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.

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.

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.

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.