Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras

Abstract: In this paper, we investigate a more general category of Θ-Yetter-Drinfeld modules ( H H Θ Aut ∈ ( )) over a Hom-Hopf algebra, which unifies two different definitions of Hom-Yetter-Drinfeld category introduced by Makhlouf and Panaite, Li and Ma, respectively. We show that the category of Θ-Yetter-Drinfeld modules with a bijective antipode S is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category. Also by the method of symmetric pairs, we prove that if a Θ-Yetter-Drinfeld category over a Hom-Hopf algebra H is symmetric, then H is trivial. Finally, we find a sufficient and necessary condition for a Θ-Yetter-Drinfeld category to be pseudosymmetric.

are obtained more unpredictable than the monoidal version. For a Hom-Hopf algebra, there exists two definitions of Yetter-Drinfeld modules in the studies of Makhlouf and Panaite [14] and Li and Ma [27], respectively. It is obvious that these two definitions of Yetter-Drinfeld modules are not equivalent. Surprisingly, such two categories of Yetter-Drinfeld modules are both braided tensor categories and provide solutions of the Yang-Baxter equation and the Hom-Yang-Baxter equation. The motivation of the definition of Yetter-Drinfeld modules by Makhlouf and Panaite relies on the main tool called "twisting principle," which was introduced by Yau [11]. But for the definition by Li and Ma, they gave it by massive calculations to satisfy the corresponding conditions. Naturally inspired by the above two definitions, the authors in [28] gave the definition of v-Yetter-Drinfeld modules, v ∈ ; see (2.15). And they showed that every category of v-Yetter-Drinfeld modules is also a braided tensor category and provides some solutions of the Yang-Baxter equation and the Hom-Yang-Baxter equation. Therefore, the v-Yetter-Drinfeld module unifies the above two definitions of Yetter-Drinfeld modules. Observing the compatibility of v-Yetter-Drinfeld modules, we find that the condition (2.16) is equivalent to the condition (2.15). In other words, we have proven that if − is even, the definition of v-Yetter-Drinfeld is favorable. One natural question to ask is whether the definition of v is favorable or not if p is odd. Furthermore, there exist more general favorable definitions of Yetter-Drinfeld module over a Hom-Hopf algebra. In this paper, we give the definition of Θ-Yetter-Drinfeld modules, H H Θ Aut ∈ ( ) where we denote the group of all Hom-Hopf algebra automorphisms of H by H H Aut ( ) and prove that even if p is odd, the definition of Yetter-Drinfeld is in the same way favorable. Meanwhile, all the definitions of Yetter-Drinfeld modules over a Hom-Hopf algebra that appeared in [14,27,28] can be regarded as special cases of the Θ-Yetter-Drinfeld module.
This paper is organized as follows. In Section 1, we recall some definitions and results which will be used later. In Section 2, after introducing the concept of Θ-Yetter-Drinfeld category H H Θ , we prove that every Θ-Yetter Drinfeld category over a Hom-Hopf algebra with a bijective antipode is a braided tensor category and that every Θ-Yetter Drinfeld category over a Hom-Hopf algebra provides a solution of the Hom-Yang-Baxter equation and the Yang-Baxter equation. In Section 3, by the method of symmetric pairs, we show that symmetric Θ-Yetter-Drinfeld categories H H Θ over a Hom-Hopf algebra are all trivial.
The results obtained in this section generalize the corresponding results in [28][29][30][31]. In Section 4, we find a necessary and sufficient condition for a Θ-Yetter-Drinfeld category H H Θ over a Hom-Hopf algebra H to be pseudosymmetric.

Preliminaries
We work over a base field k. All algebras, linear spaces etc. will be over k, and unadorned ⊗ means k ⊗ . For a comultiplication C C C Δ : → ⊗ on a vector space C, we use a Sweedler-type notation c c c Δ In what follows, we assume that k-linear maps α − are bijective, although some notions are not supposed to be bijective.
α a a a aa α a a a α a , 1 1 are satisfied for a a a A , , ′ ″ ∈ . Here we use the notation μ a a aa ⊗ ′ = ′ ( ) .
A morphism g C α D α : , Δ , , Δ , ) is a Hom-coassociative coalgebra (called a tensor product Hom-coassociative coalgebra) with the comultiplication c d then we call H μ ε α S , , 1 , Δ, , , If where the multiplication and comultiplication of H αH were denoted by h h α hh In [28], the authors gave a more general compatibility condition including the definitions in [14,27] as the special cases.
and v ∈ , there holds the following condition: Remark 2.11. Being similar to Remark 2.9, we readily see that the condition (2.15) is equivalent to

Θ-Yetter-Drinfeld category
In this section, we introduce a more general Θ-Yetter-Drinfeld category ( H H Θ Aut ∈ ( )), which unifies two different definitions of Hom-Yetter-Drinfeld category of Makhlouf and Panaite [14] and Li and Ma [27], respectively. We show that the category of Θ-Yetter-Drinfeld modules with a bijective antipode is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category.
In case that H μ α S , ,Δ , , ) is a Hom-Hopf algebra, we denote the group of all Hom-Hopf algebra automorphisms of H by H H Aut ( ). Inspired by the above three definitions of Hom-Yetter-Drinfeld modules, we can give a more general definition of Hom-Yetter-Drinfeld modules as follows.
∈ and m M ∈ , there holds the following condition: We find an interesting fact that every definition that appeared in references can be regarded as the special cases of Θ-Yetter-Drinfeld module.
and a left H -Hom-coaction by the Hom-comultiplication Δ.
) is a Θ-Yetter-Drinfeld module in Definition 3.1 with a left H -Hom-action by the Hom-multiplication μ and a left H -Hom-coaction Proof. We only establish the first part of the above example here. First, it is easy to see that H α , Δ, )-module. In fact, for any h g l H , , which proves the equality (2.8). For proving (2.9), we observe that Finally, we show the compatible condition (3.1) of Θ-Yetter-Drinfeld module. For this end, the left-hand side of (3.1) can be expressed as and also its right-hand side can be given by yielding the compatible condition (3.1) of Θ-Yetter-Drinfeld module and the proof is complete.

This completes the proof. □
In the following, we give solutions of the Hom-Yang-Baxter introduced and studied by Yau [12].
The property (i) can be easily proven. Now we claim the property (ii): given m M ∈ , n N ∈ and p P ∈ , using the conditions (2.10), ( and the proof is complete. □ Now we will state the main result in this section. Proof. ( ,∘ = − ⊗ , the pentagon axiom and hexagonal relation are satisfied, and so on. We think that it is a good exercise to a reader. For example, now we will prove the Θ-Yetter-Drinfeld compatibility condition (3.1)     In this section, we extend some interesting results to Θ- and it then follows that h α ε h ε h Θ 1 1 This means H k = , as desired. The converse is straightforward. □ The following corollary is easily obtained from the above theorem. Remark 4.4.
(1) If α id = and id Θ = , Theorem 4.2 is exactly the famous conclusion in [30], namely, the symmetric Yetter-Drinfeld category H H over a Hopf algebra is trivial.

Pseudosymmetry of Θ-Yetter-Drinfeld categories over a Hom-Hopf algebra
In this section, we will find a necessary and sufficient condition for a Θ- In this case, is called a pseudosymmetric braided tensor category.
Obviously, a symmetric braided tensor category must be a pseudosymmetric braided tensor category.