h (g) = 0, if h = g while f g := f g (g) ∈ C × from the invertibility of F . Finally F is an algebra map: implies f gh = f g f h , for any g, h ∈ G. Thus we re-obtain that Aut H (A) ≃ Hom(G, C × ). Note that the requirement F (v e = 1 A ) = 1 = F e implies that F g −1 = (F g ) −1 .
In contrast to this, the group Aut ext H (A) and then B ext (C(A, H)) can be quite big. If F ∈ Aut ext H (A), that is one does not require F to be an algebra map, the corresponding f g can take any value in C × with the only condition that f e = 1.
6.3. Taft algebras. Let N ≥ 2 be an integer and let q be a primitive N-th root of unity. The Taft algebra T N , introduced in [25], is a Hopf algebra which is neither commutative nor cocommutative. Firstly, T N is the N 2 -dimensional unital algebra generated by generators x, g subject to the relations: It is a Hopf algebra with coproduct: counit: ε(x) := 0, ε(g) := 1, and antipode: S(x) := −xg −1 , S(g) := g −1 . The four dimensional algebra T 2 is also known as the Sweedler algebra.
For any s ∈ C, let A s be the unital algebra generated by elements X, G with relations: The algebra A s is a right T N -comodule algebra, with coaction defined by The algebra of corresponding coinvariants is just the ground field C and so A s is a T N -Galois object. It is known (see [13], Prop. 2.17 and Prop. 2.22) that any T N -Galois object is isomorphic to A s for some s ∈ C and that any two such Galois objects A s and A t are isomorphic if and only if s = t. Thus the equivalence classes of T N -Galois objects are in bijective correspondence with the abelian group C. It is easy to see that the translation map of the coaction (6.6) is given on generators by Proof. Again we give a sketch of the explicit proof. It is easy to see that the elements are coinvariants for the right diagonal coaction of T N on A s ⊗ A s and that they generate C(A s , T N ) = (A s ⊗ A s ) co T N as an algebra. These elements satisfy the relations: The last two relations are easy to see. For the first one, one finds when shifting powers of G −1 to the right, for coefficients c r depending on q. Then, using the same methods as in [25], being q a primitive N-th root of unity, all coefficients c r vanish and so Ξ N = X N ⊗ G −N + (−1) N ⊗ (XG −1 ) N which then vanishes from X N = 0. Thus Ξ and Γ generate a copy of the algebra T N and the isomorphism Φ maps Ξ to x and Γ to g. The map Φ is a coalgebra map. Indeed, ∆(Φ(Γ)) = ∆(g) = g ⊗ g, Thus (Φ ⊗ Φ)(∆ C (Ξ)) = 1 ⊗ x + x ⊗ g = ∆(Φ(Ξ)). Finally: ε C (Γ) = 1 = ε(g) and ε C (Ξ) = 0 = ε(x). This concludes the proof.
The group of characters of the Taft algebra T N is the cyclic group Z N : indeed any character φ must be such that φ(x) = 0, while φ(g) N = φ(g N ) = φ(1) = 1. Then for the group of gauge transformations of the Galois object A s -the same as the group of bisections of the bialgebroid C(A s , T N ) -due to Proposition 6.5 we have, (1) for any a ∈ A s , can be given as a block diagonal matrix All matrices M j have in common the diagonal elements a j (ciclic permuted) which are all different from zero for the invertibility of M j . For the subgroup Aut T N (A s ) the M j are diagonal as well with a k = (a 1 ) k and (a 1 ) N = 1 so that M j ∈ Z N . The reason all M j share the same diagonal elements (up to permutation) is the following: firstly, the 'diagonal' form of the coaction of G in (6.6) implies that the image F (G k ) is proportional to G k , say F (G k ) = α k G k for some constant α k . Then, due to the first term in the coaction of X in (6.6), the 'diagonal' component along the basis element X l G k of the image F (X l G k ) is given again by α k for any possible value of the index l. Let us illustrate the construction for the cases of N = 2, 3. Firstly, F (1) = 1 since F is unital. When N = 2, on the basis {1, X, G, XG}, the equivariance F (a) (0) ⊗ F (a) (1) = F (a (0) ) ⊗ a (1) for the coaction (6.6) becomes . Then comparing generators, the equivariance gives while the remaining coefficients are related by the system of equations One readily finds solutions with α, β, γ arbitrary complex numbers. Thus a generic element F of Aut ext T 2 (A s ) can be represented by the matrix: Asking F to be invertible requires α = 0. On the other hand, any F ∈ Aut T 2 (A s ) is an algebra map and so is determined by its values on the generators G, X. From F (G) = αG and F (X) = γG + X: requiring s = F (X 2 ) = (γG + X) 2 = γ + (GX + XG) + s yields γ = 0; then β + αXG = F (XG) = αXG yields β = 0; and 1 = F (G 2 ) = (αG) 2 leads to re-obtain that Aut T 2 (A s ) ≃ Z 2 .
When N = 3 a similar, if longer computation, gives for Aut exp T 3 (A s ) an eight parameter group with its elements of the following block diagonal form One needs α j = 0, j = 1, 2 for invertibility. By going as before, for any F ∈ Aut T N (A s ) one starts from it values on the generators, F (G) = α 1 G and F (X) = γG + X, to conclude that F is a diagonal matrix (in particular F (X) = X) with α 2 = (α 1 ) 2 and 1 = (α 1 ) 3 ; thus Aut T 3 (A s ) ≃ Z 3 .

Crossed module structures on bialgebroids
Automorphisms of a groupoid with its natural transformations form a strict 2-group or, equivalently, a crossed module (see [14,Def. 3.21]). The crossed module combines automorphisms and bisections. A bisection σ is the 2-arrow from the identity morphism to an automorphism Ad σ and the composition of bisections can be viewed as the horizontal composition of 2-arrows. Then the crossed module involves the product on bisections and the composition on automorphisms, and the group homomorphism from bisections to automorphisms together with the action of automorphisms on bisections by conjugation.
In this section we quantise this construction for the Ehresmann-Schauenburg bialgebroid of a Hopf-Galois extension. We construct a crossed module for the bisections and the automorphisms of the bialgebroid, thus giving a generalization of a crossed module on a groupoid. Notice that the antipode of the bialgebroid is not needed in the construction. 7.1. Crossed modules and bisections. A crossed module is the data (M, N, µ, α) of two groups M, N together with group homomorphisms µ : M → N and α : N → Aut(M) such that, denoting α n : M → M for every n ∈ N, the following conditions are satisfied: (1) µ(α n (m)) = nµ(m)n −1 , for any n ∈ N and m ∈ M , We aim at proving the following. We give the proof in a few lemmas.
for any a ⊗ã ∈ C(A, H), is a vertical automorphism of C(A, H).
Proof. Since F σ is an algebra automorphism so is Ad σ . Then, for any b ∈ B it is immediate to show that Ad σ (t(b)) = t(b) and Ad σ (s(b)) = s(b). So conditions (2.6) are satisfied. The second one also shows that Ad σ is vertical, that is ε • Ad σ • s = id B , and then ε • Ad σ = ε. For the first condition in (2.7) the H-equivariance of F σ yields for any a ⊗ã ∈ C(A, H). On the other hand, Now, for any F ∈ Aut H (A), given h ∈ H, one has as can be seen by applying the canonical map χ (an isomorphism) and using equivariance of F . Using it for the right hand sides of (7.2) and (7.3) shows that they coincide. Thus the left hand side expressions coincide and the first of (2.7) is satisfied as well.
Proof. Recall that Ad τ is vertical. With X ∈ C(A, H), using definition (7.4) we compute: where we used the definition (4.6) for the product.
Taken together the previous lemmas establish the content of Theorem 7.1 that is a crossed module structure for B(C(A, H)), Aut(C(A, H)), Ad, ⊲ . We have seen that for a Galois object A of a Hopf algebra H, the bialgebroid C(A, H) is a Hopf algebra. Also, the group of gauge transformations of the Galois object, which is the same as the group of bisections B(C(A, H), is the group of characters of C(A, H) (see (6.2)). It turns out that these groups are also isomorphic to CoInn(C(A, H)). If φ ∈ Char(C(A, H)), substituting φ −1 = φ • S C in (7.4), for X = a ⊗ a ′ ∈ C(A, H), we get Ad φ (X) = φ(X (1) ) X (2) (φ • S C )(X (3) ) = coinn(φ)(X). (7.10) As a particular case, let us consider again the Taft algebra T N . We know from Sect. 6.3 that for any T N -Galois object A s the bialgebroid C(T N , A s ) is isomorphic to T N . And bisections of C(T N , A s ) are the same as characters of T N the group of which is isomorphic to Z N . A generic character is a map φ r : T N → C, given on generators x and g by φ r (x) = 0 and φ r (g) = r for r a N-root of unity r N = 1. The corresponding automorphism Ad φr = coinn(φ r ) is easily found to be on generators given by coinn(φ r )(g) = g, coinn(φ r )(x) = r −1 x .
It is known [20, Lem. 2.1] that Aut(T N ) ≃ Aut Hopf (T N ) ≃ C × : Indeed, given r ∈ C × , one defines an automorphism F r : T N → T N by F r (x) := rx and F r (g) := g. Thus Ad : Char(T N ) → Aut(T N ) is the injection sending φ r to F r −1 . Moreover, for F ∈ Aut(T N ) and φ ∈ Char(T N ), one checks that Ad F ⊲φ (x) = Ad φ (x) and Ad F ⊲φ (g) = Ad φ (g). Thus, 29 as a crossed module, the action of Aut(T N ) on Char(T N ) is the identity and the crossed module (Char(C(T N , A s )), Aut(C(T N , A s )), Ad, id) is isomorphic to (Z N , C × , j, id), with inclusion j : Z N → C × given by j(r) := e −i2rπ/N and C × acting trivially on Z N . 7.2. More general bisections. In parallel with the crossed module structure on bialgebroid automorphisms and bisections, there is a similar structure on extended bisections as in Definition 4.7 and 'extended bialgebroid automorphisms'. These are pairs (Φ, ϕ) with ϕ : B → B an algebra automorphism and Φ : L → L an unital invertible linear map, not required in general to be an algebra map, satisfying the equivariance properties in (2.7), while (2.6) is replaced by the bimodule property: Φ(b ⊲ a ⊳b) = ϕ(b) ⊲ Φ(a) ⊳ ϕ(b), for b,b ∈ B. They form a group Aut ext (L) by map composition. We sketch the construction that goes in the lines of Theorem 7.1. Given any bisection σ ∈ B ext (C (A, H)), definition (7.4) gives a map Ad σ : C(A, H) → C(A, H) that we repeat, Ad σ (a ⊗ã) = σ((a ⊗ã) (1) ) ⊲ (a ⊗ã) (2) ⊳ σ −1 ((a ⊗ã) (3) ). (7.11) This still covers the identity of B, that is ε • Ad σ • s = id B , but is not an algebra map in general; it is an extended automorphism of C(A, H). Indeed, it satisfies the properties (2.6) and (2.7), and for two extended bisections σ and τ one shows as before that Ad σ • Ad τ = Ad τ * σ , and thus Ad σ is invertible with inverse Ad σ −1 . Moreover, if Φ is an extended automorphism of C(A, H) with inverse Φ −1 the formula (7.5) is an action of Φ on B ext (C (A, H)), a group automorphism of B ext (C(A, H)). One really needs only to check that Φ ⊲ σ is well defined as an extended bisection since the rest goes as in the previous section. And with similar computations as those of Lemmas 7.5 and 7.6 one shows that Ad Φ⊲σ = Φ −1 • Ad σ • Φ, for any extended automorphism Φ and any σ ∈ B ext (C(A, H)), and that Ad τ ⊲ σ = τ * σ * τ −1 , with any σ, τ ∈ B ext (C (A, H)).
We sum up the above with an analogous of Theorem 7.1: for any X = a ⊗ã ∈ C(A, H). This reduces to (7.10) when σ = φ is a character.
Example 7.10. In Example 7.7 we gave an abelian crossed module for the Taft algebras. The use of extended characters and extended automorphisms yields a non-abelian crossed module. As we know the bialgebroid C(A s , T N ) of a Galois object A s for the Taft algebra T N is isomorphic to T N . Thus Aut ext (C(A s , T N )) ≃ Aut ext (T N ) is the group of unital invertible maps Φ : T N → T N such that Φ(h (1) ) ⊗ Φ(h (2) ) = Φ(h) (1) ⊗ Φ(h) (2) for h ∈ T N .

30
Let us illustrate this for the case N = 2. The coproduct of T 2 on the generators x, g will then require the following condition for an automorphism Φ: A little algebra then shows that Φ(g) = g , Φ(x) = c (g − 1) + a 2 x , Φ(xg) = b (1 − g) + a 1 xg for arbitrary parameters b, c ∈ C and a 1 , a 2 ∈ C × (for Φ to be invertible). As in (6.8) we can represent Φ as a matrix: Φ : One checks that matrices M Φ of the form above form a group: Aut ext (T N ) ≃ Aut Hopf (T N ).
Given σ ∈ Char ext (T 2 ) we shall denote σ a = σ(a) ∈ C for a ∈ {1, x, g, xg}. For the convolution inverse σ −1 , from the condition σ * σ −1 = ε we get on the basis that            Then, computing Ad σ (h) = σ(h (1) )h (2) σ −1 (h (3) ) leads to We see that the matrix (7.13) is of the form (7.12) with the restriction that a 2 = a −1 1 so that Ad φ has determinant 1. Clearly, the image of Char ext (T 2 ) form a subgroup of Aut ext (C(A s , T 2 )) ≃ Aut ext (T N ). Moreover, Ad : Char ext (T 2 ) → Aut ext (T 2 ) is an injective map. Finally, the action Ad Φ⊲σ is represented by the matrix product: We conclude that as a crossed module the action on Char ext (T 2 ) is not trivial.