The Braiding Structure and Duality of the Category of Left–Left BiHom–Yetter–Drinfeld Modules

: Let ( H , µ H , ∆ H , α H , β H , ψ H , ω H , S H ) be a BiHom–Hopf algebra. First, we provide a non-trivial example of a left–left BiHom–Yetter–Drinfeld module and show that the category HH BHYD is a braided monoidal category. We also study the connection between the category HH BHYD and the category H M of the left co-modules over a coquasitriangular BiHom–bialgebra ( H , σ ) . Secondly, we prove that the category of ﬁnitely generated projective left–left BiHom–Yetter–Drinfeld modules is closed for left and right duality.


Introduction
In the 1990s, Hom-type algebras appeared in physics literature in the context of the quantum deformations of some algebras, such as the Witt and Virasoro algebras, in connection with oscillator algebras [1,2]. A quantum deformation replaced the usual derivation with a σ-derivation. The algebras obtained in such a way satisfy a modified Jacobi identity involving a homomorphism. Hartwig, Larsson, and Silvestrov in [3,4] called this kind of algebra a Hom-Lie algebra. Considering the enveloping algebras of the Hom-Lie algebras, the Hom-associative algebra was introduced in [5]. Another way to study Hom-type algebras was considered by categorical approach in [6], these were called monoidal Hom-algebras. In order to unify these two kinds of Hom-type algebras, a generalization has been provided in [7], where a construction of a Hom-category, including a group action, led to the concept of BiHom-type algebras. Hence, BiHom-associative algebras and BiHom-Lie algebras involving two linear structure maps were introduced. The main axioms for these types of algebras (BiHom-associativity, BiHom-skew-symmetry, and the BiHom-Jacobi condition) were dictated by categorical considerations.
Joyal and Street [8] introduced the definition of a braided monoidal category (also known as a braided tensor category) to formalize the characteristic properties of the tensor categories of modules over braided bialgebras as well as the ideas of crossing in link and tangle diagrams. Since the braiding structure may be considered to be the categorical version of the famous Yang-Baxter equation (see [9]), it is worth constructing more braided monoidal categories. Moreover, it is well-known that the category of Yetter-Drinfeld modules is a braided monoidal category ( [10]).
The main aim of this paper is to conduct more studies of left-left BiHom-Yetter-Drinfeld modules over BiHom-Hopf algebras. The definition of left-left BiHom-Yetter-Drinfeld modules was introduced in [11 for all x, y, z ∈ A. The maps α and β (in this order) are called the structure maps of A, and condition (1) is called the BiHom-associativity condition.

Definition 3.
A BiHom-coassociative coalgebra is a 4-tuple (C, ∆, ψ, ω), in which C is a linear space, and ψ, ω : C → C, and ∆ : The maps ψ and ω (in this order) are called the structure maps of C, and condition (3) is called the BiHom-coassociativity condition.
Let us record the formula expressing the BiHom-coassociativity of ∆: A BiHom-coassociative coalgebra (C, ∆, ψ, ω) is called counital if there exists a linear map ε : C → k (called a counit) such that Similar to Definition 4.3 in [7], we define If (M, ψ M , ω M ) and (N, ψ N , ω N ) are left C-co-modules with coactions ρ M and ρ N , respectively, a morphism of left C-co-modules f : M → N is a linear map satisfying the conditions
We can obtain some properties of the antipode.

The Braiding Structure of the Category of BiHom-Yetter-Drinfeld Modules
In this section, we show that the monoidal category H H BHY D of a left-left BiHom-Yetter-Drinfeld module over a BiHom-Hopf algebra is braided and find that, if (H, σ) is a coquasitriangular BiHom-bialgebra, then the category of left H-co-modules with bijective structure maps turns out to be a subcategory of the category H H BHY D.
Proof. We only check the conditions (2) and (9). For all h, h , g, m ∈ H, we have The proof is finished.
The multiplication is as follows: is an unital BiHom-associative algebra with α H , β H bijective. Next, we construct a counital BiHom-coassociative coalgebra (H, ∆ H , ε H , ω H , ψ H ), which is defined as is a left-left BiHom-Yetter-Drinfeld module over H with the coaction ∆ H and the action: Proof. Equation (9)=⇒ Equation (10). We performed a calculation as follows:
From [11], we know the category H H BHY D is a monoidal category. (N, α N , β N , ψ N , ω N ) be two left-left Yetter-Drinfeld modules over H and define the linear maps · and ρ as follows: , these structures become a left-left BiHom-Yetter-Drinfeld module over H, denoted by M⊗N.
We discuss the braiding structure for the monoidal category H H BHY D in the following theorem.
Proof. We will first show that the braiding c is natural.
For all m ∈ M, n ∈ N, since the morphism g is left H-linear and f is left H-colinear, we obtain This follows (g ⊗ f ) • c M,N = c M ,N • ( f ⊗ g), and the diagram commutes. Next, we prove the H-linear of c M,N : and H-colinear of c M,N : Now, we prove c M,N is an isomorphism with an inverse map For all m ∈ M and n ∈ N, we compute Similarly, we can prove c M,N • c −1 M,N = id N⊗M . Finally, let us verify the hexagon axioms from [9], XIII.1.1. For any (U, α U , β U , ψ U , ω U ), The proof is finished.