Twistings, crossed coproducts and Hopf-Galois coextensions

Let $H$ be a Hopf algebra. Ju and Cai introduced the notion of twisting of an $H$-module coalgebra. In this note, we study the relationship between twistings, crossed coproducts and Hopf-Galois coextensions. In particular, we show that a twisting of an $H$-Galois coextension remains $H$-Galois if the twisting is invertible.


Introduction
A fundamental result in Hopf-Galois theory is the Normal Basis Theorem, stating that, for a finitely generated cocommutative Hopf algebra H over a commutative ring k, the set of isomorphism classes of Galois H-objects that are isomorphic to H as an H-comodule is a group, and this group is isomorphic to the second Sweedler cohomology group H 2 (H, k) (see [16]). The Galois object corresponding to a 2-cocycle is then given by a crossed product construction. The crossed product construction can be generalized to arbitrary Hopf algebras, and plays a fundamental role in the theory of extensions of Hopf algebras, see [3] and [12]. Also in this more general situation, it turns out that there is a close relationship between crossed products on one side, and Hopf-Galois extensions and cleft extensions, cf. [3], [3], [12]. A survey can be found in [15]. An alternative way to deform the multiplication on an H-comodule algebra A has been proposed in [1], using a so-called twisting of A, and it was shown that the crossed product construction can be viewed as a special case of the twisting construction. The relation between twistings and H-Galois extensions was studied in [2]. Now there exists a coalgebra version of the Normal Basis Theorem (see [6]). In this situation, one tries to deform the comultiplicationon a commutative Hopf algebra H, using this time a Harrison cocycle instead of a Sweedler cocycle. Crossed coproducts, cleft coextensions and Hopf-Galois coextensions have been introduced and studied in [9] and [11]. Ju and Cai [13] have introduced the notion twisting of an Hmodule coalgebra, which can be viewed as dual version of the twistings introduced in [1]. The aim of this paper is to study the relationship between twistings, crossed coproducts and Hopf-Galois coextensions. Our main result is the fact that the twisting of a Hopf-Galois coextension by an invertible twist map is again a Hopf-Galois coextension (and conversely). Our paper is set up as follows: in Section 1.1, we recall the twistings introduced in [13], and in Section 1.2 the definition of a Harrison cocycle and the crossed coproduct construction from [9] and [11]. In Section 2, we introduce an alternative version of 2-cocycles, called twisted 2-cocycles, and discuss the relation with Harrison cocycles (Proposition 2.3). In Section 3, we introduce an equivalence relation on the set of twistings of an H-module coalgebra, and we show that a twisting in an equivalence class is invertible if and only if all the other twistings in this equivalence class are invertible (Theorem 3.4). Two twistings are equivalentif and only if their corresponding crossed coproducts are isomorphic (Proposition 1.1). In Section 4 the relationship between twistings and Hopf-Galois coextensions is investigated. For the general theory of Hopf algebras, we refer to the literature, see for example [10], [15], [17].

Notation and preliminary results
We work over a field k. All maps are assumed to be k-linear. For the comultiplication on a k-coalgebra C, we use the Sweedler-Heyneman notation ∆ C (c) = c 1 ⊗ c 2 with the summation implicitely understood. We use a similar notation for a (right) coaction of a coalgebra on a comodule: Let A be a k-algebra, then Hom(C, A) is also an algebra, with convolution product Reg(C, A) will denote the set of convolution invertible elements in Hom(C, A). M C A will be the category of modules with a right A-action and a right C-coaction, such that the C-coaction is A-linear.
1.1. Twistings of a coalgebra. We recall some definitions and results from [13]. Let H be a Hopf algebra over a field k, with bijective antipode S. The composition inverse of the antipode will be denoted byS.
Recall that a right H-module coalgebra is a coalgebra C which is also a right H-module such that for all c ∈ C and h ∈ H. M C H is the category whose objects are right H-modules and right Ccomodules M such that the following compatibility relation is satisfied: Recall from [13] that we have the following associative multiplication on Hom(C, H ⊗ C): Remark that we have an algebra isomorphism For τ : C → H ⊗C, we define the corresponding α(τ ) = f τ : H ⊗C → H ⊗ C by f τ (h ⊗ c) = hτ (c) = hc −1 ⊗ c 0 Assume that τ satisfies the following normality conditions: If we write τ (c) = c −1 ⊗ c 0 (summation understood), then (1) takes the following form We can then define a new (in general non-coassociative) comultiplication ∆ τ on C as follows: (2) is equivalent to If τ has an inverse λ, then the functor F is an equivalence of categories. Left hand twistings are defined in a similar way. Consider the vector space isomorphism The composition on the right hand side is transported into the following associative multiplication on Hom(C, C ⊗ H): Here T is the usual twist map. The unit σ ′ on Hom(C, C ⊗ H) is given by σ ′ (c) = c ⊗ 1. If λ ∈ Hom(C, C ⊗ H) satisfies the normalizing conditions then we can twist the comultiplication on C as follows: write λ(c) = c 0 ⊗ c 1 , and define λ ∆ by λ ∆(c) = c 1.0 ⊗ c 2 · c 1.1 λ C will be C as a right H-module, with the comultiplication λ ∆. The C-coaction M ∈ C M H can also be twisted: (5) is a left hand twisting if and only if for all h ∈ H and c ∈ C, For τ ∈ Hom(C, H ⊗ C) with inverse λ, we write We then have Let T (C) and L(C) be the sets of respectively twistings and left hand twistings of C, and U(T (C)), U(L(C)) the sets of invertible twistings and left hand twistings.
Proof. It is shown in [13] that ℓ(τ ) ∈ U(L(C)) with inverse given by In [13], it is also shown that r(γ) ∈ U(T (C)), and it is straightforward to verify that the * -inverse of r(γ) is given by A routine verification similar to the one above then shows that for all c ∈ C. It is easy to show that ℓ(σ) = σ ′ and r(σ ′ ) = σ.
1.2. The crossed coproduct. We recall the following definitions from [9] and [11]. Definition 1.2. Let C be a coalgebra and H a Hopf algebra. We say that H coacts weakly on C if there is a k-linear map ρ : satisfying the following conditions, for all c ∈ C: Assume that H coacts weakly on C, and let α : C −→ H ⊗ H; α(c) = α 1 (c) ⊗ α 2 (c) be a linear map. Let C >⊳ α H be the coalgebra whose underlying vector space is C ⊗ H, with comultiplication and counit given by It was pointed out in [11] that ε α (c >⊳ h) satisfies the counit property if and only if (14) ( In [11], (15) is called the cocycle condition, and (16) is called the twisted comodule condition. Following [7], we call α satisfying (14-16), a Harrison 2-cocycle. Now consider two weak H-coactions ρ, ρ ′ : C −→ H ⊗ C, and write Also consider two 2-cocycles α, α ′ : C −→ H ⊗ H corresponding respectively to ρ and ρ ′ , and write In the next Lemma, we discuss when these are isomorphic.
Proof. The proof is a dual version of a similar statement for crossed products, see [15].
It was shown in [13] that the crossed coproduct construction can be viewed as a special case of the twisting construction from Section 1.1. Let H be a Hopf algebra, and C a right H-module coalgebra, and view C ⊗ H as a right H-module coalgebra, with the right H-action is induced by the multiplication in H. It was proved in [13] that there is a bijective correspondence between crossed coproduct structures on C ⊗ H and twistings of C ⊗ H. Let us recall the description of this bijection. Consider a weak coaction ρ and a 2-cocycle α giving rise to the crossed coproduct C >⊳ α H, and write The corresponding twisting τ : with weak coaction ρ and 2-cocycle α given by

Twisted 2-cocycles
Let H be a Hopf algebra with bijective antipode S; letS be the composition inverse of S. Take an H-module coalgebra C, and let B = C/CH + .
is called a twisted 2-cocycle if the following conditions are satisfied, for all h ∈ H and c ∈ C: Our first result is the fact that twisted 2-cocycles can be used to define twistings on C.
Proposition 2.2. With notation as above, if α : C → H ⊗ H is a twisted 2-cocycle, then the map is a twisting of C.
Proof. It follows easily from (23) that τ α satisfies the normalizing condition (1). Next we compute that and (2) follows easily. Finally we compute the left and right hand side of (3) (3) follows, and τ α is a twisting.
There is also a relation between twisted 2-cocycles and Harrison 2cocycles. Let C be a right H-module coalgebra. Consider the trivial weak coaction ρ(c) = 1⊗c, and α : C → H ⊗H. The cocycle condition (15) and the twisted comodule condition (16) of Definition 1.2 then take the following form: For all c ∈ C and h ∈ H, we have . It is easy to see that α satisfies (14) and (27). Using (28), we compute and it follows that α also satisfies (26). Conversely, let α ∈ Z 2 Harr (H, C), and define α t : We can easily show that α t satisfies conditions (23) and (24) of Definition 2.1. A straightforward computation shows that (25) is also satisfied: so it follows that α t is a twisted 2-cocycle. We leave it to the reader to show that the maps between Z 2 Harr (H, C) and Z 2 tw (H, C ⊗ H) defined above are inverses to each other.

Equivalence of twistings
In this Section, we will define an equivalence relation on the set of twistings of an H-module coalgebra C. If a twisting is invertible, then all other twistings in the same equivalence class are also invertible.
Proof. Using the second identity in (29) and B = C/CH + , we can easily prove that ψ is left B-colinear and right H-linear. Using the first identity in (29), we obtain that ψ induces a well-defined map B → B, which is the identity. In order to prove that ψ is a coalgebra map, we need to check that Again, we compute the left and right hand side, and see that they are equal: If v ∈ Reg(C, H), then its inverse w also satisfies (29), and ϕ : is the inverse of ψ.
Definition 3.2. We call τ, λ ∈ T (C) equivalent if there exists v ∈ Reg(C, H) satisfying the conditions of Proposition 3.1. We then write τ ∼ λ.
(30) is equivalent to The inverse u of v satisfies (29). It also satisfies (29) since and it follows that λ ∼ τ . Now assume that τ ∼ λ, λ ∼ γ, and take the corresponding maps v, u ∈ Reg(C, H). Set w = u * v, and write It is easily shown that w satisfies (29). v satisfies (31), and u satisfies We compute that and this proves that τ ∼ γ.
Proof. Take v ∈ Reg(C, H) satisfying the conditions in Proposition 3.1, and let ψ : C τ −→ C λ be the coalgebra isomorphism given by Let τ −1 be the inverse to τ , and write Using the temporary notation ψ −1 (c) <−1> = a and ψ −1 (c) <0> = b, it is not hard to prove that µ is a left inverse of λ. Indeed, The proof of the fact that µ is also a right inverse of λ is much more technical. From the fact that v is invertible, and using (30), we obtain and it follows that λ is convolution invertible.
Theorem 3.5. Let C be a right H-comodule algebra, and consider τ, λ ∈ T (C ⊗ H). τ and λ are equivalent in the sense of Definition 3.2 if and only if there is a left C-colinear, right H-linear coalgebra isomorphism between the crossed coproducts C >⊳ α H, ρ and C >⊳ ′ α ′ H, ρ ′ corresponding to τ and λ.

Twisting Hopf-Galois coextensions
Let H be a Hopf algebra with bijective antipode S, and C a right H-module coalgebra. As before, we use the following notation For τ ∈ T (C), we have that C τ /I τ = C/I = B. Now assume that C/B is an H-Galois coextension (see [5]). This means that the canonical map is a bijection.
Lemma 4.1. With notation as above, consider the map If the antipode S is bijective, then β is bijective (resp. injective, surjective) if and only if β ′ is bijective (resp. injective, surjective).
Proof. The map is a bijection with inverse The statement then follows from the fact that β ′ = β • φ. Proof. Let λ be the inverse of τ . As before, we use the notation (9). Let β τ be the canonical map corresponding to the coextension C τ /B, that is, This shows that (33) is commutative, and it follows that β is bijective if and only if β τ is bijective. where v ∈ Hom(C, H) satisfies the conditions (29-30) of Proposition 3.1. If ψ is an isomorphism, then v ∈ Reg(C, H).