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FRT Construction for Dynamical Yang-Baxter Maps

Notions of an (H, X)-bialgebroid and of its dynamical representation are proposed. The dynamical representations of each (H, X)-bialgebroid form a tensor category. Every dynamical Yang-Baxter map R(lambda) satisfying suitable conditions, a generalization of the set-theoretical solution to the quantum Yang-Baxter equation, gives birth to an (H, X)-bialgebroid A_R. The categories of L-operators for R(lambda) and of dynamical representations of A_R are isomorphic as tensor categories.


Introduction
Representations of every bialgebra form a tensor category. Faddeev, Reshetikhin, and Takhtajan [10] constructed some bialgebras, called FRT bialgebras, by means of constant R-matrices, solutions to the quantum Yang-Baxter equation (QYBE for short) [1,2,27,28]. This FRT construction is categorically characterized as follows. The tensor category of L-operators for the R-matrix R is isomorphic to that of representations of the FRT bialgebra associated with R.
The FRT construction for the dynamical R-matrix, a solution to the quantum dynamical Yang-Baxter equation (QDYBE for short) [7,11,12], was developed in [9,Sections 4.1,4.2,and 4.4]. This construction yields bialgebroids [3,16,26] called dynamical quantum groups, each of which provides a tensor category consisting of its dynamical representations. This tensor category is isomorphic to that of L-operators.
The other maps are similarly defined.
A generalization of the bijective 1-cocycle [8,17,22] gives birth to the dynamical Yang-Baxter map. In [19], we constructed dynamical Yang-Baxter maps by means of the groups and loops [18,Definition I.1.10]. Moreover, the paper [20] established a characterization of the dynamical Yang-Baxter maps with some conditions by making use of left quasigroups and ternary operations (See Section 3).
It is natural to try to apply the FRT construction to the dynamical Yang-Baxter map.
In this paper, we propose notions of an (H, X)-bialgebroid and of its dynamical representation. The dynamical representations of each (H, X)-bialgebroid form a tensor category; the proof is based on some concepts in category theory. We construct an (H, X)-bialgebroid A R associated with a bijective dynamical Yang-Baxter map R(λ). This is an analogue of the FRT construction. We show that the tensor category of dynamical representations of A R is isomorphic to that of L-operators for R(λ).
The organization of this paper is as follows. In Section 2, we introduce a C-linear tensor category Vect H , which plays a basic role in this paper (cf. the category V h [9, Section 3.1] and the dynamical category [5,Section 4.2]).
We assume that every translation map ·x : H ∋ λ → λ · x ∈ H (x ∈ X) is bijective. This assumption produces a group W H X anti-isomorphic to the subgroup of Aut(H) generated by the set {·x; x ∈ X}. In Section 4, we propose a notion of an (H, X)algebra (Definition 4.2), which is similar to that of a W H X × W H X -graded algebra. A major difference between them is that the (H, X)-algebra is not always a direct sum of its homogeneous components (See the end of Section 7).
The category Alg (H,X) consisting of (H, X)-algebras and their homomorphisms is a pre-tensor category (Definition 6.1), a generalization of the tensor category. We define an (H, X)-bialgebroid as an analogue of the coalgebra object of the tensor category. Definition 1.2 (Definition 6.8). An (H, X)-bialgebroid A is an object of the category Alg (H,X) , together with morphisms ∆ : A → A ⊗A and ε : A → I H,X of Alg (H,X) such that: For the object I H,X , the functor ⊗, and the natural transformations a l , a r , l, r, see Sections 5 and 6. This (H, X)-bialgebroid is a bialgebroid (Remark 6.9).
We assume that the set X is finite. In Section 7, we construct an (H, X)-bialgebroid A R associated with a bijective dynamical Yang-Baxter map R(λ) satisfying the weight zero condition (3.4).
Let D H,X (V ) (V ∈ Ob(Vect H )) denote the (H, X)-algebra defined in Section 4 (See Proposition 4.7). By a dynamical representation of an (H, X)-algebra A, we mean a pair (V, π) of an object V of Vect H and an (H, X)-algebra homomorphism π : A → D H,X (V ) (See Section 8).
In Section 9, we first define a crucial (H, X)-algebra homomorphism D H,X (V ) ⊗D H,X (W ) → D H,X (V⊗W ) (V, W ∈ Ob(Vect H )), whose definition is a bit technical, because we make use of seemingly infinite sum operations (Proposition 9.2). With the aid of this canonical homomorphism, we show that dynamical representations of each (H, X)-bialgebroid A form a tensor category DR(A) (Theorem 9.7); Section 10 describes its proof, in which the pre-tensor categories, categorical bialgebroids (Definition 6.7), and pre-tensor functors (Definition 10.4) play an essential role. The tensor category Vect H , together with a bijective dynamical Yang-Baxter map R(λ) satisfying the weight zero condition (3.4), yields another tensor category Rep R, called a centralizer category (cf. [23,Proof of Theorem 3.7]), consisting of L-operators for R(λ) (See Section 3). In the final section, Section 11, we prove our main result. To end Introduction, it is worth pointing out that we do not need (1.1) in Sections 7 and 11. A role of (1.1), a version of the QDYBE, is to imply that the tensor categories Rep R and DR(A R ) are non-trivial in general (See Remark 3.9). Conventions. For a set V , we will denote by CV the C-vector space whose basis is V . Its element g will be written as with complex coefficients g v . For two sets V and W , a C-linear map f : CV → CW will be identified with its matrix (f wv ) (w,v)∈W ×V ; we will use the notation with complex coefficients f wv .
2 C-linear tensor category Vect H Let H be a nonempty set. A C-linear tensor category Vect H , which we will introduce in this section, plays a basic role in this paper. We follow the notation of [14,Chapter XI]. An object of Vect H is, by definition, a pair (V, · V ) of a nonempty set V and a map For simplicity of notation, we write (·) and V instead of · V and (V, · V ), respectively. Furthermore, we denote λ · v (λ ∈ H, v ∈ V ) briefly by λv. Ob(Vect H ) is the class of all objects of Vect H .
Let V and W be objects of Vect Let f and g be morphisms of Vect H satisfying b(f ) = s(g). We define the composition Proposition 2.1. Vect H is a category of Ob(Vect H ) and Hom(Vect H ), together with the identity id, the source s, the target b, and the composition •.
The next task is to explain a tensor product⊗ : Vect H × Vect H → Vect H . Let V and W be objects of the category Vect H . We define the set V⊗W and the map · V⊗W : H × (V⊗W ) → H by: Clearly, the pair V⊗W := (V⊗W, · V⊗W ) is an object of Vect H .
We may identify Let f and g be morphisms of the category Vect H . We denote by f⊗g the following map from H to Hom C (C(s(f )⊗s(g)), C(b(f )⊗b(g))) (See (2.2)).
The associativity constraint a :⊗(⊗ × id) →⊗(id ×⊗) is given by Let I Vect H denote the set consisting of one element, together with the map · IVect H : The left and the right unit constraints l : This tensor category Vect H is also a C-linear category.
In addition, both the composition and the tensor product are C-bilinear.
We will show some objects and morphisms of Vect H . Let V = (V, ·) be an object of Vect H . Proposition 2.5. For any nonempty subset S ⊂ V , S := (S, · S ) is again an object of Vect H . Here · S is the restriction of (·) on the set H × S; that is, λ · S s = λ · s (λ ∈ H, s ∈ S).
For v ∈ V , we denote by i Here δ vw is Kronecker's delta symbol. These morphisms satisfy (2.6) In fact, the following ι V W : V → W is an isomorphism, if V and W satisfy (2.6).
3 Centralizer categories Let (C, ⊗, I C , a, l, r) be a tensor category [14,Definition XI.2.1]. This section clarifies that this tensor category C, together with an object X C and a morphism σ : We will first introduce a category Rep σ. An object of Rep σ is, by definition, a pair (V, L V ), called an L-operator, of an object V of C and an isomorphism L V : We write Hom(Rep σ) for the class of all morphisms of Rep σ.
For a morphism f : L V → L W of Rep σ, the objects L V and L W are called the source s(f ) and the target b(f ) of the morphism f , respectively.
We define the identity id of Rep σ by id LV = id V . The composition g • f : s(f ) → b(g) of morphisms f and g of Rep σ is the same as that of the category C. (3.3) To simplify notation, we use the same symbol L V ⊠ L W for the pair (V ⊗ W, L V ⊠ L W ). Let f : L V → L W and g : We write L IC = r −1 XC • l XC : I C ⊗ X C → X C ⊗ I C ∈ Hom(C). Here, l XC : I C ⊗ X C → X C and r XC : X C ⊗ I C → X C are the isomorphisms given by the left and the right unit constraints l, r of C. Let I Rep σ denote the pair (I C , L IC ).
The associativity constraint a : ⊠(⊠ × id) → ⊠(id × ⊠), and the left and the right unit constraints l, r are defined by those of the tensor category C: a LU ,LV ,LW = a U,V,W , l LV = l V , and r LV = r V .  3. An automorphism σ on X C ⊗ X C is a Yang-Baxter operator on X C , iff σ satisfies (id⊗σ)a(σ⊗id)a −1 (id⊗σ)a = a(σ⊗id)a −1 (id⊗σ)a(σ⊗id).
Here, a = a XC ,XC ,XC and id = id XC . We now explain the category Rep R. Fix a nonempty set H. Let X be a nonempty set, together with a map (·) : H × X ∋ (λ, x) → λ · x ∈ H. It is clear that Proposition 3.5. X = (X, ·) ∈ Ob(Vect H ).
This dynamical Yang-Baxter map R(λ) produces a Yang-Baxter operator on X in the tensor category Vect H . Let σ R denote the map from H to Hom C (CX⊗X, CX⊗X) defined by σ R (λ)(x, y) = R(λ)(y, x) (λ ∈ H, (x, y) ∈ X⊗X).  Remark 3.9. If ♯(X) ≥ 2, then the objects I Rep R and (X, σ R ) are not isomorphic; hence, Rep R is non-trivial.
In [19,Section 6], we constructed bijective dynamical Yang-Baxter maps satisfying the weight zero condition in the case that H = X (See Remark 3.6). The following is one more example, which is useful in Section 7.
Let Q 5 denote the set {0, 1, 2, 3, 4} of 5 elements with the binary operation (·) given by Table 1. Here, 0 · 2 = 2. Since each element of Q 5 appears once and only once in each row and in each column of Definition 3.10. A nonempty set Q with a binary operation (·) is a quasigroup, iff the translation maps a· and ·a on Q are bijective for all a ∈ Q: (3.6) Let µ denote the ternary operation on Z/5Z defined by The isomorphism π : Q 5 ∋ a →ā ∈ Z/5Z, together with the quasigroup Q 5 and the ternary system (Z/5Z, µ), gives birth to a bijective dynamical Yang-Baxter map R Q5 (λ) associated with Q 5 , Q 5 , and (·) satisfying the weight zero condition [20,Section 3]. In fact, the dynamical Yang-Baxter map R Q5 (λ) is defined as follows: for λ, a, b ∈ Q 5 , Here we denote by a\ :

(H, X)-algebras
Let H and X be nonempty sets, together with a map (·) : In order to define (H, X)-bialgebroids, this section is devoted to introducing (H, X)-algebras (cf. h-algebras [9,15]), which form a category Alg (H,X) .
From now on we make the following assumption. For any x ∈ X, the translation map is bijective. For example, every quasigroup Q (Definition 3.10) satisfies the assumption in the case that H = X = Q. This assumption produces a group W H X acting faithfully on the set H from the right. By the assumption, the translation map ·x (4.1) is an element of the group Aut a (H) of all bijections on the set H, together with the product f g = g • f (f, g ∈ Aut a (H)); hence, there exists a unique homomorphism F from the free group W X on the set X to Aut a (H) such that F (x)(λ) = λ · x for λ ∈ H and x ∈ X (This symbol F is a temporary notation). Let λ ∈ H and α ∈ W X . We denote by λ · α the element F (α)(λ) ∈ H. This (·) is a right action of W X on H, and consequently the factor group W H X of W X by the kernel {β ∈ W X ; λ · β = λ (∀λ ∈ H)} acts faithfully on H. For simplicity of notation, we write the induced right action of W H X as λα (λ ∈ H, α ∈ W H X ).
Here, 1 is the unit element of the factor group W H In this paper, C-algebra homomorphisms always respect the unit element. We note that the sum in (1) is not always a direct sum (See the end of Section 7).
We will denote by Ob(Alg (H,X) ) and Hom(Alg (H,X) ) the classes of all (H, X)-algebras and of all (H, X)-algebra homomorphisms, respectively.
We give a construction which assigns an (H, X)-algebra A W H X to any unital C-algebra For g(λ) v ∈ C, see (1.2). Because these homomorphisms satisfy (4.2), we obtain an (H, X)-algebra E(V ) W H X . To end this section, we assign to any object V of the category Vect H an object D H,X (V ) of Alg (H,X) , which produces a dynamical representation in Section 8.
. Write u * V v for the following morphism of Vect H whose source is V⊗{βδ} and whose target is {αγ}⊗V .
Let g be an element of M H . We assume that γ, δ(∈ W H X ) satisfy that λγ = λδ unless g(λ) = 0. A simple computation shows Remark 4.9. What we have discussed and we will discuss in the following holds by replacing the group W H X with an arbitrary group G acting on H from the right. In this case the notion in Definition 4.2 will be called a G-algebra, and they form a category Alg G . Just as Proposition 4.7, we can construct a G-algebra D G (V ) for any object V of Vect H .

Properties of Alg (H,X)
In this section, we will proceed with the study of the category Alg (H,X) , which is essential in defining (H, X)-bialgebroids.
Let us first explain an object I H,X of Alg (H,X) , which is relevant to the object D H,X (I VectH ) (See Propositions 2.3 and 4.7).
For T α , see (4.4). We set We can construct a morphism ϕ 0 : To simplify notation, we continue to write ϕ 0 for ϕ 0 | IH,X .
Let A and B be objects of Alg (H,X) . The next task is to define an object A ⊗B of Alg (H,X) , called a matrix tensor product (cf. [9, Section 4]), which produces a functor ⊗ : Alg (H,X) × Alg (H,X) → Alg (H,X) .
Write I 2 = I 2 (A, B) for the right ideal of A ⊗ C B generated by the elements Let α and β be elements of the group W H X . We denote by (A ⊗B) α,β the C-subspace .
Although I 2 is a right ideal, this definition of the product makes sense. The By taking account of (4.2) and (5.2), these maps are C-algebra homomorphisms from M H to A ⊗B.
, and the unit Remark 5.4. On account of (4.3) and (5.2), the matrix tensor product A ⊗B of (H, X)- .
Two morphisms f : A → C and g : B → D of Alg (H,X) induce a morphism f ⊗g : Here, I 2 (A, B) ⊂ A ⊗ C B is the right ideal generated by the elements (5.2). Since Let A, B, and C be objects of Alg (H,X) . The final task of this section is to introduce a functor − ⊗ − ⊗− : Alg (H,X) × Alg (H,X) × Alg (H,X) → Alg (H,X) . We first define (H, X)-algebras A ⊗B ⊗C l and A ⊗B ⊗C r , whose definitions are different from those of the (H, X)-algebras (A ⊗B) ⊗C and A ⊗(B ⊗C), respectively.
We use the symbol I l 3 = I l 3 (A, B, C) for the right ideal of (A ⊗ C B) ⊗ C C generated by the elements . This product is well defined, though I l 3 is a right ideal. We denote by µ A e Because of (4.2) and (5.3), these maps are C-algebra homomorphisms from M H to A ⊗B ⊗C l .
The definition of the (H, X)-algebra A ⊗B ⊗C r is similar. Let I r 3 denote the right ideal of A ⊗ C (B ⊗ C C) generated by For α, β ∈ W H X , we write (A ⊗B ⊗C r ) α,β for the C-subspace of A⊗ C (B⊗ C C)/I r 3 generated by {ξ 1 ⊗ (ξ 2 ⊗ ξ 3 ) + I r 3 ; ξ 1 ∈ A α,γ , ξ 2 ∈ B γ,δ , ξ 3 ∈ C δ,β (∃γ, δ ∈ W H X )}; in addition, set A ⊗B ⊗C r = α,β∈W H X (A ⊗B ⊗C r ) α,β . For x = ξ x + I r 3 , y = ξ y + I r 3 ∈ A ⊗B ⊗C r , we define a product on A ⊗B ⊗C r by xy = ξ x ξ y + I r 3 . We denote by µ A e ⊗B e ⊗C r l and µ A e ⊗B e ⊗C r r the following maps from M H to (A ⊗B ⊗C r ) 1,1 . For f ∈ M H : This isomorphismΓ implies an isomorphism Γ : A ⊗B ⊗C r → A ⊗B ⊗C l of Alg (H,X) . Here and subsequently, we let A ⊗B ⊗C stand for the (H, X)-algebra A ⊗B ⊗C l (= Γ(A ⊗B ⊗C r )).
Let f : A → A ′ , g : B → B ′ , and h : C → C ′ be morphisms of Alg (H,X) . Define the map (f ⊗g ⊗h) e : . This map is C-bilinear; for this reason, there exists a unique C-linear map (f ⊗g ⊗h)¯: c).
We denote by f ⊗g ⊗h the following map from Here, I l 3 = I l 3 (A, B, C). By The pre-tensor category is a generalization of the tensor category.
Proposition 6.2. The category Alg (H,X) is a pre-tensor category.
Taking Propositions 5.1, 5.5, and 5.8 into account, we need only construct natural transformations a l , a r , l, and r for the proof.
Let A, B, and C be objects of Alg (H,X) . Let us introduce the temporary notation F for the C-bilinear map from . We writeF for the following map from For  A, B), C) ⊂ I l 3 . By abuse of notation, we use the same symbolF for the mapF | A e ⊗B×C : A ⊗B ×C → ((A⊗ C B)⊗ C C)/I l 3 . SinceF is C-bilinear, there uniquely exists a C-linear mapΦ 1 : We will use the temporary notation Φ ′ 2 : A ⊗(B ⊗C) → A ⊗B ⊗C r for the morphism of Alg (H,X) defined similarly to a l A,B,C :  Let A be an object of Alg (H,X) . We writel A (resp.r A ) for the following map from I H,X × A to A (resp. from A × I H,X to A). For U ∈ I H,X and a ∈ A, Here, 1 MH ∈ M H is the unit element of the C-algebra M H : Becausel A (resp.r A ) is C-bilinear, there exists a unique C-linear map l A : I H,X ⊗ C A → A (resp. r A : A⊗ C I H,X → A) such that l A (U ⊗a) =l A (U, a) (resp. r A (a⊗U ) =r A (a, U )). Proof. We will only show that l A (I 2 (I H,X , A)) ⊂ {0}. It suffices to prove

By the definition of l
Simple calculation shows that (µ l is a C-algebra homomorphism by Definition 4.2, (RHS of (6.3))= 0. This is the desired conclusion.
We define the C-linear maps l A : I H,X ⊗ C A/I 2 (I H,X , A) → A and r A : A ⊗ C I H,X /I 2 (A, I H,X ) → A by l A (ξ + I 2 (I H,X , A)) = l A (ξ) (ξ ∈ I H,X ⊗ C A) and r A (ξ ′ + I 2 (A, I H,X )) = r A (ξ ′ ) (ξ ′ ∈ A ⊗ C I H,X ). Lemma 6.5 makes these definitions allowable. For simplicity of notation, we continue to write l A and r A for the C-linear maps l A | IH,X e ⊗A : I H,X ⊗A → A and r A | A e ⊗IH,X : A ⊗I H,X → A, respectively. Proposition 6.6. l A , r A ∈ Hom(Alg (H,X) ). Moreover, l : ⊗(I H,X × id) → id and r : ⊗(id × I H,X ) → id are natural transformations.
The proof of Proposition 6.2 is therefore complete; that is, Alg (H,X) , together with ⊗, − ⊗ − ⊗−, I H,X , a l , a r , l, r, is a pre-tensor category.
A categorical bialgebroid of the pre-tensor category is an analogue of the coalgebra object of the tensor category. Definition 6.7. A categorical bialgebroid A of a pre-tensor category C is an object of C, together with morphisms ∆ : A → A ⊗A, called coproduct, and ε : A → I, called counit, of C such that:

(H, X)-bialgebroid A R
In this section, we construct an (H, X)-bialgebroid A σ associated with a morphism σ : X⊗X → X⊗X of Vect H (cf. [9]). As a result, every bijective dynamical Yang-Baxter map R(λ) satisfying the weight zero condition (3.4) gives birth to the (H, X)-bialgebroid A R := A σR (For σ R , see Proposition 3.7). For the definition, we require the set X to be finite; and we will assume that X is finite, whenever we deal with the (H, X)-bialgebroid A σ .
Let HX denote the set Here M H is the C-algebra of all C-valued functions on H. Moreover, we write C{HX} for the free C-algebra on the set HX [14, Section I.2].
Definition 7.1. A σ is the quotient of C{HX} by the two-sided ideal I σ whose generators are the following (1)-(5).
Here the symbol + in ξ + ξ ′ means the addition in the algebra C{HX}, while the symbol + in (ξ + ξ ′ )(∈ HX) is the addition in the algebra M H ⊗ C M H . The notations of the scalar products and products in the other generators are similar.
The sums in the generators (2) and (4) make sense, because the set X is finite. Let HX denote the set of all words x 1 x 2 · · · x n in the alphabet HX(∋ x i ) including the empty word ∅. We define the map w l : HX → W H X as follows (For W H X , see Section 4): if x ∈ HX, then if w = x 1 x 2 · · · x n ∈ HX, then w l (w) = w l (x 1 )w l (x 2 ) · · · w l (x n ); and we set w l (∅) = 1.
The map w r : HX → W H X is similarly defined: if x ∈ HX, then if w = x 1 x 2 · · · x n ∈ HX, then w r (w) = w r (x 1 )w r (x 2 ) · · · w r (x n ); and we write w r (∅) = 1.
Let α and β be elements of W H X . We write C{HX} α,β and (A σ ) α,β for the subspaces of C{HX} and A σ generated by the sets {w; w ∈ HX, w l (w) = α, w r (w) = β} and {w + I σ ; w ∈ HX, w l (w) = α, w r (w) = β}, respectively. Obviously, C{HX} = ⊕ α,β∈W H X C{HX} α,β , while the algebra A σ is not always a direct sum of the components {(A σ ) α,β ; α, β ∈ W H X }. For more details, see the end of this section. The maps µ Aσ l , µ Aσ r : The next task is to define the coproduct ∆ : A σ → A σ ⊗A σ . Let ∆ HX denote the following map from HX to A σ ⊗ C A σ : This map uniquely induces a C-algebra homomorphism∆ : Proof. Since the right ideal I 2 satisfies that∆(C{HX})I 2 ⊂ I 2 , it suffices to prove that ∆(x) ∈ I 2 for every generator x in Definition 7.1.
We define the C-linear map ∆ : By virtue of Proposition 7.3, the map ∆ is well defined. The next is to construct the counit ε : A σ → I H,X (For I H,X , see Proposition 5.1). Let ε HX denote the following map from HX to I H,X : Here, m : M H ⊗ C M H → M H is a temporary notation of the multiplication defined by m(f ⊗ g) = f g (f, g ∈ M H ); for m(ξ) ⋆ , see (5.1). This map ε HX uniquely induces a C-algebra homomorphismε : C{HX} → I H,X such thatε(v) = ε HX (v) (v ∈ HX).
Proof. We shall have established this proposition, if we prove thatε = 0 on every generator of the ideal I σ .
We will show the proof only for the generator (4) in Definition 7.1. Let g be an element of M H . By the definition ofε, Because σ is a morphism of Vect H , σ bd ac (λ) = 0 unless (λd)b = (λc)a. Hence, which is the desired conclusion.
We denote by ε the following C-linear map from A σ to I H,X .

Proposition 7.5 makes this definition allowable.
Proposition 7.6. The map ε is a morphism of Alg (H,X) .
It is immediate that (6.4) and (6.5) hold on the generators f ⊗ g + I σ (f, g ∈ M H ), L ab + I σ , and (L −1 ) ab + I σ (a, b ∈ W H X ) of A σ . From Definitions 6.7 and 6.8, Let R(λ) be a bijective dynamical Yang-Baxter map satisfying the weight zero condition (3.4). If the morphism σ = σ R (3.5), then the generators (4) of the ideal I σR are Here R xy ac ∈ M H is the following function.
We will use the notations I R and A R instead of I σR and A σR , respectively. From Proposition 7.7, We now recall the bijective dynamical Yang-Baxter map R Q5 (λ) (λ ∈ Q 5 ) constructed at the end of Section 3. To shorten notation, we write R(λ) for R Q5 (λ). Since Q 5 is a quasigroup (See Definition 3.10), Q 5 satisfies the assumption in the beginning of Section 4; that is, for any x ∈ Q 5 , the translation map (4.1) ·x : Q 5 ∋ λ → λ · x ∈ Q 5 is bijective. Before ending this section, we show that the (Q 5 , Q 5 )-bialgebroid A R is not a direct sum of the components {(A R ) α,β ; α, β ∈ W Q5 Q5 }. This property is a characteristic feature that distinguish the (H, X)-bialgebroid A R from the dynamical quantum group [9,Section 4.4].

Dynamical representations
A dynamical representation of an (H, X)-algebra A is, by definition, a pair π V = (V, π V ) of V ∈ Ob(Vect H ) and π V : A → D H,X (V ) ∈ Hom(Alg (H,X) ) (cf. [9, Section 4.2]). For D H,X (V ), see Proposition 4.7. This section deals with two dynamical representations: one is the trivial representation of an arbitrary (H, X)-bialgebroid produced by its counit ε; and the other is the basic representation of the (H, X)-bialgebroid A σ given by a Yang-Baxter operator σ on X in Vect H . We also explain a category of dynamical representations before studying a tensor product of the dynamical representations in the next section.
Write Hom(DR(A)) for the class of all morphisms of DR(A). For a morphism f : π V → π W ∈ Hom(DR(A)), we call the objects π V and π W the source s(f ) and the target b(f ) of the morphism f , respectively. Let id πV : π V → π V denote the morphism id V : V → V ∈ Hom(Vect H ). Let g be an element of Map(H, C(V⊗W )). By using g(λ) (v1,v2) (1.2), we define the map g (v) ∈ Map(H, CW ) (v ∈ V ) by Let v ∈ V . For simplicity of notation, we use the same letter v for the constant map from H to CV whose value is always v; that is, v(λ) = v (λ ∈ H).
Let U be an element of End C (Map(H, CV )). We denote by U (1) the following element of End C (Map(H, C(V⊗W ))).
For abbreviation, we continue to write f⊗g for the tensor product of the morphisms f and g of DR(A). Let I DR(A) denote the object (I Vect H , π IVect H ) ∈ Ob(DR(A)) (See Proposition 8.1). The final task of this section is to explain the associativity constraint a :⊗(⊗ × id) → ⊗(id ×⊗), the left unit constraint l :⊗(I DR(A) × id) → id, and the right unit constraint r :⊗(id × I DR(A) ) → id. Let π i = (V i , π i ) (i = 1, 2, 3) be objects of DR(A). We denote by a π1,π2,π3 , l π1 , and r π1 the following morphisms of Vect H : Theorem 9.7. If A is an (H, X)-bialgebroid (Definition 6.8), then DR(A), together with ⊗, I DR(A) , a, l, and r, is a tensor category.
Because of Proposition 7.7 and Corollary 7.8, Corollary 9.8. DR(A σ ) is a tensor category, and so is DR(A R ).
We will prove Theorem 9.7 in the next section.
10 Proof of Theorem 9.7 Before starting the proof of Theorem 9.7, let us introduce the category C is called the groupoid of isomorphisms of the category C [14, XI.1.1 Example 2]. The notation C is means the (broad) subcategory of C whose objects are those of C and whose morphisms are isomorphisms of C. If C is a tensor category, then C is is a tensor subcategory of C; in other words, C is is a tensor category with respect to ⊗, I, a, l, and r of the tensor category C.
We will first explain a functor D H,X : Vect H is → Alg (H,X) , which plays an essential role in the proof of Theorem 9.7 (For Vect H and Alg (H,X) , see Sections 2 and 4).
For an object V of the category Vect H is , we have already defined the object D H,X (V ) of Alg (H,X) in Proposition 4.7. Let f : V → W be a morphism of the category Vect H is . Define a C-linear map D H,X (f ) from End C (Map(H, CV )) to End C (Map(H, CW )) by (Map(H, CV ))).
The morphisms ϕ 2 (V, W ) in Proposition 9.2 (V, W ∈ Ob(Vect H )) are relevant to this functor D H,X . To be more precise, Definition 10.4. Let C = (C,⊗, I C , a, l C , r C ) be a tensor category, and let D = (D, ⊗, − ⊗− ⊗−, I D , a l , a r , l D , r D ) be a pre-tensor category (Definition 6.1). A pre-tensor functor from C to D is a quintet (F, ϕ 0 , ϕ 2 , ϕ l 3 , ϕ r 3 ) where F : C → D is a functor, ϕ 0 is a morphism from I D to F (I C ), ϕ 2 : ⊗(F × F ) → F⊗ is a natural transformation, and are families of morphisms (U, V, W ∈ Ob(C)) such that for all objects U, V, W of C. We now prove Theorem 9.7. The proof is based on the concepts in category theory: the pre-tensor category, the categorical bialgebroid, and the pre-tensor functor. Let C be a tensor category, F a pre-tensor functor from C is to a pre-tensor category D, and A a categorical bialgebroid of the category D. We follow the notations of Definitions 6.7 and 10.4.
For the proof, we first introduce a tensor category PDR F (A), of which DR(A) will be a tensor subcategory, if F = D H,X : Vect H is → Alg (H,X) .
An object of PDR F (A) is, by definition, a pair (V, π V ) of V ∈ Ob(C)(= Ob(C is )) and π V : A → F (V ) ∈ Hom(D).
Let π V = (V, π V ) and π W = (W, π W ) be objects of PDR F (A). A morphism f : π V → π W of PDR F (A) is a morphism f : V → W of the category C. The objects π V and π W are called the source s(f ) and the target b(f ) of the morphism f , respectively.
Let id πV : π V → π V denote the morphism id V : V → V ∈ Hom(C). We define the composition • of PDR F (A) by that of C.
Proposition 10.6. PDR F (A) is a category.
Note that we have actually proved that PDR F (A) is a category, if A is an object of D.
The next task is to define a tensor product⊗ : PDR F (A) × PDR F (A) → PDR F (A). Let (V, π V ) and (W, π W ) be objects of PDR F (A). Set For f, g ∈ Hom(PDR F (A)), we define f⊗g : s(f )⊗s(g) → b(f )⊗b(g) by f⊗g ∈ Hom(C).
The proof of the above proposition strongly depends on the assumption (4) of the subcategory DR F (A).
Let A be an (H, X)-bialgebroid (Definition 6.8). We set F = D H,X : Vect H is → Alg A ⊗A Tensor categories Rep R and DR(A R ) Let σ : X⊗X → X⊗X be a morphism of Vect H . In this section, we show that the tensor categories Rep σ in Proposition 3.2 and DR(A σ ) in Corollary 9.8 are isomorphic; in fact, we will construct tensor functors F : DR(A σ ) → Rep σ and G : Rep σ → DR(A σ ) such that F G = id Rep σ and GF = id DR(Aσ ) as tensor functors. In particular, Rep R is isomorphic to DR(A R ) (See Proposition 3.8 and Corollary 9.8). As was mentioned in Section 7, the set X is required to be finite in this section. The first task is to introduce a (strict) tensor functor F [14, Definition XI.4.1]. Let π = (V, π) be an object of DR(A σ ). For a, b ∈ X, we define the morphism π(L ab +I σ ) [ Proposition 11.1. L F (π) is an isomorphism.