aa r X i v : . [ m a t h . QA ] F e b ON THE GAUGE GROUP OF GALOIS OBJECTS
XIAO HAN, GIOVANNI LANDI
Abstract.
We study the Ehresmann–Schauenburg bialgebroid of a noncommutativeprincipal bundle as a quantization of the classical gauge groupoid of a principal bundle.When the base algebra is in the centre of the total space algebra, the gauge group of thenoncommutative principal bundle is isomorphic to the group of bisections of the bial-gebroid. In particular we consider Galois objects (non-trivial noncommutative bundlesover a point in a sense) for which the bialgebroid is a Hopf algebra. For these we give acrossed module structure for the bisections and the automorphisms of the bialgebroid.Examples include Galois objects of group Hopf algebras and of Taft algebras.
Contents
1. Introduction 12. Algebraic preliminaries 32.1. Algebras, coalgebras and all that 32.2. Hopf–Galois extensions 52.3. The gauge groups 73. Ehresmann–Schauenburg bialgebroids 93.1. Ehresmann corings 93.2. The groups of bisections 103.3. Bisections and gauge groups 143.4. Extended bisections and gauge groups 154. Bisections and gauge groups of Galois objects 174.1. Galois objects 174.2. Hopf algebras as Galois objects 184.3. Cocommutative Hopf algebras 194.4. Group Hopf algebras 204.5. Taft algebras 215. Crossed module structures on bialgebroids 255.1. Automorphisms and crossed modules 255.2. CoInner authomorphisms of bialgebroids 285.3. Crossed module structures on extended bisections 29Appendix A. The classical gauge groupoid 32References 331.
Introduction
The study of groupoids on one hand and gauge theories on the other hand is importantin different area of mathematics and physics. In particular these subjects meet in thenotion of the gauge groupoid associated to a principal bundle. In view of the considerableamount of recent work on noncommutative principal bundles it is important to come
Date : 14 February 2020. p with a noncommutative version of groupoids and their relations to noncommutativeprincipal bundles, for which one needs to have a better understanding of bialgebroids.In this paper, we will consider the Ehresmann–Schauenburg bialgebroid associated witha noncommutative principal bundle as a quantization of the classical gauge groupoid.Classically, bisections of the gauge groupoid are closely related to gauge transformations.Here we show that under some conditions on the base space algebra of the noncommu-tative principal bundle, the gauge group of the principal bundle is group isomorphic tothe group of bisections of the corresponding Ehresmann–Schauenburg bialgebroid.For a bialgebroid there is a notion of coproduct and counit but in general not of anantipode. An antipode can be defined for the Ehresmann–Schauenburg bialgebroid of aGalois object which (the bialgebroid that is) is then a Hopf algebra. Now, with H a Hopfalgebra, a H -Galois object is a noncommutative principal bundle over a point in a sense:a H -Hopf–Galois extension of the ground field C . In contrast to the classical situationwere a bundle over a point is trivial, for the isomorphism classes of noncommutativeprincipal bundles over a point this need not be the case. Notable examples are groupHopf algebras C [ G ], whose corresponding principal bundle are C [ G ]-graded algebras andare classified by the cohomology group H ( G, C × ), and Taft algebras T N ; the equivalenceclasses of T N -Galois objects are in bijective correspondence with the abelian group C .Thus the central part of the paper is dedicated to the Ehresmann–Schauenburg bial-gebroid of a Galois object and to the study of the corresponding groups of bisections, bethey algebra maps from the bialgebroid to the ground field (and thus characters) or moregeneral transformations. These are in bijective correspondence with the group of gaugetransformations of the Galois object. We study in particular the case of Galois objects for H a general cocommutative Hopf algebra and in particular a group Hopf algebra. A niceclass of examples comes from Galois objects and corresponding Ehresmann–Schauenburgbialgebroids for the Taft algebras T N an example that we work out in full details.Automorphisms of a (usual) groupoid with natural transformations form a strict 2-group or, equivalently, a crossed module. The crossed module involves the productof bisections and the composition of automorphisms, together with the action of au-tomorphisms on bisections by conjugation. Bisections are the 2-arrows from the identitymorphisms to automorphisms, and the composition of bisections can be viewed as thehorizontal composition of 2-arrows. In the present paper this construction is extendedto the Ehresmann–Schauenburg bialgebroid of a Hopf–Galois extension by constructinga crossed module for the bisections and the automorphisms of the bialgebroid.The paper is organised as follow: In § §
3, we first have Ehresmann–Schauenburg bialgebroids and thegroup of their bisections, then we show that when the base algebra belongs to the centreof the total space algebra, the gauge group of a noncommutative principal bundle is groupisomorphic to the group of bisections of the corresponding Ehresmann–Schauenburg bial-gebroid. In § § his can generate the representation theory of 2-groups (or crossed modules) on Hopfalgebras. We work out in details this construction for the Taft algebras.2. Algebraic preliminaries
We recall here know facts from algebras and coalgebras and corersponding modulesand comodules. We also recall the more general notions of rings and corings over analgebra as well as the associated notion of bialgebroid. We move then to Hopf–Galoisextensions, as noncommutative principal bundles and the definitions of gauge groups.2.1.
Algebras, coalgebras and all that.
To be definite we work over the field C ofcomplex numbers but in the following this could be substituted by any field k . Alge-bras are assumed to be unital and associative with morphisms of algebras taken to beunital, and co-algebras are assumed to be counital and coassociative with morphism ofco-algebras taken to be co-unital. For the coproduct of a coalgebra ∆ : H → H ⊗ H we use the Sweedler notation ∆( h ) = h (1) ⊗ h (2) (sum understood), and its iterations:∆ n = (id ⊗ ∆ H ) ◦ ∆ n − H : h h (1) ⊗ h (2) ⊗ · · · ⊗ h ( n +1) . We denote by ∗ the convolutionproduct in the dual vector space H ′ := Hom( H, C ), ( f ∗ g )( h ) := f ( h (1) ) g ( h (2) ).Given an algebra A , a left A -module is a vector space V carrying a left A -action, thatis with a C -linear map ⊲ V : A ⊗ V → V such that( ab ) ⊲ V v = a ⊲ V ( b ⊲ V v ) , ⊲ V v = v . (2.1)Dually, with a bialgebra H , a right H -comodule is a vector space V carrying a right H -coaction, that is with a C -linear map δ V : V → V ⊗ H such that(id ⊗ ∆) ◦ δ V = ( δ V ⊗ id) ◦ δ V , (id ⊗ ǫ ) ◦ δ V = id . (2.2)In Sweedler notation, v δ V ( v ) = v (0) ⊗ v (1) , and the right H -comodule properties read, v (0) ⊗ ( v (1) ) (1) ⊗ ( v (1) ) (2) = ( v (0) ) (0) ⊗ ( v (0) ) (1) ⊗ v (1) =: v (0) ⊗ v (1) ⊗ v (2) ,v (0) ǫ ( 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 δ V ⊗ W : V ⊗ W −→ V ⊗ W ⊗ H , v ⊗ w v (0) ⊗ w (0) ⊗ v (1) w (1) . (2.3)a H -comodule map ψ : V → W between two H -comodules is a C -linear map ψ : V → W which is H -equivariant (or H -colinear), that is, δ W ◦ ψ = ( ψ ⊗ id) ◦ δ V .In particular, a right H -comodule algebra is an algebra A which is a right H -comodulesuch that the multiplication and unit of A are morphisms of H -comodules. This isequivalent 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 morphismsare 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 H -comodules. Explicitly, this means that, for each c ∈ C ,( c (1) ) (0) ⊗ ( c (2) ) (0) ⊗ ( c (1) ) (1) ( c (2) ) (1) = ( c (0) ) (1) ⊗ ( c (0) ) (2) ⊗ c (1) ,ǫ ( c (0) ) c (1) = ǫ ( c )1 H . Corresponding morphisms are H -comodule maps which are also coalgebra maps. Clearly,there are right A -modules and left H -comodule versions of the above ones. ext, let H be a bialgebra 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 ⊲ V : A ⊗ V → V is a morphism of H -comodules such that δ V ◦ ⊲ V = ( ⊲ V ⊗ id) ◦ δ A ⊗ V . Explicitly, for all a ∈ A and v ∈ V ,( a ⊲ V v ) (0) ⊗ ( a ⊲ V v ) (1) = a (0) ⊲ V v (0) ⊗ a (1) v (1) . (2.4)A morphism of ( A, H )-relative Hopf modules is a morphism of right H -comodules whichis also a morphism of left A -modules. In a similar way one can consider the case for thealgebra A to be acting on the right, or with a left and right A -actions.Finally, we denote by ⊗ the tensor product over C . If A is an algebra and M is a A -bimodule, the corresponding balanced tensor product M ⊗ A M is given by M ⊗ A M := M ⊗ M/ h m ⊗ am ′ − ma ⊗ m ′ i m,m ′ ∈ M, a ∈ A . We end this part on preliminaries by recalling the notions of bialgebroids (cf. [6], [5]).For an algebra B a B -ring is a triple ( A, µ, η ). Here A is a B -bimodule with B -bimodulemaps µ : A ⊗ B A → A and η : B → A , satisfying the following associativity µ ◦ ( µ ⊗ B id A ) = µ ◦ (id A ⊗ B µ ) (2.5)and unit conditions, µ ◦ ( η ⊗ B id A ) = id A = µ ◦ (id A ⊗ B η ) . (2.6)A morphism of B -rings f : ( A, µ, η ) → ( A ′ , µ ′ , η ′ ) is an B -bimodule map f : A → A ′ ,such that f ◦ µ = µ ′ ◦ ( f ⊗ B f ) and f ◦ η = η ′ .From [6, Lemma 2.2] there is a bijective correspondence between B -rings ( A, µ, η )and algebra morphisms η : B → A . Starting with a B -ring ( A, µ, η ), one obtains amultiplication 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 -bilinearassociative multiplication µ : A ⊗ B A → A is obtained from the universality of thecoequaliser A ⊗ A → A ⊗ B A which identifies an element ar ⊗ a ′ with a ⊗ ra ′ .Dually, for an algebra B a B -coring is a triple ( C, ∆ , ǫ ). Here C is a B -bimodule with B -bimodule maps ∆ : C → C ⊗ B C and ǫ : C → B , satisfying the following coassociativityand counit conditions,(∆ ⊗ B id C ) ◦ ∆ = (id C ⊗ B ∆) ◦ ∆ , ( ǫ ⊗ B id C ) ◦ ∆ = id C = (id C ⊗ B ǫ ) ◦ ∆ . (2.7)A morphism of B -corings f : ( C, ∆ , ǫ ) → ( C ′ , ∆ ′ , ǫ ′ ) is a B -bimodule map f : C → C ′ ,such that ∆ ′ ◦ f = ( f ⊗ B f ) ◦ ∆ and ǫ ′ ◦ f = ǫ .Finally, let B be an algebra. A left B -bialgebroid L consists of an ( B ⊗ B op )-ringtogether with a B -coring structures on the same vector space L with mutual compatibilityconditions. From what said above, an ( B ⊗ B op )-ring L is the same as an algebra map η : B ⊗ B op → L . Equivalently, one may consider the restrictions s := η ( · ⊗ B B ) : B → L and t := η (1 B ⊗ B · ) : B op → L 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.Thus, for a left B -bialgebroid L the compatibility conditions are required to be i) The bimodule structures in the B -coring ( L , ∆ , ǫ ) are related to those of the B ⊗ B op -ring ( L , s, t ) via b ⊲ a ⊳ b ′ := s ( b ) t ( b ′ ) a for b, b ′ ∈ B, a ∈ L . (2.8)(ii) Considering L as a B -bimodule as in (2.8), the coproduct ∆ corestricts to analgebra map from L to L × B L := n X j a j ⊗ B a ′ j | X j a j t ( b ) ⊗ B a ′ j = X j a j ⊗ B a ′ j s ( b ) , for all b ∈ B o , (2.9)where L × B L is an algebra via component-wise multiplication.(iii) The counit ǫ : L → B is a left character on the B -ring ( L , s, t ), that is it satisfiesthe properties, for b ∈ B and a, a ′ ∈ L ,(1) ǫ (1 L ) = 1 B , (unitality)(2) ǫ ( s ( b ) a ) = bǫ ( a ), (left B -linearity)(3) ǫ ( as ( ǫ ( a ′ ))) = ǫ ( aa ′ ) = ǫ ( at ( ǫ ( a ′ ))), (associativity) .We finish this part with an additional concept that we shall use later on in Section 5.1. Definition 2.1.
Let ( L , ∆ , ǫ, s, t ) be a left bialgebroid over the algebra B . An auto-morphism of the bialgebroid L is a pair (Φ , ϕ ) of algebra automorphisms, Φ :
L → L , ϕ : B → B such that:(i) Φ ◦ s = s ◦ ϕ ;(ii) Φ ◦ t = t ◦ ϕ ;(iii) (Φ ⊗ B Φ) ◦ ∆ = ∆ ◦ Φ ;(iv) ǫ ◦ Φ = ϕ ◦ ǫ . In fact the map ϕ is uniquely determined by Φ via ϕ = ǫ ◦ Φ ◦ s and one can just saythat Φ is a bialgebroid automorphism. Automorphisms of a bialgebroid L form a groupby composition that we simply denote Aut( L ). Remark . Here the pair of algebra maps (Φ , ϕ ) can be viewed as a bialgebroid map(cf. [15], § L with different source and target map (and so B -bimodule structure). If s, t are the source and target maps on L , one defines new sourceand target maps on L by s ′ := s ◦ ϕ and t ′ := t ◦ ϕ with the new bimodule structure givenby b ⊲ ϕ c ⊳ ϕ ˜ b := s ′ ( b ) t ′ (˜ b ) a , for any b, ˜ b ∈ B and a ∈ L (see (2.8)). Therefore we get anew left bialgebroid with product, unit, coproduct and counit not changed.Clearly, from conditions (i) and (ii) Φ is a B -bimodule map: Φ( b ⊲ c ⊳ ˜ b ) = b ⊲ ϕ Φ( c ) ⊳ ϕ ˜ b .The condition (iii) is well defined once the conditions (i) and (ii) are satisfied (the balancedtensor product in (iii) is induced by s ′ and t ′ ). Condition (iii) and (iv) imply Φ is a coringmap, therefore (Φ , ϕ ) is an isomorphism between the starting and the new bialgebroids.2.2. Hopf–Galois extensions.
In this section we will give a brief recall of Hopf–Galoisextensions, as noncommutative principal bundles. These are H -comodule algebras A witha canonically defined map χ : A ⊗ B A → A ⊗ H which is required to be invertible. Definition 2.3.
Let H be a Hopf algebra and let A be a H -comodule algebra with coaction δ A . Consider the subalgebra B := A coH = (cid:8) b ∈ A | δ A ( b ) = b ⊗ H (cid:9) ⊆ A of coinvariantelements and the corresponding balanced tensor product A ⊗ B A . The extension B ⊆ A is called a H - Hopf–Galois extension if the canonical Galois map χ := ( m ⊗ id) ◦ (id ⊗ B δ A ) : A ⊗ B A −→ A ⊗ H , a ′ ⊗ B a a ′ a (0) ⊗ a (1) (2.10) s bijective.Remark . In the following, we shall always implicitly assume that for the Hopf Galoisextension B ⊆ A , the algebra A is faithfully flat as a left B -module. This means thattaking the tensor product ⊗ B A with a sequence of right B -modules produces an exactsequence if and only if the original sequence is exact. Finite-rank, free or projectivemodules are examples of faithfully flat modules.The canonical map χ is a morphism of relative Hopf modules for A -bimodules and right H -comodules (cf. [1]). Both A ⊗ B A and A ⊗ H are A -bimodules. The left A -modulestructures are the left multiplication on the first factors while the right A -actions are:( a ⊗ B a ′ ) a ′′ := a ⊗ B a ′ a ′′ and ( a ⊗ h ) a ′ := aa ′ (0) ⊗ ha ′ (1) . As for the H -comodule structure, the natural right tensor product H -coaction as in (2.3): δ A ⊗ A : A ⊗ A → A ⊗ A ⊗ H, a ⊗ a ′ a (0) ⊗ a ′ (0) ⊗ a (1) a ′ (1) (2.11)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 oneregards the Hopf algebra H as a right H -comodule with the right adjoint H -coactionAd : h h (2) ⊗ S ( h (1) ) h (3) . (2.12)The right H -coaction on A ⊗ H is then given, for all a ∈ A, h ∈ H by δ A ⊗ H ( a ⊗ h ) = a (0) ⊗ h (2) ⊗ a (1) S ( h (1) ) h (3) ∈ A ⊗ H ⊗ H . (2.13)Since the canonical Galois map χ is left A -linear, its inverse is determined by therestriction τ := χ − | A ⊗ H , named translation map , τ = χ − | A ⊗ H : H → A ⊗ B A , h h < > ⊗ B h < > . The translation map enjoys a number of properties that we list here for later use. Firstly,it was shown in [4, Prop. 3.6] that,(id ⊗ B δ A ) ◦ τ = ( τ ⊗ id) ◦ ∆ , ( τ ⊗ S ) ◦ flip ◦ ∆ = (id ⊗ flip) ◦ ( δ A ⊗ B id) ◦ τ . On an element h ∈ H these respectively read h < > ⊗ B h < > (0) ⊗ h < > (1) = h (1) < > ⊗ B h (1) < > ⊗ h (2) , (2.14) h < > (0) ⊗ B h < > ⊗ h < > (1) = h (2) < > ⊗ B h (2) < > ⊗ S ( h (1) ) . (2.15)Furthermore, from [5, Lemma 34.4], for any a ∈ A and h, k ∈ H , we have the following: h < > h < > (0) ⊗ h < > (1) = 1 A ⊗ h , (2.16) h < > h < > = ǫ ( h )1 A , (2.17)( hk ) < > ⊗ B ( hk ) < > = k < > h < > ⊗ B h < > k < > , (2.18) h (1) < > ⊗ B h (1) < > h (2) < > ⊗ B h (2) < > = h < > ⊗ B A ⊗ B h < > , (2.19) a (0) a (1) < > ⊗ B a (1) < > = 1 A ⊗ B a , (2.20)for any h, k ∈ H and a ∈ A .Two H -Hopf–Galois extensions A, A ′ of a fixed algebra B are isomorphic provided thereexists an isomorphism of H -comodule algebras A → A ′ . This is the algebraic counterpartfor noncommutative principal bundles of the geometric notion of isomorphism of principal G -bundles with fixed base space. As in the geometric case this notion is relevant in theclassification of noncommutative principal bundles, (cf. [8]). n the present paper we shall be mainly interested in Galois object : given a Hopf algebra H , a H -Galois object is a H -Hopf–Galois extension of C . These could be thought of anoncommutative principal bundle over a point. It is well known (cf. [8]) that the setGal H ( C ) of isomorphic classes of H -Galois objects need not be trivial. This is in contrastto the fact that any (usual) fibre bundle over a point is trivial.2.3. The gauge groups.
In [4] gauge transformations for a noncommutative principalbundles were defined to be invertible and unital comodule maps, with no additionalrequirement. In particular they were not asked to be algebra morphisms. A drawback ofthis approach is that the resulting gauge group might be very big, even in the classicalcase; for example the gauge group of the a G -bundle over a point would be much biggerthan the structure group G . On the other hand, in [2] gauge transformations were requiredto be algebra homomorphisms. This implies in particular that they are invertible.In the line of the paper [2] we are lead to the following definition. Definition 2.5.
Given a Hopf–Galois extension B = A coH ⊆ A . Consider the collection Aut H ( A ) := { F ∈ Hom A H ( A, A ) | F | B ∈ Aut( B ) } , (2.21) of right H -comodule unital algebra morphisms of A which restrict to automorphisms of B , and the sub-collection Aut ver ( A ) := { F ∈ Aut H ( A ) | F | B = id B } . (2.22) of ‘vertical’ ones, that is that in addition are left B -module morphisms. Thus elements F ∈ Aut H ( A ) preserve the (co)-action of the structure quantum groupsince they are such that δ A ◦ F = ( F ⊗ id) δ A (or F ( a ) (0) ⊗ F ( a ) (1) = F ( a (0) ) ⊗ a (1) ). And ifin Aut ver ( A ) they also preserve the base space algebra B . These will be called the gaugegroup and the vertical gauge group respectively: in parallel with [2, Prop. 3.6], Aut H ( A )and Aut ver ( A ) are groups when B is restricted to be in the centre of A by the followingproposition: Proposition 2.6.
Let B = A coH ⊆ A be a H -Hopf–Galois extension with B in the centreof A . Then Aut H ( A ) is a group with respect to the composition of maps F · G := G ◦ F for all F, G ∈ Aut H ( A ) . For F ∈ Aut H ( A ) its inverse F − ∈ Aut H ( A ) is given by F − := m ◦ (( F | B ) − ⊗ id) ◦ ( m ⊗ id) ◦ (id ⊗ F ⊗ B id) ◦ (id ⊗ τ ) ◦ δ A (2.23) where τ is the translation map, that is for all a ∈ A , F − ( a ) := ( F | B ) − (cid:0) a (0) F ( a (1) < > ) (cid:1) a (1) < > . (2.24) In particular the vertical homomorphisms
Aut ver ( A ) form a subgroup of Aut H ( A ) .Proof. The group multiplication is clearly well defined with unit the identity map on A .Next, we compute that somewhat ‘implicitly’, a (0) F ( a (1) < > ) ⊗ B a (1) < > ∈ B ⊗ B A . Indeed( δ A ⊗ B id A )( a (0) F ( a (1) < > ) ⊗ B a (1) < > ) = a (0)(0) F ( a (1) < > ) (0) ⊗ a (0)(1) F ( a (1) < > ) (1) ⊗ B a (1) < > = a (0) F ( a (2) < > (0) ) ⊗ a (1) a (2) < > (1) ⊗ B a (2) < > = a (0) F ( a (2)(2) < > ) ⊗ a (1) S ( a (2)(1) ) ⊗ B a (2)(2) < > = a (0) F ( a (1) < > ) ⊗ H ⊗ B a (1) < > , here the 2nd step uses that F is H -equivalent map, the 3rd step uses (2.15); since δ A isright B –linear, everything is well defined. Now, being B the coinvariant subalgebra of A for the coaction, we have the exact sequence,0 −→ B i −→ A δ A − id A ⊗ id H −→ A ⊗ H −→ A is faithful flat as left B -module, we also have exactness of the sequence,0 −→ B ⊗ B A i ⊗ B id A −→ A ⊗ B A ( δ A − id A ⊗ id H ) ⊗ B id A −→ A ⊗ H ⊗ B A −→ . Thus, the equality ( δ A ⊗ B id A )( a (0) F ( a (1) < > ) ⊗ B a (1) < > ) = a (0) F ( a (1) < > ) ⊗ H ⊗ B a (1) < > shows that a (0) F ( a (1) < > ) ⊗ B a (1) < > ∈ B ⊗ B A and thus F − in (2.24) is well defined. Letus check that F − is an algebra map: F − ( aa ′ ) = ( F | B ) − (cid:0) ( aa ′ ) (0) F (( aa ′ ) (1) < > ) (cid:1) ( aa ′ ) (1) < > = ( F | B ) − (cid:0) a (0) a ′ (0) F ( a ′ (1) < > ) F ( a (1) < > ) (cid:1) a (1) < > a ′ (1) < > = F − ( a ) F − ( a ′ ) , where the 2nd step uses (2.18), and the last step uses the fact that a ′ (0) F ( a ′ (1) < > ) ⊗ B a ′ (1) < > ∈ B ⊗ B A and B belongs to the centre of A , thus F − is an algebra map. Also forany b ∈ B , F − ( b ) = ( F | B ) − ( b ), so F − | B ∈ Aut( B ). Then, for any a ∈ AF − ( F ( a )) = ( F | B ) − (cid:0) F ( a ) (0) F ( F ( a ) (1) < > ) (cid:1) F ( a ) (1) < > = ( F | B ) − (cid:0) F ( a (0) ) F ( a (1) < > ) (cid:1) a (1) < > = ( F | B ) − (cid:0) F ( a (0) a (1) < > ) (cid:1) a (1) < > = a, where the 2nd step uses the H -equivariance of F , and the last step uses (2.20). Finally, F ( F − ( a )) = F (cid:0) ( F | B ) − ( a (0) F ( a (1) < > )) a (1) < > (cid:1) = a (0) F ( a (1) < > ) F ( a (1) < > )= a. Thus F − is the inverse map of F ∈ Aut H ( A ). The map F − is H -equivariant as well soit belongs to Aut H ( A ). Indeed, for any a ∈ A we have a (0) ⊗ a (1) = F ( F − ( a )) (0) ⊗ F ( F − ( a )) (1) = F ( F − ( a ) (0) ) ⊗ F − ( a ) (1) , where the last step uses the H -equivariance of F . Applying F − ⊗ id H on both sides ofthe last equation we get F − ( a (0) ) ⊗ a (1) = F − ( a ) (0) ⊗ F − ( a ) (1) so F − is H -equivariant. We conclude that Aut H ( A ) is a group.As for the vertical automorphisms, clearly Aut ver ( A ) is closed for map compositions andone sees that F − ∈ Aut ver ( A ) when F ∈ Aut ver ( A ). Thus Aut ver ( A ) is also a group. (cid:3) Remark . A similar proposition was first given in [2], for a Hopf algebra H which is acoquasitriangular Hopf algebra, and A is a quasi-commutative H -comodule algebra. Asa consequence, B belongs to the centre of A . In the present paper, we only require B tobelongs to the centre of A without assuming H to be a coquasitriangular Hopf algebra.For the sake of the present paper, where we are concerned mainly with Galois objects,and seek to study their gauge groups with relations to bisections of suitable groupoids,there is no restriction in assuming that the base space algebra B be in the centre. . Ehresmann–Schauenburg bialgebroids
To any Hopf–Galois extension B = A co H ⊆ A one associates a B -coring [5, § § Ehresmann corings.
The coring can be given in few equivalent ways. Let B = A co H ⊆ A be a Hopf–Galois extension with right coaction δ A : A → A ⊗ H . Recall thediagonal coaction (2.11), given for all a, a ′ ∈ A by δ A ⊗ A : A ⊗ A → A ⊗ A ⊗ H, a ⊗ a ′ a (0) ⊗ a ′ (0) ⊗ a (1) a ′ (1) , with corresponding B -bimodule of coinvariant elements,( A ⊗ A ) coH = { a ⊗ ˜ a ∈ A ⊗ A ; a (0) ⊗ ˜ a (0) ⊗ a (1) ˜ a (1) = a ⊗ ˜ a ⊗ H } . (3.1) Lemma 3.1.
Let τ be the translation map of the Hopf–Galois extension. Then the B -bimodule of coinvariant elements in (3.1) is the same as the B -bimodule, C := { a ⊗ ˜ a ∈ A ⊗ A : a (0) ⊗ τ ( a (1) )˜ a = a ⊗ ˜ a ⊗ B A } . (3.2) Proof.
Let a ⊗ ˜ a ∈ ( A ⊗ A ) coH . By applying (id A ⊗ χ ) on a (0) ⊗ a (1) < > ⊗ B a (1) < > ˜ a , we get a (0) ⊗ a (1) < > a (1) < > (0) ˜ a (0) ⊗ a (1) < > (1) ˜ a (1) = a (0) ⊗ ˜ a (0) ⊗ a (1) ˜ a (1) = a ⊗ ˜ a ⊗ H = a ⊗ χ (˜ a ⊗ B A )= (id A ⊗ χ )( a ⊗ ˜ a ⊗ B A ) , where the first step uses (2.16). This shows that ( A ⊗ A ) coH ⊆ C .Conversely, let a ⊗ ˜ a ∈ C . By applying (id A ⊗ χ − ) on a (0) ⊗ ˜ a (0) ⊗ a (1) ˜ a (1) and using thefact that χ − is left A -linear and (2.18), we get a (0) ⊗ ˜ a (0) ˜ a (1) < > a (1) < > ⊗ B a (1) < > ˜ a (1) < > = a (0) ⊗ a (1) < > ⊗ B a (1) < > ˜ a = a ⊗ ˜ a ⊗ B A = (id A ⊗ χ − )( a ⊗ ˜ a ⊗ H ) , where in the first step (2.20) is used. This shows that C ⊆ ( A ⊗ A ) coH . (cid:3) We have then the following definition [5, § Definition 3.2.
Let B = A co H ⊆ A be a Hopf–Galois extension with translation map τ .If A is faithful flat as a left B -module, then the B -bimodule C in (3.2) is a B -coring withcoring coproduct ∆( a ⊗ ˜ a ) = a (0) ⊗ τ ( a (1) ) ⊗ ˜ a = a (0) ⊗ a (1) < > ⊗ B a (1) < > ⊗ ˜ a, (3.3) and counit ǫ ( a ⊗ ˜ a ) = a ˜ a. (3.4)By applying the map m A ⊗ id H to elements of (3.1), it is clear that a ˜ a ∈ B . The above B -coring is called the Ehresmann or gauge coring ; we denote it C ( A, H ).Whenever the structure Hopf algebra H has an invertible antipode, the Ehresmanncoring can also be given as an equaliser (see [7]). Indeed, let H be a Hopf algebra withinvertible antipode. And let B = A co H ⊆ A be a H -Hopf–Galois extension, with rightcoaction δ A : A → A ⊗ H , a δ A ( a ) = a (0) ⊗ a (1) . Via the inverse of S one has alsoa left H coaction A δ : A → H ⊗ A , A δ ( a ) := S − ( a (1) ) ⊗ a (0) . One also shows that := coH A = (cid:8) b ∈ A | A δ ( b ) = 1 H ⊗ b (cid:9) . Using the left B -linearity of δ A and the right B -linearity of A δ one has a B -bimodule, A H (cid:3) H A = ker( δ A ⊗ id A − id A ⊗ A δ )= (cid:8) a ⊗ ˜ a ∈ A ⊗ A : a (0) ⊗ a (1) ⊗ ˜ a = a ⊗ S − (˜ a (1) ) ⊗ ˜ a (0) (cid:9) (3.5) Lemma 3.3.
The bimodule A H (cid:3) H A is the same as the bimodules C and ( A ⊗ A ) coH .Proof. Let a ⊗ ˜ a ∈ C . Then, by applying (id A ⊗ id H ⊗ m A ) ◦ (id A ⊗ A δ ⊗ id A ) on a (0) ⊗ a (1) < > ⊗ B a (1) < > ˜ a = a ⊗ ˜ a ⊗ B A we get, for the left hand side, a (0) ⊗ S − ( a (1) < > (1) ) ⊗ a (1) < > (0) a (1) < > ˜ a = a (0) ⊗ S − ( S ( a (1)(1) )) ⊗ a (1)(2) < > a (1)(2) < > ˜ a = a (0) ⊗ a (1) ⊗ ˜ a, using (2.15) in the first step and (2.17) in the second one. As for the right hand side, weget a ⊗ S − (˜ a (1) ) ⊗ ˜ a (0) . Thus a (0) ⊗ a (1) ⊗ ˜ a = a ⊗ S − (˜ a (1) ) ⊗ ˜ a (0) , and a ⊗ ˜ a ∈ A H (cid:3) H A .Conversely, assume a ⊗ ˜ a ∈ A H (cid:3) H A . By applying (id A ⊗ id A ⊗ m A ) ◦ (id A ⊗ τ ⊗ id A )on a (0) ⊗ a (1) ⊗ ˜ a = a ⊗ S − (˜ a (1) ) ⊗ ˜ a (0) , we get a (0) ⊗ a (1) < > ⊗ B a (1) < > ˜ a = a ⊗ S − (˜ a (1) ) < > ⊗ B S − (˜ a (1) ) < > ˜ a (0) . (3.6)Now, using (2.16) in the second step, we have χ ( S − (˜ a (1) ) < > ⊗ B S − (˜ a (1) ) < > ˜ a (0) ) = S − (˜ a (1) ) < > S − (˜ a (1) ) < > (0) ˜ a (0)(0) ⊗ S − (˜ a (1) ) < > (1) ˜ a (0)(1) = ˜ a (0) ⊗ S − (˜ a (2) )˜ a (1) = ˜ a ⊗ χ (˜ a ⊗ B A ) , From this S − (˜ a (1) ) < > ⊗ B S − (˜ a (1) ) < > ˜ a (0) = ˜ a ⊗ B a (0) ⊗ a (1) < > ⊗ B a (1) < > ˜ a = a ⊗ ˜ a ⊗ B A . Thus a ⊗ ˜ a ∈ C . (cid:3) Finally the coproduct (3.3) translates to the coproduct on A H (cid:3) H A written as,∆( a ⊗ ˜ a ) = a ⊗ τ ( S − (˜ a (1) )) ⊗ ˜ a (0) , (3.7)The Ehresmann coring of a Hopf–Galois extension is in fact a bialgebroid, called the Ehresmann–Schauenburg bialgebroid (cf. [5, 34.14]). One see that C ( A, H ) = ( A ⊗ A ) coH is a subalgebra of A ⊗ A op ; indeed, given a ⊗ ˜ a, a ′ ⊗ ˜ a ′ ∈ ( A ⊗ A ) coH , one computes δ A ⊗ A ( aa ′ ⊗ ˜ a ′ ˜ a ) = a (0) a ′ (0) ⊗ ˜ a ′ (0) ˜ a (0) ⊗ a (1) a ′ (1) ˜ a ′ (1) ˜ a (1) = a (0) a ′ ⊗ ˜ a ′ ˜ a (0) ⊗ a (1) ˜ a (1) = aa ′ ⊗ ˜ a ′ ˜ a ⊗ H . Definition 3.4.
Let C ( A, H ) be the coring of a Hopf–Galois extension B = A co H ⊆ A ,with A faithful flat as a left B -module. Then C ( A, H ) is a (left) B -bialgebroid with product ( a ⊗ ˜ a ) • C ( A,H ) ( a ′ ⊗ ˜ a ′ ) = aa ′ ⊗ ˜ a ′ ˜ a, (3.8) for all a ⊗ ˜ a, a ′ ⊗ ˜ a ′ ∈ C ( A, H ) (and unit ⊗ ∈ A ⊗ A ). The target and the source mapsare given by t ( b ) = 1 ⊗ b, and s ( b ) = b ⊗ . (3.9)We refer to [5, 34.14] for the checking that all defining properties are satisfied.3.2. The groups of bisections.
The bialgebroid of a Hopf–Galois extension can beview as a quantization (of the dualization) of the classical gauge groupoid, recalled inAppendix A, of a (classical) principal bundle. Dually to the notion of a bisection on theclassical gauge groupoid there is the notion of a bisection on the Ehresmann–Schauenburgbialgebroid. And in particular there are vertical bisections. These bisections correspondto automorphisms and vertical automorphisms (gauge transformations) respectively. efinition 3.5. Let C ( A, H ) be the left Ehresmann–Schauenburg bialgebroid associate toa Hopf–Galois extension B = A coH ⊆ A . A bisection of C ( A, H ) is a unital algebra map σ : C ( A, H ) → B , such that σ ◦ t = id B and σ ◦ s ∈ Aut( B ) . In general the collections of all bisections do not have additional structure. As aparticular case that parallels Proposition 2.6 we have the following.
Proposition 3.6.
Consider the left Ehresmann–Schauenburg bialgebroid C ( A, H ) asso-ciate to a Hopf–Galois extension B = A coH ⊆ A . If B belong to the centre of A , then theset of all bisections of C is a group, denoted B ( C ( A, H )) , with product defined by σ ∗ σ ( a ⊗ ˜ a ) := ( σ ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) σ ( a (1) < > ⊗ ˜ a ) , = σ (cid:0) σ ( a (0) ⊗ a (1) < > ) a (1) < > ⊗ ˜ a (cid:1) = σ (cid:0) σ (( a ⊗ ˜ a ) (1) ) ( a ⊗ ˜ a ) (2) (cid:1) (3.10) for any bisections σ , σ and any element a ⊗ ˜ a ∈ C ( A, H ) . The unit of this group is thecounit of the bialgebroid. And for any bisection σ , its inverse is given by σ − ( a ⊗ ˜ a ) = ( σ ◦ s ) − (cid:0) aσ (˜ a (0) ⊗ ˜ a (1) < > ) ˜ a (1) < > (cid:1) . (3.11) Here ( σ ◦ s ) − is the inverse of σ ◦ s ∈ Aut( B ) .Proof. The second equality in (3.10) follows from the fact that bisections are taken tobe algebra maps. The expressions on the right hand side of (3.10) and (3.11) are welldefined. For any bisection σ and any b ∈ B , a ∈ A the condition σ ◦ t = id B yields: σ ( a (0) ⊗ a (1) < > b ) a (1) < > = σ ( a (0) ⊗ a (1) < > ) b a (1) < > . (3.12)As for the multiplication in (3.10): for bisections σ , σ and any b ∈ B , we have σ ∗ σ ( s ( b )) = σ ∗ σ ( b ⊗
1) = σ ( s ( σ ( b ⊗ σ ◦ s ) ◦ ( σ ◦ s )( b ) . Being both σ ◦ s and σ ◦ s automorphisms of B , we have ( σ ∗ σ ) ◦ s ∈ Aut( B ).Similarly one shows that σ ∗ σ ( t ( b )) = b for b ∈ B , that is ( σ ∗ σ ) ◦ t = id B . Also, themultiplication is associative: let σ , σ , σ be bisections, and let a ⊗ ˜ a ∈ C ( A, H ). From((∆ ⊗ B id C ( A,H ) ) ◦ ∆)( a ⊗ ˜ a ) = a (0) ⊗ a (1) < > ⊗ B a (1) < > ⊗ a (2) < > ⊗ B a (2) < > ⊗ ˜ a, we have (using in the second step that σ ◦ s is an algebra map):(( σ ∗ σ ) ∗ σ )( a ⊗ ˜ a )= ( σ ◦ s ) (cid:16) ( σ ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) σ ( a (1) < > ⊗ a (2) < > ) (cid:17) σ ( a (2) < > ⊗ ˜ a )= ( σ ◦ s ) (cid:16) ( σ ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1)(cid:17) ( σ ◦ s ) (cid:0) σ ( a (1) < > ⊗ a (2) < > ) (cid:1) σ ( a (2) < > ⊗ ˜ a )= (( σ ∗ σ ) ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) ( σ ∗ σ )( a (1) < > ⊗ ˜ a )= ( σ ∗ ( σ ∗ σ ))( a ⊗ ˜ a ) . The assumption that B belongs to the centre of A implies that the product σ ∗ σ is analgebra map. Indeed, for a ⊗ ˜ a and a ′ ⊗ ˜ a ′ ∈ C one has, σ ∗ σ ( aa ′ ⊗ ˜ a ′ ˜ a )= ( σ ◦ t )( σ (( aa ) ′ (0) ⊗ ( aa ′ ) (1) < > )) ( σ (( aa ′ ) (1) < > ⊗ ˜ a ′ ˜ a ))= ( σ ◦ t )( σ ( a (0) a ′ (0) ⊗ a ′ (1) < > a (1) < > )) ( σ ( a (1) < > a ′ (1) < > ⊗ ˜ a ′ ˜ a ))= ( σ ◦ t )( σ ( a (0) ⊗ a (1) < > )) ( σ ◦ t )( σ ( a ′ (0) ⊗ a ′ (1) < > )) ( σ ( a ′ (1) < > ⊗ ˜ a ′ ))( σ ( a (1) < > ⊗ ˜ a ))= ( σ ∗ σ )( a ⊗ ˜ a ) ( σ ∗ σ )( a ′ ⊗ ˜ a ′ ) he 2nd step uses (2.18), the 3rd step uses the fact that σ and σ are both algebra maps,the last step uses that B belongs to the centre.Thus σ ∗ σ is a well defined algebra map. Next, we check ǫ is the unit of thismultiplication. Firstly, since B is taken to belong to the centre of A the counit ǫ is analgebra map. Indeed, for any a ⊗ ˜ a ∈ C ( A, H ), ǫ ( aa ′ ⊗ ˜ a ′ ˜ a ) = aa ′ ˜ a ′ ˜ a = a ′ ˜ a ′ a ˜ a = ǫ ( a ⊗ ˜ a ) ǫ ( a ′ ⊗ ˜ a ′ ) , the 2nd step using that B belongs to the centre. Then, σ ∗ ǫ ( a ⊗ ˜ a ) = ( ǫ ◦ t ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) ǫ ( a (1) < > ⊗ ˜ a ) = ( ǫ ◦ t ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) a (1) < > ˜ a = ( ǫ ◦ t ) (cid:0) σ ( a ⊗ ˜ a ) (cid:1) = σ ( a ⊗ ˜ a ) , where the 3rd step uses the definition of C . Similarly, for any a ⊗ ˜ a ∈ C ( A, H ): ǫ ∗ σ ( a ⊗ ˜ a ) = ( σ ◦ t ) (cid:0) ǫ ( a (0) ⊗ a (1) < > ) (cid:1) σ ( a (1) < > ⊗ ˜ a )= ( σ ◦ t ) (cid:0) a (0) a (1) < > (cid:1) σ ( a (1) < > ⊗ ˜ a )= σ ( a ⊗ ˜ a ) , and for the last equality we use a (0) a (1) < > ⊗ B a (1) < > = 1 ⊗ B a . Thus ǫ is the unit.Next, let us check that the inverse of a bisection σ as given in (3.11), is well defined.The quantity aσ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > , the argument of ( σ ◦ s ) − in (3.11), belongs to B .Indeed, with δ A the coaction as in 2.3, one has δ A ( aσ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > ) = a (0) σ (˜ a (0) ⊗ ˜ a (1) < > )(˜ a (1) < > ) (0) ⊗ a (1) (˜ a (1) < > ) (1) = a (0) σ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > ⊗ a (1) ˜ a (2) = a (0) σ (˜ a (0)(0) ⊗ ˜ a (0)(1) < > )˜ a (0)(1) < > ⊗ a (1) ˜ a (1) = aσ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > ⊗ H , where the 1st step uses that σ is valued in B , the 2nd use (2.14), the last one uses (3.1).And for any b ∈ B , σ − ( s ( b )) = ( σ ◦ s ) − ( b ), so σ − ◦ s = ( σ ◦ s ) − ∈ Aut( B ); also σ − ( t ( b )) = ( σ ◦ s ) − ( σ ( b ⊗ σ ◦ s ) − (( σ ◦ s )( b ))) = b , so σ − ◦ t = id B .Next, let us show σ − is indeed the inverse of σ . For a ⊗ ˜ a ∈ C , we have( σ − ∗ σ )( a ⊗ ˜ a )= ( σ ◦ s )( σ − ( a (0) ⊗ a (1) < > )) σ ( a (1) < > ⊗ ˜ a )= ( σ ◦ s ) (cid:16) ( σ ◦ s ) − (cid:0) a (0) σ ( a (1) < > (0) ⊗ a (1) < > (1) < > ) a (1) < > (1) < > (cid:1)(cid:17) σ ( a (1) < > ⊗ ˜ a )= a (0) σ ( a (1) < > (0) ⊗ a (1) < > (1) < > ) a (1) < > (1) < > σ ( a (1) < > ⊗ ˜ a )= a (0) σ (cid:0) a (2) < > ⊗ S ( a (1) ) < > ) S ( a (1) ) < > σ ( a (2) < > ⊗ ˜ a )= a (0) σ ( a (2) < > a (2) < > ⊗ ˜ aS ( a (1) ) < > (cid:1) S ( a (1) ) < > = a (0) ˜ a S ( a (1) ) < > S ( a (1) ) < > = a ˜ a = ǫ ( a ⊗ ˜ a ) , here the 4th step uses (2.15), the 5th step uses that B belongs to the centre of A , the6th and 7th steps use (2.17). On the other hand,( σ ∗ σ − )( a ⊗ ˜ a ) = ( σ − ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) σ − ( a (1) < > ⊗ ˜ a )= ( σ ◦ s ) − (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) ( σ ◦ s ) − ( a (1) < > σ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > )= ( σ ◦ s ) − (cid:0) σ ( a (0) ˜ a (0) ⊗ ˜ a (1) < > a (1) < > (cid:1) a (1) < > ˜ a (1) < > )= ( σ ◦ s ) − (cid:0) σ (( a ˜ a ) (0) ⊗ ( a ˜ a ) (1) < > (cid:1) ( a ˜ a ) (1) < > )= ( σ ◦ s ) − ( σ ( a ˜ a ⊗ σ ◦ s ) − (cid:0) ( σ ◦ s )( a ˜ a ) (cid:1) = a ˜ a = ǫ ( a ⊗ ˜ a ) , where the second step uses σ − ( s ( b )) = ( σ ◦ s ) − ( b ), the 3rd step uses that B belongs tothe centre of A , the 4th step uses (2.18), and the 5th step uses that a ˜ a ∈ B .Finally, the map σ − is an algebra map: σ − ( aa ′ ⊗ ˜ a ′ ˜ a ) = ( σ ◦ s ) − (cid:16) aa ′ σ (cid:0) (˜ a ′ ˜ a ) (cid:1) (0) ⊗ (˜ a ′ ˜ a ) (1) < > (cid:17) (˜ a ′ ˜ a ) (1) < > = ( σ ◦ s ) − (cid:0) aa ′ σ (˜ a ′ (0) ˜ a (0) ⊗ ˜ a (1) < > ˜ a ′ (1) < > )˜ a ′ (1) < > ˜ a (1) < > (cid:1) = ( σ ◦ s ) − (cid:0) aa ′ σ (˜ a (0) ⊗ ˜ a (1) < > ) σ (˜ a ′ (0) ⊗ ˜ a ′ (1) < > )˜ a ′ (1) < > ˜ a (1) < > (cid:1) = ( σ ◦ s ) − (cid:0) aa ′ σ (˜ a ′ (0) ⊗ ˜ a ′ (1) < > )˜ a ′ (1) < > σ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > (cid:1) = ( σ ◦ s ) − (cid:0) a ′ σ (˜ a ′ (0) ⊗ ˜ a ′ (1) < > )˜ a ′ (1) < > aσ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > (cid:1) = ( σ ◦ s ) − (cid:0) a ′ σ (˜ a ′ (0) ⊗ ˜ a ′ (1) < > )˜ a ′ (1) < > (cid:1) ( σ ◦ s ) − (cid:0) aσ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > (cid:1) = σ − ( a ⊗ ˜ a ) σ − ( a ′ ⊗ ˜ a ′ );the second step uses (2.18), the 3rd step uses σ is an algebra map, the 5th one uses thatthe image of σ and a ′ σ (˜ a ′ (0) ⊗ ˜ a ′ (1) < > )˜ a ′ (1) < > are in B , which is in the centre of A . (cid:3) Remark . Having asked that bisections are algebra maps, they are B -linear in thesense of the coring bimodule structure in (2.8). That is, for any bisection σ and b ∈ B , σ (cid:0) ( a ⊗ ˜ a ) ⊳ b (cid:1) = σ (cid:0) t ( b ) • C a ⊗ ˜ a (cid:1) = σ ( a ⊗ ˜ a ) σ ( t ( b )) = σ ( a ⊗ ˜ a ) b and σ (cid:0) b ⊲ ( a ⊗ ˜ a ) (cid:1) = σ (cid:0) s ( b ) • C a ⊗ ˜ a (cid:1) = σ ( a ⊗ ˜ a ) σ ( s ( b )) = σ ( a ⊗ ˜ a ) ( σ ◦ s )( b ) . Among all bisections an important role is played by the vertical ones.
Definition 3.8.
Let C ( A, H ) be the left Ehresmann–Schauenburg bialgebroid associate toa Hopf–Galois extension B = A co H ⊆ A . A vertical bisection is a bisection of C which isalso a left inverse for the target map s , that is σ ◦ s = id B . Then the following statement is immediate.
Corollary 3.9.
Consider the left Ehresmann–Schauenburg bialgebroid C ( A, H ) associateto a Hopf–Galois extension B = A co H ⊆ A . If B belong to the centre of A , then theset B ver ( C ( A, H )) of all vertical bisections of C is a group, a subgroup of the group of allbisections B ( C ( A, H )) , with the restricted product given by σ ∗ σ ( a ⊗ ˜ a ) := σ ( a (0) ⊗ a (1) < > ) σ ( a (1) < > ⊗ ˜ a ) (3.13) or any vertical bisections σ , σ . Moreover, the inverse of a vertical bisection is given by σ − ( a ⊗ ˜ a ) = aσ (˜ a (0) ⊗ ˜ a (1) < > ) ˜ a (1) < > = σ (˜ a (0) ⊗ ˜ a (1) < > ) a ˜ a (1) < > . (3.14) Proof.
The right hand side of both (3.13) and (3.14) is seen to be a vertical bisection. (cid:3)
Remark . We notice that the product (3.13) on vertical bisections is just the convo-lution product due to the second expression for the coproduct in (3.3), σ ∗ σ ( a ⊗ ˜ a ) = ( σ ⊗ B σ ) ◦ ∆( a ⊗ ˜ a )= σ ( a (0) ⊗ a (1) < > ) σ ( a (1) < > ⊗ ˜ a ) . (3.15)We show directly that the product in (3.15) is well defined, Indeed, for b ∈ B we have σ ( a (0) ⊗ a (1) < > b ) σ ( a (1) < > ⊗ ˜ a ) = σ (cid:0) ( a (0) ⊗ a (1) < > ) • C (1 ⊗ b ) (cid:1) σ ( a (1) < > ⊗ ˜ a )= σ ( a (0) ⊗ a (1) < > ) σ (1 ⊗ b ) σ ( a (1) < > ⊗ ˜ a )= σ ( a (0) ⊗ a (1) < > ) b σ ( a (1) < > ⊗ ˜ a )= σ ( a (0) ⊗ a (1) < > ) σ ( b ⊗ σ ( a (1) < > ⊗ ˜ a )= σ ( a (0) ⊗ a (1) < > ) σ ( b a (1) < > ⊗ ˜ a )with the 4th step coming from σ being vertical.3.3. Bisections and gauge groups.
Recall the Definition 2.5 and the Proposition 2.6concerning the gauge group of a Hopf–Galois extension. We have the following results.
Proposition 3.11.
Let B = A coH ⊆ A be a Hopf–Galois extension, and let C ( A, H ) be the corresponding left Ehresmann–Schauenburg bialgebroid. If B is in the centre of A , then there is a group isomorphism α : Aut H ( A ) → B ( C ( A, H )) . The isomorphism α restricts to an isomorphism between vertical subgroups α : Aut ver ( A ) → B ver ( C ( A, H )) .Proof. Let F ∈ Aut H ( A ) and define σ F ∈ B ( C ( A, H )) by σ F ( a ⊗ ˜ a ) := F ( a )˜ a, (3.16)for any a ⊗ ˜ a ∈ C ( A, H ). This is well defined since δ A ( F ( a )˜ a ) = ( F ( a )˜ a ) (0) ⊗ ( F ( a )˜ a ) (1) = F ( a ) (0) ˜ a (0) ⊗ F ( a ) (1) ˜ a (1) = F ( a (0) )˜ a (0) ⊗ a (1) ˜ a (1) = F ( a )˜ a ⊗ H , where the last equality use (3.1), thus F ( a )˜ a ∈ B . And σ F is an algebra map, since σ F (( a ′ ⊗ ˜ a ′ ) • C ( A,H ) ( a ⊗ ˜ a )) = σ F ( a ′ a ⊗ ˜ a ˜ a ′ ) = F ( a ′ a )˜ a ˜ a ′ = F ( a ′ ) F ( a )˜ a ˜ a ′ = F ( a ′ )( F ( a )˜ a )˜ a ′ = F ( a ′ )˜ a ′ σ F ( a ⊗ ˜ a )= σ F ( a ′ ⊗ ˜ a ′ ) σ F ( a ⊗ ˜ a ) , where the 5th equality uses that B is in the centre of A . It is clear that σ F ◦ t = id B and σ F ◦ s = F | B ∈ Aut( B ). Thus σ F is a well defined bisection. By the definition (3.16), σ id A ( a ⊗ ˜ a ) = a ˜ a = ǫ ( a ⊗ ˜ a ) nd for any a ⊗ ˜ a ∈ C ( A, H ) we have σ G ∗ σ F ( a ⊗ ˜ a ) = ( σ F ◦ s ) (cid:0) σ G ( a (0) ⊗ a (1) < > ) (cid:1) σ F ( a (1) < > ⊗ ˜ a )= σ F (cid:0) G ( a (0) ) a (1) < > ⊗ (cid:1) σ F ( a (1) < > ⊗ ˜ a )= F ( G ( a (0) ) a (1) < > ) F ( a (1) < > )˜ a = F (cid:0) G ( a (0) ) a (1) < > a (1) < > (cid:1) ˜ a = F ( G ( a )) ˜ a = σ ( G · F ) ( a ⊗ ˜ a ) , where the 5th step uses (2.17).Conversely, given a bisection σ , one can define an algebra map F σ : A → A by F σ ( a ) := σ ( a (0) ⊗ a (1) < > ) a (1) < > . (3.17)We have already seen (cf. (3.12)) that the right hand side of (3.17) is well defined. Clearly F σ ( b ) = ( σ ◦ s )( b ) for any b ∈ B , so F σ | B ∈ Aut( B ). Moreover, F σ ( aa ′ ) = σ (( aa ′ ) (0) ⊗ ( aa ′ ) (1) < > )( aa ′ ) (1) < > = σ ( a (0) a ′ (0) ⊗ ( a (1) a ′ (1) ) < > )( a (1) a ′ (1) ) < > = σ ( a (0) a ′ (0) ⊗ a ′ (1) < > a (1) < > ) a (1) < > a ′ (1) < > = σ ( a ′ (0) ⊗ a ′ (1) < > ) σ ( a (0) ⊗ a (1) < > ) a (1) < > a ′ (1) < > = F σ ( a ) F σ ( a ′ ) , where in the third step we use (2.18). Also, F σ is H -equivalent: F σ ( a ) (0) ⊗ F σ ( a ) (1) = σ ( a (0) ⊗ a (1) < > ) a (1) < > (0) ⊗ a (1) < > (1) = σ ( a (0) ⊗ a (1) < > ) a (1) < > ⊗ a (2) = F σ ( a (0) ) ⊗ a (1) , where the 2nd step uses (2.14), thus F σ ∈ Aut H ( A ).The map α is an isomorphism. Indeed for any a ⊗ ˜ a ∈ C ( A, H ) and any σ ∈ B ( C ( A, H )): σ F σ ( a ⊗ ˜ a ) = F σ ( a )˜ a = σ ( a (0) ⊗ a (1) < > ) a (1) < > ˜ a = σ ( a ⊗ ˜ a ) , where the last step uses (3.2). On the other hand, for any a ∈ A and any F ∈ Aut H ( A ): F σ F ( a ) = σ F ( a (0) ⊗ a (1) < > ) a (1) < > = F ( a (0) ) a (1) < > a (1) < > = F ( a ) . Finally, for a vertical automorphism F ∈ Aut ver ( A ), it is clear that the corresponding σ F ∈ B ver ( C ( A, H )), and conversely from σ ∈ B ver ( C ( A, H )) we have F σ ∈ Aut ver ( A ). (cid:3) Extended bisections and gauge groups.
We have already mentioned that gaugetransformations for a noncommutative principal bundles could be defined without askingthem to be algebra homomorphisms [4]. Mainly for the sake of completeness we recordhere a version of them via bialgebroid and bisections. To distinguish them from theanalogous concepts introduced in the previous section, and for lack of a better name, wecall the extended gauge transformation and extended bisections.In the same vein of [4] we have the following definition.
Definition 3.12.
Given a Hopf–Galois extension B = A coH ⊆ A . Its extended gaugegroup Aut extH ( A ) consists of invertible H -comodule unital maps F : A → A such that theirrestrictions F | B ∈ Aut( B ) and such that F ( ba ) = F ( b ) F ( a ) for any b ∈ B and a ∈ A .The extended vertical gauge group Aut extver ( A ) is made of elements F ∈ Aut extH ( A ) whoserestrictions F | B = id B . The group structure is map composition. n parallel with this we have then the following. Definition 3.13.
Let C ( A, H ) be the left Ehresmann–Schauenburg bialgebroid of theHopf–Galois extension B = A co H ⊆ A . An extended bisection is a unital algebra map σ : C ( A, H ) → B , such that σ ◦ t = id B and σ ◦ s ∈ Aut( B ) , which in additionis B -linear in the sense of the B -coring structure on C (cf. Remark 3.7). That is, σ (cid:0) ( a ⊗ ˜ a ) ⊳ b (cid:1) = σ ( a ⊗ ˜ a ) b , and σ (cid:0) b ⊲ ( a ⊗ ˜ a ) (cid:1) = σ ( a ⊗ ˜ a ) ( σ ◦ s )( b ) . The set of all extendedbisections which are invertible for the product (3.10) : ( σ ∗ σ )( a ⊗ ˜ a ) := ( σ ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) σ ( a (1) < > ⊗ ˜ a ) . (3.18) will be denote by B ext ( C ( A, H )) , while B extver ( C ( A, H )) will denote those which are invertibleand vertical, that is such that σ ◦ s = id B as well. Lemma 3.14.
The product (3.18) is well defined.Proof.
We need to check that σ ∗ σ is B -linear in the sense of the definition. Now, forany a ⊗ a ′ ∈ C and b ∈ B we have( σ ∗ σ )(( a ⊗ ˜ a ) ⊳ b ) = ( σ ∗ σ )(( a ⊗ ˜ ab )= ( σ ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) ( σ ( a (1) < > ⊗ ˜ ab ))= ( σ ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) σ (cid:0) ( a (1) < > ⊗ ˜ a ) ⊳ b ) (cid:1) = ( σ ∗ σ )( a ⊗ a ′ ) b. Similarly,( σ ∗ σ )( b ⊲ ( a ⊗ ˜ a )) = ( σ ∗ σ )( ba ⊗ ˜ a )= ( σ ◦ s ) (cid:0) σ ( ba (0) ⊗ a (1) < > ) (cid:1) ( σ ( a (1) < > ⊗ ˜ a ))= ( σ ◦ s ) (cid:0) σ ( b ⊲ ( a (0) ⊗ a (1) < > )) (cid:1) ( σ ( a (1) < > ⊗ ˜ a ))= ( σ ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) ( σ ◦ s )( b ) (cid:1) ( σ ( a (1) < > ⊗ ˜ a ))= ( σ ◦ s )( σ ◦ s )( b ) ( σ ◦ s ) (cid:0) σ ( a (0) ⊗ a (1) < > ) (cid:1) σ ( a (1) < > ⊗ ˜ a )= ( σ ∗ σ )( a ⊗ ˜ a ) (cid:0) ( σ ∗ σ ) ◦ s (cid:1) ( b ) . Were the last step uses the identity (cid:0) ( σ ∗ σ ) ◦ s (cid:1) ( b ) = ( σ ◦ s )( σ ◦ s )( b ), and the lastbut one one the fact that σ ◦ s ∈ Aut( B ) and that B is in the centre. (cid:3) Remark . We remark that (3.11) is now not the inverse for the product in (3.18)since, in contrast to Proposition 3.6 we are not asking the bisections be algebra maps.Finally, in analogy with Proposition 3.11 we have the following.
Proposition 3.16.
Let B = A coH ⊆ A be a Hopf–Galois extension, and let C ( A, H ) bethe corresponding left Ehresmann–Schauenburg bialgebroid. If B belongs to the centre of A , there is a group isomomorphism b α : Aut extH ( A ) → B ext ( C ( A, H )) . The isomorphismrestricts to an isomorphism b α v : Aut extver ( A ) → B extver ( C ( A, H )) between vertical subgroups.Proof. This uses the same methods as Proposition 3.11. Given F ∈ Aut extH ( A ), define itsimage as in (3.16): σ F ( a ⊗ ˜ a ) = F ( a )˜ a . Being B in the centre, for all b ∈ B we have, σ F ( a ⊗ ˜ ab ) = F ( a )˜ a b = σ F ( a ⊗ ˜ a ) b ,σ F ( ba ⊗ ˜ a ) = F ( ba )˜ a = F ( b ) F ( a )˜ a = σ F ( a ⊗ ˜ a ) ( σ F ◦ s )( b ) , that is, σ F (cid:0) ( a ⊗ ˜ a ) ⊳b (cid:1) = σ F ( a ⊗ ˜ a ) b , and σ F (cid:0) b⊲ ( a ⊗ ˜ a ) (cid:1) = σ F ( a ⊗ ˜ a ) ( σ F ◦ s )( b ). Conversely,for σ ∈ B ext ( C ( A, H )), define its image as in (3.17): F σ ( a ) = σ ( a (0) ⊗ a (1) < > ) a (1) < > . Then σ ( ba ) = σ ( ba (0) ⊗ a (1) < > ) a (1) < > = ( σ ◦ s )( b ) F σ ( a ) = F σ ( b ) F σ ( a ), due to B in the centreof A . The rest of the proof goes as that of Proposition 3.11. (minus the algebra mapparts). (cid:3) Bisections and gauge groups of Galois objects
From now on we shall concentrate on
Galois objects of a Hopf algebra H . These arenoncommutative principal bundles over a point. In contrast to the classical result thatany fibre bundle over a point is trivial, the set Gal H ( C ) of isomorphic classes of H -Galoisobjects need not be trivial (cf. [3], [8]). We shall illustrate later on this non-trivialitywith examples coming from group algebras and Taft algebras.4.1. Galois objects.Definition 4.1.
Let H be a Hopf algebra, a H -Galois object of H is an H -Hopf–Galoisextension A of the ground field C . Thus for a Galois object the coinvariant subalgebra is the ground field C . Now if A isfaithfully flat over C , then bijectivity of the canonical Galois map implies C = A co H (cf.[17, Lem. 1.11]) and H is faithfully flat over C since A ⊗ A is faithfully flat over A . Recallfrom Section 2.1 that an ( A, H )-relative Hopf module M is a right H -comodule with acompatible right A -module structure. That is the action is a morphism of H -comodulessuch that δ M ( ma ) = m (0) a (0) ⊗ m (1) a (1) for all a ∈ A , m ∈ M . We have the following [13]: Lemma 4.2.
Let M be an ( A, H ) -relative Hopf module. If A is faithfully flat over C ,the multiplication induces an isomorphism M co H ⊗ A → M, whose inverse is M ∋ m m (0) m (1) < > ⊗ m (1) < > ∈ M co H ⊗ A . With coaction δ A : A → A ⊗ H , δ A ( a ) = a (0) ⊗ a (1) , and translation map τ : H → A ⊗ A , τ ( h ) = h < > ⊗ h < > , for the Ehresmann–Schauenburg bialgebroid of a Galois object, being B = C one has (see also [13, Def. 3.1]): C ( A, H ) = { a ⊗ ˜ a ∈ A ⊗ A : a (0) ⊗ ˜ a (0) ⊗ a (1) ˜ a (1) = a ⊗ ˜ a ⊗ H } (4.1)= { a ⊗ ˜ a ∈ A ⊗ A : a (0) ⊗ a (1) < > ⊗ a (1) < > ˜ a = a ⊗ ˜ a ⊗ A } . (4.2)The coproduct (3.3) and counit (3.4) become ∆ C ( a ⊗ ˜ a ) = a (0) ⊗ a (1) < > ⊗ a (1) < > ⊗ ˜ a ,and ǫ C ( a ⊗ ˜ a ) = a ˜ a ∈ C respectively, for any a ⊗ ˜ a ∈ C ( A, H ). But now there is also anantipode [13, Thm. 3.5] given, for any a ⊗ ˜ a ∈ C ( A, H ), by S C ( a ⊗ ˜ a ) := ˜ a (0) ⊗ ˜ a (1) < > a ˜ a (1) < > . (4.3)Thus the Ehresmann–Schauenburg bialgebroid of a Galois object is a Hopf algebra.Now, given that C ( A, H ) = ( A ⊗ A ) coH , Lemma 4.2 yields an isomorphism A ⊗ A ≃ C ( A, H ) ⊗ A , e χ ( a ⊗ ˜ a ) = a (0) ⊗ a (1) < > ⊗ a (1) < > ˜ a . (4.4)We finally collect some results of [13] (cf. Lemma 3.2 and Lemma 3.3) in the following: Lemma 4.3.
Let H be a Hopf algebra, and A a (faithfully flat) H -Galois object of H .There is a right H -equivariant algebra map δ C : A → C ( A, H ) ⊗ A given by δ C ( a ) = a (0) ⊗ a (1) < > ⊗ a (1) < > hich is universal in the following sense: Given an algebra M and a H -equivariant algebramap φ : A → M ⊗ A , there is a unique algebra map Φ : C ( A, H ) → M such that φ = (Φ ⊗ id A ) ◦ δ C . Explicitly, Φ( a ⊗ ˜ a ) ⊗ A = φ ( a )˜ a . The ground field C being undoubtedly in the centre, for the bisections of the Ehresmann–Schauenburg bialgebroid C ( A, H ) of a Galois object A , we can use all results of previoussections. Clearly, any bisection of C ( A, H ) is now vertical as it is vertical any auto-morphism of the principal bundle A . In fact bisections, being algebra maps, are justcharacters of the Hopf algebra C ( A, H ) with convolution product in (3.13) and inverse in(3.14) that, with the antipode in (4.3) can be written as σ − = σ ◦ S C , as is the case forcharacters. From Proposition 3.11 we have then the isomorphismAut H ( A ) ≃ B ( C ( A, H )) = Char( C ( A, H )) . (4.5)As for extended bisections and automorphisms as in Section 3.4 we have analogously fromProposition 3.16 the isomorphism,Aut extH ( A ) ≃ B ext ( C ( A, H )) = Char ext ( C ( A, H )) , (4.6)with Char ext ( C ( A, H )) the group of convolution invertible unital maps φ : C ( A, H ) → C .4.2. Hopf algebras as Galois objects.
Any Hopf algebra H is a H -Galois objectwith coaction given by its coproduct. Then H is isomorphic to the corresponding leftbialgebroid C ( H, H ).Let H be a Hopf algebra with coproduct ∆( h ) = h (1) ⊗ h (2) . For the correspondingcoinvariants: h (1) ⊗ h (2) = h ⊗
1, we have ǫ ( h (1) ) ⊗ h (2) = ǫ ( h ) ⊗
1, this imply h = ǫ ( h ) ∈ C and H co H = C . Moreover, the canonical Galois map χ : g ⊗ h gh (1) ⊗ h (2) is bijectivewith inverse given by χ − ( g ⊗ h ) := g S ( h (1) ) ⊗ h (2) . Thus H is a H -Galois object.With A = H , the corresponding left bialgebroid becomes C ( H, H ) = { g ⊗ h ∈ H ⊗ H : g (1) ⊗ h (1) ⊗ g (2) h (2) = g ⊗ h ⊗ H } = { g ⊗ h ∈ H ⊗ H : g (1) ⊗ S ( g (2) ) ⊗ g (3) h = g ⊗ h ⊗ A } . (4.7)We have a linear map φ : C ( H, H ) → H given by φ ( g ⊗ h ) := g ǫ ( h ). The map φ hasinverse φ − : H → C ( H, H ), defined by φ − ( h ) := h (1) ⊗ S ( h (2) ). This is well defined since∆ H ⊗ H ( h (1) ⊗ S ( h (2) )) = h (1) ⊗ S ( h (4) ) ⊗ h (2) S ( h (3) ) = h (1) ⊗ S ( h (2) ) ⊗ H , showing that h (1) ⊗ S ( h (2) ) ∈ C ( H, H ). Moreover, φ ( φ − ( h )) = φ ( h (1) ⊗ S ( h (2) )) = h, and φ − ( φ ( g ⊗ h )) = ǫ ( h ) φ − ( g ) = ǫ ( h ) g (1) ⊗ S ( g (2) ) = g ⊗ h. Here the last equality is obtained from the condition g (1) ⊗ S ( g (2) ) ⊗ g (3) h = g ⊗ h ⊗ H (for any g ⊗ h ∈ C ( H, H ), as in the second line of (4.7)) by applying id H ⊗ id H ⊗ ǫ onboth sides and then multiplying the second and third factors: g (1) ⊗ S ( g (2) ) ǫ ( g (3) ) ǫ ( h ) = g ⊗ h ǫ (1 H ) = ⇒ ǫ ( h ) g (1) ⊗ S ( g (2) ) = g ⊗ h. The map φ is an algebra map: φ (( g ⊗ h ) • C ( g ′ ⊗ h ′ )) = φ ( gg ′ ⊗ h ′ h ) = gg ′ ǫ ( h ′ ) ǫ ( h )= φ ( g ⊗ h ) • C φ ( g ′ ⊗ h ′ ) . t is also a coalgebra map:( φ ⊗ φ )(∆ C ( g ⊗ h )) = ( φ ⊗ φ )( g (1) ⊗ g (2) < > ⊗ g (2) < > ⊗ h )= ( φ ⊗ φ )( g (1) ⊗ S ( g (2) ) ⊗ g (3) ⊗ h )= g (1) ⊗ g (2) ǫ ( h )= ∆ H ( φ ( g ⊗ h )); ǫ C ( g ⊗ h ) = gh = ǫ H ( gh ) = ǫ H ( g ) ǫ H ( h ) = ǫ H ( φ ( g ⊗ h )) . Cocommutative Hopf algebras.
We start with a class of examples coming fromcocommutative Hopf algebras. From [13] (Remark 3.8 and Theorem 3.5.) we have:
Lemma 4.4.
Let H be a cocommutative Hopf algebra, and let A be a H -Galois object.Then the bialgebroid C ( A, H ) is isomorphic to H as Hopf algebra.Proof. We give a sketch of the proof that uses Lemma 4.3. Start with the coaction δ A : A → A ⊗ H , δ A ( a ) = a (0) ⊗ a (1) , and translation map τ ( h ) = h < > ⊗ h < > . Firstly,the image of τ is in C ( A, H ); indeed, for any h ∈ H , we get h < > (0) ⊗ h < > (0) ⊗ h < > (1) h < > (1) = h (1) < > (0) ⊗ h (1) < > ⊗ h (1) < > (1) h (2) = h (1)(2) < > ⊗ h (1)(2) < > ⊗ S ( h (1)(1) ) h (2) = h < > ⊗ h < > ⊗ H , where the first step uses (2.14), and the second step uses (2.15). While τ is not an algebramap, being H cocommutative, it is a coalgebra map. Indeed, for any h ∈ H ,∆ C ( τ ( h )) = h < > (0) ⊗ τ ( h < > (1) ) ⊗ h < > = h (2) < > ⊗ τ ( S ( h (1) )) ⊗ h (2) < > = h (2) < > ⊗ h (2) < > h (3) < > τ ( S ( h (1) )) h (3) < > h (4) < > ⊗ h (4) < > = h (3) < > ⊗ h (3) < > h (2) < > τ ( S ( h (1) )) h (2) < > h (4) < > ⊗ h (4) < > = h (1) < > ⊗ h (1) < > ⊗ h (2) < > ⊗ h (2) < > = ( τ ⊗ τ )(∆ H ( h )) , where the 2nd step uses (2.15): h (2) < > ⊗ h (2) < > ⊗ S ( h (1) ) = h < > (0) ⊗ h < > ⊗ h < > (1) , the3nd step uses twice (2.19): h (1) < > ⊗ h (1) < > h (2) < > ⊗ h (2) < > = h < > ⊗ A ⊗ h < > ; the 4rdstep uses H is cocommutative: we change the lower indices 2 and 3; and the 5th one uses: ǫ ( h ) ⊗ A = τ ( S ( h (1) ) h (2) ) = h (2) < > τ ( S ( h (1) )) h (2) < > . Also, ǫ C ( τ ( h )) = h < > h < > = ǫ ( h )1 A . On the other hand, since H is cocommutative, A is also a left H -Galois object withcoaction δ L ( a ) = a (1) ⊗ a (0) and bijective canonical map χ L ( a ⊗ ˜ a ) = a (1) ⊗ a (0) ˜ a . Thecorresponding translation map is then τ L = τ ◦ S where S = S − (since H is cocommu-tative) is the antipode of H . The map τ L is a coalgebra map being the composition oftwo such maps (for S this is the case again due to H cocommutative).From the universality of Lemma 4.3, there is a unique algebra map Φ : C ( A, H ) → H such that δ L = (Φ ⊗ id A ) ◦ δ C ; where δ C : H → C ( A, H ) ⊗ H as in the lemma. Explicitly,Φ( a ⊗ ˜ a ) ⊗ A = δ L ( a )˜ a = χ L | C ( a ⊗ ˜ a ) for a ⊗ ˜ a ∈ C ( A, H ). Indeed, with the isomorphism e χ in (4.4), the map Φ is such that χ L = (Φ ⊗ id A ) ◦ e χ , thus is an isomorphism since χ L and e χ are such. The map Φ has inverse Φ − = τ L and thus is a coalgebra map. (cid:3) onsequently, the isomorphisms 4.5 and 4.6 for a cocommutative Hopf algebra H are:Aut H ( A ) ≃ B ( C ( A, H )) = Char( H ) (4.8)and Aut extH ( A ) ≃ B ext ( C ( A, H )) = Char ext ( H ) , (4.9)with Char ext ( H ) the group of convolution invertible unital maps φ : H → C and Char( H )the subgroup of those which are algebra maps (the characters of H ).4.4. Group Hopf algebras.
Let G be a group, with neutral element e , and H = C [ G ]be its group algebra. Its elements are finite sums P λ g g with λ g complex number. Weassume that { g , g ∈ G } is a vector space basis. The product in C [ G ] follows from thegroup product in G , with algebra unit 1 C [ G ] = e . The coproduct, counit, and antipode,making C [ G ] a Hopf algebra are defined by ∆( g ) = g ⊗ g , ǫ ( g ) = 1, S ( g ) = g − .An algebra A is G -graded, that is A = ⊕ g ∈ G A g and A g A h ⊆ A gh for all g, h ∈ G ,if and only if A is a right C [ G ]-comodule algebra with coaction δ A : A → A ⊗ C [ G ], a P a g ⊗ g for a = P a g , a g ∈ A g . Moreover, the algebra A is strongly G -graded, thatis A g A h = A gh , if and only if A e = A co C [ G ] ⊆ A is Hopf–Galois (see e.g. [12, Thm.8.1.7]).Thus C [ G ]-Hopf–Galois extensions are the same as G -strongly graded algebras.In particular, if A is a C [ G ]-Galois object, that is A e = C , each component A g isone-dimensional. If we pick a non-zero element u g in each A g , the multiplication of A isdetermined by the products u g u h for each pair g, h of elements of G. We then have u g u h = λ ( g, h ) u gh (4.10)for a non vanishing λ ( g, h ) ∈ C . We get then a map λ : G × G → C × which is in fact atwo cocycle λ ∈ H ( G, C × ). Indeed, associativity of the product requires that λ satisfiesa 2-cocycle condition, that is for any g, h ∈ G , λ ( g, h ) λ ( gh, k ) = λ ( h, k ) λ ( g, hk ) . (4.11)If we choose a different non-zero element v g ∈ A g , we shall have v g = µ ( g ) u g , for somenon-zero µ ( g ) ∈ C . The multiplication (4.10) will become v g v h = λ ′ ( g, h ) v gh with λ ′ ( g, h ) = µ ( g ) µ ( h )( µ ( gh )) − λ ( g, h ) , (4.12)that is the two 2-cocycles λ ′ and λ are cohomologous. It is easy to check that for anymap µ ( g ) : G → C × the assignment ( g, h ) µ ( g ) µ ( h )( µ ( gh )) − , is a coboundary, that isa ‘trivial’ 2-cocycle which is cohomologous to λ ( g, h ) = 1. Thus the multiplication in A depends only on the second cohomology class of λ ∈ H ( G, C × ), the second cohomologygroup of G with values in C × . We conclude that the equivalence classes of C [ G ]-Galoisobjects are in bijective correspondence with the cohomology group H ( G, C × ). Example . From [8, Ex. 7.13] we have the following. For any cyclic group G (infiniteor not) one has H ( G, C × ) = 0. Thus any corresponding C [ G ]-Galois object is trivial.On the other hand, for the free abelian group of rank r ≥
2, one has H ( Z r , C × ) = ( C × ) r ( r − / . Hence, there are infinitely many isomorphism classes of C [ Z r ]-Galois objects.Since H = C [ G ] is cocommutative, we know from above that the correspondingbialgebroids C ( A, H ) are all isomorphic to H as Hopf algebra. It is instructive toshow this directly. Clearly, for any u g ⊗ u h ∈ C ( A, H ) the coinvariance condition g ⊗ u h ⊗ gh = u g ⊗ u h ⊗ H , requires h = g − so that C ( A, H ) is generated as vec-tor space by elements u g ⊗ u g − , g ∈ G , with multiplication( u g ⊗ u g − ) • C ( u h ⊗ u h − ) = λ ( g, h ) λ ( h − , g − ) u gh ⊗ u ( gh ) − , (4.13) Lemma 4.6.
The cocycle Λ( g, h ) = λ ( g, h ) λ ( h − , g − ) is trivial in H ( G, C × ) .Proof. Firstly, we can always rescale u e to λ ( e, e ) u e so to have λ ( e, e ) = 1. Then thecocycle condition (4.11) yields λ ( g, e ) = λ ( e, g ) = λ ( e, e ) = 1, for any g ∈ G . Next,with u g u h = λ ( g, h ) u gh and u h − u g − = λ ( h − , g − ) u ( gh ) − , on the one hand we have u g u h u h − u g − = λ ( g, h ) λ ( h − , g − ) λ ( gh, ( gh ) − ) u e . On the other hand u g u h u h − u g − = λ ( g, g − ) λ ( h, h − ) u e . ThusΛ( g, h ) = λ ( g, g − ) λ ( h, h − ) /λ ( gh, ( gh ) − )showing Λ( g, h ) is trivial since Λ( g, h ) = µ ( g ) µ ( h )( µ ( gh )) − with µ ( g ) = λ ( g, g − ). (cid:3) Consequently, by rescaling the generators u g → v g = λ ( g, g − ) − u g the multiplicationrule (4.10) becomes v g v h = λ ′ ( g, h ) v gh , with λ ′ ( g, h ) = Λ( g, h ) − λ ( g, h ) that we renameback to λ ( g, h ) from now on. As for the bialgebroid product in (4.13) one has,( v g ⊗ v g − ) • C ( v h ⊗ v h − ) = v gh ⊗ v ( gh ) − , (4.14)and the isomorphism Φ − : H → C ( A, H ) is simply Φ − ( g ) = τ L ( g ) = v g ⊗ v g − .As in (4.8), the group of bisections B ( C ( A, H )) of C ( A, H ), and the gauge groupAut H ( A ) of the Galois object A coincide with the group of characters on C [ G ], whichis in turn the same as Hom( G, C × ) the group (for point-wise multiplication) of groupmorphisms 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 F ( v g ) = X h ∈ G f h ( g ) v h , for complex numbers, f h ( g ) ∈ C . Then, the H -equivariance of F , F ( v g ) (0) ⊗ F ( v g ) (1) = F ( v g (0) ) ⊗ v g (1) = F ( v g ) ⊗ g, requires F ( v g ) belongs to A g and we get f h ( g ) = 0, if h = g while f g := f g ( g ) ∈ C × fromthe invertibility of F . Finally F is an algebra map: λ ( g, h ) f gh v gh = F ( λ ( g, h ) v gh ) = F ( v g v h ) = F ( v g ) F ( v h ) = λ ( g, h ) f g f h v gh , 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 − = ( F g ) − .On the other hand, the group Aut extH ( A ) and then B ext ( C ( A, H )) can be quite big. If F ∈ Aut extH ( 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 = 1.4.5. Taft algebras.
Let N ≥ q be a primitive N -th roots ofunity. The Taft algebra T N is the N -dimensional unital algebra generated by generators x , g subject to relations: x N = 0 , g N = 1 , xg − q gx = 0 . It is a Hopf algebra with coproduct:∆( x ) := 1 ⊗ x + x ⊗ g, ∆( g ) := g ⊗ g ; ounit: ǫ ( x ) := 0 , ǫ ( g ) := 1 ;and antipode: S ( x ) := − g − x, S ( g ) := g − . This Hopf algebra is neither commutative nor cocommutative. The four dimensionalalgebra T 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: X N = s , G N = 1 , XG − q GX = 0 . The algebra A s is a right T N -comodule algebra, with coaction defined by δ A ( X ) := 1 ⊗ x + X ⊗ g, δ A ( G ) := G ⊗ g. (4.15)Clearly, the corresponding coinvariants are just the ground field C and so A s is a T N -Galois object. It is known (cf. [9], Prop. 2.17, Prop. 2.22, as well as [14]) that any T N -Galois object is isomorphic to A s for some s ∈ C and that any two such Galoisobjects A s and A t are isomorphic if and only if s = t . Thus the equivalence classesof T N -Galois objects are in bijective correspondence with the abelian group C . For thecorresponding Ehresmann–Schauenburg bialgebroid C ( A s , T N ) = ( A s ⊗ A s ) co T N . Lemma 4.7.
The translation map of the coaction (4.15) is given on generators by τ ( g ) = G − ⊗ G,τ ( x ) = 1 ⊗ X − XG − ⊗ G. Proof.
We just apply the corresponding canonical map to obtain: χ ◦ τ ( g ) = G − G ⊗ g = 1 ⊗ g,χ ◦ τ ( x ) = 1 ⊗ x + X ⊗ g − XG − G ⊗ g = 1 ⊗ x. as it should be. (cid:3) We have then the following:
Proposition 4.8.
For any complex number s there is a Hopf algebra isomorphism Φ : C ( A s , T N ) ≃ T N . Proof.
It is easy to see that the elementsΞ = X ⊗ G − − ⊗ XG − , Γ = G ⊗ G − (4.16)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:Ξ N = 0 , Γ N = 1 , Ξ • C Γ = q Ξ • C Γ . (4.17)Indeed, the last two relations are easy to see. As for the first one, shifting powers of G − to the right one findsΞ N = X N ⊗ G − N + N − X r =1 c r X N − r ⊗ X r G − N + ( − N ⊗ ( XG − ) N = (cid:2) X N ⊗ N − X r =1 c r X N − r ⊗ X r + ( − N q n ( n − ⊗ X N (cid:3) G − N or explicit coefficients c r depending on q . Then, using the same methods as in [16]one shows that, being q a primitive N -th roots of unity, all coefficients c r vanish and soΞ N = X N ⊗ G − N + ( − N ⊗ ( XG − ) N which then vanishes from X N = 0.Thus the elements Ξ and Γ generate a copy of the algebra T N and the isomorphism Φmaps Ξ to x and Γ to g . The map Φ is also a coalgebra map. Indeed,∆(Φ(Γ)) = ∆( g ) = g ⊗ g, while, using Lemma 4.7,∆ C (Γ) = G (0) ⊗ G (1) < > ⊗ G (1) < > ⊗ G − = G ⊗ G − ⊗ G ⊗ G − = Γ ⊗ Γ . Thus (Φ ⊗ Φ)(∆ C (Γ)) = g ⊗ g = ∆(Φ(Γ)). Similarly,∆(Φ(Ξ)) = ∆( x ) = 1 ⊗ x + x ⊗ g, while, using Lemma 4.7 in the third step,∆ C (Ξ) = ∆ C ( X ⊗ G − ) − ∆ C (1 ⊗ XG − )= X (0) ⊗ X (1) < > ⊗ X (1) < > ⊗ G − − ⊗ ⊗ ⊗ XG − = 1 ⊗ x < > ⊗ x < > ⊗ G − + X ⊗ g < > ⊗ g < > ⊗ G − − ⊗ ⊗ ⊗ XG − = 1 ⊗ (cid:16) ⊗ X − XG − ⊗ G (cid:17) ⊗ G − + X ⊗ G − ⊗ G ⊗ G − − ⊗ ⊗ ⊗ XG − = 1 ⊗ ⊗ (cid:16) X ⊗ G − − ⊗ XG − (cid:17) + (cid:16) X ⊗ G − − ⊗ XG − (cid:17) ⊗ G ⊗ G − = 1 ⊗ Ξ + Ξ ⊗ Γ . Thus (Φ ⊗ Φ)(∆ C (Ξ)) = 1 ⊗ x + x ⊗ g = ∆(Φ(Ξ)). Finally: ǫ C (Γ) = 1 = ǫ ( g ) and ǫ C (Ξ) = 0 = ǫ ( x ). This concludes the proof. (cid:3) The group of characters of the Taft algebra T N is the cyclic group Z N : indeed anycharacter φ must be such that φ ( x ) = 0, while φ ( g ) N = φ ( g N ) = φ (1) = 1. Then forthe group of gauge transformations of the Galois object A s , the same as the group ofbisections of the bialgebroid C ( A s , T N ), due to Proposition 4.8 we have,Aut T N ( A s ) ≃ B ( C ( A s , T N )) = Char( T N ) = Z N . (4.18)On the other hand, elements F of Aut extT N ( A s ) ≃ B ext ( C ( A s , T N ), due to equivariance F ( a ) (0) ⊗ F ( a ) (1) = F ( a (0) ) ⊗ a (1) for any a ∈ A s , can be given as a block diagonal matrix F = diag( M , M , . . . , M N − , M N )with each M j a N × N invertible lower triangular matrix M j = . . . b a N − . . . b b a N − . . . . . . ...... . . . . . . . . . 0 0 b N − , b N − , . . . . . . a b N b N . . . b N,N − b N,N − a All matrices M j have in common the diagonal elements a j (ciclic permuted) which are alldifferent from zero for the invertibility of M j . For the subgroup Aut T N ( A s ) the M j arediagonal as well with a k = ( a ) k and ( a ) N = 1 so that M j ∈ Z N . The reason all M j sharethe same diagonal elements (up to permutation) is the following: firstly, the ‘diagonal’ orm of the coaction of G in (4.15) imply 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, do to the first term in the coaction of X in (4.15), 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 { , X, G, XG } , the equivariance F ( a ) (0) ⊗ F ( a ) (1) = F ( a (0) ) ⊗ a (1) for the coaction (4.15) becomes F ( X ) (0) ⊗ F ( X ) (1) = 1 ⊗ x + F ( X ) ⊗ g,F ( G ) (0) ⊗ F ( G ) (1) = F ( G ) ⊗ g,F ( XG ) (0) ⊗ F ( XG ) (1) = F ( G ) ⊗ xg + F ( XG ) ⊗ . Next, write F ( a ) = f ( a ) + f ( a ) X + f ( a ) G + f ( a ) XG , for complex numbers f k ( a ).And compute F ( a ) (0) ⊗ F ( a ) (1) = f ( a ) 1 ⊗ f ( a ) (1 ⊗ x + X ⊗ g ) + f ( a ) G ⊗ g + f ( a ) ( G ⊗ xg + XG ⊗ f ( X ) = f ( X ) = 0 f ( G ) = f ( G ) = f ( G ) = 0 f ( XG ) = f ( XG ) = 0 , while the remaining coefficients are related by the system of equations f ( X ) (1 ⊗ x + X ⊗ g ) + f ( X ) G ⊗ g = 1 ⊗ x + F ( X ) ⊗ g,f ( G ) G ⊗ g = F ( G ) ⊗ g,f ( XG ) 1 ⊗ f ( XG ) ( G ⊗ xg + XG ⊗
1) = F ( G ) ⊗ xg + F ( XG ) ⊗ . One readily finds solutions f ( X ) = 1 , f ( X ) = γ, f ( XG ) = βf ( G ) = f ( XG ) = α with α, β, γ arbitrary complex numbers. Thus a generic element F of Aut extT ( A s ) can berepresented by the matrix: F : XGGX β α α
00 0 γ XGGX . (4.19)Asking F to be invertible requires α = 0. On the other hand, any F ∈ Aut T ( A s ) is analgebra 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 ) = ( γG + X ) = γ + ( GX + XG ) + s yields γ = 0; then β + αXG = F ( XG ) = αXG yields β = 0; and 1 = F ( G ) = ( αG ) leadsto α = 1. Thus F ( X ) = X and F ( G ) = αG , with α = 1 and we conclude thatAut T ( A s ) ≃ Z . hen N = 3, a similar, if longer computation, gives for Aut expT ( A s ) an eight parametergroup with its elements F of the form F : XG X GG XGX GXX G − β α η − qδ α α δ α λ − qγ α γ θ − qβ α XG X GG XGX GXX G − . (4.20)One needs α j = 0, j = 1 , F ∈ Aut T N ( A s )one starts from it values on the generators, F ( G ) = α G and F ( X ) = γG + X , to concludethat F is a diagonal matrix (in particular F ( X ) = X ) with α = ( α ) and 1 = ( α ) ;thus Aut T ( A s ) ≃ Z .5. Crossed module structures on bialgebroids
Isomorphisms of (a usual) groupoid with natural transformations between them forma strict 2-groupoid. In particular, automorphisms of the groupoid with its natural trans-formations, form a strict 2-group or, equivalently, a crossed module (cf. [11], Definition3.21). The crossed module combines automorphisms of the groupoid and bisections sincethe latter are the natural transformations from the identity functor to automorphisms.The crossed module involves the product on bisections and the composition on automor-phisms, and the group homomorphism from bisections to automorphisms together withthe action of automorphisms on bisections by conjugation. Any bisection σ is the 2-arrowfrom the identity morphism to an automorphism Ad σ , and the composition of bisectionscan be viewed as the horizontal composition of 2-arrows.In this section we quantise this construction for the Ehresmann–Schauenburg bialge-broid of a Hopf–Galois extension. We construct a crossed module for the bisections andthe automorphisms of the bialgebroid. Notice that we do not need the antipode of bial-gebroid, that is we do not need to defined the crossed module on Hopf algebroid and thecrossed module on bialgebroid is a generalization of the crossed module on groupoid. Inthe next section, The construction can also be repeated for extended bisections.5.1. Automorphisms and crossed modules.
Recall that a crossed module is the data(
M, N, µ, α ) of two groups M , N together with a group morphism µ : M → N and agroup morphism 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 − , for any n ∈ N and m ∈ M ;(2) α µ ( m ) ( m ′ ) = mm ′ m − , for any m, m ′ ∈ M .Then, with the definition of the automorphism group of a bialgebroid as given inDefinition 2.1, we aim at proving the following. Theorem 5.1.
Given a Hopf–Galois extension B = A coH ⊆ A , let C ( A, H ) be thecorresponding left Ehresmann–Schauenburg bialgebroid, and assume B is in the cen-tre of A . Then there is a group morphism Ad : B ( C ( A, H )) → Aut( C ( A, H )) and n action ⊲ of Aut( C ( A, H )) on B ( C ( A, H )) that give a crossed module structure to (cid:0) B ( C ( A, H )) , Aut( C ( A, H )) (cid:1) . We give the proof in a few lemmas.
Lemma 5.2.
Given a Hopf–Galois extension B = A coH ⊆ A , let C ( A, H ) be the cor-responding left Ehresmann–Schauenburg bialgebroid. Assume B belongs to the centre of A . For any bisection σ ∈ B ( C ( A, H )) , denote ad σ = σ ◦ s ∈ Aut( B ) and let F σ be theassociated gauge element in Aut H ( A ) (see (3.17) ). Define Ad σ : C ( A, H ) → C ( A, H ) by Ad σ ( a ⊗ ˜ a ) := F σ ( a ) ⊗ F σ (˜ a )= σ ( a (0) ⊗ a (1) < > ) a (1) < > ⊗ σ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > . (5.1) Then the pair ( Ad σ , ad σ ) is an automorphism of C ( A, H ) .Proof. Since F σ is an algebra automorphism, so is Ad σ . Then, for any b ∈ B , Ad σ ( t ( b )) = Ad σ (1 ⊗ b ) = 1 ⊗ σ ( b ⊗
1) = t ( ad σ ( b ))and Ad σ ( s ( b )) = Ad σ ( b ⊗
1) = σ ( b ⊗ ⊗ s ( ad σ ( b )) . So conditions (i) and (ii) of Definition 2.1 are satisfied. For condition (iii), using H -equivariance of F σ , with a ⊗ ˜ a ∈ C ( A, H ) we get(∆ C ( A,H ) ◦ Ad σ )( a ⊗ ˜ a ) = F σ ( a (0) ) ⊗ a (1) < > ⊗ B a (1) < > ⊗ F σ (˜ a ) . (5.2)On the other hand, (cid:0) ( Ad σ ⊗ B Ad σ ) ◦ ∆ C ( A,H ) (cid:1) ( a ⊗ ˜ a ) = F σ ( a (0) ) ⊗ F σ ( a (1) < > ) ⊗ B F σ ( a (1) < > ) ⊗ F σ (˜ a ) . (5.3)Now, for any F ∈ Aut H ( A ), given h ∈ H , one has F ( h < > ) ⊗ B F ( h < > ) = h < > ⊗ B h < > , for any h ∈ H. (5.4)By applying the canonical map χ and using equivariance of F we compute, χ ( F ( h < > ) ⊗ B F ( h < > )) = F ( h < > ) F ( h < > ) (0) ⊗ F ( h < > ) (1) = F ( h < > ) F ( h < > (0) ) ⊗ h < > (1) = F (1 A ) ⊗ h = 1 A ⊗ h using (2.16). Being χ an isomorphism we get the relation (5.4). Using this for the righthand sides of (5.2) and (5.3) shows that they coincide and condition (iii) is satisfied.Finally,( ǫ ◦ Ad σ )( a ⊗ ˜ a ) = ǫ ( F σ ( a ) ⊗ F σ (˜ a )) = F σ ( a ˜ a ) = ( σ ◦ s )( a ˜ a ) = ad σ ◦ ǫ ( a ⊗ ˜ a ) . This finishes the proof. (cid:3)
Remark . The map Ad σ in (5.1) can also be written in the following useful ways: Ad σ ( a ⊗ ˜ a ) = σ ( a (0) ⊗ a (1) < > ) a (1) < > ⊗ a (2) < > σ − ( a (2) < > ⊗ ˜ a )= σ (( a ⊗ ˜ a ) (1) ) ( a ⊗ ˜ a ) (2) ( σ ◦ s ) ◦ σ − (( a ⊗ ˜ a ) (3) )) (5.5) ndeed, for a ⊗ a ′ ∈ C ( A, H ), by inserting (2.17) and using the definition of the inverse φ − , we compute, Ad σ ( a ⊗ ˜ a ) = σ ( a (0) ⊗ a (1) < > ) a (1) < > ⊗ σ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > = σ ( a (0) ⊗ a (1) < > ) a (1) < > ⊗ a (2) < > a (2) < > σ (˜ a (0) ⊗ ˜ a (1) < > )˜ a (1) < > = σ ( a (0) ⊗ a (1) < > ) a (1) < > ⊗ a (2) < > σ (˜ a (0) ⊗ ˜ a (1) < > ) a (2) < > ˜ a (1) < > = σ ( a (0) ⊗ a (1) < > ) a (1) < > ⊗ a (2) < > ( σ ◦ s ) ◦ σ − ( a (2) < > ⊗ ˜ a ) (cid:1) = σ (( a ⊗ ˜ a ) (1) ) ( a ⊗ ˜ a ) (2) ( σ ◦ s ) ◦ σ − (( a ⊗ ˜ a ) (3) )It is easy to see that Ad σ ◦ Ad τ = Ad τ ∗ σ for any σ , σ ∈ B ( C ( A, H )), while ( Ad σ ) − = Ad σ − and Ad ǫ = id C ( A,H ) . And, of course ad σ ◦ ad τ = ad τ ∗ σ , with ( ad σ ) − = ad σ − and ad ǫ = id B . Thus Ad is a group morphism Ad : B ( C ( A, H )) → Aut( C ( A, H )).Next, given an automorphism (Φ , ϕ ) of C ( A, H ) with inverse (Φ − , ϕ − ), we define anaction of (Φ , ϕ ) on the group of bisections B ( C ( A, H )) as follow:Φ ⊲ σ := ϕ − ◦ σ ◦ Φ , (5.6)for any σ ∈ B ( C ( A, H )). The result is an algebra map since it is a composition of algebramap. Moreover, for any b ∈ B , (Φ ⊲ σ )( t ( b )) = ϕ − ( σ ( t ( ϕ ( b )))) = ϕ − ( ϕ ( b )) = b , so that(Φ ⊲ σ ) ◦ t = id B ; while (Φ ⊲ σ )( s ( b )) = ϕ − (cid:0) ( σ ◦ s )( ϕ ( b )) (cid:1) , so that (Φ ⊲ σ ) ◦ s ∈ Aut( B ).And one checks that (Φ ⊲ σ ) − = Φ ⊲ σ − = ϕ − ◦ σ − ◦ Φ . (5.7) Lemma 5.4.
Given any automorphism (Φ , ϕ ) , the action defined in (5.6) is a groupautomorphism of B ( C ( A, H )) .Proof. Let σ, τ ∈ B ( C ( A, H )), and c ∈ C ( A, H ), we compute:(Φ ⊲ τ ) ∗ (Φ ⊲ σ )( c ) = (Φ ⊲ σ )( s (Φ ⊲ τ ( c (1) )))(Φ ⊲ σ )( c (2) )= (cid:0) ϕ − ◦ σ ◦ Φ ◦ s ◦ ϕ − ◦ τ ◦ Φ( c (1) ) (cid:1)(cid:0) ϕ − ◦ σ ◦ Φ( c (2) ) (cid:1) = (cid:0) ϕ − ◦ σ ◦ s ◦ τ ◦ Φ( c (1) ) (cid:1)(cid:0) ϕ − ◦ σ ◦ Φ( c (2) ) (cid:1) = ϕ − (cid:0) σ ◦ s ◦ τ (Φ( c (1) )) σ (Φ( c (2) )) (cid:1) = ϕ − ◦ ( τ ∗ σ ) ◦ Φ( c )= Φ ⊲ ( τ ∗ σ )( c ) , where the last but one step uses condition (iii) of Definition 2.1. Also,Φ ⊲ ǫ = ϕ − ◦ ǫ ◦ Φ = ϕ − ◦ ϕ ◦ ǫ = ǫ. Finally, for any two automorphism (Φ , ϕ ) and (Ψ , ψ ) of C ( A, H ), we haveΦ ⊲ (Ψ ⊲ ( σ )) = ϕ − ◦ ψ − ◦ σ ◦ Ψ ◦ Φ = ( ψϕ ) − ◦ σ ◦ Ψ ◦ Φ = (Ψ ◦ Φ) ⊲ σ. In particular Φ − ⊲ (Φ ⊲ ( σ )) = σ and so the action is an automorphism of B ( C ( A, H )). (cid:3)
Lemma 5.5.
For any automorphism (Φ , ϕ ) , and any σ ∈ B ( C ( A, H )) we have Ad Φ ⊲σ = Φ − ◦ Ad σ ◦ Φ . Proof.
With a ⊗ ˜ a ∈ C ( A, H ), from (5.5) we get( Ad σ ◦ Φ)( a ⊗ ˜ a ) = σ ((Φ( a ⊗ ˜ a )) (1) ) (Φ( a ⊗ ˜ a )) (2) ( σ ◦ s ) ◦ σ − ((Φ( a ⊗ ˜ a )) (3) ) , (5.8) hile, using (5.6) and (5.7), we have Ad Φ ⊲σ ( a ⊗ ˜ a ) = (Φ ⊲ σ )(( a ⊗ ˜ a ) (1) ) ( a ⊗ ˜ a ) (2) (cid:0) (Φ ⊲ σ ) ◦ s (cid:1) ◦ (Φ ⊲ σ ) − (( a ⊗ ˜ a ) (3) )= ϕ − (cid:0) σ (Φ(( a ⊗ ˜ a ) (1) )) (cid:1) ( a ⊗ ˜ a ) (2) (cid:0) (Φ ⊲ σ ) ◦ s (cid:1) ◦ ( ϕ − ◦ σ − (cid:1) (Φ(( a ⊗ ˜ a ) (3) )) . Since Φ is a bimodule map: Φ( b ( a ⊗ ˜ a )˜ b ) = ϕ ( b )Φ( a ⊗ ˜ a ) ϕ (˜ b ), for all b, ˜ b ∈ B , we get,(Φ ◦ Ad Φ ⊲σ )( a ⊗ ˜ a ) = σ (Φ(( a ⊗ ˜ a ) (1) )) Φ(( a ⊗ ˜ a ) (2) ) ( σ ◦ s ) ◦ σ − (Φ(( a ⊗ ˜ a ) (3) )) . (5.9)That the right hand sides of (5.8) and (5.9) are equal follows from the equavariancecondition (iii) of Definition 2.1. (cid:3) Lemma 5.6.
Let σ , τ ∈ B ( C ( A, H )) , then Ad τ ⊲ σ = τ ∗ σ ∗ τ − .Proof. With a ⊗ ˜ a ∈ C ( A, H ), using the definition (5.5) we compute Ad τ ⊲ σ ( a ⊗ ˜ a ) = ( ad − τ ◦ σ )( Ad τ ( a ⊗ ˜ a ))= (( τ ◦ s ) − ◦ σ ) (cid:0) τ (( a ⊗ ˜ a ) (1) ) ( a ⊗ ˜ a ) (2) ( τ ◦ s ) ◦ τ − (( a ⊗ ˜ a ) (3) ) (cid:1) = ( τ − ◦ s ) (cid:16) ( σ ◦ s ) (cid:0) τ (( a ⊗ ˜ a ) (1) ) (cid:1) σ (( a ⊗ ˜ a ) (2) ) ( σ ◦ t ) ◦ ( τ ◦ s ◦ τ − )(( a ⊗ ˜ a ) (3) ) (cid:17) = ( τ − ◦ s ) (cid:16) ( σ ◦ s ) (cid:0) τ (( a ⊗ ˜ a ) (1) ) (cid:1) σ (( a ⊗ ˜ a ) (2) ) ( τ ◦ s ) ◦ τ − (( a ⊗ ˜ a ) (3) ) (cid:17) = ( τ − ◦ s ) (cid:16) ( τ ∗ σ )(( a ⊗ ˜ a ) (1) ) (cid:17) τ − (( a ⊗ ˜ a ) (2) )= τ ∗ σ ∗ τ − ( a ⊗ ˜ a ) , where we used ( τ ◦ s ) − = τ − ◦ s , σ ◦ t = id B and definition (3.10) for the product. (cid:3) Taken together the previous lemmas establish that a crossed module structure to (cid:0) B ( C ( A, H )) , Aut( C ( A, H )) , Ad, ⊲ (cid:1) , which is the content of Theorem 5.1.5.2. CoInner authomorphisms of bialgebroids.
Given a Hopf algebra H and a char-acter φ : H → C , one defines a Hopf algebra automorphisms (see [13, page 3807]) bycoinn( φ ) : H → H, coinn( φ )( h ) := φ ( h (1) ) h (2) φ ( S ( h (3) )) , (5.10)for any h ∈ H . Recall that for a character φ − = φ ◦ S . The set CoInn( H ) of co-innerauthomorphisms of H is a normal subgroup of the group Aut Hopf ( H ) of Hopf algebraautomorphisms (this is just Aut( H ) if one sees view H as a bialgebroid over C ).We know from the previous sections that for a Galois object A of a Hopf algebra H , the corresponding bialgebroid C ( A, H ) is a Hopf algebra. Also, the group of gaugetransformations of the Galois object which is the same as the group of bisections canbe identified with the group of characters of C ( A, H ) (see (4.5)). It turns out that thesegroups are also isomorphic to CoInn( C ( A, H )). We have the following lemma:
Lemma 5.7.
For a Galois object A of a Hopf algebra H , let C ( A, H ) be the correspondingbialgebroid of A . If φ ∈ B ( C ( A, H )) = Char( C ( A, H )) , then Ad φ = coinn( φ ) .Proof. Let φ ∈ Char( C ( A, H )); then φ − = φ ◦ S C . Substituting the latter in (5.5), for a ⊗ a ′ ∈ C ( A, H ), we get Ad φ ( a ⊗ ˜ a ) = φ (( a ⊗ ˜ a ) (1) ) ( a ⊗ ˜ a ) (2) ( φ ◦ S C )(( a ⊗ ˜ a ) (3) )= coinn( φ )( a ⊗ ˜ a ) , as claimed. (cid:3) xample . Let us consider again the Taft algebra T N of Section 4.5. We know fromProposition 4.8 that for any T N -Galois object A s the bialgebroid C ( T N , A s ) is isomorphicto T N and bisections of C ( T N , A s ) are the same as characters of T N the group of whichis 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 correspondingautomorphism Ad φ r = coinn( φ r ) is easily found to be on generators given bycoinn( φ r )( g ) = g, coinn( φ r )( x ) = r − x . It is known (cf. [14], Lemma 2.1) that Aut( T N ) ≃ Aut
Hopf ( T N ) ≃ C × : Indeed, given r ∈ C × , one defines an authomorphism 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 − .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, as a crossed module, the action of Aut( T N ) on Char( T N )is trival and the crossed module (Char( C ( T N , A s )) , Aut( C ( T N , A s )) , Ad, id) is isomorphicto ( Z N , C × , i, id), with inclusion i : Z N → C × given by i ( r ) := e − i rπ/N and C × actingtrivially on Z N .5.3. Crossed module structures on extended bisections.
In parallel with the crossedmodule structure on bialgebroid automorphisms and bisections, there is a similar struc-ture on the set of ‘extended’ bialgebroid automorphisms and extended bisections.Given a left bialgebroid ( L , ∆ , ǫ, s, t ) be a left bialgebroid over the algebra B . Anextended automorphism of L is a pair (Φ , ϕ ) with ϕ : B → B an algebra map and aunital invertible linear map Φ : L → L , obeying the properties ( i ) − ( iv ) of Definition 2.1So, an extended automorphism is not required in general to be an algebra map whilestill satisfying all other properties of an automorphism. In particular we still have thebimodule property: Φ( ba ˜ b ) = ϕ ( b )Φ( a ) ϕ (˜ b ). We denote by Aut ext ( L ) the group (bycomposition) of extended automorphisms of L . There is an analogous of Theorem 5.1: Theorem 5.9.
Given a Hopf–Galois extension B = A coH ⊆ A , let C ( A, H ) be the corre-sponding left Ehresmann–Schauenburg bialgebroid, and assume B be in the centre of A .Then there is a group morphism Ad : B ext ( C ( A, H )) → Aut ext ( C ( A, H )) and an action ⊲ of Aut ext ( C ( A, H )) on B ext ( C ( A, H )) such that the group of extended automorphisms Aut ext ( C ( A, H )) and of extended bisections B ext ( C ( A, H )) , form a crossed module. This result is established in parallel and similarly to the proof of Theorem 5.1. Herewe shall only point to the differences in the definitions and the proofs.Thus, under the hypothesis of Theorem 5.9, for any bisection σ ∈ B ext ( C ( A, H )), define Ad σ : C ( A, H ) → C ( A, H ), for any a ⊗ ˜ a ∈ C ( A, H ), by Ad σ ( a ⊗ ˜ a ) := ( σ ( a (0) ⊗ a (1) < > ) a (1) < > ) ⊗ (cid:0) a (2) < > ( σ ◦ s ) ◦ σ − ( a (2) < > ⊗ ˜ a ) (cid:1) , = σ (( a ⊗ ˜ a ) (1) ) ( a ⊗ ˜ a ) (2) ( σ ◦ s ) ◦ σ − (( a ⊗ ˜ a ) (3) ) (5.11)in parallel with (5.5). Then the pair of map ( Ad σ , ad σ = σ ◦ s ) is an extended automor-phism of C ( A, H ). In particular we have, for any c = a ⊗ ˜ a ∈ C ( A, H ),∆ C ( Ad σ ( c )) = σ ( c (1) ) c (2) ⊗ B c (3) ( σ ◦ s ) ◦ σ − ( c (4) )= σ ( c (1) ) c (2) ( σ ◦ s ) ◦ σ − ( c (3) ) ⊗ B σ ( c (4) ) c (5) ( σ ◦ s ) ◦ σ − ( c (6) ) here the 2nd step use ( σ ◦ s ) ◦ σ − ( c (1) ) σ ( c (2) ) = ǫ ( c ). With the latter, we have also,( ǫ ◦ Ad σ )( c ) = σ ( c (1) ) ǫ ( c (2) ) ( σ ◦ s ) ◦ σ − ( c (3) )= σ ( c (1) ) ( σ ◦ s ) ◦ σ − ( c (2) )= ( σ ◦ s ) ǫ ( c )= ad σ ( ǫ ( c )) . Moreover, for two extended bisections σ and τ we have, for c ∈ C ( A, H ), Ad σ ◦ Ad τ ( c ) = Ad σ (cid:0) τ ( c (1) ) c (2) ( τ ◦ s ) ◦ τ − ( c (3) ) (cid:1) = s (cid:0) ad σ ( τ ( c (1) )) (cid:1) Ad σ ( c (2) ) t (cid:0) ad σ ◦ ( τ ◦ s ) ◦ τ − ( c (3) ) (cid:1) = s (cid:16) σ (cid:0) s ( τ ( c (1) )) (cid:1)(cid:17)(cid:16) σ ( c (2) ) c (3) (cid:0) σ ( s ( σ − ( c (4) ))) (cid:1)(cid:17) t ◦ (cid:0) σ ◦ s ) ◦ ( τ ◦ s ) ◦ τ − ( c (5) ) (cid:1) = (cid:16) σ (cid:0) s ( τ ( c (1) )) (cid:1) σ (cid:0) c (2) (cid:1)(cid:17) c (3) (cid:16) ( σ ◦ s ◦ τ ◦ s ) (cid:0) ( τ − ◦ s ) ◦ σ − ( c (4) ) τ − ( c (5) ) (cid:1)(cid:17) = ( τ ∗ σ )( c (1) ) c (2) (cid:0) ( τ ∗ σ ) ◦ s )( σ − ∗ τ − ( c (3) ) (cid:1) = Ad τ ∗ σ ( c ) , with the 2nd step using Ad σ is a B -bimodule map. One also shows ad σ ◦ ad τ = ad τ ∗ σ and( Ad ǫ , ad ǫ ) = (id C ( A,H ) , id B ). Therefore ( Ad σ , ad σ ) is invertible with inverse ( Ad σ − , ad σ − ).When σ is an algebra map, (5.11) reduces to (5.1) (or equivalently to(5.5)).If (Φ , ϕ ) is an extended automorphism of C ( A, H ) with inverse (Φ − , ϕ − ) the formula(5.6) is an action of (Φ , ϕ ) on B ext ( C ( A, H )), a group automorphism of B ext ( C ( A, H )).We only check
F ⊲ σ is well defined as an extended bisection since the rest goes as inthe previous section. For a ⊗ ˜ a ∈ C ( A, H ) and b ∈ B , a direct computation yields:(Φ ⊲ σ )(( a ⊗ ˜ a ) ⊳ b ) = (Φ ⊲ σ )( a ⊗ ˜ a ) b (Φ ⊲ σ )( b ⊲ ( a ⊗ ˜ a )) = (Φ ⊲ σ )( s ( b )) (Φ ⊲ σ )( a ⊗ ˜ a ) . Finally, with similar computation as those of Lemma 5.5 and Lemma 5.6 one shows thatfor any extended automorphism (Φ , ϕ ), and any σ ∈ B ext ( C ( A, H )) one has Ad Φ ⊲σ = Φ − ◦ Ad σ ◦ Φ . And that, with σ , τ ∈ B ext ( C ( A, H )), one has Ad τ ⊲ σ = τ ∗ σ ∗ τ − . Example . Consider a H -Galois object A and let C ( A, H ) be the correspondingbialgebroid, a Hopf algebra itself. Given an extended bisection σ ∈ B ext ( C ( A, H )) ≃ Char ext ( H ), the expression (5.11) reduces to Ad σ ( a ⊗ ˜ a ) = σ (( a ⊗ ˜ a ) (1) ) ( a ⊗ ˜ a ) (2) σ − (( a ⊗ ˜ a ) (3) )In analogy with (5.10) to which it reduces when φ is a character (and Lemma 5.7)we may thing of this unital invertible coalgebra map as defining an extended coinnerauthomorphism of C ( A, H ), coinn( σ )( c ) := Ad σ ( c ) = σ ( c (1) ) c (2) σ − ( c (3) ).In Example 5.8 we constructed an Abelian crossed module for the Taft algebras. Thefollowing example present a non-Abelian crossed module for Taft algebras with respectto the extended characters and extended automorphisms. xample . We know from Section 4.5 that the Schauenburg bialgebroid C ( A s , T N )of any Galois object A s for the Taft algebra T N , is isomorphic to T N itself. ThusAut ext ( C ( A s , T N )) ≃ Aut ext ( T N ) is the group of unital invertible coalgebra maps: mapsΦ : T N → T N such that Φ( h (1) ) ⊗ Φ( h (2) ) = Φ( h ) (1) ⊗ Φ( h ) (2) for any h ∈ T N with Φ(1) = 1.Let us illustrate this for the case N = 2. The coproduct of T on the generators x, g will then require the following condition for an automorphism Φ:Φ( g ) (1) ⊗ Φ( g ) (2) = Φ( g ) ⊗ Φ( g )Φ( x ) (1) ⊗ Φ( x ) (2) = 1 ⊗ Φ( x ) + Φ( x ) ⊗ g Φ( xg ) (1) ⊗ Φ( xg ) (2) = g ⊗ Φ( xg ) + Φ( xg ) ⊗ . (5.12)A little algebra then shows that Φ( g ) = g Φ( x ) = c ( g −
1) + a x Φ( xg ) = b (1 − g ) + a xg (5.13)for arbitrary parameters b, c ∈ C and a , a ∈ C × (for Φ to be invertible). As in (4.19)we can represent Φ as a matrix:Φ : xggx b a − b
00 0 1 0 − c c a xggx . (5.14)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 ) we shall denote σ a = σ ( a ) ∈ C for a ∈ { , x, g, xg } . For theconvolution inverse σ − , from the condition σ ∗ σ − = ǫ we get on the basis that σ = ( σ − ) = 1 ,σ g ( σ − ) g = 1 ,σ g ( σ − ) x + σ x = 0 ,σ g ( σ − ) xg + σ xg = 0 (5.15)from which we solve ( σ − ) g = ( σ g ) − , ( σ − ) x = − σ x ( σ g ) − , ( σ − ) xg = − σ xg ( σ g ) − (5.16)Then computing Ad σ ( h ) = σ ( h (1) ) h (2) σ − ( h (3) ) leads to Ad σ xggx = σ xg σ g − σ xg
00 0 1 0 − σ x ( σ g ) − σ x ( σ g ) − ( σ g ) − . (5.17)We see that the matrix (5.17) is of the form (5.14) with the restriction that a = a − so that Ad φ has determinant 1. Clearly, the image of Char ext ( T ) form a subgroup ofAut ext ( C ( A s , T )) ≃ Aut ext ( T N ). Moreover, Ad : Char ext ( T ) → Aut ext ( T ) is an injective ap. Finally, the action Ad Φ ⊲σ will have as matrix just the product: M Ad Φ ⊲σ = M Φ − M Ad σ M Φ (5.18)= a − [ σ xg + b ( σ g − σ g − a − [ σ xg + b ( σ g − − a − [ σ x ( σ g ) − + c (( σ g ) − − a − [ σ x ( σ g ) − + c (( σ g ) − − σ g ) − We conclude that as a crossed module the action on Char ext ( T ) is not trivial. Appendix A. The classical gauge groupoid
We recall in this section some basic facts about the gauge groupoid associated to aprincipal bundle and to the corresponding group of bisections; we shall mostly follow thebook [10]. Let π : P → M be a principal bundle over the manifold M with structureLie group G . Consider the diagonal action of G on P × P given by ( u, v ) g := ( ug, vg );denote by [ u, v ] the orbit of ( u, v ) and by Ω = P × G P the collection of orbits. Then Ωis a groupoid over M , — the gauge or Ehresmann groupoid of the principal bundle, withsource and target projections given by s ([ u, v ]) := π ( v ) , t ([ u, v ]) := π ( u ) . (A.1)The object inclusion map is M → P × G P is given by m id m := [ u, u ] (A.2)for m ∈ M and u any element in π − ( m ). And the partial multiplication [ u, v ′ ] · [ v, w ],defined when π ( v ′ ) = π ( v ) is given by[ u, v ] · [ v ′ , w ] = [ u, wg ] , (A.3)for the unique g ∈ G such that v = v ′ g . One can always choose representatives suchthat v = v ′ and the multiplication is then simply [ u, v ] · [ v, w ] = [ u, w ]. The inverse is[ u, v ] − = [ v, u ] . (A.4)A bisection of the groupoid Ω is a map σ : M → Ω, which is right-inverse to thesource projection, s ◦ σ = id M , and is such that t ◦ σ : M → M is a diffeomorphism.The collection of bisections, denoted B (Ω), form a group: given two bisections σ and σ ,their multiplication is defined by σ ∗ σ ( m ) := σ (cid:0) ( t ◦ σ )( m ) (cid:1) σ ( m ) , for m ∈ M. (A.5)The identity is the object inclusion m id m , simply denoted id, with inverse given by σ − ( m ) = (cid:16) σ (cid:0) ( t ◦ σ ) − ( m ) (cid:1)(cid:17) − ; (A.6)here ( t ◦ σ ) − as a diffeomorphism of M , while the outer inversion is the one in (A.4).The subset B P/G (Ω) of ‘vertical’ bisections, that is those bisections that are right-inverseto the target projection as well, t ◦ σ = id M , form a subgroup of B (Ω). Here one is really using the classical translation map t : P × M P → G , ( ug, u ) g . t is a classical result [10] that there is a group isomorphism between B (Ω) and thegroup of principal ( G -equivariant) bundle automorphisms of the principal bundle,Aut G ( P ) := { ϕ : P → P ; ϕ ( pg ) = ϕ ( p ) g } , (A.7)while B P/G (Ω) is group isomorphic to the group of gauge transformations, that is thesubgroup of principal bundle automorphisms which are vertical (project to the identityon the base space):Aut
P/G ( P ) := { ϕ : P → P ; ϕ ( pg ) = ϕ ( p ) g , π ( ϕ ( p )) = π ( p ) } . (A.8) Acknowledgment:
We are most grateful to Chiara Pagani for many useful discussions.GL was partially supported by INFN, Iniziativa Specifica GAST and INDAM-GNSAGA.
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Xiao Han
SISSA, via Bonomea 265, 34136 Trieste, Italy
E-mail address : [email protected] Giovanni Landi
Universit`a di Trieste, Via A. Valerio, 12/1, 34127 Trieste, ItalyInstitute for Geometry and Physics (IGAP) Trieste, Italyand INFN, Trieste, Italy
E-mail address : [email protected]@units.it