aa r X i v : . [ m a t h . QA ] S e p TWISTED EHRESMANN SCHAUENBURG BIALGEBROIDS
XIAO HAN*
Abstract.
We construct an invertible normalised 2 cocycle on the Ehresmann Schauen-burg bialgebroid of a cleft Hopf Galois extension under the condition that the cor-responding Hopf algebra is cocommutative and the image of the unital cocycle cor-responding to this cleft Hopf Galois extension belongs to the centre of the coinvari-ant subalgebra. Moreover, we show that any Ehresmann Schauenburg bialgebroid ofthis kind is isomorphic to a 2-cocycle twist of the Ehresmann Schauenburg bialgebroidcorresponding to a Hopf Galois extension without cocycle, where comodule algebra isan ordinary smash product of the coinvariant subalgebra and the Hopf algebra (i.e. C ( B σ H, H ) ≃ C ( B H, H ) ˜ σ ). We also study the theory in the case of a Galois objectwhere the base is trivial but without requiring the Hopf algebra to be cocommutative. Contents
1. Introduction 12. Algebraic preliminaries 22.1. Cleft Hopf Galois extensions 32.2. Bialgebroids 53. Ehresmann-Schauenburg bialgebroids 73.1. 2-cocycles on Ehrenmann-Schauenburg bialgebroids 7References 141.
Introduction
The study of principal bundles and groupoids are important in different areas of math-ematics and physics. In the area of noncommutative geometry, we are always interestedin Hopf Galois extensions, which can be viewed as quantisation of principal bundles.For any principal bundle, we can construct a gauge groupoid. In the ‘quantum’ case,the ‘quantum’ gauge groupoid can also be constructed for any Hopf Galois extension B = A coH ⊆ A (quantum principal bundles). This kind of ‘quantum’ gauge groupoids C ( A, H ) are called Ehresmann Schauenburg bialgebroids.In this paper, we will study cleft Hopf Galois extensions, which can be viewed asthe quantisation of trivial principal bundles. Since any cleft Hopf Galois extension isisomorphic to the crossed product B σ H of the coinvariant subalgebra B = A coH andthe Hopf algebra H for a unital cocycle σ : H ⊗ H → B , so instead of studying thecomodule algebra A directly, we can work on the crossed product algebra B σ H .As it was shown in [6] that given a cocommutative Hopf algebra H , there is a bijectivecorrespondence between the equivalence classes of H -cleft Hopf Galois extensions andthe second cohomology group H ( H, Z ( B )), where Z ( B ) is the centre of the coinvariantsubalgebra B . So in this paper we will mainly consider cleft Hopf Galois extensions withcocommutative Hopf algebra and its second cohomology group H ( H, Z ( A )). Under this pecial assumption, we can show that there is an invertible normalised 2-cocycle ˜ σ onthe Ehresmann Schauenburg bialgebroid associated to the cleft Hopf Galois extension.Moreover, the Ehresmann Schauenburg bialgebroid is isomorphic to the 2 cocycle twistedalgebra C ( B H, H ) ˜ σ , where B H is the smash product of B and H . We can also showthat if the action of H on B is trivial, then the corresponding Ehresmann Schauenburgbialgebroid associated to the cleft Hopf Galois extension is isomorphic to C ( B H, H ) ˜ σ even if the Hopf algebra H is not cocommutative. In particular, we will study the caseof Galois objects for any Hopf algebras.In Section §
2, we will give a brief introduction to Hopf Galois extension, and in particu-lar the cleft Hopf Galois extension with its properties. Then we will also recall bialgebroidsand 2-cocycles of them. In Section §
3, we will first study Ehresmann Schauenburg bialge-broid, then we show under some conditions there is an invertible normailsed 2 cocycle on C ( B H, H ), such that C ( A, H ) ≃ C ( B σ H, H ) ≃ C ( B H, H ) ˜ σ . Finally, we will applythe general theory on Galois objects.2. Algebraic preliminaries
In this section we will first recall the definitions of comodule algebras and modulealgebras of Hopf algebras, then we will study Hopf Galois extensions which can be viewedas the quantisation of principal bundles. In particular, we will recall some properties ofcleft Hopf Galois extensions, which can be viewed as trivial noncommutative principalbundles. We also recall the more general notions of rings and corings over an algebraas well as the associated notion of bialgebroids. In this paper, we will assume all thealgebras, comodules and modules are vector spaces over C .Let H be a Hopf algebra with coproduct ∆, counit ǫ and antipode S . We use thesumless Sweedler notation to denote the image of the coproduct, i.e. ∆( h ) = h (1) ⊗ h (2) for all h ∈ H . The convolution algebra of the dual space H ′ := Hom( H, C ) is an unitalassociative algebra with the product given by φ ⋆ ψ ( h ) := φ ( h (1) ) ψ ( h (2) ), for all φ, ψ ∈ H ′ .Given a Hopf algebra H , a left H module algebra is an algebra B , such that it is a left H module. Moreover, it satisfies: h ⊲ ( ab ) = ( h (1) ⊲ a )( h (2) ⊲ b ) , h ⊲ ǫ ( h )1 , (2.1)for all a, b ∈ A and h, g ∈ H , where ⊲ : H ⊗ B → B is the action of the left module B .Given a left H module algebra A , we can define a new algebra A H , which as a vectorspace is equal to A ⊗ H , and the product is given by( a h )( a ′ g ) = a ( h (1) ⊲ a ′ ) h (2) g, (2.2)for all a, a ′ ∈ A and g, h ∈ H , where we write a h for the tensor product a ⊗ h . We callthis algebra the smash product of A and H .Dually, given a Hopf algebra H , the right H -comodule is vector space V , together witha coaction, which is a linear map δ : V → V ⊗ H such that(id ⊗ ∆) ◦ δ = ( δ ⊗ id) ◦ δ, (id ⊗ ǫ ) ◦ δ = id . (2.3)We also use the sumless Sweedler notation to denote the image of coaction, i.e. δ ( v ) = v (0) ⊗ v (1) for all v ∈ V . The morphism between two right H -comodules V and W is a inear map f : V → W , such that f ( v (0) ) ⊗ v (1) = f ( v ) (0) ⊗ f ( v ) (1) (2.4)for any v ∈ V .Given two right H -comodules V and W , the tensor product V ⊗ W is also a right H -comodule with the coaction given by δ diag ( v ⊗ w ) := v (0) ⊗ w (0) ⊗ v (1) w (1) (2.5)for all v ∈ V and w ∈ W .Given a Hopf algebra H , a right H -comodule algebra is an algebra A , such that A isa right H -comodule, and the comodule map δ : A → A ⊗ H is an algebra map. Themorphism between two comodule algebras is both a comodule map and an algebra map,where A ⊗ H has the usual tensor product algebra structure.2.1. Cleft Hopf Galois extensions.
In this section, we will recall the definition of HopfGalois extensions. In particular, we will study cleft Hopf Galois extensions, which couldbe thought of as trivial noncommutative principal bundles.
Definition 2.1.
Let A be a comodule algebra of a Hopf algebra H , with the coinvariantsubalgebra B := (cid:8) b ∈ A | δ ( b ) = b ⊗ H (cid:9) ⊆ A , the extension B = A coH ⊆ A is called a Hopf Galois extension , if the canonical map χ : A ⊗ B A → A ⊗ H, a ⊗ B a ′ aa ′ (0) ⊗ a (1) (2.6) is bijective, where ⊗ B is the balanced tensor product over B (i.e. ab ⊗ B a ′ = a ⊗ B ba ′ forall a, a ′ ∈ A and b ∈ B ). In the following, we will always assume that for any Hopf Galois extension B = A coH ⊆ A , A is a faithful flat left B module.Given a Hopf Galois extension B = A coH ⊆ A , the translation map is τ := χ − | ⊗ H : H → A ⊗ B A, h h < > ⊗ B h < > . (2.7)Since the canonical map χ is left B linear, then the inverse of the canonical map can bedetermined by the translation map. It is shown in [3, Prop. 3.6] and [4, Lemma 34.4]that the translation map of a Hopf Galois extension satisfy the following properties: h < > ⊗ B h < > (0) ⊗ h < > (1) = h (1) < > ⊗ B h (1) < > ⊗ h (2) , (2.8) h (2) < > ⊗ B h (2) < > ⊗ S ( h (1) ) = h < > (0) ⊗ B h < > ⊗ h < > (1) , (2.9) h < > h < > (0) ⊗ h < > (1) = 1 A ⊗ h , (2.10) a (0) a (1) < > ⊗ B a (1) < > = 1 A ⊗ B a , (2.11)for all h ∈ H and a ∈ A .Now we recall cleft Hopf Galois extensions. Definition 2.2.
A Hopf–Galois extension B = A co H ⊆ A is cleft if there is a convolutioninvertible right H comodule map γ : H → A . A Hopf Galois extension has normal basis property , if A ≃ B ⊗ H as left B -modulesand right H -comodules. efinition 2.3. Given a Hopf algebra H and an algebra B , we call H measures B , ifthere is a linear map H ⊗ B → B , given by h ⊗ b h ⊲ b , such that • h ⊲ ǫ ( h )1 , • h ⊲ ( bb ′ ) = ( h (1) ⊲ b )( h (2) ⊲ b ′ ) ,for all h ∈ H and b, b ′ ∈ A . It was shown in [7] that we can construct an algebra by H and A , if H measures A . Lemma 2.4.
Let H be a Hopf algebra, B be an algebra and H measures B . If there isa convolution invertible linear map σ : H ⊗ H → B (i.e. there is a linear map σ − : H ⊗ H → B , such that σ ( h (1) , h ′ (1) ) σ − ( h (2) , h ′ (2) ) = ǫ ( h ) ǫ ( h ′ )1 = σ − ( h (1) , h ′ (1) ) σ ( h (2) , h ′ (2) ) for all h, h ′ ∈ H ), such that (1) 1 ⊲ b = b , (2) h ⊲ ( k ⊲ b ) = σ ( h (1) , k (1) )( h (2) k (2) ⊲ b ) σ − ( h (3) , k (3) ) , (3) σ ( h,
1) = σ (1 , h ) = ǫ ( h )1 , (4) ( h (1) ⊲ σ ( k (1) , m (1) )) σ ( h (2) , k (2) m (2) ) = σ ( h (1) , k (1) ) σ ( h (2) k (2) , m ) ,for all h, k, m ∈ H and b ∈ B . Then there is an algebra structure on B σ H , which isequal to B ⊗ H as vector space, with the product given by ( b σ h )( b ′ σ h ′ ) = b ( h (1) ⊲ b ′ ) σ ( h (2) , h ′ (1) ) σ h (3) h ′ (2) , (2.12) for all h, h ′ ∈ H and b, b ′ ∈ B . Here we have written b σ h for the tensor product b ⊗ h .Conversely, assume H measures B and σ : H ⊗ H → B is a convolution invertible linearmap, if the vector space B σ H (equal to B ⊗ H as vector space) with a product definedby (2.12) is an associative algebra with identity element σ , then we have the fourconditions above. The algebra B σ H given above is called the crossed product of B and H . Let H measures B , we call σ an unital cocycle , if σ ( h,
1) = σ (1 , h ) = ǫ ( h )1 and( h (1) ⊲ σ ( k (1) , m (1) )) σ ( h (2) , k (2) m (2) ) = σ ( h (1) , k (1) ) σ ( h (2) k (2) , m ) , (2.13)for any all h, k, m ∈ H .If σ is trivial (i.e. σ ( h, h ′ ) = ǫ ( hh ′ )1), then from Lemma 2.4 we know that B is a leftmodule algebra of H and B H is the smash product of B and H given by (2.2).It is known (cf. [14], Proposition 7.2.7) that: Proposition 2.5.
Let H measures B with a convolution invertible linear map σ : H ⊗ H → B , such that B σ H is an unital associative algebra, then we have the followingproperties of σ : (1) σ − ( h (1) , k (1) m (1) )( h (2) ⊲ σ − ( k (2) , m (2) )) = σ − ( h (1) k (1) , m ) σ − ( h (2) , k (2) ) , (2) h ⊲ σ ( k, m ) = σ ( h (1) , k (1) ) σ ( h (2) k (2) , m (1) ) σ − ( h (3) , k (3) m (2) ) , (3) h ⊲ σ − ( k, m ) = σ ( h (1) , k (1) m (1) ) σ − ( h (2) k (2) , m (2) ) σ − ( h (3) , k (3) ) , (4) ( h (1) ⊲ σ − ( S ( h (4) ) , h (5) )) σ ( h (2) , S ( h (3) )) = ǫ ( h )1 . roof. We can see that (1) and (2) can be derived from Lemma 2.4, and (3) can bederived from (1). Now let’s check (4):( h (1) ⊲ σ − ( S ( h (4) ) , h (5) )) σ ( h (2) , h (3) )= σ ( h (1) , S ( h (8) ) h (9) ) σ − ( h (2) S ( h (7) ) , h (10) ) σ − ( h (3) , S ( h (6) )) σ ( h (4) , S ( h (5) ))= σ ( h (1) , S ( h (4) ) h (5) ) σ − ( h (2) S ( h (3) ) , h (6) )= ǫ ( h )1 , for any h ∈ H . (cid:3) We can see that B = ( B σ H ) coH ⊆ B σ H is a Hopf Galois extension with thecoaction δ ( b σ h ) := b σ h (1) ⊗ h (2) for all b σ h ∈ B σ H . The inverse of its canonicalmap is given by [14] χ − ( b σ g ⊗ h ) = ( b σ g )( σ − ( S ( h (2) ) , h (3) ) σ S ( h (1) )) ⊗ B σ h (4) , (2.14)for all b ∈ B and g, h ∈ H . Indeed, we can check that χ (cid:0) ( b σ g )( σ − ( S ( h (2) ) , h (3) ) σ S ( h (1) )) ⊗ B σ h (4) (cid:1) = ( b σ g ) (cid:0) σ − ( S ( h (3) ) , h (4) ) σ ( S ( h (2) ) , h (5) ) σ S ( h (1) ) h (6) (cid:1) ⊗ h (7) = b σ g ⊗ h. We also have χ − ( χ ( b σ g ⊗ B b ′ σ h ))= ( b σ g )( b ′ σ h (1) )( σ − ( S ( h (3) ) , h (4) ) σ S ( h (2) )) ⊗ B σ h (5) = ( b σ g ) (cid:0) b ′ ( h (1) ⊲ σ − ( S ( h (6) ) , h (7) )) σ ( h (2) , S ( h (5) )) σ h (3) S ( h (4) ) (cid:1) ⊗ B σ h (8) = ( b σ g )( b ′ σ ⊗ B σ h = b σ g ⊗ B b ′ σ h, for all b, b ′ ∈ B and g, h ∈ H , where the third step uses Proposition 2.5. It is known (cf.[14], Theorem 8.2.4) that: Theorem 2.6.
Let B = A co H ⊆ A be a Hopf Galois extension. Then the following areequivalent: • B = A co H ⊆ A is cleft. • B = A co H ⊆ A has normal basis property. • A ≃ B σ H as left B -modules and right H -comodule algebras. Bialgebroids.
Here we also give an introduction of bialgebroids (cf. [4], [5]). Foran algebra B , a B -ring is a triple ( A, µ, η ). Here A is a B -bimodule and µ : A ⊗ B A → A and η : B → A are B -bimodule maps, satisfying the associativity and unit conditions µ ◦ ( µ ⊗ B id A ) = µ ◦ (id A ⊗ B µ ) and µ ◦ ( η ⊗ B id A ) = id A = µ ◦ (id A ⊗ B η ) . (2.15)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 ′ . Remark . Let B and A be algebras, if there is an algebra map η : B → A , then A is a B -bimodule with b ⊲ a ⊳ b ′ = η ( b ) aη ( b ′ ). Moreover, A is a B -ring with the productobtained from the universality of the coequaliser A ⊗ A → A ⊗ B A which identifies anelement ab ⊗ a ′ with a ⊗ ba ′ . or an algebra B , a B -coring is a triple ( C, ∆ , ǫ ). Here C is an B -bimodule and∆ : C → C ⊗ B C and ǫ : C → B are B -bimodule maps, satisfying the coassociativity andcounit conditions.(∆ ⊗ B id C ) ◦ ∆ = (id C ⊗ B ∆) ◦ ∆ and ( ǫ ⊗ B id C ) ◦ ∆ = id C = (id C ⊗ B ǫ ) ◦ ∆ . (2.16)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 . Definition 2.8.
Let B be an algebra. A left B -bialgebroid L consists of a B ⊗ B op -ring L with the unit η : B ⊗ B op → L . The restrictions of ηs := η ( · ⊗ B B ) : B → L and t := η (1 B ⊗ B · ) : B op → L are called source and target map, with their ranges commute in B .Moreover, L is a B -coring ( L , ∆ , ǫ ) on the same vector space L . They are subject tothe following compatibility axioms. (i) The bimodule structure in the B -coring ( L , ∆ , ǫ ) is related to the B ⊗ B op -ring L via b ⊲ a ⊳ b ′ := s ( b ) t ( b ′ ) a, for b, b ′ ∈ B, a ∈ L . (2.17)(ii) Considering L as an B -bimodule as in (2.17) , the coproduct ∆ corestricts to analgebra map from L to L × B L := { X i a i ⊗ B a ′ i | X i a i t ( b ) ⊗ B a ′ i = X i a i ⊗ B a ′ i s ( b ) , ∀ b ∈ B } , (2.18) where L × B L is an algebra via factorwise multiplication. (iii) The counit ǫ is a left character on the B -ring ( L , s ) : (1) ǫ (1 L ) = 1 B . (2) ǫ ( s ( b ) a ) = bǫ ( a ) , (3) ǫ ( as ( ǫ ( a ′ ))) = ǫ ( aa ′ ) = ǫ ( at ( ǫ ( a ′ ))) ,for all a, a ′ ∈ L and b ∈ B . Morphisms between left B bialgebroids are B -coring maps which are also algebra maps.Given a left B -bialgebroid L , then there is an algebra structure on B Hom B ( L⊗ B ⊗ B op L , B )with the (convolution) product given by f ⋆ g ( a, a ′ ) := f ( a (1) , a ′ (1) ) g ( a (2) , a ′ (2) ) , (2.19)for all a, a ′ ∈ L and f, g ∈ B Hom B ( L ⊗ B ⊗ B op L , B ). The unit of this algebra is ˜ ǫ : a ⊗ a ′ ǫ ( aa ′ ). Moreover, the B -bimodule structure on L ⊗ B ⊗ B op L is give by b ⊲ ( a ⊗ a ′ ) ⊳ b ′ = s ( b ) t ( b ′ ) a ⊗ a ′ , for all b, b ′ ∈ B , and the balanced tensor product L ⊗ B ⊗ B op L is inducedby the algebra map η : B ⊗ B op → L . Definition 2.9.
Let L be a left B -bialgebroid, an invertible normalised 2-cocycle on L is a convolution invertible element ˜ σ ∈ B Hom B ( L ⊗ B ⊗ B op L , B ) , such that (1) ˜ σ ( s ( b ) t ( b ′ ) a, a ′ ) = b ˜ σ ( a, a ′ ) b ′ (bilinearity), (2) ˜ σ ( a, s (˜ σ ( a ′ (1) , a ′′ (1) )) a ′ (2) a ′′ (2) ) = ˜ σ ( s (˜ σ ( a (1) , a ′ (1) )) a (2) a ′ (2) , a ′′ ) (cocycle condition). (3) σ (1 L , a ) = ǫ ( a ) = σ ( a, L ) (normalisation),for all a, a ′ , a ′′ ∈ L and b, b ′ ∈ B . It is known in [5], we can define a new left bialgebroid by an invertible normalised2-cocycle. roposition 2.10. Let L be a left B -bialgebroid and ˜ σ ∈ B Hom B ( L ⊗ B ⊗ B op L , B ) be aninvertible normalised 2-cocycle, with inverse ˜ σ − , then the B -coring L with the twistedproduct a · ˜ σ a ′ := s (˜ σ ( a (1) , a ′ (1) )) t (˜ σ − ( a (3) , a ′ (3) )) a (2) a ′ (2) , (2.20) for all a, a ′ ∈ L , constitute a left B -bialgebroid L ˜ σ . The proposition above can also apply to Hopf algebras, which are left bialgebroid over C . 3. Ehresmann-Schauenburg bialgebroids
Ehresmann Schauenburg Bialgebroids can be viewed as the quantisation of Gaugegroupoids associated to principal bundles. Here we will first give the definition of Ehres-mann Schauenburg Bialgebroids [4, § Definition 3.1.
Let B = A coH ⊆ A be a Hopf Galois extension such that A is a faithfulflat left B -module, the B -bimodule C ( A, H ) := { a ⊗ ˜ a ∈ A ⊗ A | a (0) ⊗ τ ( a (1) )˜ a = a ⊗ ˜ a ⊗ B } . (3.1) is a B -coring with the coring product and counit given by ∆( a ⊗ ˜ a ) = a (0) ⊗ τ ( a (1) ) ⊗ ˜ a, (3.2) ǫ ( a ⊗ ˜ a ) = a ˜ a. (3.3) Moreover, C ( A, H ) is a B ⊗ B op -ring with the product given by ( a ⊗ ˜ a ) • C ( a ′ ⊗ ˜ a ′ ) = aa ′ ⊗ ˜ a ′ ˜ a, (3.4) for all a ⊗ ˜ a , a ′ ⊗ ˜ a ′ ∈ C . The source and target maps are s ( b ) = b ⊗ , (3.5) t ( b ) = 1 ⊗ b. (3.6) All of the structures given above form a left B -bialgebroid, which is called EhresmannSchauenburg bialgebroid . It was shown in [4], [8] that the B -bimodule C ( A, H ) is isomorphic to the B -bimoduleof coinvariant elements, that is C ( A, H ) ≃ ( A ⊗ A ) coH := { a ⊗ ˜ a ∈ A ⊗ A | a (0) ⊗ ˜ a (0) ⊗ a (1) ˜ a (1) = a ⊗ ˜ a ⊗ H } . (3.7)3.1. Let B = A coH ⊆ A bea cleft Hopf Galois extension, then by Theorem 2.6 we know C ( A, H ) ≃ C ( B σ H, H ). Inparticular, B = ( B H ) coH ⊆ B H is a cleft Hopf Galois extension. From (3.7) we cansee C ( B H, H ) = { b h (1) ⊗ b ′ S ( h (2) ) | ∀ b, b ′ ∈ B and ∀ h ∈ H } . (3.8)We can also see the coproduct is given by∆( b h (1) ⊗ b ′ S ( h (2) )) = b h (1) ⊗ S ( h (2) ) ⊗ B h (3) ⊗ b ′ S ( h (4) ) , (3.9) ince from (2.14) we know the translation map τ : H → B H ⊗ B B H is τ ( h ) = 1 S ( h (1) ) ⊗ B h (2) , (3.10)for all h ∈ H . Lemma 3.2.
For a crossed product B σ H , with H being cocommutative and the imageof σ belonging to the centre of B , then the linear map ˜ σ ∈ B Hom B ( C ( B H, H ) ⊗ B ⊗ B op C ( B H, H ) , B ) given by ˜ σ ( b h (1) ⊗ b ′ S ( h (2) ) , c g (1) ⊗ c ′ S ( g (2) )) := b ( h (1) ⊲ c ) σ ( h (2) , g (1) )(( h (3) g (2) ) ⊲ c ′ )( h (4) ⊲ b ′ ) , (3.11) for all b h (1) ⊗ b ′ S ( h (2) ) , c g (1) ⊗ c ′ S ( g (2) ) ∈ C ( B H, H ) is an invertible normalised2-cocycle.Proof. Since H is cocommutative and the image of σ belongs to the centre of B , then byLemma 2.4, B is a left H -module algebra, which ensures the smash product B H is welldefined. Let X = b h (1) ⊗ b ′ S ( h (2) ), Y = c g (1) ⊗ c ′ S ( g (2) ) and Z = d k (1) ⊗ d ′ S ( k (2) )be arbitrary three elements in C ( B H, H ). First we can show this cocycle is well definedover the balanced tensor over B ⊗ B op : On the one hand we have˜ σ ( Xη ( d ⊗ d ′ ) , Y )) = ˜ σ (( b h (1) ⊗ b ′ S ( h (2) )) η ( d ⊗ d ′ ) , c g (1) ⊗ c ′ S ( g (2) ))= b ( h (1) ⊲ d )( h (2) ⊲ c ) σ ( h (3) , g (1) )(( h (4) g (2) ) ⊲ c ′ )( h (5) ⊲ ( d ′ b ′ )) , for all d, d ′ ∈ B ; On the other hand˜ σ ( X, η ( d ⊗ d ′ ) Y )) = ˜ σ (( b h (1) ⊗ b ′ S ( h (2) )) , η ( d ⊗ d ′ ) c g (1) ⊗ c ′ S ( g (2) ))= ˜ σ (( b h (1) ⊗ b ′ S ( h (2) )) , dc g (1) ⊗ c ′ ( S ( g (3) ) ⊲ d ′ ) S ( g (2) ))= b ( h (1) ⊲ ( dc )) σ ( h (2) , g (1) )(( h (3) g (2) ) ⊲ ( c ′ ( S ( g (3) ) ⊲ d ′ )))( h (4) ⊲ b ′ )= b ( h (1) ⊲ d )( h (2) ⊲ c ) σ ( h (3) , g (1) )(( h (4) g (2) ) ⊲ c ′ )( h (5) ⊲ ( d ′ b ′ )) . The inverse ˜ σ − is given by˜ σ − ( X, Y ) := b ( h (1) ⊲ c ) σ − ( h (2) , g (1) )(( h (3) g (2) ) ⊲ c ′ )( h (4) ⊲ b ′ ) . (3.12)We can see that˜ σ − ⋆ ˜ σ ( X, Y ) = ˜ σ − ⋆ ˜ σ ( b h (1) ⊗ b ′ S ( h (2) ) , c g (1) ⊗ c ′ S ( g (2) ))= b ( h (1) ⊲ c ) σ − ( h (2) , g (1) ) σ ( h (3) , g (2) )(( h (4) g (3) ) ⊲ c ′ )( h (5) ⊲ b ′ )= ǫ (( b h (1) ⊗ b ′ S ( h (2) ))( c g (1) ⊗ c ′ S ( g (2) )))= ˜ ǫ ( X, Y ) , where ˜ ǫ is the unit in the algebra B Hom B ( L ⊗ B ⊗ B op L , B ). Similarly, we can also see˜ σ ⋆ ˜ σ − = ˜ ǫ .It is clear that ˜ σ is left B -linear, we can also show that ˜ σ is also right B -linear:˜ σ ( t ( b ′′ ) X, Y ) = ˜ σ ( t ( b ′′ )( b h (1) ⊗ b ′ S ( h (2) )) , c g (1) ⊗ c ′ S ( g (2) ))= ˜ σ ( b h (1) ⊗ b ′ ( S ( h (3) ) ⊲ b ′′ ) S ( h (2) ) , c g (1) ⊗ c ′ S ( g (2) ))= b ( h (1) ⊲ c ) σ ( h (2) , g (1) )(( h (3) g (2) ) ⊲ c ′ )( h (4) ⊲ ( b ′ ( S ( h (5) ) ⊲ b ′′ )))= b ( h (1) ⊲ c ) σ ( h (2) , g (1) )(( h (3) g (2) ) ⊲ c ′ )( h (4) ⊲ b ′ ) b ′′ = ˜ σ ( X, Y ) b ′′ , for all b ′′ ∈ B . Now let’s show the cocycle condition of ˜ σ . n the one hand we have:˜ σ ( X, s (˜ σ ( Y (1) , Z (1) )) Y (2) Z (2) )= ˜ σ ( b h (1) ⊗ b ′ S ( h (2) ) , c ( g (1) ⊲ d ) σ ( g (2) , k (1) ) g (3) k (2) ⊗ d ′ ( S ( k (4) ) ⊲ c ′ ) S ( k (3) ) S ( g (4) )))= b (cid:16) h (1) ⊲ (cid:0) c ( g (1) ⊲ d ) σ ( g (2) , k (1) ) (cid:1)(cid:17) σ ( h (2) , g (3) k (2) ) (cid:0) ( h (3) g (4) k (3) ) ⊲ ( d ′ ( S ( k (4) ) ⊲ c ′ )) (cid:1) ( h (4) ⊲ b ′ ) . On the other hand,˜ σ ( s (˜ σ ( X (1) , Y (1) )) X (2) Y (2) , Z )= ˜ σ ( b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) h (3) g (2) ⊗ c ′ ( S ( g (4) ) ⊲ b ′ ) S ( g (3) ) S ( h (4) ) , d k (1) ⊗ d ′ S ( k (2) ))= b ( h (1) ⊲ c ) σ ( h (2) , g (1) )(( h (3) g (2) ) ⊲ d ) σ ( h (4) g (3) , k (1) )(( h (5) g (4) k (2) ) ⊲ d ′ ) (cid:0) ( h (6) g (5) ) ⊲ ( c ′ ( S ( g (6) ) ⊲ b ′ )) (cid:1) . Compare the results on both hand sides, it is sufficient to show h (1) ⊲ (( g (1) ⊲ d ) σ ( g (2) , k (1) )) σ ( h (2) , g (3) k (2) ) = σ ( h (1) , g (1) )(( h (2) g (2) ) ⊲ d ) σ ( h (3) g (3) , k ) . We can see the left hand side is h (1) ⊲ (( g (1) ⊲ d ) σ ( g (2) , k (1) )) σ ( h (2) , g (3) k (2) )= ( h (1) ⊲ ( g (1) ⊲ d ))( h (2) ⊲ σ ( g (2) , k (1) )) σ ( h (3) , g (3) k (2) )= ( h (1) ⊲ ( g (1) ⊲ d )) σ ( h (2) , g (2) ) σ ( h (3) g (3) , k )= σ ( h (1) , g (1) )(( h (2) g (2) ) ⊲ d ) σ ( h (3) g (3) , k ) , where the second step uses (2.13), and the last step uses Lemma 2.4. Finally, we canshow the normalisation condition:˜ σ ( X,
1) = b ( h (1) ⊲ b ′ ) = ǫ ( X )˜ σ (1 , X ) = b ( h (1) ⊲ b ′ ) = ǫ ( X ) . (cid:3) Lemma 3.3.
For a crossed product B σ H , with H being cocommutative and the imageof σ belonging to the centre of B , then the map φ : C ( B H, H ) → C ( B σ H, H ) is a B -coring map, which is given by φ ( b h (1) ⊗ b ′ S ( h (2) )) := b σ h (1) ⊗ b ′ σ − ( S ( h (3) ) , h (4) ) σ S ( h (2) ) (3.13) for any b h (1) ⊗ b ′ S ( h (2) ) ∈ C ( B H, H ) .Proof. First we check ǫ = ǫ σ ◦ φ , where ǫ σ is the counit of C ( B σ H, H ). Let X = b h (1) ⊗ b ′ S ( h (2) ) ∈ C ( B H, H ), then ǫ σ ( φ ( X )) = ( b σ h (1) )( b ′ σ − ( S ( h (3) ) , h (4) ) σ S ( h (2) ))= b ( h (1) ⊲ b ′ )( h (2) ⊲ σ − ( S ( h (5) ) , h (6) )) σ ( h (3) , S ( h (4) )) σ b ( h ⊲ b ′ ) σ ǫ ( b h (1) ⊗ b ′ S ( h (2) )) , where in the third step we use Proposition 2.5. Here we always identify B with its imagein B H and B σ H by b b b b σ φ is left -module map. We can also check it is right B -linear: φ ( X ⊳ b ′′ ) = φ ( b h (1) ⊗ b ′ ( S ( h (3) ) ⊲ b ′′ ) S ( h (2) ))= b σ h (1) ⊗ b ′ ( S ( h (5) ) ⊲ b ′′ ) σ − ( S ( h (3) ) , h (4) ) σ S ( h (2) )= b σ h (1) ⊗ b ′ σ − ( S ( h (4) ) , h (5) )( S ( h (3) ) ⊲ b ′′ ) σ S ( h (2) )= φ ( X ) ⊳ b ′′ , for all b ′′ ∈ B , where the third step uses the fact that H is cocommutative and the imageof σ belongs to the centre of B . Recall that the translation map of the Hopf Galoisextension B = ( B σ H ) coH ⊆ B σ H is given by τ ( h ) = σ − ( S ( h (2) ) , h (3) ) σ S ( h (1) ) ⊗ B σ h (4) , for all h ∈ H . So we have( φ ⊗ B φ )(∆( X ))= ( φ ⊗ B φ )( b h (1) ⊗ S ( h (2) ) ⊗ B h (3) ⊗ b ′ S ( h (4) ))= b σ h (1) ⊗ σ − ( S ( h (3) ) , h (4) ) σ S ( h (2) ) ⊗ B σ h (5) ⊗ b ′ σ − ( S ( h (7) ) , h (8) ) σ S ( h (6) )= ∆ σ ( b σ h (1) ⊗ b ′ σ − ( S ( h (3) ) , h (4) ) σ S ( h (2) ))= ∆ σ ( φ ( X )) , where ∆ σ is the coproduct of C ( B σ H, H ). (cid:3) Theorem 3.4.
For a crossed product B σ H , with H being cocommutative and the imageof σ belonging to the centre of B , then there is an invertible normalised 2-cocycle ˜ σ on C ( B H, H ) , such that φ : C ( B H, H ) ˜ σ → C ( B σ H, H ) is an isomorphism of left B -bialgebroids, where ˜ σ is given by (3.11) and φ is given by (3.13).Proof. Since φ is a coring map, so we only need to show φ is an algebra map. Let X = b h (1) ⊗ b ′ S ( h (2) ), Y = c g (1) ⊗ c ′ S ( g (2) ) ∈ C ( B H, H ). On the one hand, φ ( X · ˜ σ Y )= φ (cid:16) ˜ σ ( b h (1) ⊗ S ( h (2) ) , c g (1) ⊗ S ( g (2) )) h (3) g (3) ⊗ (cid:0) ( S ( g (5) ) S ( h (5) )) ⊲ ˜ σ − (1 h (6) ⊗ b ′ S ( h (7) ) , g (6) ⊗ c ′ S ( g (7) )) (cid:1) S ( g (4) ) S ( h (4) ) (cid:17) = φ (cid:16) b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) h (3) g (2) ⊗ ( S ( g (4) ) S ( h (5) )) ⊲ (cid:0) σ − ( h (6) , g (5) )(( h (7) g (6) ) ⊲ c ′ )( h (8) ⊲ b ′ )) (cid:1) S ( g (3) ) S ( h (4) ) (cid:17) = b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ ( S ( g (6) ) S ( h (7) )) ⊲ (cid:0) σ − ( h (8) , g (7) )(( h (9) g (8) ) ⊲ c ′ )( h (10) ⊲ b ′ )) (cid:1) σ − ( S ( g (4) ) S ( h (5) ) , h (6) g (5) )) σ S ( g (3) ) S ( h (4) )= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ ( S ( g (6) ) S ( h (7) )) ⊲ σ − ( h (8) , g (7) ) c ′ ( S ( g (8) ) ⊲ b ′ ) σ − ( S ( g (4) ) S ( h (5) ) , h (6) g (5) )) σ S ( g (3) ) S ( h (4) )= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ σ ( S ( g (6) ) S ( h (7) ) , h (8) g (7) ) σ − ( S ( g (8) ) S ( h (9) ) h (10) , g (9) ) σ − ( S ( g (10) ) S ( h (11) ) , h (12) ) c ′ ( S ( g (11) ) ⊲ b ′ ) σ − ( S ( g (4) ) S ( h (5) ) , h (6) g (5) )) σ S ( g (3) ) S ( h (4) )= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ σ − ( S ( g (4) ) , g (5) ) σ − ( S ( g (6) ) S ( h (5) ) , h (6) ) c ′ ( S ( g (7) ) ⊲ b ′ ) σ S ( g (3) ) S ( h (4) ) , here in the 4th, 5th and 6th step we use that H is cocommutative and the image of σ belongs to the centre of B , and the 5th step also uses Proposition 2.5. On the otherhand, φ ( X ) φ ( Y )= ( b σ h (1) ⊗ b ′ σ − ( S ( h (3) ) , h (4) ) σ S ( h (2) ))( c σ g (1) ⊗ c ′ σ − ( S ( g (3) ) , g (4) ) σ S ( g (2) ))= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ c ′ σ − ( S ( g (6) ) , g (7) ) (cid:0) S ( g (5) ) ⊲ ( b ′ σ − ( S ( h (6) ) , h (7) )) (cid:1) σ ( S ( g (4) ) , S ( h (5) )) σ S ( g (3) ) S ( h (4) )= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ c ′ σ − ( S ( g (5) ) , g (6) )( S ( g (7) ) ⊲ b ′ )( S ( g (8) ) ⊲ σ − ( S ( h (6) ) , h (7) )) σ ( S ( g (4) ) , S ( h (5) )) σ S ( g (3) ) S ( h (4) )= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ c ′ ( S ( g (5) ) ⊲ b ′ ) σ − ( S ( g (6) ) , g (7) ) σ ( S ( g (8) ) , S ( h (6) ) h (7) ) σ − ( S ( g (9) ) S ( h (8) ) , h (9) ) σ − ( S ( g (10) ) , S ( h (10) )) σ ( S ( g (4) ) , S ( h (5) )) σ S ( g (3) ) S ( h (4) )= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ c ′ ( S ( g (4) ) ⊲ b ′ ) σ − ( S ( g (5) ) , g (6) ) σ − ( S ( g (7) ) S ( h (5) ) , h (6) ) σ S ( g (3) ) S ( h (4) ) , where in the 3rd, 4th and 5th steps we use that H is cocommutative and the image of σ belongs to the centre of B , and the 4th step also uses Proposition 2.5. Since H iscocommutative, we have φ ( X · ˜ σ Y ) = φ ( X ) φ ( Y ). (cid:3) Similarly, we can also show the following theorem:
Theorem 3.5.
For a crossed product B σ H , with the image of σ belonging to the centreof B , if the action of H on B is trivial (i.e. h ⊲ b = ǫ ( h ) b , for all h ∈ H and b ∈ B ), then there is an invertible normalised 2-cocycle ˜ σ on C ( B H, H ) , such that φ : C ( B H, H ) ˜ σ → C ( B σ H, H ) is an isomorphism of left B -bialgebroids, where ˜ σ is givenby (3.11) and φ is given by (3.13).Proof. If H measures B with trivial action, and B σ H is an unital associative algebrawith the image of σ belongs to the centre of B , then we can also get Lemma 3.2, Lemma3.3 and Theorem 3.4 without asking H being cocommutative. Indeed, in this case B H is equal to B ⊗ H with factorwise multiplication, so by Lemma 3.2˜ σ ( b h (1) ⊗ b ′ S ( h (2) ) , c g (1) ⊗ c ′ S ( g (2) )) := bcσ ( h, g ) c ′ b ′ , (3.14)for all b h (1) ⊗ b ′ S ( h (2) ), c g (1) ⊗ c ′ S ( g (2) ) ∈ C ( B H, H ). In this case, Lemma 3.3can be also shown, since even without the cocommutativity of H we can show that φ isright B -linear. Moreover, in the proof of Theorem 3.4 we can see on the one hand φ ( X · ˜ σ Y )= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ ( S ( g (6) ) S ( h (7) )) ⊲ (cid:0) σ − ( h (8) , g (7) )(( h (9) g (8) ) ⊲ c ′ )( h (10) ⊲ b ′ )) (cid:1) σ − ( S ( g (4) ) S ( h (5) ) , h (6) g (5) )) σ S ( g (3) ) S ( h (4) )= bcσ ( h (1) , g (1) ) σ h (2) g (2) ⊗ σ − ( h (7) , g (8) ) c ′ b ′ σ − ( S ( g (5) ) , g (6) ) σ − ( S ( h (5) ) , h (6) g (7) ) σ ( S ( g (4) ) , S ( h (4) )) σ S ( g (3) ) S ( h (3) )= bcσ ( h (1) , g (1) ) σ h (2) g (2) ⊗ c ′ b ′ σ − ( S ( g (5) ) , g (6) ) σ − ( S ( h (5) ) , h (6) ) σ ( S ( g (4) ) , S ( h (4) )) σ S ( g (3) ) S ( h (3) ) , here in the 2nd and 3rd steps we use the fact that action is trivial, and the image of σ belongs to the centre of B , and the 2nd step also uses σ − ( S ( g (1) ) S ( h (1) ) , h (2) g (2) ) = σ − ( S ( g (2) ) , g (3) ) σ − ( S ( h (2) ) , h (3) g (4) ) σ ( S ( g (1) ) , S ( h (1) )) , which can be derived from Proposition 2.5, the 3rd step also uses Proposition 2.5. Onthe other hand, φ ( X ) φ ( Y )= b ( h (1) ⊲ c ) σ ( h (2) , g (1) ) σ h (3) g (2) ⊗ c ′ σ − ( S ( g (6) ) , g (7) ) (cid:0) S ( g (5) ) ⊲ ( b ′ σ − ( S ( h (6) ) , h (7) )) (cid:1) σ ( S ( g (4) ) , S ( h (5) )) σ S ( g (3) ) S ( h (4) )= bcσ ( h (1) , g (1) ) σ h (2) g (2) ⊗ c ′ b ′ σ − ( S ( g (5) ) , g (6) ) σ − ( S ( h (5) ) , h (6) ) σ ( S ( g (4) ) , S ( h (4) )) σ S ( g (3) ) S ( h (3) ) , where in the last step we use the fact that image of σ belongs to the centre of B and theaction is trivial. (cid:3) Recall that a
Galois object of a Hopf algebra H is a comodule algebra A , such thatthe canonical Galois map is bijective, and A coH = C . As a result of Theorem 2.6 andTheorem 3.5 we have: Corollary 3.6.
Let A be a cleft Galois object of H , then C ( A, H ) is isomorphic to H γ as Hopf algebra, where γ : H ⊗ H → C is an invertible normalised 2-cocycle on H .Proof. By Theorem 2.6, we know A ≃ C σ H , then by Theorem 3.5 we know C ( C σ H, H ) ≃C ( C H, H ) ˜ σ . Since in [8] we know H ≃ C ( H, H ) ≃ C ( C H, H ), where the isomorphicmap f : H ≃ C ( H, H ) is given by f : h h (1) ⊗ S ( h (2) ), and its inverse is given by f − : g ⊗ h gǫ ( h ). Then there is a 2-cocycle γ on H such that C ( A, H ) ≃ H γ . As weknow in [15] that C ( A, H ) is a Hopf algebra, then we get the corollary. (cid:3)
Corollary 3.6 is also shown in [15], here we can view Theorem 3.5 as a generalisationof it.Let γ : H ⊗ H → C be an invertible normalised 2-cocycle on a Hopf algebra H , and B = A coH ⊆ A be a cleft Hopf Galois extension. It is shown in [1] that we can define a H γ -comodule algebra A γ on the same underlying H -comodule A (i.e. δ A =: δ A γ : A γ → A γ ⊗ H γ ) with a new product given by a · γ a ′ := a (0) a ′ (0) γ − ( a (1) , a ′ (1) ) , (3.15)for all a, a ′ ∈ A . Moreover, B = A coH γ γ ⊆ A γ is a Hopf Galois extension, with thetranslation map given by τ γ ( h ) := h (3) < > ⊗ B h (3) < > γ ( h (1) , S ( h (2) )) , (3.16)for all h ∈ H . Indeed, since the canonocal map χ γ is a left A γ -module map, it is sufficientto show( χ γ ◦ τ γ )( h ) = h (3) < > (0) h (3) < > (0) ⊗ γ − ( h (3) < > (1) , h (3) < > (1)(1) ) h (3) < > (1)(2) γ ( h (1) , S ( h (2) ))= h (4) < > h (4) < > (0) ⊗ γ − ( S ( h (3) ) , h (4) < > (1)(1) ) h (4) < > (1)(2) γ ( h (1) , S ( h (2) ))= 1 ⊗ γ − ( S ( h (3) ) , h (4) ) h (5) γ ( h (1) , S ( h (2) ))= 1 ⊗ h, or all h ∈ H , where the 2nd step uses (2.9), the 3rd step uses (2.10), and the last stepuses the fact that γ is a 2-cocycle on the Hopf algebra H . We can also see( χ − γ ◦ χ γ )( a ′ ⊗ B a ) = a ′ · γ a (0) · γ a (3) < > ⊗ B a (3) < > γ ( a (1) , S ( a (2) ))= a ′ · γ ( a (0) a (4) < > (0) ) ⊗ B a (4) < > γ − ( a (1) , a (4) < > (1) ) γ ( a (2) , S ( a (3) ))= a ′ · γ ( a (0) a (5) < > ) ⊗ B a (5) < > γ − ( a (1) , S ( a (4) )) γ ( a (2) , S ( a (3) ))= a ′ · γ ( a (0) a (1) < > ) ⊗ B a (1) < > = a ′ ⊗ B a, for all a, a ′ ∈ A γ , where the 3rd step use (2.9), and the last step uses (2.11). Therefore, B = A coH γ γ ⊆ A γ is a Hopf Galois extension. Lemma 3.7.
Let B = A coH ⊆ A be a cleft Hopf Galois extension, then B = A coH γ γ ⊆ A γ is also a cleft Hopf Galois extension.Proof. Since B = A coH ⊆ A is a cleft Hopf Galois extension, from Theorem 2.6 thereis an isomorphic map F : A → B ⊗ H between left B -modules and right H -comodules.Define F γ = F on the underlying vector space of A . Then we can see F γ is left B -linearand right H -colinear: F γ ( b · γ a ) = F ( ba ) = bF ( a ) = b · γ F γ ( a ) , for all b ∈ B and a ∈ A . Thus F γ is left B linear. We can also see F γ ( a ) (0) ⊗ F γ ( a ) (1) = F ( a ) (0) ⊗ F ( a ) (1) = F ( a (0) ) ⊗ a (1) = F γ ( a (0) ) ⊗ a (1) , thus F γ is a H comodule map. Therefore, B = A coH γ γ ⊆ A γ is a cleft Hopf Galoisextension. (cid:3) Corollary 3.8.
Let A be a cleft Galois object of H , and γ : H ⊗ H → C be an invertiblenormalised 2-cocycle on H , then C ( A γ , H γ ) is isomorphic to of C ( A, H ) ω , where ω is aninvertible normalised 2-cocycle on C ( A, H ) .Proof. Since A is a cleft Galois object, from the Lemma 3.7 we know A γ is a cleft Galoisobject of H γ .Therefore, by Corollary 3.6 there is an invertible normalised 2-cocycle ρ on H , suchthat C ( A γ , H γ ) ≃ ( H γ ) ρ , Since C ( A, H ) ≃ H σ , we have C ( A γ , H γ ) ≃ C ( A, H ) σ − ⋆γ⋆ρ , (3.17)and ω = σ − ⋆ γ ⋆ ρ . (cid:3) Acknowledgment:
I would like to thank Prof. Dr. Giovanni Landi and Prof. Dr. Shahn Majid for manyuseful discussion. I am glad to thank Dr. Song Cheng for the proofreading. eferences [1] P. Aschieri, P. Bieliavsky, C. Pagani, A. Schenkel, Noncommutative principal bundles through twistdeformation . Commun. Math. Phys. 352 (2017) 287–344.[2] J. Bichon,
Hopf-Galois objects and cogroupoids . Rev. Un. Mat. Argentina, 5 (2014) 11–69.[3] T. Brzezi´nski,
Translation map in quantum principal bundles . J. Geom. Phys. 20 (1996) 349–370.[4] T. Brzezi´nski, R. Wisbauer,
Corings and comodules . London Mathematical Society Lecture Notes309, CUP 2003.[5] G. B¨ohm,
Hopf algebroids . Handbook of Algebra, Vol. 6, North-Holland, 2009, Pages 173–235.[6] Y. Doi, equivalent crossed products for a Hopf algebra . Comm. Alg. 17 (1989), 3053-3085.[7] Y. Doi, M. Takauchi,
Cleft comodule algebras for a bialgebra . Comm. Alg. 14 (1986), 801-818.[8] X. Han, G. Landi,
On the gauge group of Galois objects . arXiv:2002.06097 [math.QA].[9] P.M. Hajac, T. Maszczyk,
Pullbacks and nontriviality of associated noncommutative vector bundles .arXiv:1601.00021 [math.KT].[10] C. Kassel,
Principal fiber bundles in non-commutative geometry . In: Quantization, geometry andnoncommutative structures in mathematics and physics, Mathemathical Physics Studies, Springer2017, pp. 75–133.[11] S. Majid,
Foundations of quantum group theory . CUP 1995 and 2000.[12] A. Masuoka,
Cleft extensions for a Hopf algebra generated by a nearly primitive element . Comm.Algebra 22 (1994) 4537–4559.[13] K. Mackenzie,
General theory of Lie groupoids and Lie algebroids . London Mathematical SocietyLecture Notes Series 213, CUP 2005.[14] S. Montgomery,
Hopf algebras and their actions on rings . AMS 1993.[15] P. Schauenburg,
Hopf bi-Galois extensions . Commun. Algebra 24 (1996) 3797–3825.[16] K. Szlachanyi,
Monoidal Morita equivalence . arXiv:math/0410407 [math.QA].
Xiao Han
SISSA, via Bonomea 265, 34136 Trieste, Italy
E-mail address : [email protected]@sissa.it