The Brauer Group of A Finite Quantum Group

Yinhuo Zhang Max-Planck Institut für Mathematik Vivatsgasse 7, Bonn 53111, Germany

Dept. of Mathematics University of Antwerp (UIA) B-2610, Belgium

Abstract

In this paper, we give a general method to compute the Brauer group of a finite quantum group, i.e., a faithfully projective coquasitriangular Hopf algebra over a commutative ring. Let (H, R) be a finite quantum group with an R-matrix R on H ⊗H. There exists a deformation H_{R} of H. This deformation H_{R} can be embedded into the Drinfel’d double D(H) as a left coideal subalgebra. We construct a group Gal(H_{R}) consisting of quantum commutative H_{R}^{∗}-bigalois objects. There is an exact sequence of group morphisms:

1 ⟶ Br(k) ⟶ BC(k, H, R) ⟶ Gal(H_{R})

where Br(k) is the usual Brauer group of k, BC(k, H, R) is the Brauer group of (H, R) with respect to the R-matrix R. The applications to the Brauer group of triangular Hopf algebras and the Brauer-Long group will be presented.

Contents:

0. Introduction

1. Preliminaries

2. The deformation H_{R}

3. The group Gal(H_{R})

4. The exact sequence

5. Examples

6. The triangular case

7. The quantum double case

8. Application to the Brauer-Long group.

Introduction

Let k be a commutative ring with identity, H a Hopf algebra over k with a bijective antipode. In [7], we introduced H-Azumaya algebras and the Brauer group BQ(k, H) classifying the H-Azumaya algebras. When H is a finite commutative and cocommutative Hopf algebra, the Brauer group BQ(k, H) turns out to be the Brauer-Long group introduced by F.W. Long in [17] which is the most proper generalization of the Brauer-Wall group BW(k) (see [34]).

In [8], we made further investigation on the basic properties of H-Azumaya algebras and studied in detail the split part of the Brauer group BQ(k, H) in order to find a non-abelian cohomological interpretation of some subgroups of BQ(k, H). This approach turns out to be difficulty as later we found that the Hopf automorphism group can be embedded into the Brauer group BQ(k, H), which showed that BQ(k, H) is not necessarily a torsion group (see [32] for detail). In a recent calculation of the Brauer group of the Sweedler 4-dimensional Hopf algebra cf.[33], we found that the Hopf automorphism group is not the only non-torsion part of BQ(k, H), the group of Galois objects plays the non-torsion role as well in the Brauer group BQ(k, H). Nevertheless, when H is commutative and cocommutative, the group of Galois objects and the group of Hopf automorphisms generate a subgroup which is isomorphic to a factor group of the group of bigalois objects cf.[5]. This fact strongly indicates that the group of bigalois objects plays the vital role in the computation of the full Brauer group of a finite quantum group. This indication was further strengthened by a beautiful exact sequence of the Brauer-Long group due to K-L. Ulbrich in [30] where the group of bigalois objects appears in the picture. However, when (H, R) is no longer commutative and cocommutative, the group of bigalois objects of H or H^{∗} does not fit into an exact sequence of the Brauer group of H. The solution found in this paper is the deformation H_{R} of a finite quantum group (H, R), which is no longer a Hopf algebra, instead a left coideal subalgebra of the quantum double D(H). The main idea of this paper is to embed the quotient group BC(k, H, R)/Br(k) of a finite quantum group (H, R) into a suitable group of ”bigalois objects” of the deformation H_{R}, which is easier to compute (or to estimate). Since the full Brauer group BQ(k, H) of any finite Hopf algebra H is equal to BC(k, D(H)^{∗}, R′), where (D(H)^{∗}, R′) is the dual of the Drinfel’d quantum double group, it is sufficient to consider the general case BC(k, H, R) for a finite quantum group (H, R).

In Section one, we recall the definition of the Brauer groups BQ(k, H) and BC(k, H, R) when (H, R) is a coquasitriangular Hopf algebra. In section 2, we define a new deformation H_{R} of a finite coquasitriangular Hopf algebra (H, R). H_{R} turns out to be a left coideal subalgebra of the quantum double D(H) though it is not a Hopf subalgebra. The main result in this section is the existence of a covariant functor from the category of Yetter-Drinfel’d H-modules to the category of H_{R}-bimodules cf.Prop.2.7.

In Section 3, we construct a group Gal(H_{R}) consisting of H_{R}^{∗}-bigalois objects which are quantum commutative. the group Gal(H_{R}) plays the significant role in the computation of the Brauer group BC(k, H, R).

In Section 4, we establish a group morphism π̃ from the Brauer group BC(k, H, R) to the group Gal(H_{R}). In order to define the morphism π̃, we have to show that any element of BC(k, H, R) is represented by an H-Azumaya algebra which is an H^{op}-Galois extension of its coinvariants. Such an H-Azumaya algebra is called a Galois R-Azumaya algebra. The centralizer of the coinvariants of a Galois R-Azumaya algebra turns out to be an H_{R}^{∗}-bigalois object. The kernel of the morphism π̃ is isomorphic to the usual Brauer group Br(k) of k. In section 5, we present two examples of the exact sequence (28). Section 6 is devoted to the special case when the finite quantum group (H, R) is triangular. Section 7 will focus on the Drinfel’d double case, that is, the full Brauer group of a finite Hopf algebra (not necessarily coquasitriangular). The last section goes to the application to the Brauer-Long group.

1 Preliminaries

Let k be a fixed commutative ring with unit. Throughout all algebras, unadorned ⊗, Hom are over k. A module (or an algebra) is said to be finite if it is faithfully projective (i.e. faithful, finitely generated and projective) as a k-module. A finite quantum group is a finite Hopf algebra over k which is co-quasitriangular (CQT), that is, there is an invertible element R ∈ (H ⊗ H)^{∗}, the convolution algebra of H ⊗ H, subject to the following conditions:

CQT1. R(h ⊗ 1) = R(1 ⊗ h) = ε(h), CQT2. R(x ⊗ yz) = ∑ R(x_{(1)} ⊗ z)R(x_{(2)} ⊗ y), CQT3. R(yz ⊗ x) = ∑ R(y ⊗ x_{(1)})R(z ⊗ x_{(2)}), CQT4. R(x_{(1)} ⊗ y_{(1)})x_{(2)}y_{(2)} = ∑ R(x_{(2)} ⊗ y_{(2)})y_{(1)}x_{(1)}.

for all x, y and z ∈ H. Since H is faithfully projective, we may identify R with an invertible element ∑ R^{1} ⊗ R^{2} in H^{∗} ⊗ H^{∗}. In this case, (H^{∗}, R) is a quasitriangular Hopf algebra, namely, R as an element in the Hopf algebra (H^{∗}, \underline{Δ}, \underline{ε}) satisfies the conditions:

QT1. ∑ \underline{ε}(R^{1})R^{2} = ∑ R^{1}\underline{ε}(R^{2}) = 1, QT2. ∑ \underline{Δ}(R^{1}) ⊗ R^{2} = ∑ R^{1} ⊗ r^{1} ⊗ R^{2}r^{2}, QT3. ∑ R^{1} ⊗ \underline{Δ}(R^{2}) = ∑ R^{1}r^{1} ⊗ r^{2} ⊗ R^{2}, QT4. R\underline{Δ}(p) = \underline{Δ}^{op}(p)R.

for all p ∈ H^{∗}. Note that the conditions CQT2 + CQT3 are equivalent to the cocycle condition, i.e,

∑ R(x_{(1)} ⊗ y_{(1)})R(x_{(2)}y_{(2)} ⊗ z) = R(y_{(1)} ⊗ z_{(1)})R(x ⊗ y_{(2)}z_{(2)})

for all x, y and z ∈ H. On the other hand, R results in two linear Hopf algebra morphisms:

\begin{matrix}Θ_{l} : & H^{cop} ⟶ H^{∗}, & Θ_{l}(h)(l) = R(h ⊗ l), \\ Θ_{r} : & H^{op} ⟶ H^{∗}, & Θ_{r}(h)(l) = R(l ⊗ h). \end{matrix}

Thus the condition CQT4 can be interpreted as

∑ x_{(2)}y_{(2)}Θ_{l}(x_{(1)})(y_{(1)}) = ∑ y_{(1)}x_{(1)}Θ_{r}(y_{(2)})(x_{(2)})

for all x, y ∈ H.

Let us now recall the definition of the Brauer group of a Hopf algebra with a bijective antipode and some related notions. Let H be a Hopf algebra with a bijective antipode (not necessarily finite). A Yetter-Drinfel’d H-module ( simply, YD H-module) M is a crossed H-bimodule [35]. That is, M is a k-module which is at once a left H-module and a right H-comodule satisfying the following equivalent compatibility conditions [18, 5.1.1]:

1). ∑ h_{(1)} · m_{(0)} ⊗ h_{(2)}m_{(1)} = ∑(h_{(2)} · m)_{(0)} ⊗ (h_{(2)} · m)_{(1)}h_{(1)}, 2). ∑(h · m)_{(0)} ⊗ (h · m)_{(1)} = ∑ h_{(2)} · m_{(0)} ⊗ h_{(3)}m_{(1)}S^{−1}(h_{(1)}),

where the sigma notations for comodules and for comultiplications can be found in the standard reference book [28]. Denote by Q^{H} the category of YD H-modules. A YD H-module algebra is a YD H-module A such that A is a left H-module algebra and a right H^{op}-comodule algebra. For the details of H-(co)module algebras we refer to [28].

In [7] we defined the Brauer group of a Hopf algebra H by considering isomorphism classes of H-Azumaya algebras. A YD H-module algebra A is called H-Azumaya if it is faithfully projective as a k-module and if the following YD H-module algebra maps are isomorphisms:

\begin{matrix}F : & A#\overline{A} & ⟶ & End(A), & F (a#\overline{b})(x) = ∑ ax_{(0)}(x_{(1)} · b), \\ G : & \overline{A}#A & ⟶ & End(A)^{op}, & G(\overline{a}#b)(x) = ∑ a_{(0)}(a_{(1)} · x)b, \end{matrix}

where \overline{A} is the H-opposite YD H-module algebra of A, that is, \overline{A} = A as a YD H-module, but with the multiplication given by

\overline{a} · \overline{b} = \overline{b_{(0)}(b_{(1)} · a)}

for \overline{a}, \overline{b} ∈ \overline{A} (see [7] for the detail). For a faithfully projective YD H-module M, the endomorphism algebra End_{k}(M) is a YD H-module algebra with H-structures given by

\begin{matrix}(h · f)(m) = ∑ h_{(1)} · f(S(h_{(2)}) · m), \\ ∑ f_{(0)}(m) ⊗ f_{(1)} = ∑ f(m_{(0)})_{(0)} ⊗ S^{−1}(m_{(1)})f(m_{(0)})_{(1)}. \end{matrix}

The elementary H-Azumaya algebra End(M)^{op} has the different H-structures from those of End(M) (see [7] for detail).

Two H-Azumaya algebras A and B are Brauer equivalent (denoted by A ∼ B) if there exist two faithfully projective YD H-modules M and N such that A#End(M) ≅ B#End(N). Note that A ∼ B if and only if A is H-Morita equivalent to B (cf.[7, Th.2.10]). The Brauer group of the Hopf algebra H is denoted by BQ(k, H). An element in BQ(k, H) represented by an H-Azumaya algebra A is indicated by [A]. The unit in BQ(k, H) is represented by End(M) for any faithfully projective YD H-module M.

If A is an H-Azumaya algebra, the left and right H-center defined by

{a ∈ A | ax = ∑ x_{(0)}(x_{(1)} · a), ∀x ∈ A},

resp.

{a ∈ A | xa = ∑ a_{(0)}(a_{(1)} · x), ∀x ∈ A},

are trivial.

Now let H be a CQT Hopf algebra with a CQT structure R. If M is a right H-(or H^{op})- comodule, The morphism Θ_{l} induces a left H-module structure on M as follows:

h ▷_{1} m = Θ_{l}(h) · m = ∑ m_{(0)}R(h ⊗ m_{(1)})  (1)

for h ∈ H and m ∈ M. The H-action (1) together with the original H-coaction makes M a YD H-module cf.[8, 18]. Denote by M_{R}^{H} the category of YD H-modules with the left H-module structure (1) coming from the right H-comodule structure. It is obvious that M_{R}^{H} is a full braided monoidal subcategory of Q^{H}.

When A = M is a right H^{op}-comodule algebra, (1) makes A a left H-module algebra and hence a YDH-module algebra. In the sequel, a YD H-module algebra A is called an R-module algebra if the H-action on A comes from the H^{op}-coaction on A through R. An R-module algebra is said to be R-Azumaya if it is H-Azumaya. The subset of BQ(k, H) consisting of the elements represented by the R-Azumaya algebras turns out to be a subgroup of BQ(k, H), denoted by BC(k, H, R). It is obvious that BC(k, H, R) contains the Brauer group Br(k).

Dually, if H is a QT Hopf algebra with a QT structure R, then a left H-module algebra A is simultaneously a YD H-module algebra with the right H^{op}-comodule structure given by

A ⟶ A ⊗ H^{op}, a ↦ ∑ R^{2} · a ⊗ R^{1}

for a ∈ A. The subset of BQ(k, H) consisting of the elements represented by the H-Azumaya algebras with right H^{op}-comodule structures stemming from left H-module structures in the above way, turns out to be a subgroup of BQ(k, H), denoted by BM(k, H, R). It is obvious that BM(k, H, R) contains the Brauer group Br(k).

The Brauer group BQ(k, H) is a special case of the Brauer group Br(C) of a braided monoidal category C as introduced in [32]. The fact that BC(k, H, R) is a subgroup of BQ(k, H) when (H, R) is a coquasitriangular Hopf algebra, can be explained in a categorical way. If D is a closed braided monoidal subcategory of a braided monoidal category C, then the Brauer group Br(D) is a subgroup of Br(C). This fact allows us to consider various subgroups of the Brauer group Br(C) of a braided monoidal category C whenever C contains certain closed braided subcategories. For example, if (H, R) is a CQT Hopf algebra, then the category MH^{op} of right H^{op}-comodules is a closed braided subcategory of the braided category of YD H^{op}-modules with the braiding φ given by:

M ⊗ N ⟶ N ⊗ M, m ⊗ n ↦ ∑ n_{(0)} ⊗ m_{(0)}R(n_{(1)} ⊗ m_{(1)}),

where m ∈ M and n ∈ N. The Brauer group Br(M_{R}^{H}) of M_{R}^{H} is indeed BC(k, H, R). When H is a finite Hopf algebra, it is well-known that the category of YD H-modules is equivalent to the category of left D(H)-modules, where D(H) is the Drinfel’d double of H. So we have that BQ(k, H) = BM(k, D(H), R) = BC(k, D(H)^{∗}, R), where R is the canonical quasi-triangular structure on D(H).

To end this section let us recall the notion of a Hopf Galois extensions by a Hopf algebra H. Let A be a right H-comodule algebra. A is said to be H-Galois if the canonical linear map

β : A ⊗_{A_{0}} A ⟶ A ⊗ H

sending element a ⊗ b to the element ∑ ab_{(0)} ⊗ b_{(1)}, is an isomorphism, where

A_{0} = {x ∈ A | ∑ x_{(0)} ⊗ x_{(1)} = x ⊗ 1}

is the coinvariant subalgebra of A. For a general Hopf Galois theory one may refer to [21, 23, 24].

2 The deformation H_{R}

In this section, we construct a deformation of a CQT Hopf algebra (H, R) which we will use through the whole paper. We start with the definition of the new product ⋆ on the vector space H:

h ⋆ l = ∑ l_{(2)}h_{(2)}R(S^{−1}(l_{(3)})l_{(1)} ⊗ h_{(1)}) = ∑ h_{(2)}l_{(1)}R(l_{(2)} ⊗ S(h_{(1)})h_{(3)}))  (2)

where h and l are in H. The second equation of (2) is evident by the definition of R. We show that ⋆ is indeed a multiplication on H.

Lemma 2.1. (H, ⋆) is an algebra with unit 1.

Proof. It is sufficent to verify that ⋆ is associative. Given elements h, l and m ∈ H, we have

(h ⋆ l) ⋆ m = ∑(l_{(2)}h_{(2)}) ⋆ mR(S^{−1}(l_{(3)})l_{(1)} ⊗ h_{(1)})

= ∑ m_{(2)}l_{(3)}h_{(3)}R(S^{−1}(m_{(3)})m_{(1)} ⊗ l_{(2)}h_{(2)})R(S^{−1}(l_{(4)})l_{(1)} ⊗ h_{(1)})
= ∑ m_{(3)}l_{(3)}h_{(4)}R(S^{−1}(m_{(5)})m_{(1)} ⊗ h_{(3)})R(S^{−1}(m_{(4)})m_{(2)} ⊗ l_{(2)})
R(S^{−1}(l_{(4)}) ⊗ h_{(1)})R(l_{(1)} ⊗ h_{(2)}).

Let a = S^{−1}(m_{(4)}), b = l_{(1)} and c = h_{(2)}. By utilizing the formula (see [19, ?]):

∑ R(a_{(1)} ⊗ c_{(2)})R(a_{(2)} ⊗ b_{(2)})R(b_{(1)} ⊗ c_{(1)}) = ∑ R(a_{(1)} ⊗ b_{(1)})R(a_{(2)} ⊗ c_{(1)})R(b_{(2)} ⊗ c_{(2)})

for all a, b and c ∈ H, we obtain

(h ⋆ l) ⋆ m = ∑ m_{(3)}l_{(3)}h_{(4)}R(S^{−1}(m_{(4)})m_{(2)} ⊗ h_{(2)})R(S^{−1}(m_{(5)})m_{(1)} ⊗ l_{(1)})
R(S^{−1}(l_{(4)}) ⊗ h_{(1)})R(l_{(2)} ⊗ h_{(3)})
= ∑ m_{(3)}l_{(3)}h_{(2)}R(S^{−1}(m_{(4)}l_{(4)})m_{(2)}l_{(2)} ⊗ h_{(1)})R(S^{−1}(m_{(5)}m_{(1)} ⊗ l_{(1)})
= ∑ h ⋆ (m_{(2)}l_{(2)})R(S^{−1}(m_{(3)})m_{(1)} ⊗ l_{(1)})
= h ⋆ (l ⋆ m).

So ⋆ is indeed a multiplication of H. □

Denote by H_{R} the algebra (H, ⋆). It is easy to see that the counit map ε of H is still an augmentation map from H_{R} to k.

In general, there is a double Hopf algebra for a CQT Hopf algebra (H, R) (not necessaily finite). This double Hopf algebra, denoted by D[H] in this paper, is H ⊗ H as a coalgebra. The multiplication of D[H] is defined by

(h ⊗ l)(h′ ⊗ l′) = ∑ hh′_{(2)} ⊗ l_{(2)}l′R(h′_{(1)} ⊗ l_{(1)})R(S(h′_{(3)}) ⊗ l_{(3)})

for h, l, h′ and l′ ∈ H. The antipode of D[H] is given by

S(h ⊗ l) = (1 ⊗ S(l))(S(h) ⊗ 1)

for all h, l ∈ H. The counit of D[H] is ε ⊗ ε.

For the detail of D[H], we refer to [19]. In the sequel, we will write h ▷◁ l for an element in D[H] and H ▷◁ H for D[H] as well. Since H is finite in this paper, the canonical Hopf algbera morphism Θ_{l} induces an Hopf algebra morphism from D[H] to D(H), the Drinfel’d quantum double H^{∗}^{op} ▷◁ H (cf. [13]):

Φ : D[H] ⟶ D(H), Φ(h ▷◁ l) = Θ_{l}(h) ▷◁ l.  (3)

When Θ_{l} is an isomorphism, we may identify D[H] with D(H). Thus any YD H-modules are automatically a left D[H]-module. Moreover, the following lemma says that H_{R} can be embedded into D[H].

Lemma 2.2. The following linear map is an algebra monomorphism:

ϕ : H_{R} ⟶ D[H], ϕ(h) = ∑ S^{−1}(h_{(2)}) ▷◁ h_{(1)}.

Proof. Given h, l ∈ H_{R}, we have

ϕ(h ⋆ l) = ∑ ϕ(l_{(2)}h_{(2)})R(S^{−1}(l_{(3)})l_{(1)} ⊗ h_{(1)})

= ∑ S^{−1}(l_{(3)}h_{(3)}) ▷◁ l_{(2)}h_{(2)}R(S^{−1}(l_{(4)})l_{(1)} ⊗ h_{(1)})
= ∑ S^{−1}(h_{(4)}S^{−1}(l_{(3)}) ▷◁ l_{(2)}h_{(3)}R(S^{−1}(l_{(4)}) ⊗ h_{(1)})R(l_{(1)} ⊗ h_{(2)})
= ∑ S^{−1}(h_{(4)}S^{−1}(l_{(3)}) ▷◁ h_{(2)}l_{(1)}R(S^{−1}(l_{(4)}) ⊗ h_{(1)})R(l_{(2)} ⊗ h_{(3)})
= ∑(S^{−1}(h_{(2)}) ▷◁ h_{(1)})(S^{−1}(l_{(2)}) ▷◁ l_{(1)})
= ϕ(h)ϕ(l).

It is obvious that ϕ is injective. □

Proposition 2.3. H_{R} is a left D[H]-comodule algebra. Proof. Define a linear map from H_{R} to D[H] ⊗ H_{R} as follows:

χ : H_{R} ⟶ D[H] ⊗ H_{R}, χ(h) = ∑(S^{−1}(h_{(3)}) ▷◁ h_{(1)}) ⊗ h_{(2)}.

It is evident that χ is a left D[H]-comodule map. We have to show that χ is an algebra map. Indeed, if h, l ∈ H_{R}, then

χ(h ⋆ l) = ∑ χ(h_{(2)}l_{(1)})R(l_{(2)} ⊗ S(h_{(1)})h_{(3)})
= ∑ S^{−1}(h_{(4)}l_{(3)}) ▷◁ h_{(2)}l_{(1)} ⊗ h_{(3)}l_{(2)}R(l_{(4)} ⊗ S(h_{(1)})h_{(5)})
= ∑ S^{−1}(h_{(4)}l_{(3)}) ▷◁ h_{(2)}l_{(1)} ⊗ h_{(3)}l_{(2)}R(l_{(4)} ⊗ h_{(5)})
R(S^{−1}(l_{(5)}) ⊗ h_{(1)})
= ∑ S^{−1}(l_{(4)}h_{(5)}) ▷◁ h_{(2)}l_{(1)} ⊗ h_{(3)}l_{(2)}R(l_{(3)} ⊗ h_{(4)})
R(S^{−1}(l_{(5)}) ⊗ h_{(1)})
= ∑ S^{−1}(l_{(4)}h_{(5)}) ▷◁ h_{(2)}l_{(1)} ⊗ l_{(3)}h_{(4)}R(l_{(2)} ⊗ h_{(3)})
R(S^{−1}(l_{(5)}) ⊗ h_{(1)})
= ∑ S^{−1}(l_{(6)}h_{(6)}) ▷◁ h_{(2)}l_{(1)} ⊗ l_{(3)}h_{(5)}R(S^{−1}(l_{(4)})l_{(2)} ⊗ h_{(4)})
R(S^{−1}(l_{(7)}) ⊗ h_{(1)})R(l_{(5)} ⊗ h_{(3)})
= ∑(S^{−1}(h_{(4)}) ▷◁ h_{(1)})(S^{−1}(l_{(5)}) ▷◁ l_{(1)}) ⊗ l_{(3)}h_{(3)}
R(S^{−1}(l_{(4)})l_{(2)} ⊗ h_{(2)})
= ∑(S^{−1}(h_{(3)}) ▷◁ h_{(1)})(S^{−1}(l_{(3)}) ▷◁ l_{(1)}) ⊗ h_{(2)} ⋆ l_{(2)}
= χ(h)χ(l).
□

Lemma 2.2 and Proposition 2.3 say that H_{R} can be embedded into D[H] as left coideal subalgebra. In fact, H_{R} can be further embedded into D(H) as a left coideal subalgebra. Corollary 2.4. The composite algebra map:

[figure: H_{R}; ϕ; D[H]; Φ; D(H)]

is injective, and H_{R} is isomorphic to a left coideal subalgebra of D(H). Proof. Since H is faithfully flat over k, we have that the kernel of Φ is Ker(Θ_{l}) ▷◁ H. If ΦΘ_{l}(h) = 0 for some h ∈ H_{R}, then

ϕ(h) = ∑ S^{−1}(h_{(2)}) ▷◁ h_{(1)} ∈ Ker(Θ_{l}) ▷◁ H.

It follows that

h = ∑ ε(S^{−1}(h_{(2)}))h_{(1)} = (ε ⊗ ι)ϕ(h) = 0.

Since Φ is a Hopf algebra map, Φϕ(H_{R}) is a left coideal subalgebra of D(H). □.

Let us now consider Yetter-Dinfel’d H-modules and H_{R}-bimodules. Let M be a Yetter-Drinfel’d module over H, or a left D(H)-module. The following composite map:

H_{R} ⊗ M ⟶^{ϕ⊗ι} D[H] ⊗ M ⟶^{Φ⊗ι} D(H) ⊗ M

makes M a left H_{R}-module. If we write −▷ for the above left action, then we have the explicit formula:

h −▷ m = ∑(h_{(2)} · m_{(0)})R(S^{−1}(h_{(4)}) ⊗ h_{(3)}m_{(1)}S^{−1}(h_{(1)}))  (4)

for h ∈ H_{R} and m ∈ M.

Since there is an augmentation map ε on H_{R}, we may define the H_{R}-invariant of a left H_{R}-module M which is

M^{H_{R}} = {m ∈ M | h −▷ m = ε(h)m, ∀h ∈ H_{R}}.

When a left H_{R}-module comes from a YD H-module we have

Lemma 2.5. Let M be a YD H-module. Then

M^{H_{R}} = {m ∈ M | h · m = h ▷_{1} m = ∑ m_{(0)}R(h ⊗ m_{(1)}), ∀h ∈ H}.

Proof. By definition of the action of H_{R} on M, we have h −▷ m = ∑ S^{−1}(h_{(2)}) ▷_{1} (h_{(1)} · m) for any h ∈ H_{R} and m ∈ M. It follows that the latter set is contained in M^{H_{R}}.

Conversely, if m ∈ M^{H_{R}}, then we have

h · m = ∑ h_{(3)} ▷_{1} (S^{−1}(h_{(2)}) ▷_{1} (h_{(1)} · m))
= ∑ h_{(2)} ▷_{1} (h_{(1)} −▷ m)
= ∑ h_{(2)} ▷_{1} (ε(h_{(1)})m)
= h ▷_{1} m

for any h ∈ H_{R} and m ∈ M. □

Following Lemma 2.5, we obtain that the invariant submodule M^{H_{R}} is the maximal submodule sitting in the subcategory M_{R}^{H}, where the left H-action is the induced one ▷_{1}. Thus we get an idempotent radical κ:

κ : Q^{H} ⟶ M_{R}^{H}, κ(M) = M^{H_{R}}.

It follows that a YD H-module is sitting in M_{R}^{H} if and only if H_{R} acts trivally on M, i.e, M = M^{H_{R}}

Now we will define a right H_{R}-module structure on a YD H-module M. Observe that the right H-comodule structure of M induces two left H-module structures. The first one is (1), and the second one is given by

h ▷_{2} m = ∑ m_{(0)}R(S(m_{(1)}) ⊗ h)  (5)

for h ∈ H and m ∈ M. With this second left H-action M, M becomes a right D[H]-module.

Lemma 2.6. Let M be a Yetter-Drinfel’d H-module. Then M is a right D[H]-module defined by

m ↽ (h ▷◁ l) = S(l) ▷_{2} (S(h) · m)  (6)

for m ∈ M and h, l ∈ H. Moreover, if A is a YD H-module algebra, then (6) makes A into a right D[H]^{cop}-module algebra.

Proof. It is sufficient to show that

m ↽ [(1 ▷◁ l)(h ▷◁ 1)] = S(h) · (S(l) ▷_{2} m)

for h, l ∈ H and m ∈ M. Indeed, we have

m ↽ [(1 ▷◁ l)(h ▷◁ 1)]
= ∑ m ↽ (h_{(2)} ▷◁ l_{(2)})R(h_{(1)} ⊗ l_{(1)})R(S(h_{(3)} ⊗ l_{(3)})
= ∑ S(l_{(2)}) ▷_{2} (S(h_{(2)}) · m)R(h_{(1)} ⊗ l_{(1)})R(S(h_{(3)} ⊗ l_{(3)})
= ∑ S(h_{(3)}) · (S(l_{(3)}) ▷_{2} m)R(h_{(4)} ⊗ l_{(4)})R(S(h_{(2)}) ⊗ l_{(2)})
R(h_{(1)} ⊗ l_{(1)})R(S(h_{(5)}) ⊗ l_{(5)})
= S(h) · (S(l) ▷_{2} m)

for h, l ∈ H and m ∈ M. The second statement is obvious. □

Unfortunately, the right D[H]-module structure (6) does not match the canonical left D[H]- module structure induced by Hopf algebra map Φ to give a D[H]-bimodule structure on M. However the right H_{R}-module structure on M given by

M ⊗ H_{R} ⟶^{ι⊗ϕ} M ⊗ D[H] ⟶ M,

precisely,

m ◁− h = ∑(h_{(3)} · m_{(0)})R(h_{(4)}m_{(1)}S^{−1}(h_{(2)}) ⊗ h_{(1)})  (7)

for m ∈ M and h ∈ H_{R}, together with the left H_{R}-module structure (4) does give an H_{R}- bimodule structure on M.

Proposition 2.7. Let M be a YD H-module. Then M is an H_{R}-bimodule via (4) and (7).

Proof. Given h, l ∈ H_{R} and m ∈ M, we have to prove

(l −▷ m) ◁− h = l −▷ (m ◁− h)

i.e.,

∑ S(h_{(1)}) ▷_{2} (h_{(2)} · (S^{−1}(l_{(2)}) ▷_{1} (l_{(1)} · m))) = ∑ S^{−1}(l_{(2)}) ▷_{1} (l_{(1)} · (S(h_{(1)}) ▷_{2} (h_{(2)} · m))).

Indeed, we have

∑ S(h_{(1)}) ▷_{2} (h_{(2)} · (S^{−1}(l_{(2)}) ▷_{1} (l_{(1)} · m)))
= ∑ S^{−1}(l_{(4)}) ▷_{1} (S(h_{(2)}) ▷_{2} (l_{(2)}h_{(4)} · m))R(l_{(1)} ⊗ h_{(3)})R(l_{(3)} ⊗ S(h_{(1)}))
= ∑ S(h_{(3)}) ▷_{2} (S^{−1}(l_{(5)}) ▷_{1} (l_{(2)}h_{(6)} · m))R(l_{(1)} ⊗ h_{(5)})
R(l_{(3)} ⊗ S(h_{(1)}))R(l_{(6)} ⊗ S(h_{(4)})R(S^{−1}(l_{(4)}) ⊗ S(h_{(2)}))
= ∑ S(h_{(1)}) ▷_{2} (S^{−1}(l_{(3)}) ▷_{1} (l_{(2)}h_{(4)} · m))R(l_{(1)} ⊗ h_{(3)})R(l_{(4)} ⊗ S(h_{(2)}))
= ∑ S(h_{(1)}) ▷_{2} (S^{−1}(l_{(3)}) ▷_{1} (h_{(3)}l_{(1)} · m))R(l_{(2)} ⊗ h_{(4)})R(l_{(4)} ⊗ S(h_{(2)}))
= ∑ S(h_{(1)}) ▷_{2} (h_{(2)} · (S^{−1}(l_{(2)}) ▷_{1} (l_{(1)} · m))).

So M is an H_{R}-bimodule. □

Remark that the right H_{R}-invariant of a YD H-module is different from the left one described in Lemma 2.5. One may apply the same argument in Lemma 2.5 to obtain the right invariant set of a YD H-module M is

{m ∈ M | h · m = h ▷_{2} m, ∀h ∈ H.}

Like the left invariants, the set of right invariants is the maximal right-right YD H-submodule of M. Combining Lemma 2.3, 2.6 and Proposition 2.7, we obtain the following:

Corollary 2.8. If A is a YD H-module algebra, then A is an H_{R}-bimodule algebra in the sense that

\begin{matrix}h −▷ (ab) = ∑(h_{(−1)} · a)(h_{(0)} −▷ b) \\ (ab) ◁− h = ∑(a · h_{(0)})(b ◁− h_{(−1)}) \end{matrix}

for a, b ∈ A and h ∈ H_{R}, where χ(h) = ∑ h_{(−1)} ⊗ h_{(0)} ∈ D[H] ⊗ H_{R}.

Now we are looking at the dual coalgebra H_{R}^{∗} which is a quotient coalgebra of the dual Hopf algebra D[H]^{∗} by Lemma 2.2. Let us view R as an invertible element ∑ R^{1} ⊗ R^{2} in H^{∗} ⊗ H^{∗}. Then the comultiplication of the Hopf algebra D[H]^{∗} reads as follows

Δ(p ⊗ q) = ∑ p_{(1)} ⊗ R(q_{(1)} ⊗ p_{(2)})R^{−1} ⊗ q_{(2)}
= ∑(p_{(1)} ⊗ R^{1}q_{(1)}S(r^{1})) ⊗ (R^{2}p_{(2)}r^{2} ⊗ q_{(2)})

where p, q ∈ H^{∗} and we use the same S to denote the antipode of H^{∗} (one may easily tell the difference from the context). Note that as an algebra D[H]^{∗} is the tensor product H^{∗} ⊗ H^{∗}.

Lemma 2.9. H_{R}^{∗} is a left D[H]^{∗}-module coalgebra with the comultiplication given by

Δ_{R}(p) = ∑ R^{2}r^{2}p_{(2)} ⊗ r^{1}p_{(1)}S^{−1}(R^{1})
= R^{2}p_{(1)}r^{2} ⊗ p_{(2)}r^{1}S^{−1}(R^{1})

where R = r, p ∈ H_{R}^{∗}. The counit ε of H is now a group-like element of H_{R}^{∗}. □

Actually H_{R}^{∗} is a quotient coalgebra in the sense that the following coalgebra map is sujective:

ϕ^{∗} : D[H]^{∗} ⟶ H_{R}^{∗}, p ▷◁ q ↦ qS^{−1}(p).  (8)

So the action of D[H]^{∗} on H_{R}^{∗} reads as follows:

p ▷◁ q ⇁ x = qxS^{−1}(p)  (9)

for p ▷◁ q ∈ D[H]^{∗} and x ∈ H_{R}^{∗}. In particular, we have

p ▷◁ q ⇁ ε = ϕ^{∗}(p ▷◁ q) = qS^{−1}(p)

for any p ▷◁ q ∈ D[H]^{∗}.

In the sequel, we will write Δ_{R}(p) = ∑ p^{(1)} ⊗ p^{(2)}, p ∈ H_{R}^{∗} in order to distinguish Δ_{R}(p) from Δ(p) = ∑ p_{(1)} ⊗ p_{(2)}, the usual comultiplication of the dual Hopf algebra H^{∗}. Let us figure out the relation between H_{R}^{∗}-comodule structure and the H^{∗}-comodule structures of a YD H-module. Let M be a YD H-module. We use the following summation notation for the dual H^{∗}-comodule structure of M

M ⟶ M ⊗ H^{∗}, m ↦ ∑ m_{[0]} ⊗ m_{[1]},

and the usual Sweedler notation ∑ m_{(0)} ⊗ m_{(1)} for the H-comodule stucture of M. The right and left H_{R}^{∗}-comodule structure will be indicated by the Sweedler ”upcase” notations.

It is not difficult to check that the right D[H]^{∗}-comodule structure of M reads as follows:

M ⟶ M ⊗ D[H]^{∗}, m ↦ ∑ m_{[0](0)} ⊗ m_{[0](1)} ▷◁ m_{[1]}.

Similarly, one may get the dual left D[H]^{∗}-comodule structure of (6). It follows from (8) that we obtain the corresponding dual right H_{R}^{∗}-comodule structure of (4) and the dual left H_{R}^{∗}-comodule structure of (7) respectively:

\begin{matrix}M ⟶ M ⊗ H_{R}^{∗}, & ∑ m^{(0)} ⊗ m^{(1)} = ∑ m_{[0](0)} ⊗ m_{[1]}S^{−1}(Θ_{r}(m_{[0](1)})), \\ M ⟶ H_{R}^{∗} ⊗ M, & ∑ m^{(−1)} ⊗ m^{(0)} = ∑ Θ_{l}(m_{[0](1)})m_{[1]} ⊗ m_{[0](0)}. \end{matrix}  (10)

Now let A be a YD H-module algebra. Then A is both a left and right D[H]-module algebra, and therefore an H_{R}-bimodule algebra in the sense of Corollary 2.8. We present the dual comodule structures of the formulas in Coro.2.8 because they will be frequently used in the sequel. Let us use the following upcase Sweedler notations for the dual left and right D[H]^{∗}- comodule structures of a YD H-module algebra A:

∑ a^{[−1]} ⊗ a^{[0]}, ∑ a^{[0]} ⊗ a^{[1]}

for an element a ∈ A. Then the dual comodule versions of the formulas in Corollary 2.8 read as follows:

\begin{matrix}∑(ab)^{(0)} ⊗ (ab)^{(1)} = ∑ a^{[0]}b^{(0)} ⊗ a^{[1]} ⇁ b^{(1)}, \\ ∑(ab)^{(−1)} ⊗ (ab)^{(0)} = ∑ a^{(0)}b^{[0]} ⊗ b^{[−1]} ⇁ a^{(−1)} \end{matrix}  (11)

for a, b ∈ A. We will call A a right (or left) H_{R}^{∗}-comodule algebra in the sense of (11).

Finally, for a YD H-module M, we will write M_{⋄} (or _{⋄}M) for the right (or left) H_{R}^{∗}- coinvariants. For instance,

M_{⋄} = {m ∈ M | ∑ m^{(0)} ⊗ m^{(1)} = m ⊗ ε.}

It is obvious that M_{0} = M^{H_{R}} cf.Lemma 2.5.

3 The group Gal(H_{R})

In this section, we construct a group Gal(H_{R}) of ”Galois” objects for the deformation H_{R}. The group Gal(H_{R}) plays the vital part in this paper.

Definition 3.1. Let A be a right D[H]^{∗}-comodule algebra. A/A_{⋄} is said to be a right H_{R}^{∗}-Galois extension if the linear map

β^{r} : A ⊗_{A_{⋄}} A ⟶ A ⊗ H_{R}^{∗}, β^{r}(a ⊗ b) = ∑ a^{(0)}b ⊗ a^{(1)}

is an isomorphism. Similarly, if A is a left D[H]^{∗}-comodule algebra, then A/_{0}A is said to be left Galois if the linear map

β^{l} : A ⊗_{_{⋄}A} A ⟶ H_{R}^{∗} ⊗ A, β^{l}(a ⊗ b) = ∑ b^{(−1)} ⊗ ab^{(0)}

is an isomorphism.

Note that the reason why we use the linear maps β^{r} and β^{l}, instead of the canonical linear maps defining the Hopf-Galois extensions (see [23, 3.7]), will be seen in the next section where our derived objects are Galois extensions of the type in Definition 3.1. However, when H_{R} is a Hopf algebra (although it is not the case in general), the definitions in 3.1 are equivalent to the classical ones. The following are characterizations of a right H_{R}^{∗}-Galois object, i.e, a right H_{R}^{∗}-Galois extension of k cf.[23, 3.7] and [16].

Lemma 3.2. Let A be a right D[H]^{∗}-comodule algebra such that A_{⋄} = k. The following statements are equivalent:

1). A/k is H_{R}^{∗}-Galois. 2). A#H_{R} is canonically isomorphic to End(A), and A is finitely projective k-module.

A right D[H]^{∗}-comodule algebra satisfying Lemma 3.2 is called a right H_{R}^{∗}-Galois object. Similarly, one may have the symmetric version of Lemma 3.2 for a left H_{R}^{∗}-Galois object. Note that a D[H]-linear algebra map between two Galois objects is an isomorphism. What we are interested in, in this section, are those H_{R}^{∗}-bigalois objects which are both left and right H_{R}^{∗}-Galois such that the left and right H_{R}^{∗}-coactions commutes.

Denote by E(H_{R}) the category of YD H-module algebras A such that A are H_{R}^{∗}-bigalois objects. The morphisms in E(H_{R}) are YD H-module algebra isomorphisms. We are going to define a product in the category E(H_{R}).

Given two objects X and Y in E(H_{R}), we define X ∧ Y as a subset of X#_{θ}Y, the latter is the braided product algebra in M_{R}^{H}:

X ∧ Y = {∑ x_{i}#y_{i} ∈ X#_{θ}Y | ∑ x_{i} ◁− h#y_{i} = ∑ x_{i}#h −▷ y_{i}, ∀h ∈ H_{R}.} (12)

Lemma 3.3. Let X, Y be two objects of E(H_{R}). Then

X ∧ Y = {∑ x_{i}#y_{i} ∈ X#_{θ}Y | ∑ h_{(1)} · x_{i}#h_{(2)} ▷_{1} y_{i} = ∑ h_{(1)} ▷_{2} x_{i}#h_{(2)} · y_{i}, ∀h ∈ H}. (13)

Proof. Let T be the set

{∑ x_{i}#y_{i} ∈ X#_{θ}Y | ∑ h_{(1)} · x_{i}#h_{(2)} ▷_{1} y_{i} = ∑ h_{(1)} ▷_{2} x_{i}#h_{(2)} · y_{i}, ∀h ∈ H}.

In order to simplify the computation, we will write x#y for an element ∑ x_{i}#y_{i} in X#Y in the sequel. Given x#y ∈ X ∧ Y and h ∈ H, we have

∑ h_{(1)} · x#h_{(2)} ▷_{1} y
= ∑ h_{(1)} ▷_{2} (S(h_{(2)} ▷_{2} (h_{(3)} · x))#h_{(4)} ▷_{1} y
= ∑ h_{(1)} ▷_{2} (x ◁− h_{(2)})#h_{(3)} ▷_{1} y
= ∑ h_{(1)} ▷_{2} x#h_{(3)} ▷_{1} (h_{(2)} −▷ y)
= ∑ h_{(1)} ▷_{2} x#h_{(4)} ▷_{1} (S^{−1}(h_{(3)}) ▷_{1} (h_{(2)} · y))
= ∑ h_{(1)} ▷_{2} x#h_{(2)} · y.

So x#y belongs to T and we have that X ∧ Y ⊆ T.

Conversely, if x#y ∈ T, and h ∈ H_{R}, we have

x ◁− h#y = ∑ S(h_{(1)}) ▷_{2} (h_{(2)} · x)#y
= ∑ S(h_{(1)}) ▷_{2} (h_{(2)} · x)#S^{−1}(h_{(4)})h_{(3)} ▷_{1} y
= ∑ S(h_{(1)}) ▷_{2} (h_{(2)} ▷_{2} x)#S^{−1}(h_{(4)}) ▷_{1} (h_{(3)} · y)
= ∑ x#S^{−1}(h_{(2)}) ▷_{1} (h_{(1)} · y)
= x#h −▷ y.

It follows that T ⊆ X ∧ Y. □

The above alternative definition (13) of X ∧Y is more practicable when we check if an element is in X ∧ Y. Moreover, it hints that we might have an left H-action on X ∧ Y given by

h · ∑(x_{i}#y_{i}) = ∑ h_{(1)} · x_{i}#h_{(2)} ▷_{1} y_{i} = ∑ h_{(1)} ▷_{2} x_{i}#h_{(2)} · y_{i}  (14)

whenever ∑ x_{i}#y_{i} ∈ X ∧ Y and h ∈ H. (14) is indeed a left H-module strucuture on X ∧ Y. To show this, one may easily check that the element h · (x#y) is sitting in X ∧ Y by Lemma 3.3 if x#y ∈ X ∧ Y.

Lemma 3.4. X ∧ Y with the H-action (14) and the H-coaction inheriting from X#Y is a YD H-module.

Proof. Let us first check that X ∧ Y is an H-subcomodule of X#_{θ}Y. Given x#y ∈ X ∧ Y, h ∈ H and p ∈ H^{∗}, we have

∑ h_{(1)} · x_{(0)}#h_{(2)} ▷_{1} y_{(0)} < p, y_{(1)}x_{(1)} >
= ∑ h_{(1)} · (p_{(2)} · x)#h_{(2)} ▷_{1} (p_{(1)} · y)
= ∑ p_{(3)} · (h_{(2)} · x)#p_{(2)} · (h_{(3)} ▷_{1} y) < p_{(1)}, h_{(4)} >< p_{(4)}, S^{−1}(h_{(1)}) >
= ∑ p_{(3)} · (h_{(2)} ▷_{2} x)#p_{(2)} · (h_{(3)} · y) < p_{(1)}, h_{(4)} >< p_{(4)}, S^{−1}(h_{(1)}) >
= ∑ h_{(1)} ▷_{2} (p_{(2)} · x)#h_{(2)} · (p_{(1)} · y)
= ∑ h_{(1)} ▷_{2} x_{(0)}#h_{(2)} · y_{(0)} < p, y_{(1)}x_{(1)} > .

It follows that X ∧ Y is a right H-subcomodule of X#_{θ}Y.

Now we show that the right H-comodule X ∧ Y together with the left H-module structure (14) is a YD H-module. Indeed, for h ∈ H and x#y ∈ X ∧ Y, we have

ρ(h · (x#y)) = ∑ ρ(h_{(1)} · x#h_{(2)} ▷_{1} y)
= ∑ h_{(2)} · x_{(0)}#h_{(5)} ▷_{1} y_{(0)} ⊗ h_{(6)}y_{(1)}S^{−1}(h_{(4)})h_{(3)}x_{(1)}S^{−1}(h_{(1)})
= ∑ h_{(2)} · (x_{(0)}#y_{(0)}) ⊗ h_{(3)}y_{(1)}x_{(1)}S^{−1}(h_{(1)})

for any h ∈ H and x#y ∈ X ∧ Y. □

Proposition 3.5. If X, Y are two objects of E(H_{R}), then X ∧ Y is an object of E(H_{R}).

Proof. First of all, we have to show that X ∧ Y is a subalgebra of X#_{θ}Y. Let x#y and x′#y′ be two elements of X ∧ Y. For any h ∈ H, we have

∑ h_{(1)} · (xx′_{(0)})#h_{(2)} ▷_{1} (y_{(0)}y′)R(x′_{(1)} ⊗ y_{(1)})
= ∑(h_{(1)} · x)(h_{(2)} · x′_{(0)})#(h_{(3)} ▷_{1} y_{(0)})(h_{(4)} ▷_{1} y′)R(x′_{(1)} ⊗ y_{(1)})
= ∑(h_{(1)} · x)(h_{(2)} · x′_{(0)})#y_{(0)}y′_{(0)}R(h_{(3)} ⊗ y_{(1)})R(h_{(4)} ⊗ y′_{(1)})R(x′_{(1)} ⊗ y_{(2)})
= ∑(h_{(1)} · x)(h_{(4)} · x′_{(0)})#y_{(0)}y′_{(0)}R(h_{(5)}x′_{(1)}S^{−1}(h_{(3)}) ⊗ y_{(1)})
R(h_{(2)} ⊗ y_{(2)})R(h_{(6)} ⊗ y′_{(1)})
= ∑(h_{(1)} · x#h_{(2)} ▷_{1} y)(h_{(3)} · x′#h_{(4)} ▷_{1} y).

Similarly, one may obtain

∑ h_{(1)} ▷_{2} (xx′_{(0)})#h_{(2)} · (y_{(0)}y′)R(x′_{(1)} ⊗ y_{(1)})
= ∑(h_{(1)} ▷_{2} x#h_{(2)} · y)(h_{(3)} ▷_{2} x′#h_{(4)} · y).

It follows that

∑ h_{(1)} · (xx′_{(0)})#h_{(2)} ▷_{1} (y_{(0)}y′)R(x′_{(1)} ⊗ y_{(1)})
= ∑(h_{(1)} · x#h_{(2)} ▷_{1} y)(h_{(3)} · x′#h_{(4)} ▷_{1} y)
= ∑(h_{(1)} ▷_{2} x#h_{(2)} · y)(h_{(3)} ▷_{2} x′#h_{(4)} · y)
= ∑ h_{(1)} ▷_{2} (xx′_{(0)})#h_{(2)} · (y_{(0)}y′)R(x′_{(1)} ⊗ y_{(1)}).

By Lemma 3.3, we have shown that X ∧ Y is a subalgebra of X#_{θ}Y s and hence a H^{op}- comodule subalgebra of X#_{θ}Y. Moreover, the previous computations actually showed that X ∧ Y is a left H-module algebra with respect to the H-action (14). It follows from Lemma 3.4 that X ∧ Y is a YD H-module algebra.

It remains to show that X ∧ Y is an H_{R}^{∗}-bigalois object. Observe that the H_{R}-bimodule structure on X ∧ Y induced by the YD H-module structure defined above is as follows:

h −▷ (∑ x_{i}#y_{i}) = ∑ h −▷ x_{i}#y_{i}, (∑ x_{i}#y_{i}) ◁− h = ∑ x_{i}#y_{i} ◁− h  (15)

whenever h ∈ H_{R} and ∑ x_{i}#y_{i} ∈ X ∧ Y. We verify the first formula and leave the second one to the reader. Indeed,

h −▷ (x_{#}y) = ∑ S^{−1}(h_{(2)}) ▷_{1} (h_{(1)} · (x#y))
= ∑ S^{−1}(h_{(3)}) ▷_{1} (h_{(1)} · x#h_{(2)} ▷_{1} y)
= ∑ S^{−1}(h_{(2)}) ▷_{1} (h_{(1)} · x)#y
= h −▷ x#y,

for any h ∈ H_{R} and x#y ∈ X ∧ Y. Thus the right (or left) H_{R}^{∗}-comodule structure of X ∧ Y only comes from the right one of X (or the left one of Y).

Applying (15), we may easily obtain that both the left and right coinvariant subalgebras of X ∧ Y are trivial. To show that X ∧ Y is a right H_{R}^{∗}-Galois object, we need a new object (X ⊗ X) ∧ Y which is not in E(H_{R}), but which makes sense with respect to the product ∧. Consider X ⊗ X as a product YD H-module algebra in the category Q^{H}. By Coro.2.8, X ⊗ X is an H_{R}-biomodule algebra. So the product defined in (12) makes sense even though X ⊗ X is not H_{R}^{∗}-Galois. For instance, H_{R} acts on X ⊗ X from the right hand side in the following way:

(x ⊗ x′) ◁− h = ∑ x ◁− h_{(0)} ⊗ x′ ↽ h_{(−1)}

for h ∈ H_{R} and x ⊗ x′ ∈ X ⊗ X.

Now one may take a while to check that the following diagram is commutative:

(X ∧ Y) ⊗ (X ∧ Y) \overline{β_{X∧Y}^{r}} (X ∧ Y) ⊗ H_{R}^{∗}
[figure: τ; δ]
(X ⊗ X) ∧ Y

where τ is the linear map given by

τ((x#y) ⊗ (x′#y′)) = ∑(x ⊗ x_{(0)})#(x_{(1)} ▷_{1} y)y′

and the linear map δ is defined by

δ((x′ ⊗ x)#y) = ∑ x^{(0)}x′#y ⊗ x^{(1)}.

However, we have to show that τ and δ are well-defined. We leave the easier verification of τ to the reader, and show that δ is well-defined.

Observe that the multiplication map of X: X ⊗ X ⟶ X is an H_{R}-bimodule map because X is an H_{R}-bimodule algebra in the sense of Coro.2.8. The multiplication map then induces an H_{R}-bimodule map:

μ : (X ⊗ X) ∧ Y ⟶ X ∧ Y.

If we can prove that the map η:

(X ⊗ X) ∧ Y ⟶ (X ⊗ X) ∧ Y ⊗ H_{R}^{∗}

given by η(x ⊗ x′)#y = ∑(x^{(0)} ⊗ x′)#y ⊗ x^{(1)}, is well-defined, then δ is well-defined because δ actually is the composite map of η with μ:

δ : (X ⊗ X) ∧ Y ⟶^{η}(X ⊗ X) ∧ Y ⊗ H_{R}^{∗} ⟶^{μ} X ∧ Y ⊗ H_{R}^{∗}.

Indeed, to show that η is well-defined, it is equivalent to show that for any l ∈ H_{R}, and (x ⊗ x′)#y ∈ (X ⊗ X) ∧ Y, the element

(l −▷ x ⊗ x′)#y

is still in (X ⊗ X) ∧ Y. This is the case since

(l −▷ x ⊗ x′) ◁− h#y = ∑((l −▷ x) ◁− h_{(0)} ⊗ x′ ↽ h_{(−1)})#y
= ∑(l −▷ (x ◁− h_{(0)}) ⊗ x′ ↽ h_{(−1)})#y
= ∑(l −▷ x ⊗ x′)#h −▷ y

for any h, l ∈ H_{R} and (x ⊗ x′)#y ∈ (X ⊗ X)#Y.

Since both τ and δ are obviously isomorphisms, we obtain that β_{X∧Y} is an isomorphism. It follows that X ∧ Y is a right H_{R}^{∗}-Galois object. Similarly, one may show that X ∧ Y is a left H_{R}^{∗}-Galois object, and X ∧ Y is an object in E(H_{R}). □

Now let H^{∗} be the covolution algebra of H. There is a canonical YD H-module stucture on H^{∗} such that H^{∗} is a YD H-module algebra:

h · p = ∑ p_{(1)} < p_{(2)}, h >, H − action
h^{∗} · p = ∑ h_{(2)}^{∗}pS^{−1}(h_{(1)}^{∗}), H − coaction (16)

for h^{∗}, p ∈ H^{∗} and h ∈ H.

Lemma 3.6. H^{∗} is an object in E(H_{R}).

Proof. It is sufficient to show that the induced H_{R}-bimodule structure (4) and (7) on H^{∗} is the same as the dual H_{R}-bimodule sructure stemming from the comultiplication Δ_{R} (see Lemma 2.9) of H_{R}^{∗}. Indeed, given p ∈ H^{∗}, h ∈ H_{R}, we have

h −▷ p = ∑ S^{−1}(h_{(2)}) ▷_{1} (h_{(1)} · p)
= ∑ Θ_{l}(S^{−1}(h_{(3)}))(h_{(1)} · p)S^{−1}(Θ_{l}(S^{−1}(h_{(2)}))
= ∑ Θ_{l}(S^{−1}(h_{(3)}))p_{(1)}Θ_{l}(h_{(2)}) < p_{(2)}, h_{(1)} >
= ∑ R^{2}p_{(1)}r^{2} < p_{(2)}, h_{(1)} >< r^{1}, h_{(2)} >< S^{−1}(R^{1}), h_{(3)} >
= ∑ R^{2}p_{(1)}r^{2} < p_{(2)}r^{1}S^{−1}(R^{1}), h >
= ∑ p^{(1)} < p^{(2)}, h > .

Similarly, one have p ◁− h = ∑ p^{(2)} < p^{(1)}, h > for any h ∈ H_{R} and p ∈ H^{∗}. Since H_{R}^{∗} is a quotient coalgebra of D[H]^{∗}, we have that H^{∗} with the H_{R}^{∗}-bicomodule stucture Δ_{R} is an H_{R}^{∗}-bigalois object. □

Denote by I the object H^{∗} described in Lemma 3.6. I is infact a unit in the category of E(H_{R}).

Proposition 3.7. The category E(H_{R}) is a monoidal category with product ∧ and the unit I.

Proof. It is sufficient to show that I ∧ X ≅ X ≅ X ∧ I for any X ∈ E(H_{R}). We show that I ∧ X ≅ X. The proof of X ∧ I ≅ X is similar. Let ρ^{+} be the composite map of the flip map with the right H_{R}^{∗}-comodule structure of X. We show that

ρ^{+} : X ⟶ I ∧ X, ρ^{+}(x) = ∑ x^{(1)}#x^{(0)}

is the desired isomorphism in E(H_{R}). By Lemma 3.6, we have

∑ x^{(1)} ◁− h#x^{(0)} = ∑ x^{(2)} < x^{(1)}, h > #x^{(0)}
= ∑ x^{(1)}#h −▷ x^{(0)}

for h ∈ H_{R} and x ∈ X. So ρ^{+} is a well-defined map. Second, we verify that ρ^{+} is an algebra map. Let x, y ∈ X and h ∈ H_{R}. we compute ρ^{+}(x)ρ^{+}(y).

ρ^{+}(x)ρ^{+}(y) = ∑(x^{(1)}#x^{(0)})(y^{(1)}#y^{(0)})

= ∑ x^{(1)}y_{(0)}^{(1)}#x_{(0)}^{(0)}R(y_{(1)}^{(1)} ⊗ x_{(1)}^{(1)})
= ∑ x^{(1)}(Θ_{l}(x_{(1)}^{(0)}) · y^{(1)})#x_{(0)}^{(0)}y^{(0)}
= ∑ x_{[1]}Θ_{r}(S(x_{[0](3)}))Θ_{r}(x_{[0](2)})y_{[1]}Θ_{r}(S(y_{[0](1)}))Θ_{r}(S(x_{[0](1)}))#x_{[0](0)}y_{[0](0)}
= ∑ x_{[1]}y_{[1]}Θ_{r}(S(y_{[0](1)}x_{[0](1)}))#x_{[0](0)}y_{[0](0)}
= ∑(xy)^{(1)} ⊗ (xy)^{(0)}
= ∑ ρ^{+}(xy).

It is obvious that ρ^{+} is a YD H-module morphism, hence a morphism in E(H_{R}). □

Denote by E(H_{R}) the set of the isomorphism classes of objects in E(H_{R}). Proposition 3.7 says that E(H_{R}) is a semigroup. In the rest of this section we show that E(H_{R}) contains a subgroup.

Let X be an object in E(H_{R}). Let \overline{X} be the opposite algebra in M_{R}^{H}. That is, \overline{X} = X as a right H^{op}-comodule, but with the multiplication given by

\overline{x} ∘ \overline{y} = ∑ \overline{y_{(0)}x_{(0)}}R(y_{(1)} ⊗ x_{(1)})

when \overline{x}, \overline{y} ∈ \overline{X}. Since the H-action on X does not give an H-module algebra structure on \overline{X}, we have to define a new H-action on \overline{X} such that \overline{X} together with the inherited H^{op}-comodule structure is a YD H-module algebra. Let H act on \overline{X} as follows

\begin{aligned}h ⇀ \overline{x} & = ∑ \overline{S(h_{(4)}) · x_{(0)}}R(h_{(6)} ⊗ x_{(2)})R(x_{(1)} ⊗ S(h_{(3)}))R(h_{(5)} ⊗ S(h_{(2)}))u^{−1}(h_{(1)}) \\ & = ∑ \overline{h_{(3)}^{u} · (h_{(2)} ▷_{2} (h_{(5)} ▷_{1} x))}R(S(h_{(4)}) ⊗ h_{(1)}) \end{aligned} (17)

where h ∈ H_{R}, \overline{x} ∈ \overline{X}, h^{u} = ∑ S(h_{(2)})u^{−1}(h_{(1)}) and u = ∑ S(R^{2})R^{1} ∈ H^{∗} is the Casimir element of H^{∗}. Since the square of the antipode of H^{∗} is an inner automorphism induced by the Casimir element u, we have the formula (see [19]):

\begin{matrix}∑ u(h_{(1)})h_{(2)} = ∑ S^{2}(h_{(1)})u(h_{(2)}), \\ ∑ u^{−1}(h_{(1)})S^{−1}(h_{(2)}) = ∑ S(h_{(1)})u^{−1}(h_{(2)}) \end{matrix}  (18)

for any h ∈ H. We will use the formulas (18) quite often in the sequel. For instance, one may change the order of the actions ·, ▷_{1} and ▷_{2} in the formula (17) in order to have an alternative formula:

h ⇀ \overline{x} = ∑ \overline{h_{(1)} ▷_{2} (h_{(5)} ▷_{1} (S^{−1}(h_{(4)}) · x))}R(S^{2}(h_{(6)}) ⊗ h_{(3)})  (19)

for h ∈ H_{R} and x ∈ \overline{X}.

Lemma 3.8. Let X be an object in E(H_{R}). Then \overline{X} together with the H-action (17) is a YD H-module algebra.

Proof. We show that \overline{X} together with (17) is a YD H-module, and leave to the reader the tedious check that \overline{X} is left H-module algebra.

First we show that (17) is a left H-module structure on \overline{X}. Given h, l ∈ H and \overline{x} ∈ \overline{X}, we have

l ⇀ (h ⇀ \overline{x})
= ∑ l ⇀ \overline{S(h_{(4)}) · x_{(0)}}R(h_{(6)} ⊗ x_{(2)})R(x_{(1)} ⊗ S(h_{(3)}))R(h_{(5)} ⊗ S(h_{(2)}))u^{−1}(h_{(1)})

= ∑ \overline{S(l_{(4)}) · (S(h_{(4)}) · x_{(0)})}R(l_{(6)} ⊗ S(h_{(4)})x_{(2)}h_{(8)})R(S(h_{(5)})x_{(1)}h_{(7)} ⊗ S(l_{(3)}))

R(l_{(5)} ⊗ S(l_{(2)}))u^{−1}(l_{(1)})R(h_{(10)} ⊗ x_{(4)})R(x_{(3)} ⊗ S(h_{(3)}))R(h_{(9)} ⊗ S(h_{(2)}))u^{−1}(h_{(1)})

= ∑ \overline{S(h_{(6)}l_{(6)}) · x_{(0)}}R(l_{(8)} ⊗ h_{(8)})R(l_{(9)} ⊗ S(h_{(3)}))R(l_{(10)} ⊗ x_{(3)})
R(h_{(5)} ⊗ l_{(5)})R(x_{(1)} ⊗ S(h_{(4)}))R(h_{(7)} ⊗ S(l_{(3)}))R(l_{(7)} ⊗ S(l_{(2)}))
R(h_{(10)} ⊗ x_{(4)})R(x_{(2)} ⊗ S(h_{(4)}))R(h_{(9)} ⊗ S(h_{(2)}))u^{−1}(h_{(1)})u^{−1}(l_{(1)})
= ∑ \overline{S(l_{(4)}h_{(4)}) · x_{(0)}}R(h_{(5)} ⊗ l_{(5)})R(l_{(8)} ⊗ h_{(8)})R(l_{(9)}h_{(9)} ⊗ S(h_{(2)}))
R(h_{(7)}l_{(7)} ⊗ S(l_{(2)}))R(x_{(1)} ⊗ S(l_{(4)}h_{(4)}))R(l_{(10)}h_{(10)} ⊗ x_{(2)})u^{−1}(h_{(1)})u^{−1}(l_{(1)})
= ∑ \overline{S^{−1}(l_{(3)}h_{(3)}) · x_{(0)}}((u^{−1} ⊗ u^{−1})R_{21}R)(l_{(5)} ⊗ h_{(5)})
R(S^{−1}(l_{(4)}h_{(4)}) ⊗ l_{(1)}h_{(1)})R(l_{(6)}h_{(6)} ⊗ x_{(2)})R(x_{(1)} ⊗ S^{−1}(l_{(2)}h_{(2)}))
= ∑ \overline{S^{−1}(l_{(3)}h_{(3)}) · x_{(0)}}u^{−1}(l_{(5)}h_{(5)})R(S^{−1}(l_{(4)}h_{(4)}) ⊗ l_{(1)}h_{(1)})
R(l_{(6)}h_{(6)} ⊗ x_{(2)})R(x_{(1)} ⊗ S^{−1}(l_{(2)}h_{(2)}))
= ∑ \overline{S(l_{(4)}h_{(4)}) · x_{(0)}}R(l_{(5)}h_{(5)} ⊗ S(l_{(2)}h_{(2)}))R(l_{(6)}h_{(6)} ⊗ x_{(2)})
R(x_{(1)} ⊗ S(l_{(3)}h_{(3)}))u^{−1}(l_{(1)}h_{(1)})
= lh ⇀ \overline{x},

where we used the identity (u^{−1} ⊗ u^{−1})R_{21}R = Δ(u^{−1}) (see [19] for the proof), and R_{21} = ∑ R^{2} ⊗ R^{1}.

Second we show that \overline{X} is a YD H-module. Given h ∈ H and \overline{x} ∈ \overline{X}, we compute ρ(h ⇀ \overline{x}).

ρ(h ⇀ \overline{x}) = ∑ \overline{(S(h_{(4)}) · x_{(0)})_{(0)}} ⊗ (S(h_{(4)}) · x_{(0)})_{(1)}R(h_{(6)} ⊗ x_{(2)})
R(x_{(1)} ⊗ S(h_{(3)}))R(h_{(5)} ⊗ S(h_{(2)}))u^{−1}(h_{(1)})
= ∑ \overline{S(h_{(5)}) · x_{(0)}} ⊗ S(h_{(4)})x_{(1)}h_{(6)}R(x_{(2)} ⊗ S(h_{(3)}))
R(h_{(8)} ⊗ x_{(3)})R(h_{(7)} ⊗ S(h_{(2)}))u^{−1}(h_{(1)})
= ∑ \overline{S(h_{(5)}) · x_{(0)}} ⊗ x_{(2)}S(h_{(3)})h_{(6)}u^{−1}(h_{(1)})
R(h_{(8)} ⊗ x_{(3)})R(x_{(1)} ⊗ S(h_{(4)})R(h_{(6)} ⊗ S(h_{(3)}))
= ∑ \overline{S(h_{(5)}) · x_{(0)}} ⊗ x_{(2)}h_{(7)}S^{−1}(h_{(1)})u^{−1}(h_{(2)})
R(h_{(8)} ⊗ x_{(3)})R(x_{(1)} ⊗ S(h_{(4)}))R(h_{(7)} ⊗ S(h_{(2)}))
= ∑ \overline{S(h_{(5)}) · x_{(0)}} ⊗ h_{(8)}x_{(3)}S^{−1}(h_{(1)})u^{−1}(h_{(2)})
R(h_{(7)} ⊗ x_{(2)})R(x_{(1)} ⊗ S(h_{(4)}))R(h_{(6)} ⊗ S(h_{(3)}))
= ∑ h_{(2)} ⇀ \overline{x_{(0)}} ⊗ h_{(3)}x_{(1)}S^{−1}(h_{(1)}).

So \overline{X} is a YD H-module algebra. □

Now we are looking at the H_{R}-bimodule structure of \overline{X} steming from the new YD H-module structure of \overline{X}.

Lemma 3.9. The H_{R}-bimodule structure on \overline{X} is given by

\begin{matrix}h −▷ \overline{x} = ∑ \overline{S^{−1}(h_{(2)}) · (h_{(1)} ▷_{2} x)} ≡ \overline{h ↠ x}, \\ \overline{x} ◁− h = ∑ \overline{S(h_{(1)}) · (h_{(2)} ▷_{1} x)} ≡ \overline{x ↞ h}. \end{matrix}  (20)

Proof. We verify the right H_{R}-action, and leave the check of the left H_{R}-action to the reader.

Indeed, if h ∈ H_{R} and \overline{x} ∈ \overline{X}, we have

\overline{x} ◁− h = ∑ S(h_{(1)}) ▷_{2} (h_{(2)} ⇀ \overline{x})
= ∑ \overline{h_{(4)} ▷_{1} (S^{−1}(h_{(3)}) · x)}u(h_{(1)})R(S(h_{(5)}) ⊗ S^{−1}(h_{(3)})) by using (19)
= ∑ \overline{S^{−1}(h_{(4)}) · (h_{(7)} ▷_{1} x)}R^{−1}(h_{(6)} ⊗ S^{−1}(h_{(5)}))
R(h_{(8)} ⊗ S^{−1}(h_{(3)}))R(S(h_{(9)} ⊗ S^{−1}(h_{(2)}))u(h_{(1)})
= ∑ \overline{S^{−1}(h_{(2)}) · (h_{(5)} ▷_{1} x)}u^{−1}(h_{(3)})u(h_{(1)})
= ∑ \overline{S(h_{(3)}) · (h_{(4)} ▷_{1} x)}u^{−1}(h_{(2)})u(h_{(1)})
= ∑ \overline{S(h_{(1)}) · (h_{(2)} ⇀ x)}.
□

From (20), one immidiately obtain that the left and right H_{R}-invariant subalgebra are trivial. Moreover, the H_{R}-bimodule structure (20) on \overline{X} can be translated into the H_{R}^{∗}-bicomodule structure on \overline{X} in the following way:

\begin{matrix}∑ \overline{x}^{(0)} ⊗ \overline{x}^{(1)} = ∑ \overline{x_{(0)[0]}} ⊗ Θ_{l}(S(x_{(1)}))S^{−1}(x_{(0)[1]}), \\ ∑ \overline{x}^{(−1)} ⊗ \overline{x}^{(0)} = ∑ S(x_{(0)[1]})Θ_{r}(x_{(1)}) ⊗ \overline{x_{(0)[0]}}. \end{matrix}  (21)

Now Let us recall that a YD H-module algebra A is said to be quantum commutative if

ab = ∑ b_{(0)}(b_{(1)} · a)  (22)

for any a, b ∈ A. That is, A is a commutative algebra in Q^{H}.

Lemma 3.10. Let A be quantum commutative YD H-module algebra. Let \overline{A} be the opposite algebra in M_{R}^{H}. Then the multiplication of \overline{A} reads as follows:

\begin{aligned}\overline{a} ∘ \overline{b} & = ∑ \overline{a_{(0)}(S(a_{(1)}) ↠ b)} \\ & = ∑ \overline{(a ↞ b_{(1)})b_{(0)}} \end{aligned}  (23)

for a, b ∈ A, where the actions ↞ and ↠ are defined in (20).

Proof. Let \overline{a} and \overline{b} be two elements in \overline{A}. By definition, we have

\overline{a} ∘ \overline{b} = ∑ \overline{b_{(0)}(b_{(1)} ▷_{1} a)}.

Since A is also quantum commutative in Q^{H}, we have

\overline{a} ∘ \overline{b} = ∑ \overline{b_{(0)}(b_{(1)} ▷_{1} a)}
= ∑ \overline{S(b_{(1)}) · (b_{(2)} ▷_{1} a)b_{(0)}}
= \overline{(a ↞ b_{(1)})b_{(0)}}.

Similarly, we have

\overline{a} ∘ \overline{b} = ∑ \overline{(S(a_{(1)}) ▷_{2} b)a_{(0)}}
= ∑ \overline{a_{(0)}(a_{(1)} · (S(a_{(2)}) ▷_{2} b))}
= \overline{a_{(0)}(S(a_{(1)}) ↠ b)}

for any \overline{a}, \overline{b} ∈ \overline{A}.

Proposition 3.11. Let X be an object in E(H_{R}) such that X is quantum commutative in Q^{H}. Then the opposite algebra \overline{X} in M_{R}^{H} is an object in E(H_{R}).

Proof. We have to show that \overline{X} is an H_{R}^{∗}-bigalois objects. Let f be the following linear map:

f : X ⊗ X ⟶ X ⊗ X

given by f(x ⊗ y) = ∑ x_{(0)} ⊗ S(x_{(1)}) ↠ y. It is easy to see that f is a linear isomorphism. Let us compute the canonical Galois linear map βl ; \overline{X} from \overline{X} ⊗ \overline{X} to H_{R}^{∗} ⊗ \overline{X}. By applying the formulas (21) and (23), we obtain

βl ; \overline{X}(\overline{y} ⊗ \overline{x}) = ∑ \overline{x}^{(−1)} ⊗ \overline{y} ∘ \overline{x}^{(0)}
= ∑ S(x_{(0)[1]})Θ_{r}(x_{(1)}) ⊗ \overline{y} ∘ \overline{x_{(0)[0]}}
= ∑ S(x_{(0)[1]})Θ_{r}(x_{(1)}) ⊗ \overline{y} ∘ \overline{x_{(0)[0]}}
= ∑ S(x_{(0)[1]})Θ_{r}(x_{(1)}) ⊗ \overline{y_{(0)}(y_{(1)} ↠ x_{(0)[0]})}
= ∑ S(x_{(0)[1]})Θ_{r}(x_{(1)}) ⊗ \overline{y_{(0)}(y_{(1)} ↠ x)_{(0)[0]}}

for x\overline{,}\overline{y} ∈ \overline{X}, where the last equation holds because \overline{X} is an H_{R}^{∗}-bicomodule. Since X is right H_{R}^{∗}-Galois, we have the Galois isomorphism β_{X}^{r}:

β_{X}^{r}(x ⊗ y) = ∑ x_{[0](0)}y ⊗ x_{[1]}S^{−1}(Θ_{r}(x_{[0](1)}))

for any x, y ∈ X. β_{X}^{r} in turn induces an isomorphism

γ^{r} : X ⊗ X ⟶ X ⊗ H_{R}^{∗}

given by γ^{r}(y ⊗ x) = ∑ yx_{(0)[0]} ⊗ S(x_{(0)[1]})Θ_{r}(x_{(1)}). It follows that βl ; \overline{X} is the following composite linear isomorphism if we identify the vector space X with \overline{X}:

[figure: X ⊗ X; f; X ⊗ X; γ^{r}; X ⊗ H_{R}^{∗}; τ; H_{R}^{∗} ⊗ X.]

where τ is the flip map.

Similarly, one may verify that βr ; \overline{X} is the composite isomorphism:

[figure: X ⊗ X; g; X ⊗ X; γ^{l}; H_{R}^{∗} ⊗ X; τ; X ⊗ H_{R}^{∗},]

where g and γ^{l} are given by

\begin{matrix}g(x ⊗ y) = ∑ x ↞ y_{(1)} ⊗ y_{(0)}, \\ γ^{l}(x ⊗ y) = ∑ S^{−1}(x_{(0)[1]}x_{[1]}) ⊗ x_{(0)[0]}y \end{matrix}

for x, y ∈ X. So \overline{X} is indeed an H_{R}^{∗}-bigalois object, and hence an object in E(H_{R}). □

Now we are able to state our main theorem in this section. Denote by Gal(H_{R}) the subset of E(H_{R}) consisting of the isomorphism classes of objects in E(H_{R}) such that the objects are quantum commutative in Q^{H}.

Theorem 3.12. The set Gal(H_{R}) is a group.

Proof. First of all we show that Gal(H_{R}) is a sub-semigroup of E(H_{R}). It is obvious that I is a quantum commutative algebra and [I] ∈ Gal(H_{R}). Suppose that [X] and [Y] are two elements of Gal(H_{R}). We have to verify that X ∧ Y is quantum commutative. For simplicity, we will write x#y for an element ∑ x_{i}#y_{i} of X ∧ Y. Given x#y, a#b ∈ X ∧ Y, we compute ∑(x_{(0)}#y_{(0)})(y_{(1)}x_{(1)} ⇀ (a#b)).

∑(x_{(0)}#y_{(0)})(y_{(1)}x_{(1)} ⇀ (a#b)
= ∑(x_{(0)}#y_{(0)})(y_{(1)}x_{(1)} ▷_{2} a#y_{(2)}x_{(2)} · b)
= ∑ x_{(0)}(S(y_{(1)})y_{(2)}x_{(1)} ▷_{2} a)#y_{(0)}(y_{(3)}x_{(2)} · b)
= ∑ x_{(0)}(x_{(1)} ▷_{2} a)#y_{(0)}(y_{(1)} · (x_{(2)} · b))
= ∑ x_{(0)}(x_{(1)} · a)#y_{(0)}(y_{(1)} · (x_{(2)} ▷_{1} b))
= ∑ ax_{(0)}#(x_{(1)} ▷_{1} b)y
= (a#b)(x#y).

This means that X ∧ Y is quantum commutative as well.

Second we show that \overline{X} is quantum commutative if X is quantum commutative. Indeed, given \overline{x}, \overline{y} ∈ \overline{X}, we have

∑ \overline{y_{(0)}} ∘ (y_{(1)} ⇀ \overline{x}) = ∑ \overline{y_{(0)}} ∘ \overline{S(y_{(4)}) · x_{(0)})}R(y_{(6)} ⊗ x_{(2)})R(x_{(1)} ⊗ S(y_{(3)}))
R(y_{(5)} ⊗ S(y_{(2)}))u^{−1}(y_{(1)})
= ∑ \overline{S(y_{(6)}) · x_{(0)})y_{(0)}}R(S(y_{(5)})x_{(1)}y_{(7)} ⊗ y_{(1)})
R(y_{(9)} ⊗ x_{(3)})R(x_{(2)} ⊗ S(y_{(4)}))R(y_{(8)} ⊗ S(y_{(3)}))u^{−1}(y_{(2)})
= ∑ \overline{S(y_{(5)}) · x_{(0)})y_{(0)}}R(S(y_{(4)})x_{(1)}y_{(7)} ⊗ y_{(1)})R(y_{(8)} ⊗ x_{(3)})
R(x_{(2)}y_{(7)} ⊗ S(y_{(3)}))u^{−1}(y_{(2)})
= ∑ \overline{S(y_{(5)}) · x_{(0)})y_{(0)}}R(S(y_{(4)})x_{(1)}y_{(7)⊗y_{(1)}} ⊗ y_{(1)})R(y_{(8)} ⊗ x_{(3)})
R(S(x_{(2)}y_{(7)}) ⊗ y_{(2)}))u^{−1}(y_{(3)})
= ∑ \overline{S(y_{(4)}) · x_{(0)})y_{(0)}}R(S(y_{(3)}) ⊗ y_{(1)})R(y_{(5)} ⊗ x_{(1)})u^{−1}(y_{(2)})
= ∑ \overline{S(y_{(4)}) · x_{(0)})y_{(0)}}u^{−1}(y_{(3)})R(S^{−1}(y_{(2)}) ⊗ y_{(1)})R(y_{(5)} ⊗ x_{(1)})
= ∑ \overline{S(y_{(1)}) · x_{(0)})y_{(0)}}R(y_{(2)} ⊗ x_{(1)})
= ∑ \overline{y_{(0)}x_{(0)}}R(y_{(1)} ⊗ x_{(1)})
= \overline{x} ∘ \overline{y}.

Thus we have proved that [\overline{X}] is an element of Gal(H_{R}) if [X] ∈ Gal(H_{R}). Finally we show that [\overline{X}] is the inverse of [X] in Gal(H_{R}).

In order to show that X ∧ \overline{X} ≅ I in E(H_{R}), it is suffcient to construct a non-zero YD Hmodule algebra morphism from I to X ∧ \overline{X}. Since X is a right H_{R}^{∗}-Galois object we have the isomorphism:

β^{r} : X ⊗ X ⟶ X ⊗ H_{R}^{∗}.

Write ∑ U_{i}^{p} ⊗ V_{i}^{p} for the inverse image β^{r}^{−1}(1 ⊗ p) of an element p ∈ H_{R}^{∗}. We claim that ∑ U_{i}^{p} ⊗ \overline{V} _{i}^{p} is in X ∧ \overline{X} for any p ∈ H_{R}^{∗}. Indeed, if p ∈ H_{R}^{∗}, we have the formulas

\begin{matrix}β^{r}(∑ U_{i}^{p} ⊗ V_{i}^{p}) = ∑ U_{i}^{p(0)}V_{i}^{p} ⊗ U_{i}^{p(1)} = 1 ⊗ p, \\ γ^{r}(∑ U_{i}^{p} ⊗ \overline{V_{i}^{p}}) = ∑ U_{i}^{p}\overline{V_{i}^{p}}^{(0)} ⊗ \overline{V_{i}^{p}}^{(−1)} = 1 ⊗ p \end{matrix}

where ∑ U_{i}^{p}\overline{V_{i}^{p}}^{(0)} ⊗ \overline{V_{i}^{p}}^{(−1)} should be understood as γ^{r}(∑ U_{i}^{p} ⊗ \overline{V_{i}^{p}}) = ∑ U_{i}^{p}V_{i(0)[0]}^{p} ⊗ S(V_{i(0)[1]}^{p})Θ_{r}(V_{i(1)}^{p}).

Write ∑ X_{j}^{p} ⊗ Y_{j}^{p} for the element β^{l}^{−1}(p ⊗ 1) whenever p ∈ H_{R}^{∗}. Then we have

∑ X_{j}^{\overline{a}^{(1)}} ⊗ Y_{j}^{\overline{a}^{(1)}} \overline{a}^{(0)} = \overline{a} ⊗ 1  (24)

for any \overline{a} ∈ \overline{X}. Applying formula (24), we obatin

∑(ι ⊗ γ^{r})(1 ⊗ U_{i}^{p} ⊗ \overline{V_{i}^{p}}) = 1 ⊗ 1 ⊗ p
= ∑ U_{i}^{p}\overline{V_{i}^{p}}^{(0)} ⊗ 1 ⊗ \overline{V_{i}^{p}}^{−1}
= ∑ U_{i}^{p}X_{j}^{q} ⊗ Y_{j}^{q}\overline{V_{i}^{p}}^{(0)} ⊗ \overline{V_{i}^{p}}^{−1}
= ∑(ι ⊗ γ^{r})(U_{i}^{p}X_{j}^{q} ⊗ Y_{j}^{q} ⊗ \overline{V_{i}^{p}}^{(0)})

where q = \overline{V_{i}^{p}}^{(1)}. Since ι ⊗ γ^{r} is an isomorphism, we obtain

∑ 1 ⊗ U_{i}^{p} ⊗ \overline{V_{i}^{p}} = ∑ U_{i}^{p}X_{j}^{q} ⊗ Y_{j}^{q} ⊗ \overline{V_{i}^{p}}^{(0)}  (25)

where again q = \overline{V_{i}^{p}}^{(1)}. Now let β^{l} ⊗ ι act on (25), we get

∑ U_{i}^{p(−1)} ⊗ U_{i}^{p(0)} ⊗ \overline{V_{i}^{p}} = ∑ \overline{V_{i}^{p}}^{(1)} ⊗ U_{i}^{p} ⊗ \overline{V_{i}^{p}}^{(0)}.

It follows that

∑ U_{i}^{p} ◁− h#V_{i}^{p} = ∑ U_{i}^{p}#h −▷ V_{i}^{p},

for any h ∈ H_{R}. Next we show that the well-defined map

Ψ : I ⟶ X ∧ \overline{X}, Ψ(p) = β^{r}^{−1}(1 ⊗ p)

is an algebra map. In order to simplify the notations, we write a#\overline{b} and c#\overline{d} for Ψ(q) and Ψ(p) respectively if q and p are two elements of I. Since

∑ a^{(0)}b ⊗ b^{(1)} = ∑ a_{[0](0)}b ⊗ a_{[1]}S^{−1}(Θ_{r}(a_{[0](1)}) = 1 ⊗ p

is equivalent to

∑ a_{[0]}b_{(0)} ⊗ a_{[1]}Θ_{r}(b_{(1)}) = 1 ⊗ p,

It is suficient to show that

∑ x_{[0]}y_{(0)} ⊗ x_{[1]}Θ_{r}(y_{(1)}) = 1 ⊗ qp,

where

x#\overline{y} = (a#\overline{b})(c#\overline{d}) = ∑ ac_{(0)}#\overline{d_{(0)}b_{(0)})}R(d_{(1)}c_{(1)} ⊗ b_{(1)}).

Indeed, we have

∑ x_{[0]}y_{(0)} ⊗ x_{[1]}Θ_{r}(y_{(1)})
= ∑ a_{[0]}c_{(0)[0]}d_{(0)}b_{(0)} ⊗ a_{[1]}c_{(0)[1]}Θ_{r}(d_{(1)})Θ_{r}(b_{(1)})R(d_{(2)}c_{(1)} ⊗ b_{(2)}).

Applying the equations:

∑ c_{(0)[0]}d_{(0)} ⊗ c_{(0)[1]}Θ_{r}(d_{(1)}) ⊗ S^{−1}(Θ_{l}(d_{(2)}c_{(1)}))
= ∑ c_{[0](0)}d_{(0)} ⊗ R^{−1}(c_{[1]}Θ_{r}(d_{(2)}) ⊗ S^{−1}(Θ_{l}(c_{[0](1)}d_{(1)}))R
= 1 ⊗ (ι ⊗ S)(R^{−1}(p ⊗ 1)R)
= ∑ 1 ⊗ S(R^{1})pr^{1} ⊗ S(R^{2}r^{2})

where R = r, we obtain

∑ x_{[0]}y_{(0)} ⊗ x_{[1]}Θ_{r}(y_{(1)})
= ∑ a_{[0]}b_{(0)} ⊗ a_{[1]}S(R^{1})pr^{1}Θ_{r}(b_{(1)}) < S(R^{2}r^{2}), b_{(2)} >
= ∑ a_{[0]}b_{(0)} ⊗ a_{[1]}(S(R^{1}) · p)Θ_{r}(b_{(1)}) < S(R^{2}, b_{(2)} >
= ∑ a_{[0]}b_{(0)} ⊗ a_{[1]}p_{(0)}Θ_{r}(b_{(1)})R(S(p_{(1)}) ⊗ S(b_{(2)}))
= ∑ a_{[0]}b_{(0)} ⊗ a_{[1]}p_{(0)}(p_{(1)} · Θ_{r}(b_{(1)}))
= ∑ a_{[0]}b_{(0)} ⊗ a_{[1]}Θ_{r}(b_{(1)})p
= 1 ⊗ qp.

So Ψ is inded an algebra map.

Finally, we show that Ψ is a YD H-module map so that it is an H_{R}-bimodule map (or a H_{R}^{∗}-bicomodule map) as well. Given p ∈ I and h ∈ H_{R}, we have

∑ U_{i[0]}^{p}V_{i(0)}^{p} ⊗ U_{i[1]}^{p}Θ_{r}(V_{i(1)}^{p}) = 1 ⊗ p.

It implies that

h · Ψ(p) = ∑ h_{(1)} · U_{i}^{p}#h_{(2)} ▷_{1} \overline{V_{i}^{p}}
= ∑ U_{i[0]}^{p}#\overline{V_{i(0)}^{p}} < h, U_{i[1]}^{p}Θ_{r}(V_{i(1)}^{p}) >
= ∑ Up_{(1)}_{i}#\overline{V p_{(1)}_{i}} < h, p_{(2)}) >
= Ψ(h · p).

To show that Ψ is H^{op}-colinear, we verify that Ψ is left H^{∗}-linear. Indeed, if p ∈ H^{∗} and q ∈ I, we have

Ψ(p · q) = ∑ Ψ(p_{(2)}qS^{−1}(p_{(1)}))
= ∑ Ψ(p_{(2)})Ψ(q)Ψ(S^{−1}(p_{(1)}))
= ∑ Ψ(q)_{(0)}(Ψ(q)_{(1)} · Ψ(p_{(2)}))Ψ(S^{−1}(p_{(1)}))
= ∑ Ψ(q)_{(0)}Ψ(p_{(2)}))Ψ(S^{−1}(p_{(1)})) < Ψ(q)_{(1)}, p_{(3)} >
= ∑ Ψ(q)_{(0)} < Ψ(q)_{(1)}, p >,

where we used the facts that X ∧ \overline{X} is quantum commutative, that Ψ is an algebra map and that Ψ is H-linear. So Ψ is indeed a YD H-module algebra map, and I ≅ X ∧ \overline{X} in E(H_{R}). This proved that [\overline{X}] is a right inverse of [X] in Gal(H_{R}). Since any element of Gal(H_{R}) has a right inverse element in Gal(H_{R}), Gal(H_{R}) is a group. □

4 The exact sequence

In this section, we investigate the R-Azumaya algebras which are Galois extensions of the coinvariants, and establish a group morphism from BC(k, H, R) to the group Gal(H_{R}) constructed in the previous section. The main result will be the exact sequence (28). In the sequel, we will write

M_{0} = {m ∈ M | ∑ m_{(0)} ⊗ m_{(1)} = m ⊗ 1}

for the coinvariant k-submodule of a right H-comodule M, in order to make a difference between H_{R}^{∗}-coinvariants and H-coinvariants. We start with a special elementary R-Azumaya algebra.

Lemma 4.1. Let M = H^{op} be the right H^{op}-comodule, and let A be the induced R-Azumaya algebra End(M). Then A ≅ H^{∗}^{op}#H^{op}, where the left H^{op}-action on H^{∗}^{op} is given by h · p = ∑ p_{(1)} < p_{(2)}, S^{−1}(h) >= S^{−1}(h) ⇀ p, whenever h ∈ H^{op} and p ∈ H^{∗}^{op}.

Proof. Let π be the representation map of the regular left H^{op}-module H^{op}:

π : H^{op} ⟶ End(M).

We claim that π is a right H^{op}-colinear algebra map. Indeed, by definition, the right H^{op}- comodule structure of π(h) for h ∈ H^{op} is given by

ρ(π(h))(x ⊗ 1) = ∑(π(h)(x_{(0)}))_{(0)} ⊗ S^{−1}(x_{(1)})(π(h)(x_{(0)}))_{(1)}
= ∑(x_{(1)}h)_{(1)} ⊗ S^{−1}(x_{(2)})(x_{(1)}h)_{(2)}
= ∑ x_{(1)}h_{(1)} ⊗ S^{−1}(x_{(3)})x_{(2)}h_{(2)}
= ∑ π(h_{(1)})(x) ⊗ h_{(2)}

for any x ∈ M. It follows that ρ(π(h)) = ∑ π(h_{(1)}) ⊗ h_{(2)}. So π is a right H^{op}-colinear algebra map. This fact implies that the right H^{op}-comodule algebra A is a smash product A_{0}#H^{op}, where A_{0} is the coinvariant subalgebra of A.

Now we show that A_{0} is isomorphic to H^{∗}^{op}. It is obvious that

A_{0} = End^{H}(M)

the subalgebra of all H^{op}-colinear endomorphisms of M. Since M is H^{op} by our choice, we know that the linear map

λ : H^{∗}^{op} ⟶ End^{H}(H^{op}), λ(p)(x) = ∑ p(x_{(1)})x_{(2)}

for p ∈ H^{∗}^{op} and x ∈ H^{op}, is an algebra isomorphism.

Finally, if p ∈ H^{∗}^{op}, h ∈ H^{op} and x ∈ M, then

π(h)(λ(p)(x)) = ∑ x_{(2)}h < p, x_{(2)} >
= ∑(S^{−1}(h_{(1)}) ⇀ p)(π(h_{(2)})(x)).

It follows that the action of H^{op} on H^{∗}^{op} is

h · p = S^{−1}(h) ⇀ p = ∑ p_{(1)} < p_{(2)}, S^{−1}(h) >

for h ∈ H^{op} and p ∈ H^{∗}^{op}. □

Corollary 4.2. Any element of BC(k, H, R) can be represented by an R-Azumaya algebra which is a smash product.

Proof. Let [A] be an element of BC(k, H, R). Since End(H^{op}) represents the unit of BC(k, H, R), we have [A#End(H^{op})] = [A]. Now the composite algebra map

H^{op} ⟶^{λ} End(H^{op}) ↪ A#End(H^{op})

is still H^{op}-colinear. It follows that A#End(H^{op}) is a smash product algebra B#H^{op} where B = (A#End(H^{op}))_{0}. □

Since any smash product algebra is a Galois extension of its coinvariants, we have that any element of BC(k, H, R) can be represented by an R-Azumaya algebra which is a H^{op}-Galois extension of its coinvariants.

Lemma 4.3. Let A be an R-Azumaya algebra. If A is an H^{op}-Galois extension of A_{0}, then \overline{A} is a H^{op}-Galois extension of A_{0}^{op}.

Proof. Since A/A_{0} is H^{op}-Galois, we have the canonical isomorphism:

β′_{A} : A ⊗_{A_{0}} A ⟶ A ⊗ H^{op}, a ⊗ b ↦ ∑ a_{(0)}b ⊗ b_{(1)}.

Since the flip map τ is an linear isomorphism from A ⊗_{A_{0}} A to \overline{A} ⊗_{A_{0}^{op}} \overline{A}, β′_{A} gives an isomorphism

η : \overline{A} ⊗_{A_{0}^{op}} \overline{A} ⟶ \overline{A} ⊗ H^{op}, η(\overline{a} ⊗ \overline{b}) = ∑ \overline{a_{(0)}b} ⊗ a_{(1)}.

Define a linear map:

ξ : \overline{A} ⊗ H^{op} ⟶ A ⊗ H^{op}

sending an element a ⊗ h to ∑ \overline{a_{(0)}} ⊗ h_{(3)}R(h_{(2)} ⊗ a_{(1)}S(h_{(1)})). We show that ξ is a linear isomorphism. As remarked in the previous section, we may view R as an element ∑ R^{1} ⊗ R^{2} in H^{∗} ⊗ H^{∗} which is a QT structure of H^{∗}. Then the element u = ∑ R^{2}S^{−1}(R^{1}) is the Casimir element of H^{∗} which is invertible. Thus we may rewrite the linear map ξ as the following composite map:

A ⊗ H^{op} ⟶^{ι⊗_{u}} A ⊗ H^{op} ⟶^{σ} A ⊗ H^{op}

where ũ is defined by ũ(h) = h ↼ u = ∑ h_{(2)}u(h_{(1)}), and σ is defined by

σ(a ⊗ h) = ∑ h_{(1)} · a ⊗ h_{(2)} = ∑ a_{(0)} ⊗ h_{(2)}R(h_{(1)} ⊗ a_{(1)})

for all a ∈ A and h ∈ H^{op}. Since ũ and σ are linear isomorphisms, we have that ξ is an isomorphism. Now we show that the canonical map β\overline{A} is a composite map of isomorphisms. We have

ξ ∘ β′_{A} ∘ τ(\overline{a} ⊗ \overline{b}) = ∑ ξ(∑ \overline{b_{(0)}a} ⊗ b_{(1)})
= ∑ \overline{b_{(0)}a_{(0)}} ⊗ b_{(4)}R(b_{(3)} ⊗ a_{(1)}b_{(1)}S(b_{(2)}))
= ∑ \overline{b_{(0)}a_{(0)}} ⊗ b_{(2)}R(b_{(1)} ⊗ a_{(1)})
= ∑ \overline{a}\overline{b_{(0)}} ⊗ b_{(1)}
= β\overline{A}(\overline{a} ⊗ \overline{b})

for any \overline{a} ∈ \overline{A} and \overline{b} ∈ \overline{B}. So β\overline{A} is an isomorphism, and \overline{A}/A_{0}^{op} is an H^{op}-Galois extension.□

In the sequel, an R-Azumaya algebra A is said to be Galois if it is a right H^{op}-Galois extension of its coinvariant subalgebra A_{0}. Let A be a Galois R-Azumaya algebra. Denote by π(A) the centralizer subalgebra C_{A}(A_{0}) of A_{0} in A. A_{0} is clearly an R-subalgebra of A. The Miyashita-Ulbrich action of H on π(A) is given by

h ⇀ a = ∑ X_{i}^{h}aY_{i}^{h},  (26)

where ∑ X_{i}^{h} ⊗ Y_{i}^{h} = β^{−1}(1 ⊗ h), for h ∈ H. It is well-known (e.g. see [29, 7]) that π(A) together with the action (26) is a new YD H-module algebra. Moreover, π(A) is quantum commutative in the sense of (22). In fact, in (22) the element b can be any element in A. By Corollary 2.8, π(A) is an H_{R}-bimodule algebra, or A is an H_{R}^{∗}-bicomodule algebra.

Lemma 4.4. Let A be a Galois R-Azumaya algebra. Then π(A)/k is an H_{R}^{∗}-biextension.

Proof. Given a ∈ π(A)_{⋄}, then by Lemma 2.5, we have

h ⇀ a = h ▷_{1} a

for any h ∈ H. Then for any element b ∈ A, we have

ab = ∑ b_{(0)}(b_{(1)} ⇀ a)
= ∑ b_{(0)}(b_{(1)} ▷_{1} a).

This means that a is an element in the left H-center of A which is trivial. So π(A)_{⋄} = k. Similarly, if a is in _{⋄}π(A), then we have

ab = ∑ b_{(0)}(b_{(1)} ⇀ a)
= ∑ b_{(0)}(b_{(1)} ▷_{2} a)
= ∑ b_{(0)}a_{(0)}R^{−1}(a_{(1)} ⊗ b_{(1)}),

for any b ∈ A. This implies that

∑ a_{(0)}b_{(0)}R(a_{(1)} ⊗ b_{(1)}) = ba

for any b ∈ A. So a is in the right H-center of A which is trivial as well. It follows that π(A) is an H_{R}^{∗}-biextension. □

Recall from [29] that when a Galois H^{op}-comodule algebra A is an Azumaya algebra, the centralizer π(A) is a right H^{∗}-Galois extension of k with respect to the Miyashita-Ulbrich action (26). This is no loger the case when A is an R-Azumaya algebra. However, we will see that A would be an H_{R}^{∗}-Galois object, instead of an H^{∗}-Galois object.

Proposition 4.5. Let A be a Galois R-Azumaya algebra, and π(A) be as above. Then π(A) is an object in Gal(H_{R}).

Proof. Let F be the canonical H-linear algebra isomorphism:

A#\overline{A} ⟶ End(A).

It is easy to see that F induces an algebra isomorphism:

π(A)#\overline{π(A)} ⟶ End_{A_{0}−A_{0}}(A)

where the latter is the subalgebra of all A_{0}-biendomorphisms of A. Since A/A_{0} is H^{op}-Galois, we have a linear isomorphism:

δ : Hom(H, π(A)) ⟶ End_{A_{0}−A_{0}}(A)

given by δ(f)(a) = ∑ a_{(0)}f(a_{(1)}) (see [12, 3.2]). Define a linear map α as follows:

α : π(A) ⊗ π(A) ⟶ π(A) ⊗ H^{∗}

by sending an element a ⊗ b to ∑ a_{[0]}b_{(0)} ⊗ a_{[1]}Θ_{r}(b_{(1)}). One may take a while to check that α and δ fit the following commutative diagram:

[figure: π(A) ⊗ π(A); F; End_{A_{0}−A_{0}}(A); π(A) ⊗ H^{∗}; α; ≅; Hom(H, π(A)).; δ]

Note that here we view F as a linear isomorphism from π(A) ⊗ π(A) to End_{A_{0}−A_{0}}(A). It follows from the above commutative diagram that α is a linear isomorphism. It is evident that the canonical Galois linear map β^{r} is now the composite isomorphism:

[figure: π(A) ⊗ π(A); α; π(A) ⊗ H^{∗}; η; π(A) ⊗ H^{∗}]

where η is given by η(a ⊗ p) = ∑ a_{(0)} ⊗ pS^{−1}(Θ_{r}(a_{(1)})) whenever a ∈ π(A) and p ∈ H^{∗}. So we obtain that π(A) is a right H_{R}^{∗}-Galois object.

Similarly, one has the commutative diagram

[figure: π(A) ⊗ π(A); G; End_{A_{0}−A_{0}}(A)^{op}; π(A) ⊗ H^{∗}; α′; ∼; Hom(H, π(A)); δ]

where α′ is given by

α′(a ⊗ b) = ∑ a_{(0)[0]}b ⊗ a_{(0)[1]}Θ_{l}(a_{(1)})

for a, b ∈ π(A). Let ζ and η′ be the linear automorphisms of π(A) ⊗ H^{∗} given by

ζ(a ⊗ p) = ∑ a_{[0]} ⊗ S(a_{[1]})p
η′(a ⊗ p) = ∑ a_{(0)} ⊗ pS(Θ_{l}(a_{(1)}))

for any a ∈ A and p ∈ H^{∗}. We have

η′ζα′(a ⊗ b) = ∑ ab_{[0](0)} ⊗ S(Θ_{l}(b_{[0](1)})b_{[1]})

for any a, b ∈ π(A). It follows that the Galois linear map β^{l} is the composite isomorphism (S^{−1} ⊗ ι)τη′ζα′, where τ is the flip map. So π(A) is a left H_{R}^{∗}-Galois object. This completes the proof. □

Now we are ready to show that π induces a group morphism from the Brauer group BC(k, H, R) to the group Gal(H_{R}).

Proposition 4.6. Let A and B be two Galois R-Azumaya algebras. Then we have π(A#B) = π(A) ∧ π(B).

Proof. It is obvious that π(A#B) ⊆ π(A)#π(B) because A_{0} ⊗ B_{0} = A_{0}#B_{0} ⊆ (A#B)_{0}. For an element h ∈ H^{op}, we let

∑ β_{A}^{−1}(1 ⊗ h) = X_{j}^{h} ⊗ Y_{j}^{h} ∈ A ⊗_{A_{0}} A
∑ β_{B}^{−1}(1 ⊗ h) = U_{i}^{h} ⊗ V_{i}^{h} ∈ B ⊗_{B_{0}} B

Then we have

∑(X_{j}^{h}#1) ⊗ (Y_{j}^{h}#1) = β_{A#B}^{−1}(1 ⊗ h) = ∑(1#U_{i}^{h}) ⊗ (1#V_{i}^{h}).

This implies that the Miyashita-Ulbirich action of H on π(A#B) can be written in two folds:

h ⇀ a#b = ∑(X_{j}^{h}#1)(a#b)(Y_{j}^{h}#1)
= ∑(1#U_{i}^{h})(a#b)(1#V_{i}^{h})

where a#b should be read as a sum of elements in A#B. Precisely, we have for a#b ∈ π(A#B),

h ⇀ (a#b) = ∑(X_{j}^{h}#1)(a#b)(Y_{j}^{h}#1)
= ∑ X_{j}^{h}aY_{j(0)}^{h}#Y_{j(1)}^{h} ▷_{1} b
= ∑ h_{(1)} ⇀ a#h_{(2)} ▷_{1} b.

On the other hand, we have

h ⇀ (a#b) = ∑(1#U_{i}^{h})(a#b)(1#V_{i}^{h})
= ∑ a_{(0)}#(a_{(1)} ▷_{1} U_{i}^{h})bV_{i}^{h}
= ∑ a_{(0)}#U_{i(0)}^{h}bV_{i}^{h}R(a_{(1)} ⊗ U_{i(1)}^{h})
= ∑ a_{(0)}R(a_{(1)} ⊗ S^{−1}(h_{(1)}))#h_{(2)} ⇀ b
= ∑ h_{(1)} ▷_{2} a#h_{(2)} ⇀ b.

This means that a#b is sitting in π(A) ∧ π(B) by Lemma 3.3. It follows that π(A#B) ⊆ π(A) ∧ π(B).

Conversely, if a#b is an element of π(A)∧π(B), we show that a#b ∈ π(A#B). Indeed, given x#y ∈ (A#B)_{0}, we have

∑ x_{(0)}#y_{0} ⊗ y_{(1)}x_{(1)} = x ⊗ y ⊗ 1,

or

∑ x_{(0)}#y ⊗ x_{(1)} = ∑ x#y_{(0)} ⊗ S(y_{(1)}).

These two formulas lead to the equations:

(a#b)(u#v) = ∑ au_{(0)}#(x_{(1)} ▷_{1} b)v

= ∑ x_{(0)}(x_{(1)} ⇀ a)#(x_{(2)} ▷_{1} b)v
= ∑ x_{(0)}(x_{(1)} ▷_{2} a)#(x_{(2)} ⇀ b)y
= ∑ x_{(0)}(x_{(1)} ▷_{2} a)#y_{(0)}(y_{(1)}x_{(2)} ⇀ b)
= ∑ x_{(0)}(x_{(1)} ▷_{2} a)#yb
= ∑ x(S(y_{(1)}) ▷_{2} a)#y_{(0)}b
= ∑ xa_{(0)}#(a_{(1)} ▷_{1} y)b
= (x#y)(a#b),

where we used the quantum commutativity (22). This implies that a#b ∈ π(A#B). So π(A) ∧ π(B) ⊆ π(A#B), and hence they are equal. □

Lemma 4.7. Let M be a finite right H^{op}-comodule, and A = End(M) be the elementary R-Azumaya algebra. If A is a Galois R-Azumaya algebra, then π(A) ≅ I.

Proof. Since M is a right H^{op}-comodule, we may view M as a left H^{∗}-module. The representation map

λ : H^{∗} ⟶ A, λ(p)(m) = p · m = ∑ m_{(0)}p(m_{(1)})

sends H^{∗} into the subalgebra π(A) because A_{0} = End_{H^{∗}}(A). Thus λ gives an algebra morphism from I (see Lemma 3.6) to π(A). If we can prove that λ is a YD H-module map, then λ becomes an H_{R}-bimodule map, and hence an isomorphism between the two Galois objects in Gal(H_{R}). By definition (16), λ is right H^{op}-colinear. We show that λ is left H-linear as well.

To show that λ(h · p) = h ⇀ λ(p) for h ∈ H and p ∈ I, it is sufficient (or equivalent) to show that

λ(p)f = ∑ f_{(0)}λ(f_{(1)} · p)

for any f ∈ A. Given m ∈ M and f ∈ A, we have

∑ f_{(0)}λ(f_{(1)} · p)(m) = f_{(0)}(m_{(0)}) < p, m_{(1)}f_{(1)} > .

Since

∑ f_{(0)}(m) ⊗ f_{(1)} = ∑ f(m_{(0)})_{(0)} ⊗ S(m_{(1)})f(m_{(0)})_{(1)},

we have

∑ f_{(0)}λ(f_{(1)} · p)(m) = ∑ f_{(0)}(m_{(0)}) < p, m_{(1)}f_{(1)} >
= ∑ f(m)_{(0)} < p, f(m)_{(1)} >
= p · f(m)
= λ(p)f(m).

This proves that λ is a YD H-module algebra map, and hence an isomorphism because I and π(A) are H_{R}^{∗}-bigalois objects. □

Lemma 4.8. π induces a group morphism π̃ from BC(k, H, R) to Gal(H_{R}), where π̃([A]) = [π(A)] and A is a Galois R-Azumaya algebra representing the class [A] in BC(k, H, R).

Proof. First we show that π̃ is well-defined. Suppose that A abd B are two Galois R-Azumaya algebras representing the same class in BC(k, H, R). Then there are two finite right H^{op}-modules M, N such that

A#End(M) ≅ B#End(N).

Let H^{op} be the regular right H^{op}-comodule. By Lemma 4.1, End(H^{op}) is a Galois R-Azumaya algebra. Since End(M)#End(H^{op}) ≅ End(M ⊗ H^{op}) is still a Galois R-Azumaya algebra, by Lemma 4.7 we have

π(End(M)#End(H^{op})) ≅ π(End(M ⊗ H^{op})) ≅ I.

This implies that

π(A) ≅ π(A) ∧ I
≅ π(A#End(M)#End(H^{op}))
≅ π(B#End(N)#End(H^{op}))
≅ π(B) ∧ I
≅ π(B).

So we obtain that π̃([A]) = π̃([B]), and π̃ is well-defined. □

In order to figure out the kernel of π̃, we need two more preparations. Recall from [2] that an action of a Hopf algebra H on an algebra A is called an inner action if there is an invertible element u in the convolution algebra Hom(H, A) such that

h · a = ∑ u(h_{(1)})au^{−1}(h_{(2)})

for any a ∈ A and h ∈ H. If in addition, u is an algebra map, then the action of H is called a strongly inner action.

Lemma 4.9. Let A be a Galois R-Azumaya algebra such that π(A) ≅ I. Then the action of H^{∗}^{cop} (or the coaction of H^{op}) on A is strongly inner.

Proof. By assumption, there is a YD H-module algebra isomorphism

ψ : I ⟶ π(A).

Thus the action and the coaction of H on π(A) are determined by the corresponding action and coactions of H on I through ψ. Namely, we have:

\begin{matrix}h ⇀ ψ(p) = ∑ ψ(p_{(1)}) < p_{(2)}, h >, \\ h^{∗} · ψ(p) = ∑ ψ(h_{(2)}^{∗})ψ(p)ψ(S^{−1}(h_{(1)}^{∗}) \end{matrix}  (27)

where p ∈ I, h^{∗} ∈ H^{∗}^{op} and h ∈ H. In particular, the H^{op}-coaction on π(A) is strongly inner. We show that this inner action extends to the inner action on A. Indeed, given a ∈ A and h^{∗} ∈ H^{∗}^{op}, we have

h^{∗} · a = ∑ a_{(0)} < h^{∗}, a_{(1)} >
= ∑ ψ(h_{(3)}^{∗}S^{−1}(h_{(2)}^{∗}))a_{(0)} < h_{(1)}^{∗}, a_{(1)} >
= ∑ ψ(h_{(3)}^{∗})a_{(0)}(a_{(1)} ⇀ ψ(S^{−1}(h_{(2)}^{∗}))) < h_{(1)}^{∗}, a_{(2)} >
= ∑ ψ(h_{(f)}^{∗})a_{(0)}ψ(S^{−1}(h_{(3)}^{∗})) < S^{−1}(h_{(2)}^{∗}), a_{(1)} >< h_{(1)}^{∗}, a_{(2)} >
= ∑ ψ(h_{(2)}^{∗})aψ(S^{−1}(h_{(1)}^{∗}).

This means that the algebra map ψ : H^{∗}^{op} ⟶ π(A) ↪ A induces a strongly inner action of H^{∗}^{op} on A. □

Lemma 4.10. Let A be a Galois R-Azumaya algebra such that π(A) ≅ I, and B any R-Azumaya algebra. Then as algebras (1). A ⊗ B ≅ A#B, (2). \overline{A} ≅ A^{op}, (3). A is an Azumaya algebra.

Proof. (1). Let ψ : I ⟶ π(A) be an isomorphism of the two Galois objects in E(H_{R}). We define a linear map ξ as follows:

ξ : A ⊗ B ⟶ A#B, ξ(a ⊗ b) = ∑ aψ(Θ_{r}(S(b_{(1)})))#b_{(0)}.

It is easy to see that ξ is an isomorphism. We verify that ξ is an algebra map as well. Indeed, If a, c ∈ A and b, d ∈ B, then

ξ((a ⊗ b)(c ⊗ d)) = ∑ acψ(Θ_{r}(S(d_{(1)}b_{(1)})))#b_{(0)}d_{(0)}
= ∑ aψ(Θ_{r}(S(b_{(3)})))ψ(Θ_{r}(b_{(2)}))cψ(Θ_{r}(S(d_{(1)})))
ψ(Θ_{r}(S(b_{(1)})))#b_{(0)}d_{(0)}
= ∑ aψ(Θ_{r}(S(b_{(2)})))[Θ_{r}(b_{(1)}) · (cψ(Θ_{r}(d_{(1)})))]#b_{(0)}d_{(0)}
= ∑(aψ(Θ_{r}(S(b_{(1)}))#b_{(0)})(cψ(Θ_{r}(d_{(1)})))#d_{(0)})
= ξ(a ⊗ b)ξ(c ⊗ d).

(2) and (3). The proof of Part (1) shows that there is a linear map ν : \overline{A} ⟶ A^{op} given by

ν(\overline{a}) = ∑ ψ(Θ_{r}(S(a_{(1)})))a_{(0)}.

We show that ν is the desired algebra isomorphism.

First ν is an algebra map. Given \overline{a}, \overline{b} ∈ \overline{A}, we have

ν(\overline{a}\overline{b}) = ∑ ν(\overline{b_{(0)}a_{(0)}})R(b_{(1)} ⊗ a_{(1)})
= ∑ ψ(Θ_{r}(S(a_{(1)}b_{(1)})))b_{(0)}a_{(0)}R(b_{(2)} ⊗ a_{(2)})
= ∑ ψ(Θ_{r}(S(b_{(2)}a_{(2)})))b_{(0)}a_{(0)}R(b_{(1)} ⊗ a_{(1)})
= ∑ ψ(Θ_{r}(S(b_{(1)}a_{(2)})))(Θ_{r}(a_{(1)}) · b)a_{(0)}
= ∑ ψ(Θ_{r}(S(b_{(1)}a_{(3)})))ψ(Θ_{r}(a_{(2)}))bψ(Θ_{r}(S(a_{(1)})))a_{(0)}
= ∑ ψ(Θ_{r}(S(b_{(1)})))b_{(0)}ψ(Θ_{r}(S(a_{(1)})))a_{(0)}
= ν(\overline{a}) ∘ ν(\overline{b}).

Now one may easily check that the following diagram commutes:

[figure: A ⊗ \overline{A}; ξ; A#\overline{A}; F; End(A)]
[figure: ι ⊗ ν; can]

A ⊗ A^{op} Since F ∘ ξ is an isomorphism, we obtain that ι ⊗ ν is injective and can is surjective. Since all the algebras involved are finite, ν and can are isomorphisms. So A is an Azumaya algebra and \overline{A} ≅ A^{op} as algebras. □

Theorem 4.11. We have an exact sequence of group morphisms:

[figure: 1; Br(k); ι; BC(k, H, R); π̃; Gal(H_{R}).]  (28)

Proof. Suppose that A, B are two Galois R-Azumaya algebras such that [A], [B] ∈ Ker(π̃). By Lemma 4.11, A and B are Azumaya algebras, and A#B ≅ A⊗B. This implies that there is a gropup morphism

ζ : Ker(π̃) ⟶ Br(k), ζ([A]) = [A]

by forgetting the H-structures on A, where [A] ∈ Ker(π̃) is represented by a Galois R-Azumaya algebra. It is evident that ζ ∘ ι = id, the identity map on Br(k). If we can show that ζ is also injective, then Ker(π̃) ≅ Br(k). Indeed, if ζ([A]) = 1 ∈ Br(k), then there is a finite k-module M such that A ≅ End(M) as an algebra. By Lemma 4.9, the coaction of H^{∗}^{op} on A is strongly inner. So there is an H^{∗}^{op}-coaction on M such that M is a right H^{∗}^{op}- comodule and A ≅ End(M) as H^{∗}^{op}-comodule algebra. This implies that [A] = [End(M)] = 1 in BC(k, H, R). It follows that the sequence (28) is exact. □

Note that the exact sequence (28) indicates that the factor group BC(k, H, R)/Br(k) is completely determined by the H_{R}^{∗}-bigalois objects. In particular, when k is an algebraic field, BC(k, H, R) is a subgroup of Gal(H_{R}).

5 Examples

In this section, we let k be a field. Let H_{4} be the Sweedler 4-dimensional Hopf algebra over k. That is, H_{4} is generated by two elements g and h satisfying

g^{2} = 1, h^{2} = 0, gh + hg = 0.

The comultiplication, the counit and the antipode are give as follows:

\begin{matrix}Δ(g) = g ⊗ g, & Δ(h) = 1 ⊗ h + h ⊗ g, \\ ε(g) = 1, & ε(h) = 0, \\ S(g) = g, & S(h) = gh. \end{matrix}

There is a family of CQT structures R_{t} on H_{4} parameterized by t ∈ k as follows:

R_{t} 1 g h gh
1 1 1 0 0
g 1 -1 0 0
h 0 0 t -t
gh 0 0 t t

It is not hard to check that the Hopf algebra morphisms Θ_{l} and Θ_{r} induced by R_{t} are as follows:

\begin{matrix}Θ_{l} : & H_{4}^{cop} ⟶ H_{4}^{∗}, & Θ_{l}(g) =, \overline{1} − \overline{g}, & Θ_{l}(h) = t(\overline{h} − \overline{gh}), \\ Θ_{r} : & H_{4}^{op} ⟶ H_{4}^{∗}, & Θ_{r}(g) = \overline{1} − \overline{g}, & Θ_{r}(h) = t(\overline{h} + \overline{gh}). \end{matrix}

When t is non-zero, Θ_{l} and Θ_{r} are isomorphisms, so that H_{4} is a self-dual Hopf algebra.

Then H_{R_{0}} is nothing else but the H_{4} itself. When t is not zero, we have that H_{R_{t}} is a 4-dimensional algebra generated by two elements u and v such that u and v satisfy the relations:

u^{2} = 1, uv + vu = 0, v^{2} = t(1 − u).

The double algebra D[H_{4}] with respect to R_{t} is generated by four elements, g_{1}, g_{2}, h_{1} and h_{2} such that

\begin{matrix}g_{i}^{2} = 1, h_{i}^{2} = 0, & g_{i}h_{j} + h_{j}g_{i} = 0, \\ g_{1}g_{2} = g_{2}g_{1}, & h_{1}h_{2} + h_{2}h_{1} = t(1 − g_{1}g_{2}). \end{matrix}

The comultiplication of D[H_{4}] is easy because the Hopf subalgebras generated by g_{i}, h_{i}, i = 1, 2, are isomorphic to H_{4}. Thus the algebra embedding ϕ reads as follows:

H_{R_{t}} ⟶ D[H_{4}], ϕ(u) = g_{1}g_{2}, ϕ(v) = g_{1}(h_{1} + h_{2}).

Let us first consider the triangular case (H_{4}, R_{0}). In this case, We have an exact and split sequence cf.[33]:

1 ⟶ (k, +) ⟶ BC(k, H, R_{0}) ⟶ BW(k, H) ⟶ 1.  (29)

where (k, +) is the addition group which is isomorphic to the group of H_{4}-bi-Galois objects [22].

On the other hand, the Brauer-Wall group BW(k) is BC(k, kZ_{2}, R′), where kZ_{2} is the sub-Hopf algebra of H_{4} generated by the group-like element g ∈ H_{4}, and R′ is the restriction of R_{0} to kZ_{2}. The following exact squence is well-known:

1 ⟶ Br(k) ⟶ BC(k, kZ_{2}) ⟶ Q_{2}(k) ⟶ 1,

where Q_{2}(k) is the group of graded quadratic extensions of k, which is nothing but Gal(Z_{R′}) where Z_{R′} ≅ kZ_{2} is the deformation of (kZ_{2}, R′). It follows that

Gal(H_{R_{0}}) = (k, +) × Q_{2}(k)

and we have an exact sequence:

1 ⟶ Br(k) ⟶ BC(H_{4}, R_{0}) ⟶ (k, +) × Q_{2}(k) ⟶ 1.

to be continued · · ·.

References

[1] M. Beattie, A direct sum decomposition for the Brauer group of H-module algebras, J. Alg. 43(1976), 686-693.

[2] R.J. Blattner, M. Cohen, and S. Montgomery, Crossed products and inner actions of Hopf algebras, Trans. AMS 298 (1986), 672-711.

[3] S. Caenepeel, Computing the Brauer-Long group of a Hopf algebra I: the cohomological theory, Israel J. Math. 72(1990), 38-83.

[4] S. Caenepeel, Computing the Brauer-Long group of a Hopf algebra II: the Skolem-Noether theory, J. Pure Appl. Alg. 84(1993),107-144.

[5] S. Caenepeel, The Brauer-Long group revisited: the multiplication rules, preprint 1998.

[6] S. Caenepeel and M. Beattie, A cohomological approach to the Brauer-Long group and the groups of Galois extensions and strongly graded rings, Trans. Amer. Math. Soc. 324(1991), 747-775.

[7] S. Caenepeel, F.Van Oystaeyen and Y.H.Zhang, Quantum Yang-Baxter module algebras, K-Theory 8(1994), 231-255.

[8] S. Caenepeel, F.Van Oystaeyen and Y.H.Zhang, The Brauer group of Yetter-Drinfel’d module algebras, Trans. Amer. Math. Soc. 349(1997), 3737-3771.

[9] L.N. Childs, G. Garfinkel and M. Orzech, The Brauer group of graded Azumaya algebras, Trans. Amer. Math. Soc. 175(1973), 299-326.

[10] L.N. Childs, The Brauer group of graded Azumaya algebras II: graded Galois extensions, Trans. Amer. Math. Soc. 204(1975), 137-160.

[11] A.P. Deegan, A subgroup of the generalized Brauer group of Γ-Azumaya algebras, J.London Math. Soc. 2(1981), 223-240.

[12] Y.Doi and M.Takeuchi, Quaternion algebras and Hopf crossed products, Comm. Alg. 23(9)(1995), 3291-3325.

[13] V.G. Drinfel’d, Quantum groups, Proc. of the Int. Congress of Math., Berkeley, Ca. (1987), 798-819.

[14] M. Koppinen, A Skolem-Noether theorem for Hopf algebra measurings, Arch. Math. 13(1981), 353-361.

[15] M.A. Knus and M. Ojanguren, Théorie de la descente et algèbres d’Azumaya, Lecture Notes in Math. 389, Springer-Verlag, Berlin, 1974.

[16] H.K.Kreimer and M.Takeuchi, Hopf algebras and Galois extensions, Indiana Univ. Math. J. 30(1981), 675-692.

[17] F.W. Long, The Brauer group of dimodule algebras, J. Alg. 31(1974), 559-601.

[18] L.A. Lame, D.E. Radford, Algebraic aspects of the quantum Yang-Baxter equation, J. Alg. 154(1992), 228-288.

[19] S. Majid, Quantum groups, Cambridge press, 199?

[20] S. Majid, Doubles of quasitriangular Hopf algebras, Comm. Alg. 19(1991), 3061-3073.

[21] S. Montgomery, Hopf algebras and their actions on rings, CBMS, 82(1992).

[22] P. Schauenburg, Bigalois objects over Taft algebras, preprint.

[23] H-J, Schneider, Pricipal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72(1990), 167-195.

[24] H-J, Schneider, Representation theory of Hopf Galois extensions, Israel J. Math. 72(1990), 196-231.

[25] M. Orzech, Brauer groups of graded algebras, Lecture Notes in Math. 549, Springer-Verlag, Berlin, 1976, 134-147.

[26] M. Orzech, On the Brauer group of algebras having a grading and an action, Can. J. Math. 32(1980), 533-552.

[27] D. E. Radford, Minimal quasitriangular Hopf algebras, J.Alg. 157(1993), 285-315.

[28] M.E. Sweedler, Hopf algebras, Benjamin, 1969.

[29] K-H. Ulbrich, Galoiserweiterungen von nicht-kommutativen Ringen, Comm. Alg. 10(1982), 655-672.

[30] K-H. Ulbrich, A exact sequence for the Brauer group of dimodule Azumaya algebras, Math. J. Okayama Univ. 35(1993), 63-88.

[31] F.Van Oystaeyen and Y.H.Zhang, The embedding of automorphism group into the Brauer group, Canadian Math. Bull. 41(1998), 359-367.

[32] F.Van Oystaeyen and Y.H.Zhang, The Brauer group of a braided monoidal category, J. Alg. 202(1998), 96-128.

[33] F.Van Oystaeyen and Y.H.Zhang, The Brauer group of the Sweedler’s Hopf algebra H_{4}. to appear in Proc. AMS.

[34] C.T.C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213(1964), 187-199.

[35] D.N.Yetter, Quantum groups and eepresentations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108(1990), 261-290