Long Dimodules and Quasitriangular Weak Hopf Monoids

: In this paper, we prove that for any pair of weak Hopf monoids H and B in a symmetric monoidal category where every idempotent morphism splits, the category of H - B -Long dimodules BH Long is monoidal. Moreover, if H is quasitriangular and B coquasitriangular, we also prove that BH Long is braided. As a consequence of this result, we obtain that if H is triangular and B cotriangular, B H Long is an example of a symmetric monoidal category.


Introduction
Let R be a commutative fixed ring with unit and let C be the non-strict symmetric monoidal category of R-Mod where ⊗ denotes the tensor product over R. The notion of Long H-dimodule for a commutative and cocommutative Hopf algebra H in C was introduced by Long [1] to study the Brauer group of H-dimodule algebras. For two arbitrary Hopf algebras H and B with bijective antipode there exists a well-known connection between the category of left-left H-B-Long dimodules, denoted by B H Long, and the category of left-left Yetter-Drinfel'd modules over the Hopf algebra H ⊗ B, denoted by H⊗B H⊗B YD. This relation can be formulated in the following way: if H is a quasitriangular and B coquasitriangular, B H Long is a braided monoidal subcategory of H⊗B H⊗B YD. As a consequence of this fact, we ensure that under the suitable conditions, Long dimodules provide nontrivial examples of solutions for the Yang-Baxter equation. On the other hand, for a commutative and cocommutative Hopf algebra H, the category of left-right H-H-Long dimodules, denoted by H Long H , is the category of left-right Yetter-Drinfel'd modules over H. Then, for all these reasons, it is not unreasonable to assume that there exists an interesting relationship between Long dimodules and the problem of find solutions for the Yang-Baxter equation. Moreover, the previous statement can be extended, as was proved by Militaru in [2], to the problem of find solutions for the D-equation.
The results about the connections between Long dimodules and Yetter-Drinfeld modules can be generalized to Hom-Hopf algebras and to non-associative Hopf structures as for example Hopf quasigroups. In [3], for two monoidal Hom-Hopf algebras (H, α) and (B, β) the authors introduce the notion of generalized Hom-Long dimodule and the category of generalized Hom-Long dimodules proving that this category is an example of autonomous category. Also, if (H, α) is quasitriangular and (B, β) is coquasitriangular they obtain that the category of generalized Hom-Long dimodules is a braided monoidal subcategory of the category of left-left Yetter-Drinfel'd modules over the monoidal Hom-Hopf algebra (H ⊗ B, α ⊗ β). On the other hand, in [4] (see also [5]) we can find the definition of Long dimodule for Hopf quasigroups and, if H is a quasitriangular Hopf quasigroup and B coquasitriangular Hopf quasigroup, as in the previous settings, the authors prove A monoidal category is called strict if the associativity, right unit and left unit constraints are identities. It is a well-known fact that every non-strict monoidal category is monoidal equivalent to a strict one (see [13]). Then, in general, we can assume without loss of generality that the category is strict and, as a consequence of the quoted equivalence, the results proved in this paper remain valid for every non-strict symmetric monoidal category, what would include for example the categories of vector spaces over a field F, or the one of left modules over a commutative ring R. In what follows, for simplicity of notation, given objects M, N, P in C and a morphism f : M → N, we write P ⊗ f for id P ⊗ f and f ⊗ P for f ⊗ id P .
A braiding for a strict monoidal category C is a natural family of isomorphisms A strict braided monoidal category is a strict monoidal category with a braiding. Braided monoidal categories were introduced by Joyal and Street (see [14]) motivated by the theory of braids and links in topology. Please note that as a consequence of the definition, the equalities t M,K = t K,M = id M hold, for all object M of C. If the braiding satisfies that t N,M • t M,N = id M⊗N , for all M, N in C, we will say that C is symmetric and the braiding will be called a symmetry.
Throughout this paper C denotes a strict symmetric monoidal category with tensor product ⊗, unit object K and natural isomorphism of symmetry c. We also assume that in C every idempotent morphism splits, i.e., for any morphism q : X → X such that q • q = q there exist an object Z, called the image of q, and morphisms i : Z → X, p : X → Z, such that q = i • p and p • i = id Z . Please note that Z, p and i are unique up to isomorphism. The categories satisfying this property constitute a broad class that includes, among others, the categories with epi-monic decomposition for morphisms and categories with equalizers or coequalizers. For example, complete bornological spaces is a symmetric monoidal closed category that is not abelian, but it has coequalizers (see [15]). On the other hand, let Hilb be the category whose objects are complex Hilbert spaces and whose morphisms are the continuous linear maps. Then, Hilb is not an abelian and closed category but it is a symmetric monoidal category (see [16]) with coequalizers.
A monoid in C is a triple A = (A, η A , µ A ) where A is an object in C and η A : A comonoid in C is a triple D = (D, ε D , δ D ) where D is an object in C and ε D : If A is a monoid, C is a comonoid and f : C → A, g : C → A are morphisms, we define the convolution product by f * g = µ A • ( f ⊗ g) • δ C . Definition 1. A weak bimonoid H is an object in C with a monoid structure (H, η H , µ H ) and a comonoid structure (H, ε H , δ H ) such that the following axioms hold: we will say that the weak bimonoid is a weak Hopf monoid.
For any weak bimonoid, if we define the morphisms , it is straightforward to show that they are idempotent and the equalities The morphisms η H L , µ H L , ε H L and δ H L are the unique morphisms satisfying  [17] for the detailed proofs).
Finally, the proof for (22) can be obtained as the previous one by reversing arrows.
If H is a weak Hopf monoid in C, the antipode λ H is unique, antimultiplicative, anticomultiplicative and leaves the unit and the counit invariant: Also, it is straightforward to show the equalities and If H and B are weak bimonoids in C, the tensor product H ⊗ B so is. In this case, the monoid-comonoid structure is the one of H ⊗ B and Then, if H and B are weak Hopf monoids in C, the tensor product H ⊗ Bso is with In the end of this section, we summarize some properties about left modules and left comodules over a weak Hopf monoid. The complete details can be found in [18][19][20].

Definition 2.
Let H be a weak Hopf monoid in C. We say that (M, ϕ M ) is a left H-module if M is an object in C and ϕ M : Given two left H-modules (M, ϕ M ) and (N, If (M, ϕ M ) and (N, ϕ N ) are left H-modules we define the morphism ϕ M⊗N : It is easy to show that ϕ M⊗N satisfies and the morphism hold. Moreover, if (M, ϕ M ), (N, ϕ N ) and (P, ϕ P ) are left H-modules, we have that also holds.
Given two left H-comodules (M, ρ M ) and (N, ρ N ), a morphism f : M → N in C is a morphism of left H-comodules if It is easy to show that ρ M⊗N satisfies and the morphism holds. Moreover, as in the case of left modules, if (M, ρ M ), (N, ρ N ) and (P, ρ P ) are left Hcomodules, we have that also holds.

The Category of Long Dimodules Over Weak Hopf Monoids
In this section, we generalize the notion of Long dimodule to the weak Hopf monoid setting.
Similarly, if H L and B L are the images of the target morphisms with the corresponding structure of monoid-comonoid, by the properties of η H and ε B , (5) for H, (7) for B, the associativity of µ H , the coassociativity of δ B and the naturality of c, we have that Let (M, ρ M ) be a left B-comodule and let τ : H ⊗ B → K be a skew pairing between H and B over K such that Then, the triple Indeed, by (24), (b3) and the B-comodule condition for M, we have that ϕ M • (η H ⊗ M) = id M . Moreover, using that M is a left B-comodule, the naturality of c, (b1) and (23), holds and, consequently, (M, ϕ M ) is a left H-module. Finally, using that M is a left B-comodule and (45), condition (44) holds because In particular, if H = B, ρ H = δ H and τ : H ⊗ H → K is a skew pairing between H and H over K such that H Long if and only if (46) holds.

Example 3.
Let H and B be weak Hopf monoids in C. We define a skew copairing between H and B over K as a morphism σ : Then, by a similar proof to the one developed for the example linked to skew pairings we have that belongs to B H Long. Indeed: First note that by the naturality of c, (c4) and the condition of lef H-module for M, we have that (ε B ⊗ M) • ρ M = id M . Moreover, using that M is a left H-module, (c2) and the naturality of c, and then, (M, ρ M ) is a left B-comodule. Finally, using (47), the naturality of c and the condition of left H-module for M, (44) holds because In particular, if H = B, ϕ H = µ H and σ : K → H ⊗ H is a skew copairing between H and H over K such that H Long if and only if (48) holds.

Example 4. Let H and B be weak Hopf monoids in
As a consequence, the morphism is idempotent and there exist two morphisms j M⊗N : Moreover, the following identities hold: Finally, (51)-(54) follows easily from (49) and the properties of j M⊗N and q M⊗N .

Lemma 3. Let H and B be weak Hopf monoids and let
Then, the idempotent morphisms ∇ M⊗N , ∇ M⊗N and Ω M⊗N , defined in (32), (40) and (50), satisfy that where ϕ M⊗N and ρ M⊗N are the morphisms defined in (30) and (38) respectively.

Lemma 4. Let H and B be weak Hopf monoids and let
Proof. The proofs for (61)-(65) follow directly from (59) and (60). The proof for (60) is similar to the one of (59). Then, we only need to show that (59) holds. Indeed: and the coaction ρ M×N by Similarly, by (54) we obtain that (ε B ⊗ M ⊗ N) • ρ M×N = id M×N and, by (39) and (58), we have that (35) and (43), we have that (57)) and Therefore, the proof is complete.
Proof. The proof for the identity (67) is: (57) and (58) for N and P)  (33) and (41) for M and N) (57) and (58) for M and N) On the other hand, (68) follows by Therefore, (69) holds because j M⊗N ⊗ P is a monomorphism and q M⊗N ⊗ P is an epimorphism. The proof for (70) is similar and we leave the details to the reader.
is a natural isomorphism in B H Long and satisfies the Pentagon Axiom.
On the other hand, (73) follows by (by the naturality of c and (44)) (13)). Finally,  (10) and (44) are natural isomorphisms in B H Long and satisfy the Triangle Axiom.
Proof. First note that it is easy to show that l M and r M are natural morphisms because (66) holds. The morphisms l M is an isomorphism with inverse Indeed, on one hand, and, on the other hand,   (15) (2) and (13) (5) (15) (58) and naturality of c)    (2) and (18)

Definition 5. Let H be a weak Hopf monoid.
Let Ω H and Ω H be the idempotent morphisms defined in (83). We will say that H is a quasitriangular weak Hopf algebra if there exists a morphism σ : K → H ⊗ H in C satisfying the following conditions: There exists a morphism σ : K → H ⊗ H such that: We will say that a quasitriangular weak Hopf monoid H is triangular if moreover σ = c H,H • σ.
For any quasitriangular weak Hopf monoid the morphism σ is unique and by [21] [Lemma 3.5] the equalities hold.

Lemma 7.
Let H be a quasitriangular weak Hopf monoid in C. Then, Proof. We will prove (103). The proof for (104) is similar and we lend the details to the reader. First note that the identities and, consequently, Finally, and, consequently, By reversing arrows in Definition 5 we get the definition of coquasitriangular weak Hopf monoid. Definition 6. Let B be a weak Hopf monoid. Let Γ B and Γ B be the idempotent morphisms defined in (94). We will say that B is a coquasitriangular weak Hopf algebra if there exists a morphism ω : B ⊗ B → K in C satisfying the following conditions: As a consequence of this definition, we obtain that ω is unique and the equalities hold.
We will say that a coquasitriangular weak Hopf monoid B is cotriangular if moreover ω = ω • c B,B .

Lemma 8.
For any coquasitriangular weak Hopf monoid B, the following equalities Proof. The proof is the same that the one given for the quasitriangular setting by reversing arrows.
The groupoid algebra of a finite groupoid is the main example of a cocommutative weak Hopf monoid. Recall that a finite groupoid G is simply a category with a finite number of objects in which every morphism is an isomorphism. The set of objects of G will be denoted by G 0 , the set of morphisms by G 1 , the identity morphism on x ∈ G 0 by id x and, for a morphism g : x → y in G 1 , we write s(g) and t(g) for the source and the target of g, respectively.
Let G be a finite groupoid, and let R be a commutative ring. The groupoid algebra is the direct product R[G] = g∈G 1 Rg where the product of two morphisms is their composition if the latter is defined and 0 otherwise, i.e., µ R is a cocommutative weak Hopf monoid in the symmetric monoidal category R-Mod, with coproduct δ R[G] , counit ε R [G] and antipode λ R[G] given by δ R[G] (g) = g ⊗ R g, ε R[G] (g) = 1 and λ R[G] (g) = g −˙1 , respectively. Moreover, the target and source morphisms are Π L R[G] (g) = id t(g) , Π R R[G] (g) = id s(g) and the morphism σ that provides the quasitriangular structure is the linear extension of If G 1 is finite, R[G] is free of a finite rank as a R-module. Hence R[G] is finite as object in the category R-Mod and R[G] * = Hom R (R[G], R) = g∈G 1 R f g is a commutative weak Hopf monoid. The weak Hopf monoid structure of R[G] * is given by the formulas Then, by the general theory, R[G] * is an example of coquasitriangular weak Hopf monoid in R-Mod where ω is defined by On the other hand, the construction of a weak Hopf monoid K(G, H) in the symmetric monoidal category of vector spaces over a field K using a matched pair of finite groupoids (G, H) was introduced in [23]. In [24] we can find a result that asserts the following: A matched pair of rotations gives rise to a quasitriangular structure for the associated weak Hopf monoid K(G, H). Also, by [24] [Theorem 5.10] we know that there is an isomorphism of quasitriangular weak Hopf monoids between the Drinfeld double of K(G, H) and the weak Hopf monoid of a suitable matched pair of groupoids.
Finally, in [22], for a weak Hopf monoid H in the symmetric monoidal category of vector spaces over an algebraically closed field, Nikshych, Turaev and Vainerman defined the Drinfeld double D(H) of H and they proved that D(H) is a quasitriangular weak Hopf monoid (see [22] [Proposition 6.2]). Now we recall the notion of left-left Yetter-Drinfeld module in the weak Hopf monoid setting.  (44)).
To get (f1) of Definition 7, note that on one hand, (by the naturality of c) and on the other hand, The condition (f2) of Definition 7 follows because, using the previous calculus, Similarly, by (98) and the condition of left H-module for M, we obtain Thus, L is injective on objects.  (17) (115) and (116) (by the naturality pf c) (14) (5) = Ω M⊗N (by (110)).
Finally, the equality for the associative constraints follows (125) and Proposition 2. Then, In the final result of this paper, we will prove that under the conditions of the previous theorem, B H Long is a braided monoidal category. of the image of the composition of two suitable idempotent morphisms associated with the module and comodule structure, respectively.
Finally, as mentioned in the Introduction, the results studied in this paper are strongly related with the theory developed by G. Militaru