数学杂志  2017, Vol. 37 Issue (6): 1161-1172   PDF    
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MA Tian-shui
WANG Yong-zhong
LIU Lin-lin
GENERALIZED RADFORD BIPRODUCT HOM-HOPF ALGEBRAS AND RELATED BRAIDED TENSOR CATEGORIES
MA Tian-shui1,2, WANG Yong-zhong3, LIU Lin-lin1    
1. Department of Mathematics, School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China;
2. Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China;
3. School of Mathematics and Information Science, Xinxiang University, Xinxiang 453003, China
Abstract: In this paper, the Hom-type of Radford biproduct is introduced. By combining generalized smash product Hom-algebra and generalized smash coproduct Hom-coalgebra, we derive necessary and suffcient conditions for them to be a Hom-bialgebra, which includes the well-known Radford biproduct.
Key words: Radford biproduct     quantum Yang-Baxter equation     Yetter-Drinfeld category    
广义Radford双积Hom-Hopf代数和相关辫子张量范畴
马天水1,2, 王永忠3, 刘琳琳1    
1. 河南师范大学数学与信息科学学院数学系, 河南 新乡 453007;
2. 河南师范大学数学与信息科学学院大数据统计分析与优化控制河南省工程实验室, 河南 新乡 453007;
3. 新乡学院数学与信息科学学院, 河南 新乡 453003
摘要:本文研究了Radford双积的Hom-型.通过把广义smash积Hom-代数和广义smash余积Hom-余代数相结合,得到了他们成为Hom-双代数的充分必要条件,这一结果推广了著名的Radford双积.
关键词Radford双积    量子Yang-Baxter方程    Yetter-Drinfeld范畴    
1 Introduction

Let $H$ be a bialgebra, $A\# H$ a smash product algebra and $A\times H$ a smash coproduct coalgebra. Radford (see [13]) gave a bialgebra structure on $A\otimes H$ (named Radford biproduct by other researchers) via $A\# H$ and $A\times H$. Later, Majid made the following conclusion: to any Hopf algebra $A$ in the braided category of Yetter-Drinfeld modules $_H^H{\mathcal{YD}}$, one can associate an ordinary Hopf algebra $A\star H$, there called the bosonization of $A$ (i.e., Radford biproduct) (see [8]). While Radford biproduct is one of the celebrated objects in the theory of Hopf algebras, which plays a fundamental role in the classification of finite-dimensional pointed Hopf algebras (see [1]). Other references related to Radford biproduct see [1, 6-8, 13, 14].

The algebra of Hom-type can be found in [2] by Hartwig, Larsson and Silvestrov, where a notion of Hom-Lie algebra in the context of $q$-deformation theory of Witt and Virasoro algebras (see [3]) was introduced. There are various settings of Hom-structures such as algebras, coalgebras, Hopf algebras, see [6, 10-12] and so on.

In [15], Yau introduced and characterized the concept of module Hom-algebras as a twisted version of usual module algebras. Based on Yau's definition of module Hom-algebras, Ma, Li and Yang [6] constructed smash product Hom-Hopf algebra $(A\natural H,\alpha\otimes \beta)$ generalizing the Molnar's smash product (see [13]), and gave the cobraided structure (in the sense of Yau's definition in [16]) on $(A\natural H,\alpha\otimes \beta)$. Makhlouf and Panaite defined and studied a class of Yetter-Drinfeld modules over Hom-bialgebras in [9] and derived the constructions of twistors, pseudotwistors, twisted tensor product and smash product in the setting of Hom-case (see [10]). Li and Ma studied the Yetter-Drinfeld category of Hom-type via Radford biproduct (see [5]). Recently, Ma, Liu and Li extend the above results in the monoidal Hom-case.

In this paper, we unify the Makhlouf-Panaite's smash product in [10] and Ma-Li-Yang's in [6], and then extend the Radford biproduct to a more general case. We also construct a class of braided tensor categories (extending the Yetter-Drinfeld category to the Hom-case), and provide a solution to the Hom-quantum Yang-Baxter equation.

2 Preliminaries

Throughout this paper, $K$ will be a field, and all vector spaces, tensor products, and homomorphisms are over $K$. We use Sweedler's notation for terminologies on coalgebras. For a coalgebra $C$, we write comultiplication $\Delta(c)=c_1\otimes c_2$ for any $c\in C$. And we denote $Id_M$ for the identity map from $M$ to $M$. Any unexplained definitions and notations can be found in [4-6, 14]. We now recall some useful definitions.

Definition 2.1 A Hom-algebra is a quadruple $(A,\mu,1_A,\alpha)$ (abbr. $(A,\alpha)$), where $A$ is a linear space, $\mu: A\otimes A \longrightarrow A$ is a linear map, $1_A \in A$ and $\alpha$ is an automorphism of $A$, such that

$\begin{eqnarray*} &(HA1)& \alpha(aa')=\alpha(a)\alpha(a'); \alpha(1_A)=1_A,\\ &(HA2)& \alpha(a)(a'a'')=(aa')\alpha(a''); a1_A=1_Aa=\alpha(a) \end{eqnarray*}$

are satisfied for $a,a',a''\in A$. Here we use the notation $\mu(a\otimes a')=aa'$.

Let $(A,\alpha)$ and $(B,\beta)$ be two Hom-algebras. Then $(A\otimes B,\alpha\otimes \beta)$ is a Hom-algebra (called tensor product Hom-algebra) with the multiplication $(a\otimes b)(a'\otimes b')=aa'\otimes bb'$ and unit $1_A\otimes 1_B$.

Definition 2.2 A Hom-coalgebra is a quadruple $(C,\Delta,\varepsilon_C,\beta)$ (abbr. $(C,\beta)$), where $C$ is a linear space, $\Delta: C \longrightarrow C\otimes C$, $\varepsilon_C: C\longrightarrow K$ are linear maps, and $\beta$ is an automorphism of $C$, such that

$\begin{eqnarray*} &(HC1)& \beta(c)_1\otimes \beta(c)_2=\beta(c_1)\otimes \beta(c_2); \varepsilon_C\circ \beta=\varepsilon_C,\\ &(HC2)& \beta(c_{1})\otimes c_{21}\otimes c_{22}=c_{11}\otimes c_{12}\otimes \beta(c_{2}); \varepsilon_C(c_1)c_2=c_1\varepsilon_C(c_2)=\beta(c) \end{eqnarray*}$

are satisfied for $c \in A$. Here we use the notation $\Delta(c)=c_1\otimes c_2$ (summation implicitly understood).

Let $(C,\alpha)$ and $(D,\beta)$ be two Hom-coalgebras. Then $(C\otimes D,\alpha\otimes \beta)$ is a Hom-coalgebra (called tensor product Hom-coalgebra) with the comultiplication $\Delta(c\otimes d)=c_1\otimes d_1\otimes c_2\otimes d_2$ and counit $\varepsilon_C\otimes \varepsilon_D$.

Definition 2.3 A Hom-bialgebra is a sextuple $(H,\mu,1_H,\Delta,\varepsilon,\gamma)$ (abbr. $(H,\gamma)$), where $(H,\mu,1_H,\gamma)$ is a Hom-algebra and $(H,\Delta,\varepsilon,\gamma)$ is a Hom-coalgebra, such that $\Delta$ and $\varepsilon$ are morphisms of Hom-algebras, i.e., $\Delta(hh')=\Delta(h)\Delta(h');$ $\Delta(1_H)=1_H\otimes 1_H,$ $\varepsilon(hh')=\varepsilon(h)\varepsilon(h');$ $\varepsilon(1_H)=1.$ Furthermore, if there exists a linear map $S: H\longrightarrow H$ such that

$ S(h_1)h_2=h_1S(h_2)=\varepsilon(h)1_H \hbox{and} S(\gamma(h))=\gamma(S(h)), $

then we call $(H,\mu,1_H,\Delta,\varepsilon,\gamma,S)$(abbr. $(H,\gamma,S)$) a Hom-Hopf algebra.

Let $(H,\gamma)$ and $(H',\gamma')$ be two Hom-bialgebras. The linear map $f: H\longrightarrow H'$ is called a Hom-bialgebra map if $f\circ \gamma=\gamma'\circ f$ and at the same time $f$ is a bialgebra map in the usual sense.

Definition 2.4 Let $(A,\beta)$ be a Hom-algebra. A left $(A,\beta)$-Hom-module is a triple $(M,\rhd,\alpha)$, where $M$ is a linear space, $\rhd: A\otimes M \longrightarrow M$ is a linear map, and $\alpha$ is an automorphism of $M$, such that

$\begin{eqnarray*} &(HM1)& \alpha(a\rhd m)=\beta(a)\rhd \alpha(m),\\ &(HM2)& \beta(a)\rhd (a'\rhd m)=(aa')\rhd \alpha(m); 1_A\rhd m=\alpha(m) \end{eqnarray*}$

are satisfied for $a,a' \in A$ and $m\in M$.

Let $(M,\rhd_M,\alpha_M)$ and $(N,\rhd_N,\alpha_N)$ be two left $(A,\beta)$-Hom-modules. Then a linear morphism $f: M\longrightarrow N$ is called a morphism of left $(A,\beta)$-Hom-modules if $f(h\rhd_M m)=h\rhd_N f(m)$ and $\alpha_N\circ f=f\circ \alpha_M$.

Definition 2.5 Let $(H,\beta)$ be a Hom-bialgebra and $(A,\alpha)$ a Hom-algebra. If $(A,\rhd,\alpha)$ is a left $(H,\beta)$-Hom-module and for all $h\in H$ and $a,a'\in A$,

$\begin{eqnarray*} &(HMA1)& \beta^{2}(h)\rhd (aa')=(h_1\rhd a)(h_2\rhd a'),\\ &(HMA2)& h\rhd 1_A=\varepsilon_H(h)1_A, \end{eqnarray*}$

then $(A,\rhd,\alpha)$ is called an $(H,\beta)$-module Hom-algebra.

Definition 2.6 Let $(C,\beta)$ be a Hom-coalgebra. A left $(C,\beta)$-Hom-comodule is a triple $(M,\rho,\alpha)$, where $M$ is a linear space, $\rho: M\longrightarrow C\otimes M$ (write $\rho(m)=m_{-1}\otimes m_0,\forall m\in M$) is a linear map, and $\alpha$ is an automorphism of $M$, such that

$\begin{eqnarray*} &(HCM1)&\alpha(m)_{-1}\otimes \alpha(m)_{0}=\beta(m_{-1})\otimes \alpha(m_{0}),\\ &(HCM2)&\beta(m_{-1})\otimes m_{0-1}\otimes m_{00}= m_{-11}\otimes m_{-12}\otimes \alpha(m_{0}); \varepsilon_C(m_{-1})m_{0}=\alpha(m) \end{eqnarray*}$

are satisfied for all $m\in M$.

Let $(M,\rho^M,\alpha_M)$ and $(N,\rho^N,\alpha_N)$ be two left $(C,\beta)$-Hom-comodules. Then a linear map $f: M\longrightarrow N$ is called a map of left $(C,\beta)$-Hom-comodules if $f(m)_{-1}\otimes f(m)_{0}=m_{-1}\otimes f(m_{0})$ and $\alpha_N\circ f=f\circ \alpha_M$.

Definition 2.7 Let $(H,\beta)$ be a Hom-bialgebra and $(C,\alpha)$ a Hom-coalgebra. If $(C,\rho,\alpha)$ is a left $(H,\beta)$-Hom-comodule and for all $c\in C$,

$\begin{eqnarray*} &(HCMC1)&\beta^{2}(c_{-1})\otimes c_{01}\otimes c_{02}=c_{1-1}c_{2-1}\otimes c_{10}\otimes c_{20},\\ &(HCMC2)&c_{-1}\varepsilon_C(c_0)=1_H\varepsilon_C(c), \end{eqnarray*}$

then $(C,\rho,\alpha)$ is called an $(H,\beta)$-comodule Hom-coalgebra.

Definition 2.8 Let $(H,\beta)$ be a Hom-bialgebra and $(C,\alpha)$ a Hom-coalgebra. If $(C,\rhd,\alpha)$ is a left $(H,\beta)$-Hom-module and for all $h\in H$ and $c\in A$,

$\begin{eqnarray*} &(HMC1)&(h\rhd c)_1\otimes (h\rhd c)_2=(h_1\rhd c_1)\otimes (h_2\rhd c_2),\\ &(HMC2)&\varepsilon_C(h\rhd c)=\varepsilon_H(h)\varepsilon_C(c), \end{eqnarray*}$

then $(C,\rhd,\alpha)$ is called an $(H,\beta)$-module Hom-coalgebra.

Definition 2.9 Let $(H,\beta)$ be a Hom-bialgebra and $(A,\alpha)$ a Hom-algebra. If $(A,\rho,\alpha)$ is a left $(H,\beta)$-Hom-comodule and for all $a,a'\in A$,

$\begin{eqnarray*} &(HCMA1)&\rho(aa')=a_{-1}a'_{-1}\otimes a_{0}a'_{0},\\ &(HCMA2)&\rho(1_A)=1_H\otimes 1_A, \end{eqnarray*}$

then $(A,\rho,\alpha)$ is called an $(H,\beta)$-comodule Hom-algebra.

3 Generalized Radford Biproduct Hom-Hopf Algebra

In this section, we first introduce the notions of generalized smash product Hom-algebra $A\sharp^m H$ and generalized Hom-smash coproduct Hom-coalgebra $A\natural_n H$. Then the necessary and sufficient conditions for $A\sharp^m H$ and $A\natural_n H$ on $A\otimes H$ to be a Hom-bialgebra structure are derived.

Proposition 3.1 Let $(H,\beta)$ be a Hom-bialgebra, $(A,\triangleright,\alpha)$ an $(H,\beta)$-module Hom-algebra and $m\in \mathcal{Z}$. Then $(A\sharp^m H,\alpha\otimes\beta)$ ($A\sharp^m H=A\otimes H$ as a linear space) with the multiplication $ (a\otimes h)(a'\otimes h')=a(\beta^{m}(h_1)\triangleright\alpha^{-1}(a'))\otimes \beta^{-1}(h_2)h', $ where $a,a'\in A,h,h'\in H$, and unit $1_A\otimes 1_H$ is a Hom-algebra. In this case, we call $(A\sharp^m H,\alpha\otimes\beta)$ generalized smash product Hom-algebra.

Proof It is straightforward by the definition of Hom-algebra.

Remarks (1) Noting that $(A\sharp^0 H,\alpha\otimes \beta)$ is exactly the Ma-Li-Yang's Hom-smash product in [5, 6] and $(A\sharp^{-2} H,\alpha\otimes \beta)$ is exactly the Makhlouf-Panaite's Hom-smash product in [10].

(2) If $\alpha=Id_A$ and $\beta=Id_H$ in $(A\sharp^m H,\alpha\otimes \beta)$, then one can obtain the usual smash product $A\# H$ in [13].

(3) Let $(H,\mu_{H},\Delta_{H})$ be a bialgebra and $(A,\alpha)$ a left $H$-module algebra in the usual sense with action denoted by $H\otimes A\rightarrow A,h\otimes a\mapsto h\cdot a$. Let $\beta: H\rightarrow H$ be a bialgebra endomorphism and $\alpha: A\rightarrow A$ an algebra endomorphism, such that $\alpha(h\cdot a)=\beta(h)\cdot\alpha(a)$ for all $h\in H$ and $a\in A$. If we consider the Hom-bialgebra $H_{\beta}=(H,\beta\circ\mu_{H},\Delta_{H}\circ\beta,\beta)$ and the Hom-associative algebra $A_{\alpha}=(A,\alpha\circ\mu_{H},\alpha)$, then $(A_{\alpha},\alpha)$ is a left $(H_{\beta},\beta)$-module Hom-algebra with action $ H_{\beta}\otimes A_{\alpha}\rightarrow A_{\alpha},h\otimes a\mapsto h\triangleright a:=\alpha(h\cdot a)=\beta(h)\cdot\alpha(a). $

Proof Straightforward.

Proposition 3.2 Let $(H,\beta)$ be a Hom-bialgebra, $(C,\rho,\alpha)$ an $(H,\beta)$-comodule Hom-coalgebra and $n\in \mathcal{Z}$. Then $(C\natural H,\alpha\otimes\beta)$ ($C\natural H=C\otimes H$ as a linear space) with the comultiplication $ \Delta_{C\natural H}(c\otimes h)=c_1\otimes \beta^{n}(c_{2(-1)})\beta^{-1}(h_1)\otimes \alpha^{-1}(c_{2(0)})\otimes h_2, $ where $c\in C,h\in H$, and counit $\varepsilon_{C}\otimes \varepsilon_{H}$ is a Hom-coalgebra. In this case, we call $(A\natural_n H,\alpha\otimes \beta)$ generalized smash coproduct Hom-coalgebra.

Proof Straightforward.

Remarks (1) $(A\natural_0 H,\alpha\otimes \beta)$ is exactly the Li-Ma's Hom-smash coproduct in [5].

(2) $(A\natural_{-2} H,\alpha\otimes \beta)$ is exactly the dual version of the Makhlouf-Panaite's Hom-smash product in [10].

(3) If $\alpha=Id_A$ and $\beta=Id_H$ in $(A\sharp^m H,\alpha\otimes \beta)$, then one can obtain the usual smash coproduct $A\times H$ in [13].

Theorem 3.3 Let $(H,\beta)$ be a Hom-bialgebra, $(A,\alpha)$ a left $(H,\beta)$-module Hom-algebra with module structure $\triangleright:H\otimes A\longrightarrow A$ and a left $(H,\beta)$-comodule Hom-coalgebra with comodule structure $\rho: A \longrightarrow H\otimes A$. Then the following are equivalent:

(i) $(A\diamondsuit^m_n H ,\mu_{A\sharp H},1_A\otimes 1_H,\Delta_{A\natural H},\varepsilon_{A}\otimes \varepsilon_{H},\alpha\otimes \beta)$ is a Hom-bialgebra, where $(A\sharp^m H,\alpha\otimes \beta)$ is a generalized smash product Hom-algebra and $(A\natural_n H,\alpha\otimes \beta)$ is a generalized smash coproduct Hom-coalgebra.

(ii) The following conditions hold:

(R1) $(A,\rho,\alpha) \hbox{ is an } (H,\beta)\hbox{-comodule Hom-algebra};$

(R2) $(A,\triangleright,\alpha) \hbox{ is an } (H,\beta)\hbox{-module Hom-coalgebra};$

(R3) $\varepsilon_{A} \hbox{ is a Hom-algebra map and } \Delta_{A}(1_A)=1_A\otimes 1_A;$

(R4) $\Delta_A(a b)=a_1(\beta^{m+n+2}(a_{2(-1)})\triangleright\alpha^{-1}(b_1))\otimes \alpha^{-1}(a_{2(0)})b_2;$

(R5) $\beta^{n+1}((\beta^{m+1}(h_1)\triangleright b)_{-1})h_2\otimes (\beta^{m+1}(h_1)\triangleright b)_{0}=h_1\beta^{n+2}(b_{(-1)})\otimes \beta^{m+2}(h_2)\triangleright b_{(0)},$

where $a,b\in B$, $h\in H$ and $m,n\in \mathcal{Z}$. In this case, we call $(A\diamondsuit^m_n H,\alpha\otimes \beta)$ generalized Radford biproduct Hom-bialgebra.

Proof By a tedious computation we can prove it.

Remarks (1) When $m=n=0$ in Theorem 3.3, we can get [5, Theorem 3.3].

(2) When $\alpha=Id_A$ and $\beta=Id_H$ in Theorem 3.3, then one can obtain [13, Theorem 1].

Proposition 3.4 Let $(H,\beta,S_{H})$ be a Hom-Hopf algebra, and $(A,\alpha)$ ba a Hom-algebra and a Hom-coalgebra. Assume that $(A\diamondsuit^m_n H,\alpha\otimes\beta)$ is a generalized Radford biproduct Hom-bialgebra defined as above, and $S_{A}:A\rightarrow A$ is a linear map such that $S_{A}(a_1)a_2=a_1S_{A}(a_2)=\varepsilon_{A}(a)1_{A}$ and $\alpha\circ S_{A}=S_{A}\circ\alpha$ hold. Then $(A\diamondsuit^m_n H,\alpha\otimes\beta,S_{A\diamondsuit^m_n H})$ is a Hom-Hopf algebra, where

$\begin{eqnarray*} S_{A\diamondsuit^m_n H}(a\otimes h)=(\beta^{m}(S_{H}(\beta^{n}(a_{(-1)})\beta^{-1}(h))_{1})\triangleright S_{A}(\alpha^{-2}(a_{(0)})))\otimes \beta^{-1}(S_{H}(\beta^{n}(a_{(-1)})\beta^{-1}(h))_{2}). \end{eqnarray*}$

Proof For all $a\in A,h\in H$, we have

$\begin{eqnarray*} &&(S_{A\diamondsuit^m_n H}\ast Id_{A\diamondsuit^m_n H})(a\otimes h)\\ &=&S_{A\diamondsuit^m_n H}(a_1\otimes \beta^{n}(a_{2(-1)})\beta^{-1}(h_1))(\alpha^{-1}(a_{(0)})\otimes h_2)\\ &=&((\beta^{m}(S_{H}(\beta^{n}(a_{1(-1)})\beta^{-1}(\beta^{n}(a_{2(-1)})\beta^{-1}(h_1)))_{1})\triangleright S_{A}(\alpha^{-2}(a_{1(0)})))\\ &&\otimes\beta^{-1}(S_{H}(\beta^{n}(a_{1(-1)})\beta^{-1}(\beta^{n}(a_{2(-1)})\beta^{-1}(h_1)))_{2}))(\alpha^{-1}(a_{2(0)})\otimes h_2)\\ &=&(\beta^{m}(S_{H}(\beta^{n}(a_{1(-1)})\beta^{-1}(\beta^{n}(a_{2(-1)})\beta^{-1}(h_1)))_{1})\triangleright S_{A}(\alpha^{-2}(a_{1(0)})))\\ &&\times(\beta^{m}(\beta^{-1}(S_{H}(\underline{\beta^{n}(a_{1(-1)})\beta^{-1}(\beta^{n}(a_{2(-1)})\beta^{-1}(h_1)}))_{2})_1)\triangleright \alpha^{-2}(a_{2(0)}))\\ &&\otimes \beta^{-1}(\beta^{-1}(S_{H}(\beta^{n}(a_{1(-1)})\beta^{-1}(\beta^{n}(a_{2(-1)})\beta^{-1}(h_1)))_{2})_2)h_2\\ &\stackrel{(HA2)}{=}&(\beta^{m}(S_{H}(\beta^{n-1}(\underline{a_{1(-1)}a_{2(-1)}})\beta^{-1}(h_1))_{1})\triangleright S_{A}(\alpha^{-2}(\underline{a_{1(0)}})))\\ &&\times (\beta^{m}(\beta^{-1}(S_{H}(\beta^{n-1}(\underline{a_{1(-1)}a_{2(-1)}})\beta^{-1}(h_1))_{2})_1)\triangleright \alpha^{-2}(\underline{a_{2(0)}}))\\ &&\otimes \beta^{-1}(\beta^{-1}(S_{H}(\beta^{n-1}(a_{1(-1)}a_{2(-1)})\beta^{-1}(h_1))_{2})_2)h_2\\ &\stackrel{(HCMC1)}{=}&(\beta^{m}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{1})\triangleright S_{A}(\alpha^{-2}(a_{(0)1})))\underline{(\beta^{m}(\beta^{-1}(S_{H}(\beta^{n+1}(a_{(-1)})}\\ &&\underline{\times \beta^{-1}(h_1))_{2})_1})\triangleright \alpha^{-2}(a_{(0)2}))\otimes\underline{\beta^{-1}(\beta^{-1}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{2})_2)}h_2\\ &\stackrel{(HC1)}{=}&(\beta^{m}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{1})\triangleright S_{A}(\alpha^{-2}(a_{(0)1})))(\beta^{m-1}(S_{H}(\beta^{n+1}(a_{(-1)})\\ &&\times \beta^{-1}(h_1))_{21})\triangleright \alpha^{-2}(a_{(0)2}))\otimes\beta^{-2}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{22})h_2\\ &\stackrel{(HC2)}{=}&\underline{(\beta^{m-1}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{11})\triangleright S_{A}(\alpha^{-2}(a_{(0)1})))(\beta^{m-1}(S_{H}(\beta^{n+1}(a_{(-1)})}\\ &&\underline{\times \beta^{-1}(h_1))_{12})\triangleright \alpha^{-2}(a_{(0)2}))}\otimes\beta^{-1}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{2})h_2\\ &\stackrel{(HMA1)}{=}&(\beta^{m+1}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{1})\triangleright (S_{A}(\alpha^{-2}(a_{(0)1}))\alpha^{-2}(a_{(0)2}))\\ &&\otimes\beta^{-1}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{2})h_2\\ &=&\beta^{m+1}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{1})\triangleright 1_{A}\varepsilon_{A}(a_{(0)})\\ &&\otimes\beta^{-1}(S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))_{2})h_2\\ &=&1_{A}\varepsilon_{A}(a_{(0)})\otimes S_{H}(\beta^{n+1}(a_{(-1)})\beta^{-1}(h_1))h_2=1_{A}\varepsilon_{A}(a)\otimes S_{H}(h_1)h_2\\ &=&(1_A\otimes 1_H)\varepsilon_{A}(a)\varepsilon_{H}(h), \end{eqnarray*}$

and the rest is direct.

4 Generalized Hom-Yetter-Drinfeld Category

In this section, we construct a class of braided tensor category, which extends the Yetter-Drinfeld category to the Hom-case. Next we give the concept of Hom-Yetter-Drinfeld module via generalized Radford biproduct Hom-Hopf algebra defined in Theorem 3.3.

Definition 4.1 Let $(H,\beta)$ be a Hom-bialgebra, $(U,\triangleright_{U},\alpha_{U})$ a left $(H,\beta)$-module with action $\triangleright_{U}: H\otimes U\rightarrow U,h\otimes u\mapsto h\triangleright_{U} u$ and $(U,\rho^{U},\alpha_{U})$ a left $(H,\beta)$-comodule with coaction $\rho^{U}: U\rightarrow H\otimes U,u\mapsto u_{(-1)}\otimes u_{(0)}$. Then we call $(U,\triangleright_{U},\rho^{U},\alpha_{U})$ a (left-left) Hom-Yetter-Drinfeld module over $(H,\beta)$ if the following condition holds:

$ h_{1}\beta^{n+2}(u_{(-1)})\otimes\beta^{m+2}(h_{2})\triangleright u_{(0)}=\beta^{n+1}((\beta^{m+1}(h_1)\triangleright u)_{(-1)})h_2\otimes(\beta^{m+1}(h_1)\triangleright u)_{(0)}\\(HYD) $

for all $h\in H$ and $u\in U$.

Proposition 4.2 When $(H,\beta)$ is a Hom-Hopf algebra, $(HYD)$ is equivalent to

$ \rho(\beta^{m+3}(h)\triangleright u)=(\beta^{-n-3}(h_{11})\beta^{-1}(u_{(-1)}))S(\beta^{-n-1}(h_{2}))\otimes \beta^{m+2}(h_{12})\triangleright u_{(0)}\\(HYD)' $

for all $h\in H,u\in U$.

Proof $(HYD)\Longrightarrow (HYD)'$. We have

$\begin{eqnarray*} &&(\beta^{-n-3}(h_{11})\beta^{-1}(u_{(-1)}))S(\beta^{-n-1}(h_{2}))\otimes \beta^{m+2}(h_{12})\triangleright u_{(0)}\\ &=&\beta^{-n-1}(\beta^{-2}\underline{(h_{11}\beta^{n+2}(u_{(-1)}))}S(h_{2}))\otimes \underline{\beta^{m+2}(h_{12})\triangleright u_{(0)}}\\ &\stackrel{(HYD)}{=}&\beta^{-n-1}(\underline{\beta^{-2}(\beta^{n+1}((\beta^{m+1}(h_{11})\triangleright u)_{(-1)})S(h_{2}))})\otimes (\beta^{m+1}(h_{11})\triangleright u)_{(0)}\\ &\stackrel{(HA1)}{=}&\beta^{-n-1}(\underline{(\beta^{n-1}((\beta^{m+1}(h_{11})\triangleright u)_{(-1)})\beta^{-2}(h_{12}))S(h_{2})})\otimes (\beta^{m+1}(h_{11})\triangleright u)_{(0)}\\ &\stackrel{(HA2)}{=}&\beta^{-n-1}(\beta^{n}((\beta^{m+1}(h_{11})\triangleright u)_{(-1)})(\beta^{-2}(h_{12})S(\beta^{-1}(h_{2}))))\otimes (\beta^{m+1}(h_{11})\triangleright u)_{(0)}\\ &\stackrel{(HC1)}{=}&\beta^{-n-1}(\beta^{n}((\beta^{m+2}(h_{1})\triangleright u)_{(-1)})(\beta^{-2}(h_{21}))S(\beta^{-2}(h_{22})))\otimes (\beta^{m+2}(h_{1})\triangleright u)_{(0)}\\ &=&(\beta^{m+3}(h)\triangleright u)_{(-1)}\otimes(\beta^{m+3}(h)\triangleright u)_{(0)}. \end{eqnarray*}$

$(HYD)'\Longrightarrow (HYD)$ is proved as follows:

$\begin{eqnarray*} &&\beta^{n+1}((\beta^{m+1}(h_1)\triangleright u)_{(-1)})h_2\otimes(\beta^{m+1}(h_1)\triangleright u)_{(0)}\\ &=&\beta^{n+1}(\underline{(\beta^{m+3}(\beta^{-2}(h_1))\triangleright u)_{(-1)}})h_2\otimes\underline{(\beta^{m+3}(\beta^{-2}(h_1))\triangleright u)_{(0)}}\\ &\stackrel{(HYD)'}{=}&((\beta^{-4}(h_{111})\beta^{n}(u_{(-1)}))S(\beta^{-2}(h_{12})))h_2\otimes\beta^{m}(h_{112})\triangleright u_{(0)}\\ &\stackrel{(HC2)}{=}&\underline{((\beta^{-2}(h_{1})\beta^{n}(u_{(-1)}))S(\beta^{-3}(h_{221})))\beta^{-2}(h_{222})}\otimes\beta^{m+1}(h_{21})\triangleright u_{(0)}\\ &\stackrel{(HA2)}{=}&(\beta^{-1}(h_{1})\beta^{n+1}(u_{(-1)}))(S(\beta^{-3}(h_{221}))\beta^{-3}(h_{222}))\otimes\beta^{m+1}(h_{21})\triangleright u_{(0)}\\ &=&h_{1}\beta^{n+2}(u_{(-1)})\otimes\beta^{m+2}(h_{2})\triangleright u_{(0)}, \end{eqnarray*}$

finishing the proof.

Definition 4.3 Let $(H,\beta)$ be a Hom-bialgebra. We denote by $^{H}_{H}\mathbb{YD}$ the category whose objects are Hom-Yetter-Drinfeld modules $(U,\triangleright_{U},\rho^{U},\alpha_{U})$ over $(H,\beta)$; the morphisms in the category are morphisms of left $(H,\beta)$-modules and left $(H,\beta)$-comodules.

In the following, we give a solution to the Hom-quantum Yang-Baxter equation introduced and studied by Yau in [16].

Proposition 4.4 Let $(H,\beta)$ be a Hom-bialgebra and $(U,\triangleright_{U},\rho^{U},\alpha_{U})$, $(V,\triangleright_{V},\rho^{V},\alpha_{V})$ $\in ^{H}_{H}\mathbb{YD}$. Define the linear map

$ \tau_{U,V}: U\otimes V\rightarrow V\otimes U,u\otimes v\mapsto\beta^{m+n+3}(u_{(-1)})\triangleright_{V}v\otimes u_{(0)}, $

where $u\in U$ and $v\in V$. Then we have $\tau_{U,V}\circ(\alpha_{U}\otimes\alpha_{V})=(\alpha_{V}\otimes\alpha_{U})\circ\tau_{U,V}$, if $(W,\triangleright_{W},\rho^{W},\alpha_{W})\in ^{H}_{H}\mathbb{YD}$, the map $\tau\_,\_$ satisfy the Hom-Yang-Baxter equation

$ (\alpha_{W}\otimes \tau_{U,V})\circ(\tau_{U,W}\otimes \alpha_{V})\circ(\alpha_{U}\otimes\tau_{V,W}) =(\tau_{V,W}\otimes\alpha_{U})\circ(\alpha_{V}\otimes\tau_{U,W})\circ(\tau_{U,V}\otimes\alpha_{W}). $

Proof It is easy to prove the first equality, so we only check the second one. For all $u\in U,v\in V$ and $w\in W$, we have

$\begin{eqnarray*} &&(\alpha_{W}\otimes \tau_{U,V})\circ(\tau_{U,W}\otimes \alpha_{V})\circ(\alpha_{U}\otimes \tau_{V,W})(u\otimes v\otimes w)\\ &=&\alpha_{W}(\beta^{m+n+3}(\alpha_{U}(u)_{(-1)}\triangleright_{W}(\beta^{m+n+3}(v_{(-1)})\triangleright_{W}w))\otimes\beta^{m+n+3}(\alpha_{U}(u)_{(0)(-1)}\\ &&\triangleright_{V}\alpha_{V}(v_{(0)})\otimes\alpha_{U}(u)_{(0)(0)}\\ &=&\beta^{m+n+5}(u_{(-1)})\triangleright_{W}(\beta^{m+n+4}(v_{(-1)})\triangleright_{W}\alpha_{W}(w))\otimes\beta^{m+n+4}(u_{(0)(-1)})\triangleright_{V}\alpha_{V} (v_{(0)})\\ &&\otimes\alpha_{U}(u_{(0)(0)})\\ &=&\beta^{m+n+4}(u_{(-1)1})\triangleright_{W}(\beta^{m+n+4}(v_{(-1)})\triangleright_{W}\alpha_{W}(w))\otimes\beta^{m+n+4}(u_{(-1)2})\triangleright_{V}\alpha_{V} (v_{(0)})\\ &&\otimes\alpha^{2}_{U}(u_{(0)})\\ &=&((\beta^{m+n+3}(u_{(-1)1})\beta^{m+n+4}(v_{(-1)}))\triangleright_{W}\alpha^{2}_{W}(w))\otimes\beta^{m+n+4}(u_{(-1)2})\triangleright_{V}\alpha_{V}(v_{(0)})\\ &&\otimes\alpha^{2}_{U}(u_{(0)})\\ &=&(\beta^{m+n+3}(u_{(-1)1}\alpha_{V}(v)_{(-1)})\triangleright_{W}\alpha^{2}_{W}(w))\otimes\beta^{m+n+4}(u_{(-1)2})\triangleright_{V}\alpha_{V}(v)_{(0)}\\ &&\otimes\alpha^{2}_{U}(u_{(0)})\\ &=&(\beta^{m+1}(\underline{\beta^{n+2}(u_{(-1)})_1\beta^{n+2}(\alpha_{V}(v)_{(-1)})})\triangleright_{W}\alpha^{2}_{W}(w))\otimes\underline{\beta^{m+2}(\beta^{n+2}(u_{(-1)})_2)}\\ &&\underline{\triangleright_{V}\alpha_{V}(v)_{(0)}}\otimes\alpha^{2}_{U}(u_{(0)})\\ &=&(\beta^{m+1}(\beta^{n+1}((\beta^{m+1}(\beta^{n+2}(u_{(-1)1}))\triangleright_{V}\alpha_{V}(v))_{(-1)})\beta^{n+2}(u_{(-1)2})) \triangleright_{W}\alpha^{2}_{W}(w))\\ &&\otimes (\beta^{m+1}(\beta^{n+2}(u_{(-1)1}))\triangleright_{V}\alpha_{V}(v))_{(0)}\otimes\alpha^{2}_{U}(u_{(0)})\\ &\stackrel{(HYD)}{=}&(\beta^{m+n+2}((\beta^{m+n+3}(u_{(-1)1})\triangleright_{V}\alpha_{V}(v))_{(-1)})\beta^{m+n+3}(u_{(-1)2}))\triangleright_{W}\alpha^{2}_{W}(w)\\ &&\otimes (\beta^{m+n+3}(u_{(-1)1})\triangleright_{V}\alpha_{V}(v))_{(0)}\otimes\alpha^{2}_{U}(u_{(0)})\\ &=&(\beta^{m+n+2}((\beta^{m+n+4}(u_{(-1)})\triangleright_{V}\alpha_{V}(v))_{(-1)})\beta^{m+n+3}(u_{(0)(-1)}))\triangleright_{W}\alpha^{2}_{W}(w))\\ &&\otimes (\beta^{m+n+4}(u_{(-1)})\triangleright_{V}\alpha_{V}(v))_{(0)}\otimes\alpha_{U}(u_{(0)(0)})\\ &=&\beta^{m+n+3}((\beta^{m+n+4}(u_{(-1)})\triangleright_{V}\alpha_{V}(v))_{(-1)})\triangleright_{W}(\beta^{m+n+3}(u_{(0)(-1)})\triangleright_{W}\alpha^{2}_{W}(w))\\ &&\otimes (\beta^{m+n+4}(u_{(-1)})\triangleright_{V}\alpha_{V}(v))_{(0)}\otimes\alpha_{U}(u_{(0)(0)})\\ &=&(\tau_{V,W}\otimes\alpha_{U})\circ(\alpha_{V}\otimes\tau_{U,W})\circ(\tau_{U,V}\otimes\alpha_{W})(u\otimes v\otimes w). \end{eqnarray*}$

The proof is completed.

Lemma 4.5 Let $(H,\beta)$ be a Hom-bialgebra, if $(U,\triangleright_{U},\rho^{U},\alpha_{U}),(V,\triangleright_{V},\rho^{V},\alpha_{V})$ are (left-left) Hom-Yetter-Drinfeld modules, then $(U\otimes V,\triangleright_{U\otimes V},\rho^{U\otimes V},\alpha_{U}\otimes \alpha_{V})$ is a Hom-Yetter-Drinfeld module with structures

$ \triangleright_{U\otimes V}: H\otimes U\otimes V\rightarrow U\otimes V,h\otimes u\otimes v\mapsto(h_1\triangleright_{U}u)\otimes(h_2\triangleright_{V}v) $

and

$ \rho^{U\otimes V}:U\otimes V\rightarrow H\otimes U\otimes V,u\otimes v\mapsto\beta^{-2}(u_{(-1)}v_{(-1)})\otimes u_{(0)}\otimes v_{(0)} $

for all $h\in H,u\in U,v\in V$.

Proof It is easy to check that $(U\otimes V,\triangleright_{U\otimes V},\alpha_{U}\otimes \alpha_{V})$ is an $(H,\beta)$-Hom module and $(U\otimes V,\rho^{U\otimes V},\alpha_{U}\otimes \alpha_{V})$ is an $(H,\beta)$-Hom comodule. Now we check the condition $(HYD)$. For all $h\in H,u\in U,v\in V$, we have

$\begin{eqnarray*} &&\beta^{n+1}((\beta^{m+1}(h_1)\triangleright(u\otimes v))_{(-1)})h_2\otimes(\beta^{m+1}(h_1)\triangleright(u\otimes v))_{(0)}\\ &=&\beta^{n+1}((\beta^{m+1}(h_{11})\triangleright u\otimes\beta^{m+1}(h_{12})\triangleright v)_{(-1)})h_2\otimes(\beta^{m+1}(h_{11})\triangleright u\\ &&\otimes\beta^{m+1}(h_{12})\triangleright v)_{(0)}\\ &=&\beta^{n-1}((\beta^{m+1}(h_{11})\triangleright u)_{(-1)}(\beta^{m+1}(h_{12})\triangleright v)_{(-1)})h_2\otimes(\beta^{m+1}(h_{11})\triangleright u)_{(0)}\\ &&\otimes(\beta^{m+1}(h_{12})\triangleright v)_{(0)}\\ &=&[\beta^{n-1}((\beta^{m+1}(h_{11})\triangleright u)_{(-1)})\beta^{n-1}((\beta^{m+1}(h_{12})\triangleright v)_{(-1)})]h_2\otimes(\beta^{m+1}(h_{11})\triangleright u)_{(0)}\\ &&\otimes(\beta^{m+1}(h_{12})\triangleright v)_{(0)}\\ &=&\beta^{n}((\beta^{m+1}(h_{11})\triangleright u)_{(-1)})[\beta^{n-1}((\beta^{m+1}(h_{12})\triangleright v)_{(-1)})\beta^{-1}(h_2)]\otimes(\beta^{m+1}(h_{11})\triangleright u)_{(0)}\\ &&\otimes(\beta^{m+1}(h_{12})\triangleright v)_{(0)}\\ &=&\beta^{n}((\beta^{m}(h_{1})\triangleright u)_{(-1)})[\beta^{n-1}((\beta^{m+1}(h_{21})\triangleright v)_{(-1)})\beta^{-2}(h_{22})]\otimes(\beta^{m}(h_{1})\triangleright u)_{(0)}\\ &&\otimes(\beta^{m+1}(h_{21})\triangleright v)_{(0)}\\ &=&\beta^{n}((\beta^{m}(h_{1})\triangleright u)_{(-1)})\beta^{-2}[\underline{\beta^{n+1}((\beta^{m+1}(h_{21})\triangleright v)_{(-1)})h_{22}}]\otimes(\beta^{m}(h_{1})\triangleright u)_{(0)}\\ &&\otimes\underline{(\beta^{m+1}(h_{21})\triangleright v)_{(0)}}\\ &\stackrel{(HYD)}{=}&\beta^{n}((\beta^{m}(h_{1})\triangleright u)_{(-1)})\beta^{-2}(h_{21}\beta^{n+2}(v_{(-1)}))\otimes(\beta^{m}(h_{1})\triangleright u)_{(0)}\otimes\beta^{m+2}(h_{22})\triangleright v_{(0)}\\ &=&[\beta^{n-1}((\beta^{m}(h_{1})\triangleright u)_{(-1)})\beta^{-2}(h_{21})]\beta^{n+1}(v_{(-1)})\otimes(\beta^{m}(h_{1})\triangleright u)_{(0)}\otimes\beta^{m+2}(h_{22})\triangleright v_{(0)}\\ &=&[\beta^{n-1}((\beta^{m+1}(h_{11})\triangleright u)_{(-1)})\beta^{-2}(h_{12})]\beta^{n+1}(v_{(-1)})\otimes(\beta^{m+1}(h_{11})\triangleright u)_{(0)}\\ &&\otimes\beta^{m+3}(h_{2})\triangleright v_{(0)}\\ &=&\beta^{-2}[\underline{\beta^{n+1}((\beta^{m+1}(h_{11})\triangleright u)_{(-1)})h_{12}}]\beta^{n+1}(v_{(-1)})\otimes\underline{(\beta^{m+1}(h_{11})\triangleright u)_{(0)}}\\ &&\otimes\beta^{m+3}(h_{2})\triangleright v_{(0)}\\ &\stackrel{(HYD)}{=}&(\beta^{-2}(h_{11})\beta^{n}(u_{(-1)}))\beta^{n+1}(v_{(-1)})\otimes\beta^{m+2}(h_{12})\triangleright u_{(0)}\otimes\beta^{m+3}(h_{2})\triangleright v_{(0)}\\ &=&(\beta^{-1}(h_{1})\beta^{n}(u_{(-1)}))\beta^{n+1}(v_{(-1)})\otimes\beta^{m+2}(h_{21})\triangleright u_{(0)}\otimes\beta^{m+2}(h_{22})\triangleright v_{(0)}\\ &=&h_{1}(\beta^{n}(u_{(-1)})\beta^{n}(v_{(-1)}))\otimes\beta^{m+2}(h_{21})\triangleright u_{(0)}\otimes\beta^{m+2}(h_{22})\triangleright v_{(0)}\\ &=&h_{1}\beta^{n}(u_{(-1)}v_{(-1)})\otimes\beta^{m+2}(h_{2})\triangleright (u_{(0)}\otimes v_{(0)})\\ &=&h_{1}\beta^{n+2}((u\otimes v)_{(-1)})\otimes\beta^{m+2}(h_{2})\triangleright (u\otimes v)_{(0)}, \end{eqnarray*}$

finishing the proof.

Lemma 4.6 Let $(H,\beta)$ be a Hom-bialgebra, and

$(U,\triangleright_{U},\rho^{U},\alpha_{U}),(V,\triangleright_{V},\rho^{V},\alpha_{V}),(W,\triangleright_{W},\rho^{W},\alpha_{W}) \in ^{H}_{H}\mathbb{YD}.$

With notation as above, define the linear map

$ a_{U,V,W}: (U\otimes V)\otimes W\rightarrow U\otimes(V\otimes W),(u\otimes v)\otimes w\mapsto\alpha_{U}^{-1}(u)\otimes(v\otimes \alpha_{W}(w)), $

where $u\in U,v\in V$ and $w\in W$. Then $a_{U,V,W}$ is an ismorphism of left $(H,\beta)$-Hom-modules and left $(H,\beta)$-Hom-comodules.

Proof Same to the proof of [9, Proposition 3.2].

Lemma 4.7 Let $(H,\beta)$ be a Hom-bialgebra and $(U,\triangleright_{U},\rho^{U},\alpha_{U}),(V,\triangleright_{V},\rho^{V},\alpha_{V}) \in {_{H}^{H}\mathbb{YD}}.$ Define the linear map

$ c_{U,V}:U\otimes V\rightarrow V\otimes U,u\otimes v\mapsto(\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v))\otimes\alpha^{-1}_{U}(u_{(0)}), $

where $u\in U$ and $v\in V$. Then $c_{U,V}$ is a morphism of left $(H,\beta)$-Hom-modules and left $(H,\beta)$-Hom-comodules.

Proof For all $h\in H,u\in U$ and $v\in V$, we have

$\begin{eqnarray*} && (\alpha_{V}\otimes\alpha_{U})\circ c_{U,V}(u\otimes v)\\ &=&\alpha_{V}(\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v))\otimes u_{(0)}\\ &=&(\beta^{m+n+3}(u_{(-1)})\triangleright_{V}v)\otimes u_{(0)}\\ &=&\beta^{m+n+2}(\alpha_{U}(u)_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(\alpha_{V}(v))\otimes\alpha^{-1}_{U}(\alpha_{U}(u)_{(0)})\\ &=&c_{U,V}\circ(\alpha_{U}\otimes\alpha_{V})(u\otimes v),\\ &&c_{U,V}(h\triangleright_{U\otimes V}(u\otimes v))\\ &=&c_{U,V}((h_1\triangleright_{U}u)\otimes(h_2\triangleright_{V}v))\\ &=&(\beta^{m+n+2}((h_1\triangleright_{U}u)_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(h_2\triangleright_{V}v))\otimes\alpha^{-1}_{U}((h_1\triangleright_{U}u)_{(0)})\\ &=&(\beta^{m+n+2}((h_1\triangleright_{U}u)_{(-1)})\triangleright_{V}(\beta^{-1}(h_2)\triangleright_{V}\alpha^{-1}_{V}(v))) \otimes\alpha^{-1}_{U}((h_1\triangleright_{U}u)_{(0)})\\ &=&((\underline{\beta^{m+n+1}((h_1\triangleright_{U}u)_{(-1)})\beta^{-1}(h_2)})\triangleright_{V}v)\otimes\alpha^{-1}_{U}(\underline{(h_1\triangleright_{U}u)_{(0)}})\\ &\stackrel{(HYD)}{=}&(\beta^{m}(\beta^{n+1}((h_1\triangleright_{U}u)_{(-1)})\beta^{-m-1}(h_2))\triangleright_{V}v)\otimes\alpha^{-1}_{U}((h_1\triangleright_{U}u)_{(0)})\\ &=&(\beta^{m}(\beta^{-m-1}(h_1)\beta^{n+2}(u_{(-1)}))\triangleright_{V}v)\otimes\alpha^{-1}_{U}(\beta^{m+2}(\beta^{-m-1}(h_2))\triangleright_{U}u_{(0)})\\ &=&((\beta^{-1}(h_1)\beta^{m+n+2}(u_{(-1)}))\triangleright_{V}v)\otimes h_2\triangleright_{U}\alpha^{-1}_{U}(u_{(0)})\\ &=&(h_1\triangleright_{V}(\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v)))\otimes h_2\triangleright_{U}\alpha^{-1}_{U}(u_{(0)})\\ &=&h\triangleright_{U\otimes V}((\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v))\otimes \alpha^{-1}_{U}(u_{(0)}))\\ &=&h\triangleright_{U\otimes V}c_{U,V}(u\otimes v) \end{eqnarray*}$

and

$\begin{eqnarray*} &&(\rho^{V\otimes U}\circ c_{U,V})(u\otimes v)\\ &=&\rho^{V\otimes U}((\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v))\otimes\alpha^{-1}_{U}(u_{(0)}))\\ &=&\beta^{-2}((\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v))_{(-1)}\alpha^{-1}_{U}(u_{(0)})_{(-1)})\\ &&\otimes(\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v))_{(0)}\otimes\alpha^{-1}_{U}(u_{(0)})_{(0)}\\ &=&\beta^{-2}((\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v))_{(-1)}\beta^{-1}(u_{(0)(-1)}))\otimes(\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha^{-1}_{V}(v))_{(0)}\\ &&\otimes\alpha^{-1}_{U}(u_{(0)(0)})\\ &=&\beta^{-2}((\beta^{m+n+1}(u_{(-1)1})\triangleright_{V}\alpha^{-1}_{V}(v))_{(-1)}\beta^{-1}(u_{(-1)2}))\otimes(\beta^{m+n+1}(u_{(-1)1}) \triangleright_{V}\alpha^{-1}_{V}(v))_{(0)}\\ &&\otimes u_{(0)} \end{eqnarray*}$
$\begin{eqnarray*} &=&\beta^{-n-3}(\beta^{n+1}((\beta^{m+1}(\beta^{n}(u_{(-1)})_1)\triangleright_{V}\alpha^{-1}_{V}(v))_{(-1)})\beta^{n}(u_{(-1)})_2)\otimes (\beta^{m+1}(\beta^{n}(u_{(-1)})_1)\\ &&\triangleright_{V}\alpha^{-1}_{V}(v))_{(0)}\otimes u_{(0)}\\ &\stackrel{(HYD)}{=}&\beta^{-n-3}(\beta^{n}(u_{(-1)1})\beta^{n+2}(\alpha^{-1}_{V}(v)_{(-1)}))\otimes\beta^{m+2}(\beta^{n}(u_{(-1)2}))\triangleright_{V} \alpha^{-1}_{V}(v)_{(0)}\otimes u_{(0)}\\ &=&\beta^{-3}(u_{(-1)1})\beta^{-2}(v_{(-1)})\otimes\beta^{m+n+2}(u_{(-1)2})\triangleright_{V}\alpha^{-1}_{V}(v_{(0)})\otimes u_{(0)}\\ &=&\beta^{-2}(u_{(-1)}v_{(-1)})\otimes\beta^{m+n+2}(u_{(0)(-1)})\triangleright_{V}\alpha^{-1}_{V}(v_{(0)})\otimes \alpha^{-1}_{U}(u_{(0)(0)})\\ &=&(Id\otimes c_{U,V})(\beta^{-2}(u_{(-1)}v_{(-1)})\otimes u_{(0)}\otimes v_{(0)})\\ &=&(Id\otimes c_{U,V})\circ\rho^{U\otimes V}(u\otimes v), \end{eqnarray*}$

finishing the proof.

Theorem 4.8 Let $(H,\beta)$ be a Hom-bialgebra. Then the Hom-Yetter-Drinfeld category $^{H}_{H}\mathbb{YD}$ is a pre-braided tensor category, with tensor product, associativity constraints, and pre-braiding in Lemmas 4.5, 4.6 and 4.7, respectively, and the unit $I=(K,Id_{K})$.

Proof The proof of the pentagon axiom for $a_{U,V,W}$ is same to the proof of [9, Theorem 3.4]. Next we prove that the hexagonal relation for $c_{U,V}$. Let $(U,\triangleright_{U},\rho^{U},\alpha_{U})$, $(V,\triangleright_{V},\rho^{V},\alpha_{V})$, $(W,\triangleright_{W},\rho^{W},\alpha_{W})$ $\in ^{H}_{H}\mathbb{YD}$. Then for all $u\in U,v\in V$ and $w\in W$, we have

$\begin{eqnarray*} &&((Id_{V}\otimes c_{U,W})\circ a_{V,U,W}\circ(c_{U,V}\otimes Id_{W}))((u\otimes v)\otimes w)\\ &=&((Id_{V}\otimes c_{U,W})\circ a_{V,U,W})((\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha_{V}^{-1}(v))\otimes\alpha_{U}^{-1}(u_{(0)})\otimes w)\\ &=&(Id_{V}\otimes c_{U,W})(\alpha_{V}^{-1}(\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha_{V}^{-1}(v))\otimes(\alpha_{U}^{-1}(u_{(0)})\otimes \alpha_{W}(w)))\\ &=&\alpha_{V}^{-1}(\beta^{m+n+2}(u_{(-1)})\triangleright_{V}\alpha_{V}^{-1}(v))\otimes\beta^{m+n+1}(u_{(0)(-1)})\triangleright_{W}w\otimes\alpha_{U}^{-2}(u_{(0)(0)})\\ &=&\alpha_{V}^{-1}(\beta^{m+n+1}(u_{(-1)1})\triangleright_{V}\alpha_{V}^{-1}(v))\otimes\beta^{m+n+1}(u_{(-1)2})\triangleright_{W}w\otimes\alpha_{U}^{-1}(u_{(0)})\\ &=&a_{V,W,U}(\beta^{m+n+1}(u_{(-1)1})\triangleright_{V}\alpha_{V}^{-1}(v)\otimes\beta^{m+n+1}(u_{(-1)2})\triangleright_{W}w\otimes\alpha_{U}^{-2}(u_{(0)}))\\ &=&a_{V,W,U}(\beta^{m+n+2}(\alpha_{U}^{-1}(u)_{(-1)})\triangleright_{V\otimes W}(\alpha_{V}^{-1}(v)\otimes w)\otimes\alpha_{U}^{-1}(\alpha_{U}^{-1}(u)_{(0)})\\ &=&(a_{V,W,U}\circ c_{U,V\otimes W})(\alpha_{U}^{-1}(u)\otimes(v\otimes \alpha_{W}(w)))\\ &=&(a_{V,W,U}\circ c_{U,V\otimes W}\circ a_{U,V,W})((u\otimes v)\otimes w) \end{eqnarray*}$

and

$\begin{eqnarray*} &&((c_{U,W}\otimes Id_{V})\circ a_{U,W,V}^{-1}\circ(Id_{U}\otimes c_{V,W}))(u\otimes(v\otimes w))\\ &=&((c_{U,W}\otimes Id_{V})\circ a_{U,W,V}^{-1})(u\otimes(\beta^{m+n+2}(v_{(-1)})\triangleright_{W}\alpha_{W}^{-1}(w))\otimes\alpha_{V}^{-1}(v_{(0)}))\\ &=&(c_{U,W}\otimes Id_{V})(\alpha_{U}(u)\otimes \beta^{m+n+2}(v_{(-1)})\triangleright_{W}\alpha_{W}^{-1}(w)\otimes\alpha_{V}^{-2}(v_{(0)}))\\ &=&\beta^{m+n+2}(\alpha_{U}(u)_{(-1)})\triangleright_{W}\alpha_{W}^{-1}(\beta^{m+n+2}(v_{(-1)})\triangleright_{W}\alpha_{W}^{-1}(w))\otimes\alpha_{U}^{-1} (\alpha_{U}(u)_{(0)})\otimes\alpha_{V}^{-2}(v_{(0)})\\ &=&\beta^{m+n+3}(u_{(-1)})\triangleright_{W}(\beta^{m+n+1}(v_{(-1)})\triangleright_{W}\alpha_{W}^{-2}(w))\otimes u_{(0)}\otimes\alpha_{V}^{-2}(v_{(0)})\\ &=&(\beta^{m+n+2}(u_{(-1)})\beta^{m+n+1}(v_{(-1)}))\triangleright_{W}\alpha_{W}^{-1}(w)\otimes u_{(0)}\otimes\alpha_{V}^{-2}(v_{(0)})\\ &=&\beta^{m+n+1}(\alpha_{U}(u)_{(-1)}v_{(-1)})\triangleright_{W}\alpha_{W}^{-1}(w)\otimes \alpha_{U}^{-1}(\alpha_{U}(u)_{(0)})\otimes\alpha_{V}^{-2}(v_{(0)})\\ &=&a_{W,U,V}^{-1}(\beta^{m+n}(\alpha_{U}(u)_{(-1)}v_{(-1)})\triangleright_{W}\alpha_{W}^{-2}(w)\otimes \alpha_{U}^{-1}(\alpha_{U}(u)_{(0)})\otimes\alpha_{V}^{-1}(v_{(0)}))\\ &=&(a_{W,U,V}^{-1}\circ c_{U\otimes V,W})((\alpha_{U}(u)\otimes v)\otimes\alpha_{W}^{-1}(w))\\ &=&(a_{W,U,V}^{-1}\circ c_{U\otimes V,W}\circ a_{U,V,W}^{-1})(u\otimes(v\otimes w)), \end{eqnarray*}$

and the rest is obvious. These complete the proof.

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