On Hom-Yetter-Drinfeld Category

Abstract

Let $(H, \beta )$ be a Hom-Hopf algebra. Recently we introduced the Hom-Yetter-Drinfeld category $_H^H{\mathcal {YD}}$ via Radford biproduct Hom-Hopf algebra, and proved that $_H^H{\mathcal {YD}}$ is a braided tensor category. Let $(H, \beta , \mathrm {R} (\hbox {or~}\sigma ))$ be a quasitriangular (or cobraided) Hom-Hopf algebra. In this paper, we prove that the category $_H{\mathcal {M}}$ (or $^H{\mathcal {M}}$) of left $(H, \beta )$-Hom-modules comodules) is a braided tensor subcategory of $_H^H{\mathcal {YD}}$. As a generalization of Radford biproduct Hom-Hopf algebra, we derive necessary and sufficient conditions for R-smash product Hom-algebra $(A\natural _R H, \alpha \otimes \beta )$ and T-smash coproduct Hom-coalgebra $(A\diamond _T H,\alpha \otimes \beta )$ to be a Hom-Hopf algebra. At last, two nontrivial examples are given.

13.1 Introduction

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been intensively investigated in the literature recently. Hom-algebras are generalizations of algebras obtained by a twisting map, which have been introduced for the first time in [6] by Makhlouf and Silvestrov. The associativity is replaced by Hom-associativity, Hom-coassociativity for a Hom-coalgebra can be considered in a similar way.

In [9, 12], Yau introduced and characterized the concept of module Hom-algebras as a twisted version of usual module algebras and the dual version (i.e. comodule Hom-coalgebras) was studied by Zhang in [13]. Based on Yau’s definition of module Hom-algebras, Ma, Li and Yang in [3] constructed smash product Hom-Hopf algebra $(A\natural H,\alpha \otimes \beta )$ generalizing the Molnar’s smash product (see [4]), and gave the cobraided structure (in the sense of Yau’s definition in [11]) on $(A\natural H,\alpha \otimes \beta )$, and also considered the case of twist tensor product Hom-Hopf algebra. Makhlouf and Panaite defined and studied a class of Yetter-Drinfeld modules over Hom-bialgebras in [5] via the “twisting principle” introduced by Yau for Hom-algebras and since then extended to various Hom-type algebras. In [2], the authors introduced the notion of Hom-Yetter-Drinfeld category $_H^H{\mathcal {YD}}$ via Radford biproduct Hom-Hopf algebra, and proved that the Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation (in the sense of Yau’s definition in [10,11,12]) and $_H^H{\mathcal {YD}}$ is a braided tensor category.

It is well-known that the category $_H{\mathcal {M}}$ (or $^H{\mathcal {M}}$) of left H-modules (or comodules) is a braided tensor subcategory of $_H^H{\mathcal {YD}}$, where $(H, \mathrm {R} ~(\hbox {or}\,\sigma ))$ is a quasitriangular (or cobraided) Hopf algebra. Radford biproduct plays an important role in the lifting method for the classification of finite dimensional pointed Hopf algebras. In [4], Ma and Wang generalized the Radford biproduct to the twist tensor biproduct.

The main purpose of this article is to consider the above results in the Hom-setting.

This article is organized as follows. In Sect. 13.2, we recall some definitions and results which will be used later. In Sect. 13.3, we construct a subcategory of the Hom-Yetter-Drinfeld category via quasitriangular Hom-Hopf algebra (see Theorem 13.1). On the other hand, we give a second braided tensor structure on $_H^H{\mathcal {YD}}$ (see Theorem 13.3). Yau’s results in [10, 11] can be obtained as a corollary (see Corollaries 13.1 and 13.2). In [2], we defined the Hom-Yetter-Drinfeld module by Radford biproduct Hom-Hopf algebra. In Sect. 13.4, we consider a generalized version of Radford’s biproduct Hom-Hopf algebra, named twisted tensor biproduct Hom-Hopf algebra (see Theorems 13.6 and 13.7). And two nontrivial examples are given (see Examples 13.1 and 13.2).

13.2 Preliminaries

Throughout this paper, we follow the definitions and terminologies in [1,2,3, 10, 11, 13], with all algebraic systems supposed to be over the field K. Given a K-space M, we write $id_M$ for the identity map on M.

We now recall some useful definitions throughout this paper. We point out that we will be using in the special contexts we consider a simplified terminology for involved Hom-algebra structures just for convenience of exposition in this article.

Definition 13.1

A Hom-algebra Hom-algebra, or more exactly, a unital multiplicative Hom-associative algebra, is a quadruple $(A,\mu ,1_A,\alpha )$ (abbr. $(A,\alpha )$), where A is a K-linear space, $\mu : A\otimes A \longrightarrow A$ is a K-linear map, $1_A \in A$ and $\alpha$ is an automorphism of A, such that

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

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 13.2

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

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

are satisfied for $c \in A$. Here we use the notation $\varDelta (c)=c_1\otimes c_2$ (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 $\varDelta (c\otimes d)=c_1\otimes d_1\otimes c_2\otimes d_2$ and counit $\varepsilon _C\otimes \varepsilon _D$.

Definition 13.3

A Hom-bialgebra is a sextuple $(H,\mu ,1_H,\varDelta ,\varepsilon ,\gamma )$ (abbr. $(H,\gamma )$), where $(H,\mu ,1_H,\gamma )$ is a Hom-algebra and $(H,\varDelta ,\varepsilon ,\gamma )$ is a Hom-coalgebra, such that $\varDelta$ and $\varepsilon$ are morphisms of Hom-algebras, i.e.

$$ \varDelta (hh')=\varDelta (h)\varDelta (h');~~\varDelta (1_H)=1_H\otimes 1_H, $$

$$ \varepsilon (hh')=\varepsilon (h)\varepsilon (h');~~\varepsilon (1_H)=1. $$

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 $(H,\mu ,1_H,\varDelta ,\varepsilon ,\gamma ,S)$(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 13.4

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

$$\begin{aligned}&(HM1) \, \, \alpha (a\triangleright m)=\beta (a)\triangleright \alpha (m),\\&(HM2) \,\, \beta (a)\triangleright (a'\triangleright m)=(aa')\triangleright \alpha (m);~~ 1_A\triangleright m=\alpha (m) \end{aligned}$$

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

Let $(M,\triangleright _M,\alpha _M)$ and $(N,\triangleright _N,\alpha _N)$ 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 $f(h\triangleright _M m)=h\triangleright _N f(m)$ and $\alpha _M\circ f=f\circ \alpha _N$.

Definition 13.5

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

$$\begin{aligned}&(HMA1)\, \, \beta ^{2}(h)\triangleright (aa')=(h_1\triangleright a)(h_2\triangleright a'),\\&(HMA2) \,\, h\triangleright 1_A=\varepsilon _H(h)1_A, \end{aligned}$$

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

Definition 13.6

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

$$\begin{aligned}&(HCM1) \, \, \alpha (m)_{-1}\otimes \alpha (m)_{0}=\beta (m_{-1})\otimes \alpha (m_{0}),\\&(HCM2) \, \, \beta (m_{-1})\otimes m_{0-1}\otimes m_{00}{=} m_{-11}\otimes m_{-12}\otimes \alpha (m_{0});~~ \varepsilon _C(m_{-1})m_{0}=\alpha (m) \end{aligned}$$

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 morphism of left $(C,\beta )$-Hom-comodules if $f(m)_{-1}\otimes f(m)_{0}=m_{-1}\otimes f(m_{0})$ and $\alpha _M\circ f=f\circ \alpha _N$.

Definition 13.7

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 $c\in C$,

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

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

Definition 13.8

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

$$\begin{aligned}&(HMC1) \, \,m (h\triangleright c)_1\otimes (h\triangleright c)_2=(h_1\triangleright c_1)\otimes (h_2\triangleright c_2),\\&(HMC2) \, \, \varepsilon _C(h\triangleright c)=\varepsilon _H(h)\varepsilon _C(c), \end{aligned}$$

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

Definition 13.9

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 $a, a'\in A$,

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

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

Definition 13.10

Let $(H,\beta )$ be a Hom-bialgebra and $(A,\triangleright ,\alpha )$ an $(H,\beta )$-module Hom-algebra. Then $(A\natural H, \alpha \otimes \beta )$ ($A\natural H=A\otimes H$ as a linear space) with the multiplication

$$ (a\otimes h)(a'\otimes h')=a(h_1\triangleright \alpha ^{-1}(a'))\otimes \beta ^{-1}(h_{2})h', $$

where $a, a'\in A, h, h'\in H$, and unit $1_A\otimes 1_H$ is a Hom-algebra, we call it smash product Hom-algebra denoted by $(A\natural H,\alpha \otimes \beta )$.

Definition 13.11

Let $(H, \beta )$ be a Hom-bialgebra, $(M,\triangleright _M, \alpha _M)$ a left $(H,\beta )$-module with action $\triangleright _M: H\otimes M\longrightarrow M, h\otimes m\mapsto h\triangleright _M m$ and $(M,\rho ^M, \alpha _M)$ a left $(H,\beta )$-comodule with coaction $\rho ^M: M\longrightarrow H\otimes M, m\mapsto m_{-1}\otimes m_{0}$. Then we call $(M,\triangleright _M$, $\rho ^M, \alpha _M)$ a (left-left) Hom-Yetter-Drinfeld module over $(H,\beta )$ if the following condition holds:

$$ (HYD)~~~h_1\beta (m_{-1})\otimes (\beta ^3(h_2)\triangleright _M m_0)=(\beta ^2(h_1)\triangleright _M m)_{-1}h_2\otimes (\beta ^2(h_1)\triangleright _M m)_{0}, $$

where $h\in H$ and $m\in M$.

Definition 13.12

Let $(A,\mu _A,1_A,\alpha )$ and $(H,\mu _H,1_H,\beta )$ be two Hom-algebras, $R: H\otimes A \longrightarrow A\otimes H$ a linear map such that for all $a\in A, h\in H$,

$$ (R)~~~~~~~~\alpha (a)_R\otimes \beta (h)_R=\alpha (a_R)\otimes \beta (h_R). $$

Then $(A\natural _R H, \alpha \otimes \beta )$ ($A\natural _R H=A\otimes H$ as a linear space) with the multiplication

$$ (a\otimes h)(b\otimes g)=a\alpha ^{-1}(b)_{R}\otimes \beta ^{-1}(h_{R})g, $$

where $a, b\in A, h, g\in H$, and unit $1_A\otimes 1_H$ becomes a Hom-algebra if and only if the following conditions hold:

$$\begin{aligned}&(RS1) \,\, a_{R}\otimes 1_{BR}=\alpha (a)\otimes 1_H;~~1_{AR}\otimes h_R=1_A\otimes \beta (h),\\&(RS2) \,\, \alpha (a)_R\otimes (hg)_R=a_{Rr}\otimes \beta ^{-1}(\beta (h)_r)g_R,\\&(RS3) \,\, \alpha ((ab)_R)\otimes \beta (h)_R=\alpha (a_R)\alpha (b)_r\otimes h_{Rr}, \end{aligned}$$

where $a, b\in A, h, g\in H$. We call this Hom-algebra R-smash product Hom-algebra and denote it by $(A\natural _R H,\alpha \otimes \beta )$.

Definition 13.13

Let $(C,\varDelta _C,\varepsilon _C,\alpha )$ and $(H,\varDelta _H,\varepsilon _H,\beta )$ be two Hom-coalgebras, $T: C\otimes H \longrightarrow H\otimes C$ (write $T(c\otimes h)=h_T\otimes c_T, \forall c\in C, h\in H$) a linear map such that for all $c\in C, h\in H$,

$$ (T)~~~~~~~\alpha (c)_T\otimes \beta (h)_T=\alpha (c_T)\otimes \beta (h_T). $$

Then $(C\diamond _T H, \alpha \otimes \beta )$ ($C\diamond _T H=C\otimes H$ as a linear space) with the comultiplication

$$ \varDelta _{C\diamond _T H}(c\otimes h)=c_1\otimes \beta ^{-1}(h_1)_T\otimes \alpha ^{-1}(c_{2T})\otimes h_2, $$

and counit $\varepsilon _C\otimes \varepsilon _H$ becomes a Hom-coalgebra if and only if the following conditions hold:

$$\begin{aligned}&(TS1) \,\, \varepsilon _H(h_T)c_T=\varepsilon _H(h)\alpha (c);~~h_T\varepsilon _C(c_T)=\beta (h)\varepsilon _C(c),\\&(TS2) \, \, h_{T1}\otimes h_{T2}\otimes \alpha (c_T)=\beta (\beta ^{-1}(h_1)_T)\otimes h_{2t}\otimes c_{Tt},\\&(TS3) \,\, \beta (h_T)\otimes \alpha (c)_{T1}\otimes \alpha (c)_{T2}=h_{Tt}\otimes \alpha (c_1)_t\otimes \alpha (c_{2T}), \end{aligned}$$

where $c\in C, h\in H$ and t is a copy of T. We call this Hom-coalgebra T-smash coproduct Hom-coalgebra and denote it by $(C\diamond _T H,\alpha \otimes \beta )$.

Definition 13.14

A quasitriangular Hom-Hopf algebra is a octuple $(H,\mu ,1_H, \varDelta ,$ $\varepsilon ,S,\beta , \mathrm {R})$ (abbr.$(H,\beta ,\mathrm {R})$) in which $(H,\mu ,1_H,\varDelta ,\varepsilon ,S,\beta )$ is a Hom-Hopf algebra and $\mathrm {R}=\mathrm {R}^1\otimes \mathrm {R}^2 \in H\otimes H$, satisfying the following axioms (for all $h\in H$ and $\mathrm {R}=\mathrm {r}$):

$$\begin{aligned}&(QHA1) \, \, \varepsilon (\mathrm {R}^1)\mathrm {R}^2=\mathrm {R}^1\varepsilon (\mathrm {R}^2)=1,\\&(QHA2)\, \, {\mathrm {R}^1}_{1}\otimes {\mathrm {R}^1}_{2}\otimes \beta (\mathrm {R}^2)=\beta (\mathrm {R}^1)\otimes \beta (\mathrm {r}^1)\otimes \mathrm {R}^2\mathrm {r}^2,\\&(QHA3) \, \, \beta (\mathrm {R}^1)\otimes {\mathrm {R}^2}_{1}\otimes {\mathrm {R}^2}_{2}=\mathrm {R}^1\mathrm {r}^1\otimes \beta (\mathrm {r}^2)\otimes \beta (\mathrm {R}^2),\\&(QHA4) \, \, h_2\mathrm {R}^1\otimes h_1\mathrm {R}^2=\mathrm {R}^1h_1\otimes \mathrm {R}^2h_2,\\&(QHA5)\, \, \beta (\mathrm {R}^1)\otimes \beta (\mathrm {R}^2)=\mathrm {R}^1\otimes \mathrm {R}^2. \end{aligned}$$

Definition 13.15

A cobraided Hom-Hopf algebra is a octuple $(H,\mu ,1_H, \varDelta ,$ $\varepsilon ,S,\beta ,\sigma )$ (abbr.$(H,\beta ,\sigma )$) in which $(H,\mu ,1_H,\varDelta ,\varepsilon ,S,\beta )$ is a Hom-Hopf algebra and $\sigma$ is a bilinear form on H (i.e., $\sigma \in Hom(H\otimes H, K)$), satisfying the following axioms (for all $h, g, l\in H$):

$$\begin{aligned}&(CHA1)\, \, \sigma (h,1_H)=\sigma (1_H,h)=\varepsilon (h),\\&(CHA2)\,\, \sigma (hg,\beta (l))=\sigma (\beta (h),l_1)\sigma (\beta (g),l_2),\\&(CHA3)\,\, \sigma (\beta (h),gl)=\sigma (h_1,\beta (l))\sigma (h_2,\beta (g)),\\&(CHA4)\,\, \sigma (h_1,g_1)h_2g_2=g_1h_1\sigma (h_2,g_2),\\&(CHA5)\, \, \sigma (\beta (h),\beta (g))=\sigma (h,g). \end{aligned}$$

13.3 A Class of Braided Tensor Category

In this section, we construct a subcategory of the Hom-Yetter-Drinfeld category. On the other hand, we give a second braided tensor structure on $_H^H{\mathcal {YD}}$. Yau’s results in [10, 11] can be obtained as a corollary.

First we recall the structure of Hom-Yetter-Drinfeld category in [2].

Proposition 13.1

([2]) Let $(H, \beta )$ be a Hom-bialgebra. Then the Hom-Yetter-Drinfeld category $_H^H{\mathcal {YD}}$ is a braided tensor category, with tensor product defined by

$$ \triangleright _{M\otimes N}: H\otimes M\otimes N\longrightarrow M\otimes N, h\otimes m\otimes n\mapsto (h_1\triangleright _M m)\otimes (h_2\triangleright _N n), $$

and

$$ \rho ^{M\otimes N}: M\otimes N\longrightarrow H\otimes M\otimes N, m\otimes n\mapsto \beta ^{-2}(m_{-1}n_{-1})\otimes m_{0}\otimes n_{0}, $$

where $h\in H$, $m\in M$ and $n\in N$, associativity constraints defined by

$$ a_{M,N,P}: (M\otimes N)\otimes P\longrightarrow M\otimes (N\otimes P),~~(m\otimes n)\otimes p\mapsto \alpha _M^{-1}(m)\otimes (n \otimes \alpha _P(p)), $$

where $m\in M$, $n\in N$ and $p\in P$, the braiding defined by

$$ c_{M,N}: M\otimes N\longrightarrow N\otimes M,~~m\otimes n\mapsto (\beta ^2(m_{-1})\triangleright _N \alpha _{N}^{-1}(n))\otimes \alpha _M^{-1}(m_{0}) , $$

where $m\in M$ and $n\in N$ and the unit $(K,id_K)$.

Proposition 13.2

Let $(H, \beta , \mathrm {R})$ be a quasitriangular Hom-Hopf algebra and $(M,\alpha _M)$ a left $(H,\beta )$-Hom-module with action $\bar{\triangleright }_M: H\otimes M\longrightarrow M, h\otimes m\mapsto h\bar{\triangleright }_M m$. Define the linear map

$$ \bar{\rho }^{M}: M\longrightarrow H\otimes M, m\mapsto \beta ^{-3}(\mathrm {R}^2)\otimes (\mathrm {R}^1\bar{\triangleright }_M m), $$

Then $(M, \bar{\triangleright }_M, \bar{\rho }^M, \alpha _M)$ is a Hom-Yetter-Drinfeld module over $(H, \beta )$.

Proof

The condition (HCM1) is easy to be proved by (QHA5) and (HAM1). We check (HCM2) as follows.

$$\begin{aligned} \begin{array}{rcl} \hbox {LHS} &{}{\mathop {=}\limits ^{}}&{}\beta ^{-2}(\mathrm {R}^2)\otimes \beta ^{-3}(\mathrm {r}^2)\otimes \mathrm {r}^1\bar{\triangleright }_M(\mathrm {R}^1\bar{\triangleright }_M m)\\ &{}{\mathop {=}\limits ^{(QHA5)}}&{}\beta ^{-2}(\mathrm {R}^2)\otimes \beta ^{-2}(\mathrm {r}^2)\otimes \beta (\mathrm {r}^1)\bar{\triangleright }_M(\mathrm {R}^1\bar{\triangleright }_M m)\\ &{}{\mathop {=}\limits ^{(HM2)}}&{}\beta ^{-2}(\mathrm {R}^2)\otimes \beta ^{-2}(\mathrm {r}^2)\otimes (\mathrm {r}^1\mathrm {R}^1)\bar{\triangleright }_M m)\\ &{}{\mathop {=}\limits ^{(QHA3)}}&{}\beta ^{-3}({\mathrm {R}^2}_{1})\otimes \beta ^{-3}({\mathrm {R}^2}_{2})\otimes (\beta (\mathrm {R}^1)\bar{\triangleright }_M \alpha _M(m))\\ &{}{\mathop {=}\limits ^{(HC1)(HM1)}}&{}\beta ^{-3}({\mathrm {R}^2})_{1}\otimes \beta ^{-3}({\mathrm {R}^2})_{2}\otimes \alpha _M(\mathrm {R}^1\bar{\triangleright }_M m)=\hbox {RHS}, \end{array} \end{aligned}$$

and it is obvious that $\varepsilon _H(m_{-1})m_{0}=\alpha _M(m)$ by (QHA1), (HC1) and (HM1). Thus $(M, \bar{\rho }^M, \alpha _M)$ is a $(H, \beta )$-Hom-comodule.

Next we check that the condition (HYD) holds.

$$\begin{aligned} \begin{array}{rcl} \hbox {LHS} &{}{\mathop {=}\limits ^{}}&{}h_1\beta ^{-2}(\mathrm {R}^2)\otimes (\beta ^3(h_2)\bar{\triangleright }_M (\mathrm {R}^1\bar{\triangleright }_M m))\\ &{}{\mathop {=}\limits ^{(HM2)}}&{}h_1\beta ^{-2}(\mathrm {R}^2)\otimes ((\beta ^2(h_2)\mathrm {R}^1)\bar{\triangleright }_M \alpha _M(m))\\ &{}{\mathop {=}\limits ^{(HA1)(HC1)}}&{}\beta ^{-2}(\beta ^2(h)_1\mathrm {R}^2)\otimes ((\beta ^2(h)_2\mathrm {R}^1)\bar{\triangleright }_M \alpha _M(m))\\ &{}{\mathop {=}\limits ^{(QHA4)}}&{}\beta ^{-2}(\mathrm {R}^2\beta ^2(h)_2)\otimes ((\mathrm {R}^1\beta ^2(h)_1)\bar{\triangleright }_M \alpha _M(m))\\ &{}{\mathop {=}\limits ^{(HA1)(HC1)}}&{}\beta ^{-2}(\mathrm {R}^2)h_2\otimes ((\mathrm {R}^1\beta ^2(h_1))\bar{\triangleright }_M \alpha _M(m))\\ &{}{\mathop {=}\limits ^{(QHA5)}}&{}\beta ^{-3}(\mathrm {R}^2)h_2\otimes ((\beta (\mathrm {R}^1)\beta ^2(h_1))\bar{\triangleright }_M \alpha _M(m))\\ &{}{\mathop {=}\limits ^{(HM2)}}&{}\beta ^{-3}(\mathrm {R}^2)h_2\otimes (\mathrm {R}^1\bar{\triangleright }_M (\beta ^2(h_1)\bar{\triangleright }_M m))=\hbox {RHS}, \end{array} \end{aligned}$$

finishing the proof.    $\square$

Proposition 13.3

Let $(H, \beta , \mathrm {R})$ be a quasitriangular Hom-Hopf algebra, $(M, \bar{\triangleright }_M, \bar{\rho }^M, \alpha _M)$ and $(N, \bar{\triangleright }_N, \bar{\rho }^N, \alpha _N)$ two Hom-Yetter-Drinfeld module over $(H, \beta )$ with the structure defined in Proposition 13.2. We regard $(M\otimes N, \bar{\triangleright }_{M\otimes N}, \alpha _M\otimes \alpha _N)$ as a left $(H,\beta )$-Hom-module via the standard action

$$ h\bar{\triangleright }_{M\otimes N} (m\otimes n)=(h_1\bar{\triangleright }_M m)\otimes (h_2\bar{\triangleright }_N n) $$

and we regard $(M\otimes N, \bar{\triangleright }_{M\otimes N}, \bar{\rho }^{M\otimes N}, \alpha _M\otimes \alpha _N)$ as a Hom-Yetter-Drinfeld module over $(H, \beta )$ with the structure defined in Proposition 13.2. Then this Hom-Yetter-Drinfeld $(M\otimes N, \bar{\triangleright }_{M\otimes N}, \bar{\rho }^{M\otimes N}, \alpha _M\otimes \alpha _N)$ coincides with the Hom-Yetter-Drinfeld module defined in Proposition 13.1.

Proof

We only need to check that the two comodule structures on $M\otimes N$ coincide, i.e., for all $m\in M$ and $n\in N$,

$$ \beta ^{-2}(m_{-1}n_{-1})\otimes (m_{0}\otimes n_{0})=\beta ^{-3}(\mathrm {R}^2)\otimes (\mathrm {R}^1\bar{\triangleright }_{M\otimes N} (m\otimes n)). $$

While

$$\begin{aligned} \begin{array}{rcl} \hbox {LHS} &{}{\mathop {=}\limits ^{}}&{}\beta ^{-2}(\beta ^{-3}(\mathrm {R}^2)\beta ^{-3}(\mathrm {r}^2))\otimes ((\mathrm {R}^1\bar{\triangleright }_{M} m)\otimes (\mathrm {r}^1\bar{\triangleright }_{N} n))\\ &{}{\mathop {=}\limits ^{(HA1)(QHA5)}}&{}\beta ^{-4}(\mathrm {R}^2\mathrm {r}^2)\otimes ((\beta (\mathrm {R}^1)\bar{\triangleright }_{M} m)\otimes (\beta (\mathrm {r}^1)\bar{\triangleright }_{N} n))\\ &{}{\mathop {=}\limits ^{(QHA2)}}&{}\beta ^{-3}(\mathrm {R}^2)\otimes (({\mathrm {R}^1}_1\bar{\triangleright }_{M} m)\otimes ({\mathrm {R}^1}_2\bar{\triangleright }_{N} n))=\hbox {RHS}, \end{array} \end{aligned}$$

finishing the proof.    $\square$

Proposition 13.4

([12]) Let $(H, \beta )$ be a Hom-bialgebra. If $(M,\bar{\triangleright }_{M},\alpha _M)$ and $(N,\bar{\triangleright }_{N},\alpha _N)$ are two $(H,\beta )$-Hom-modules, then $(M\otimes N, \bar{\triangleright }_{M\otimes N}, \alpha _M\otimes \alpha _N)$ is a $(H,\beta )$-Hom-module with the action defined by

$$ \bar{\triangleright }_{M\otimes N}: H\otimes (M\otimes N)\longrightarrow M\otimes N, ~~~~h\bar{\triangleright }_{M\otimes N} (m\otimes n)=(h_1\bar{\triangleright }_{M} m)\otimes (h_2\bar{\triangleright }_{N} n). $$

By Propositions 13.1–13.4, we have

Theorem 13.1

Let $(H, \beta , \mathrm {R})$ be a quasitriangular Hom-Hopf algebra. Denote by $_H{\mathcal {M}}$ the category whose objects are left $(H,\beta )$-Hom-modules $(M,\bar{\triangleright }_M,\alpha _M)$ and morphisms are morphisms of left-$(H,\beta )$-Hom-modules. Then $_H{\mathcal {M}}$ is a braided tensor subcategory of $_H^H{\mathcal {YD}}$, with tensor product defined as in Proposition 13.4, associativity constraints defined by the formula $a_{M,N,P}: (M\otimes N)\otimes P\longrightarrow M\otimes (N\otimes P),~~(m\otimes n)\otimes p\mapsto \alpha _M^{-1}(m)\otimes (n \otimes \alpha _P(p)),$ where $m\in M$, $n\in N$ and $p\in P$, the braiding defined by $c_{M,N}: M\otimes N\longrightarrow N\otimes M,~~m\otimes n\mapsto \alpha _{N}^{-1}(\mathrm {R}^2\bar{\triangleright }_N n)\otimes \alpha _{M}^{-1}(\mathrm {R}^1\bar{\triangleright }_M n),$ where $m\in M$ and $n\in N$ and the unit $(K,id_K)$.

Remark 13.1

Let $m_{-1}\otimes m_{0}=\beta ^{-3}(\mathrm {R}^2)\otimes (\mathrm {R}^1\bar{\triangleright }_M m)$ in Proposition 13.1, we can get the the braiding in Theorem 13.1.

Proposition 13.5

([2]) Let $(H, \beta )$ be a Hom-bialgebra and $(M,\triangleright _M, \rho ^M, \alpha _M)$, $(N,\triangleright _N,$ $\rho ^N, \alpha _N)$ $\in _H^H{\mathcal {YD}}$. Define the linear map

$$ \tau _{M,N}: M\otimes N\longrightarrow N\otimes M, ~~m\otimes n\mapsto \beta ^{3}(m_{-1})\triangleright _N n \otimes m_{0}, $$

where $m\in M$ and $n\in N$. Then, we have $\tau _{M,N}\circ (\alpha _M\otimes \alpha _N)=(\alpha _N\otimes \alpha _M)\circ \tau _{M,N}$ and, if $(P,\triangleright _P, \rho ^P, \alpha _P) \in _H^H{\mathcal {YD}}$, the maps $\tau _{\underline{~~},~\underline{~~}}$ satisfy the Hom-Yang-Baxter equation:

$$ (\alpha _P\otimes \tau _{M,N})\circ (\tau _{M,P}\otimes \alpha _N)\circ (\alpha _M\otimes \tau _{N,P})=(\tau _{N,P}\otimes \alpha _M)\circ (\alpha _N\otimes \tau _{M,P})\circ (\tau _{M,N}\otimes \alpha _P). $$

Corollary 13.1

([10]) Let $(H, \beta , \mathrm {R})$ be a quasitriangular Hom-Hopf algebra and $(M,\bar{\triangleright }_M,\alpha _M)$ a left $(H,\beta )$-Hom-module. Then the linear map

$$ B: M\otimes M\longrightarrow M\otimes M,~~~~B(m\otimes m')=(\mathrm {R}^2\bar{\triangleright }_M m')\otimes (\mathrm {R}^1\bar{\triangleright }_M m) $$

is a solution of the Hom-Yang-Baxter equation for $(M,\bar{\triangleright }_M,\alpha _M)$.

Proof

By Theorem 13.1, and let $m_{-1}\otimes m_{0}=\beta ^{-3}(\mathrm {R}^2)\otimes (\mathrm {R}^1\bar{\triangleright }_M m)$ in Proposition 13.5, we can obtain the result.    $\square$

We have seen that Hom-modules over quasitriangular Hom-Hopf algebras become Hom-Yetter-Drinfeld modules. Similarly, Hom-comodules over cobraided Hom-Hopf algebras become Hom-Yetter-Drinfeld modules. In the following, we introduce another braided tensor category structure on Hom-Yetter-Drinfeld category.

Similar to [2, Lemma 4.4], we have

Proposition 13.6

With notations as above. Let $(H, \beta )$ be a Hom-bialgebra and $(M,\bullet _M, \psi ^M, \alpha _M)$, $(N,\bullet _N, \psi ^N, \alpha _N)$ $\in _H^H{\mathcal {YD}}$. Define the linear maps

$$ \bullet _{M\otimes N}: H\otimes M\otimes N\longrightarrow M\otimes N, h\otimes (m\otimes n)\mapsto (\beta ^{-2}(h_1)\bullet _M m)\otimes (\beta ^{-2}(h_2)\bullet _N n), $$

and

$$ \psi ^{M\otimes N}: M\otimes N\longrightarrow H\otimes M\otimes N, m\otimes n\mapsto m_{-1}n_{-1}\otimes m_{0}\otimes n_{0}, $$

where $h\in H$, $m\in M$ and $n\in N$. Then $(M\otimes N, \bullet _{M\otimes N}, \psi ^{M\otimes N}, \alpha _M\otimes \alpha _N)$ is a Hom-Yetter-Drinfeld module.

Theorem 13.2

Let $(H, \beta )$ be a Hom-bialgebra. Then the Hom-Yetter-Drinfeld category $_H^H{\mathcal {YD}}$ is a braided tensor category, with tensor product defined as in Proposition 13.6, associativity constraints defined by

$$ \mathbf {a}_{M,N,P}: (M\otimes N)\otimes P\longrightarrow M\otimes (N\otimes P),~~(m\otimes n)\otimes p\mapsto \alpha _M(m)\otimes (n \otimes \alpha _P^{-1}(p)), $$

where $m\in M$, $n\in N$ and $p\in P$, the braiding defined by

$$ \mathbf {c}_{M,N}: M\otimes N\longrightarrow N\otimes M,~~m\otimes n\mapsto (\beta ^2(m_{-1})\bullet _N \alpha _{N}^{-1}(n))\otimes \alpha _M^{-1}(m_{0}) , $$

where $m\in M$ and $n\in N$ and the unit $(K,id_K)$.

Proof

Same to [2, Theorem 4.7].    $\square$

Similar to Propositions 13.2, 13.3, we have

Proposition 13.7

Let $(H, \beta , \sigma )$ be a cobraided Hom-Hopf algebra.

(1) Let $(M, \alpha _M)$ a left $(H,\beta )$-Hom-comodule with coaction $\bar{\psi }^M: M\longrightarrow H\otimes M, m\mapsto m_{-1}\otimes m_{0}$. Define the linear map

$$ \bar{\bullet }_{M}: H\otimes M\longrightarrow M, h\bar{\bullet }_M m=\sigma (m_{-1},\beta ^{-3}(h))m_{0} $$

Then $(M, \bar{\bullet }_M, \bar{\psi }^M, \alpha _M)$ is a Hom-Yetter-Drinfeld module over $(H, \beta )$.

(2) Let $(N,\bar{\psi }^N, \alpha _N)$ be another left $(H,\beta )$-Hom-comodule with coaction $\bar{\psi }^N: M\longrightarrow H\otimes N, n\mapsto n_{-1}\otimes n_{0}$, regarded as a Hom-Yetter-Drinfeld module over $(H, \beta )$ with the structure defined as above, via the map $\bar{\bullet }_N: H\otimes N\longrightarrow N, h\otimes n\mapsto h\bar{\bullet }_N n=\sigma (n_{-1},\beta ^{-3}(h))n_{0}$. We regard $(M\otimes N,\bar{\psi }^{M\otimes N}, \alpha _M\otimes \alpha _N)$ as a left $(H,\beta )$-Hom-comodule via the standard coaction $M\otimes N\longrightarrow H\otimes (M\otimes N),~~m\otimes n\mapsto m_{-1}n_{-1}\otimes (m_0\otimes n_0)$ and then we get $(M\otimes N, \bar{\bullet }_{M\otimes N}, \bar{\psi }^{M\otimes N}, \alpha _M\otimes \alpha _N)$ as a Hom-Yetter-Drinfeld module defined as above, then this Yetter-Drinfeld module coincides with the Hom-Yetter-Drinfeld module defined in Theorem 13.2.

Proposition 13.8

([12]) Let $(H, \beta )$ be a Hom-bialgebra. If $(M,\bar{\psi }^{M},\alpha _M)$ and $(N,\bar{\psi }^{N},\alpha _N)$ are two $(H,\beta )$-Hom-comodules, then $(M\otimes N, \bar{\psi }^{M\otimes N}, \alpha _M\otimes \alpha _N)$ is a $(H,\beta )$-Hom-comodule with the coaction defined by

$$ \bar{\psi }^{M\otimes N}: M\otimes N\longrightarrow H\otimes (M\otimes N), ~~~m\otimes n\mapsto m_{-1}n_{-1}\otimes (m_0\otimes n_0). $$

By Propositions 13.6, 13.7, 13.8 and Theorem 13.2, we have

Theorem 13.3

Let $(H, \beta , \sigma )$ be a cobraided Hom-Hopf algebra. Denote by $^H{\mathcal {M}}$ the category whose objects are left $(H,\beta )$-Hom-comodules $(M,\bar{\psi }^M,\alpha _M)$ and morphisms are morphisms of left-$(H,\beta )$-Hom-comodules. Then $^H{\mathcal {M}}$ is a braided tensor subcategory of $_H^H{\mathcal {YD}}$, with tensor product defined as in Proposition 13.8, associativity constraints defined by the formula $\bar{\mathbf {a}}_{M,N,P}: (M\otimes N)\otimes P\longrightarrow M\otimes (N\otimes P),~~(m\otimes n)\otimes p\mapsto \alpha _M(m)\otimes (n \otimes \alpha _P^{-1}(p)),$ where $m\in M$, $n\in N$ and $p\in P$, the braiding defined by $\bar{\mathbf {c}}_{M,N}: M\otimes N\longrightarrow N\otimes M,~~m\otimes n\mapsto \sigma (n_{-1},m_{-1})\alpha _N^{-1}(n_{0})\otimes \alpha _M^{-1}(m_{0}),$ where $m\in M$ and $n\in N$ and the unit $(K,id_K)$.

Corollary 13.2

([11]) Let $(H, \beta , \sigma )$ be a cobraided Hom-Hopf algebra. If $(M,\bar{\psi }^M,\alpha _M)$ and $(N,\bar{\psi }^N,\alpha _N)$ are two $(H, \beta )$-Hom-comodules, we define the linear map

$$ B_{M,N}: M\otimes N\longrightarrow N\otimes M,~~~~ m\otimes n\mapsto \sigma (n_{-1},m_{-1})(n_{0}\otimes m_{0}). $$

Then, we have $B_{M,N}\circ (\alpha _M\otimes \alpha _N)=(\alpha _N\otimes \alpha _M)\circ B_{M,N}$ and, if $(P,\bar{\psi }^P,\alpha _P)$ is another $(H, \beta )$-Hom-comodule, the maps $B_{\underline{~~},~\underline{~~}}$ satisfy the Hom-Yang-Baxter equation:

$$ (\alpha _P\otimes B_{M,N})\circ (B_{M,P}\otimes \alpha _N)\circ (\alpha _M\otimes B_{N,P})=(B_{N,P}\otimes \alpha _M)\circ (\alpha _N\otimes B_{M,P})\circ (B_{M,N}\otimes \alpha _P). $$

Proof

By Theorem 13.3 and let $h\bullet _N n=\sigma (n_{-1},\beta ^{-3}(h))n_0$ in Proposition 13.5, we can obtain the result.    $\square$

Theorem 13.4

Let $(H, \beta , \sigma )$ be a cobraided Hom-Hopf algebra. Assume that $(A, \rho ^A, \alpha )$ is a Hom-Hopf algebra in the category $^H{\mathcal {M}}$. Define $\triangleright _A: H\otimes A\longrightarrow A$ by

$$ h\triangleright _A a=\sigma (a_{-1}, \beta ^{-3}(h))a_{0}, $$

where $h\in H$, $a\in A$ and $\rho ^A(a)=a_{-1}\otimes a_{0}$. Then $(A^{\natural }_{\diamond } H, \alpha \otimes \beta )$ is a Radford biproduct Hom-Hopf algebra.

Proof

By Theorem 13.3, we only need to prove that the conditions (HM1), (HM2), (HMA1), (HMA2) and (HYD) hold. And (HM1) and (HMA2) are easy. While

$$\begin{aligned} \begin{array}{rcl} \beta (h)\triangleright _A (g\triangleright _A a) &{}{\mathop {=}\limits ^{}}&{}\sigma (a_{-1},\beta ^{-3}(g))\sigma (a_{0-1},\beta ^{-2}(h))a_{00}\\ &{}{\mathop {=}\limits ^{(HCM2)}}&{}\sigma (\beta ^{-1}(a_{-11}),\beta ^{-3}(g))\sigma (a_{-12},\beta ^{-2}(h))\alpha (a_{0})\\ &{}{\mathop {=}\limits ^{(CHA5)}}&{}\sigma (a_{-11},\beta ^{-2}(g))\sigma (a_{-12},\beta ^{-2}(h))\alpha (a_{0})\\ &{}{\mathop {=}\limits ^{(CHA3)}}&{}\sigma (\beta (a_{-1}),\beta ^{-3}(h)\beta ^{-3}(g))\alpha (a_{0})\\ &{}{\mathop {=}\limits ^{(HCM1)(HA1)}}&{}\sigma (\alpha (a)_{-1}),\beta ^{-3}(hg))\alpha (a)_{0}\\ &{}{\mathop {=}\limits ^{}}&{}hg\triangleright _A \alpha (a), \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{rcl} \beta ^2(h)\triangleright _A (ab) &{}{\mathop {=}\limits ^{}}&{}\sigma ((ab)_{-1},\beta ^{-1}(h))(ab)_{0}\\ &{}{\mathop {=}\limits ^{(HCMA1)}}&{}\sigma (a_{-1}b_{-1},\beta ^{-1}(h))a_{0}b_{0}\\ &{}{\mathop {=}\limits ^{(CHA2)(HC1)}}&{}\sigma (\beta (a_{-1}),\beta ^{-2}(h_{1}))\sigma (\beta (b_{-1}),\beta ^{-2}(h_{2}))a_{0}b_{0}\\ &{}{\mathop {=}\limits ^{(CHA5)}}&{}\sigma (a_{-1},\beta ^{-3}(h_{1}))\sigma (b_{-1},\beta ^{-3}(h_{2}))a_{0}b_{0}\\ &{}{\mathop {=}\limits ^{}}&{}(h_1\triangleright _A a)(h_2\triangleright _A b), \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{rcl} (\beta ^2(h_1)\triangleright _A a)_{-1}h_2\otimes (\beta ^2(h_1)\triangleright _A a)_{0} &{}{\mathop {=}\limits ^{}}&{}\sigma (a_{-1},\beta ^{-1}(h_1))a_{0-1}h_2\otimes a_{00}\\ &{}{\mathop {=}\limits ^{(HCM2)}}&{}\sigma (\beta ^{-1}(a_{-11}),\beta ^{-1}(h_1))a_{-12}h_2\otimes \alpha (a_{0})\\ &{}{\mathop {=}\limits ^{(CHA5)}}&{}\sigma (a_{-11},h_1)a_{-12}h_2\otimes \alpha (a_{0})\\ &{}{\mathop {=}\limits ^{(CHA4)}}&{}h_1a_{-11}\sigma (a_{-12},h_2)\otimes \alpha (a_{0})\\ &{}{\mathop {=}\limits ^{(HCM2)}}&{}h_1\beta (a_{-1})\otimes \sigma (a_{0-1},h_2)a_{00}\\ &{}{\mathop {=}\limits ^{}}&{}h_1\beta (a_{-1})\otimes (\beta ^3(h_2)\triangleright _A a_{0}), \end{array} \end{aligned}$$

finishing the proof.    $\square$

Dually, we have

Theorem 13.5

Let $(H, \beta , \mathrm {R})$ be a quasitriangular Hom-Hopf algebra. Assume that $(A, \triangleright _A, \alpha )$ is a Hom-Hopf algebra in the category $_H{\mathcal {M}}$. Define $\rho ^A: A\longrightarrow H\otimes A$ by

$$ \rho ^A(a)=\beta ^{-3}(\mathrm {R}^2)\otimes (\mathrm {R}^1\triangleright _A a), $$

where $a\in A$. Then $(A^{\natural }_{\diamond } H, \alpha \otimes \beta )$ is a Radford biproduct Hom-Hopf algebra.

13.4 Twisted Tensor Biproduct Hom-Hopf Algebra

In this section, we consider the twisted tensor biproduct Hom-Hopf algebra generalizing the Radford’s biproduct Hom-Hopf algebra. And two nontrivial examples are given.

Theorem 13.6

Let $(H, \beta )$ be a Hom-bialgebra, $(A, \alpha )$ a Hom-algebra and a Hom-coalgebra. Let $R: H\otimes A \longrightarrow A\otimes H$ and $T: A\otimes H \longrightarrow H\otimes A$ be two linear maps such that the conditions (R) and (T) hold. Assume that $(A\natural _R H,\alpha \otimes \beta )$ is a R-smash product Hom-algebra and $(A\diamond _T H,\alpha \otimes \beta )$ is a T-smash coproduct Hom-coalgebra. Then the following are equivalent:

• $(A^{\natural _R}_{\diamond _T} H, \mu _{\natural _R H}, 1_A\otimes 1_H, \varDelta _{A\diamond _T H}, \varepsilon _A\otimes \varepsilon _H, \alpha \otimes \beta )$ is a Hom-bialgebra.

• The following conditions hold ($\forall ~a,b\in A$ and $h,g\in H$):    

(B1) $1_{AT}\otimes 1_{HT}=1_{A}\otimes 1_{H}$ and $\varDelta _A(1_A)=1_A\otimes 1_A$,    

(B2) $(ab)_1\otimes 1_{HT}\otimes (ab)_{2T}=a_1\alpha ^{-1}(b_1)_R\otimes \beta ^{-1}(1_{HTR})1_{Ht}\otimes a_{2T}b_{2t}$,    

(B3) $h_{T}\otimes a_T=1_{HT}\beta ^{-1}(h)_t\otimes \alpha ^{-1}(a)_T1_{At}$,

(B4) $(h_1g_1)_T\otimes 1_{AT}\otimes \beta (h_2)\beta (g_2)$  

$~~~~~~~~~~~~~=h_{1T}g_{1t}\otimes 1_{AT}\alpha (\alpha ^{-2}(1_{At})_R)\otimes \beta ^{-1}(\beta (h_2)_R)\beta (g_2)$,

(B5) $\alpha ^{-1}(a)_{R1}\otimes \beta ^{-1}(h_{R1})_T\otimes \alpha (\alpha ^{-1}(a)_{R2})_T\otimes h_{R2}$

$~~~~~~~~~~~~~=\alpha ^{-1}(a_1)_R\otimes \beta ^{-1}(\beta ^{-1}(h_1)_{TR})1_{Ht}\otimes 1_{AT}\alpha (\alpha ^{-2}(a_{2t})_r)\otimes h_{2r}$.    (B6)

$\varepsilon _A(a_R)\varepsilon _H(h_R)=\varepsilon _A(a)\varepsilon _H(h)$ and $\varepsilon _A$ is a Hom-algebra map.

In this case, we call this Hom-bialgebra twisted tensor biproduct Hom-bialgebra and denote it by $(A^{\natural _R}_{\diamond _T} H,\alpha \otimes \beta ).$

Proof

($\Longleftarrow$) It is easy to prove that $\varepsilon _{A^{\natural _R}_{\diamond _T} H}=\varepsilon _A\otimes \varepsilon _H$ is a morphism of Hom-algebras. Next we check $\varDelta _{A^{\natural _R}_{\diamond _T} H}=\varDelta _{A\diamond _T H}$ is a morphism of Hom-algebras as follows. For all $a,b\in A$ and $h,g\in H$, we have

$$\begin{aligned}&\varDelta _{A^{\natural _R}_{\diamond _T} H}((a\otimes h)(b\otimes g))\\&~~~~~{\mathop {=}\limits ^{}}(a\alpha ^{-1}(b)_{R})_{1}\otimes \beta ^{-1}((\beta ^{-1}(h_{R})g)_{1})_{T}\otimes \alpha ^{-1}((a\alpha ^{-1}(b)_{R})_{2T})\otimes (\beta ^{-1}(h_{R})g)_{2}\\&~~~{\mathop {=}\limits ^{(B3)}}(a\alpha ^{-1}(b)_{R})_{1}\otimes 1_{HT}\beta ^{-2}((\beta ^{-1}(h_{R})g)_{1})_{t}\otimes \alpha ^{-1}(\alpha ^{-1}((a\alpha ^{-1}(b)_{R})_{2})_{T}1_{At})\\&~~~~~~~~~\otimes (\beta ^{-1}(h_{R})g)_{2}\\&~~~{\mathop {=}\limits ^{(T)}}(a\alpha ^{-1}(b)_{R})_{1}\otimes \beta ^{-1}(1_{HT})\beta ^{-2}((\beta ^{-1}(h_{R})g)_{1})_{t}\otimes \alpha ^{-1}(\alpha ^{-1}((a\alpha ^{-1}(b)_{R})_{2T})1_{At})\\&~~~~~~~~~\otimes (\beta ^{-1}(h_{R})g)_{2}\\&~~~{\mathop {=}\limits ^{(B2)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes \beta ^{-1}(\beta ^{-1}(1_{H\bar{T}r})1_{H\bar{t}})\beta ^{-2}((\beta ^{-1}(h_{R})g)_{1})_{t}\\&~~~~~~~~~\otimes \alpha ^{-1}((\alpha ^{-1}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-1}(b)_{R2\bar{t}}))1_{At})\otimes (\beta ^{-1}(h_{R})g)_{2}\\&~~~{\mathop {=}\limits ^{(HA1)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes \beta ^{-1}(\beta ^{-1}(1_{H\bar{T}r})1_{H\bar{t}})(\beta ^{-3}(h_{R})_{1}\beta ^{-2}(g)_{1})_{t}\\&~~~~~~~~~\otimes \alpha ^{-1}((\alpha ^{-1}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-1}(b)_{R2\bar{t}}))1_{At})\otimes \beta (\beta (\beta ^{-3}(h_{R})_{2})\beta (\beta ^{-2}(g)_{2}))\\&~~~{\mathop {=}\limits ^{(B4)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes \beta ^{-1}(\beta ^{-1}(1_{H\bar{T}r})1_{H\bar{t}})(\beta ^{-3}(h_{R})_{1T}\beta ^{-2}(g)_{1t})\\&~~~~~~~~~\otimes \alpha ^{-1}((\alpha ^{-1}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-1}(b)_{R2\bar{t}}))(1_{AT}\alpha (\alpha ^{-2}(1_{At})_{\bar{R}})))\\&~~~~~~~~~\otimes \beta (\beta ^{-1}(\beta (\beta ^{-3}(h_{R})_{2})_{\bar{R}})\beta (\beta ^{-2}(g)_{2}))\\&~~~{\mathop {=}\limits ^{(HA1)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes (\beta ^{-2}(1_{H\bar{T}r})\beta ^{-1}(1_{H\bar{t}}))(\beta ^{-3}(h_{R})_{1T}\beta ^{-2}(g)_{1t})\\&~~~~~~~~~\otimes \alpha ^{-1}((\alpha ^{-1}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-1}(b)_{R2\bar{t}}))(1_{AT}\alpha (\alpha ^{-2}(1_{At})_{\bar{R}})))\\&~~~~~~~~~\otimes \beta (\beta ^{-1}(\beta (\beta ^{-3}(h_{R})_{2})_{\bar{R}})\beta (\beta ^{-2}(g)_{2}))\\&~~~{\mathop {=}\limits ^{(HA2)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes (\beta ^{-2}(1_{H\bar{T}r})\beta ^{-1}(\beta ^{-1}(1_{H\bar{t}})\beta ^{-3}(h_{R})_{1T}))\beta (\beta ^{-2}(g)_{1t})\\&~~~~~~~~~\otimes \alpha ^{-1}((\alpha ^{-1}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-1}(\alpha ^{-1}(b)_{R2\bar{t}})1_{AT}))\alpha ^2(\alpha ^{-2}(1_{At})_{\bar{R}})))\\&~~~~~~~~~\otimes \beta (\beta ^{-1}(\beta (\beta ^{-3}(h_{R})_{2})_{\bar{R}})\beta (\beta ^{-2}(g)_{2}))\\&~~~{\mathop {=}\limits ^{(HA1)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes (\beta ^{-2}(1_{H\bar{T}r})(\beta ^{-2}(1_{H\bar{t}})\beta ^{-1}(\beta ^{-3}(h_{R})_{1T})))\beta (\beta ^{-2}(g)_{1t})\\&~~~~~~~~~\otimes (\alpha ^{-2}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-2}(\alpha ^{-1}(b)_{R2\bar{t}})\alpha ^{-1}(1_{AT})))\alpha (\alpha ^{-2}(1_{At})_{\bar{R}})\\&~~~~~~~~~\otimes \beta (\beta ^{-1}(\beta (\beta ^{-3}(h_{R})_{2})_{\bar{R}})\beta (\beta ^{-2}(g)_{2}))\\&~~~{\mathop {=}\limits ^{(T)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes (\beta ^{-2}(1_{H\bar{T}r})(1_{H\bar{t}}\beta ^{-1}(\beta ^{-3}(h_{R})_{1})_{T}))\beta (\beta ^{-2}(g)_{1t})\\&~~~~~~~~~\otimes (\alpha ^{-2}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-2}(\alpha ^{-1}(b)_{R2})_{\bar{t}}1_{AT}))\alpha (\alpha ^{-2}(1_{At})_{\bar{R}})\\&~~~~~~~~~\otimes \beta (\beta ^{-1}(\beta (\beta ^{-3}(h_{R})_{2})_{\bar{R}})\beta (\beta ^{-2}(g)_{2}))\\&~~~{\mathop {=}\limits ^{(B3)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes (\beta ^{-2}(1_{H\bar{T}r})\beta ^{-3}(h_{R})_{1T})\beta (\beta ^{-2}(g)_{1t})\\&~~~~~~~~~\otimes (\alpha ^{-2}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-1}(\alpha ^{-1}(b)_{R2})_{T}))\alpha (\alpha ^{-2}(1_{At})_{\bar{R}})\\&~~~~~~~~~\otimes \beta (\beta ^{-1}(\beta (\beta ^{-3}(h_{R})_{2})_{\bar{R}})\beta (\beta ^{-2}(g)_{2}))\\&~~~{\mathop {=}\limits ^{(HC1)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes (\beta ^{-2}(1_{H\bar{T}r})\beta ^{-3}(h_{R1})_{T})\beta (\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes (\alpha ^{-2}(a_{2\bar{T}})\alpha ^{-1}(\alpha ^{-1}(\alpha ^{-1}(b)_{R2})_{T}))\alpha (\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta (\beta ^{-1}(\beta ^{-2}(h_{R2})_{\bar{R}})\beta ^{-1}(g_{2}))\\&~~~{\mathop {=}\limits ^{(T)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b)_{R1})_{r}\otimes (\beta ^{-2}(1_{H\bar{T}r})\beta ^{-2}(\beta ^{-1}(h_{R1})_{T}))\beta (\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes (\alpha ^{-2}(a_{2\bar{T}})\alpha ^{-3}(\alpha (\alpha ^{-1}(b)_{R2})_{T}))\alpha (\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta (\beta ^{-1}(\beta ^{-2}(h_{R2})_{\bar{R}})\beta ^{-1}(g_{2}))\\ \end{aligned}$$

$$\begin{aligned}&~{\mathop {=}\limits ^{(B5)(HA1)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b_{1})_{R})_{r}\otimes (\beta ^{-2}(1_{H\bar{T}r})\beta ^{-2}(\beta ^{-1}(\beta ^{-1}(h_{1})_{TR})1_{H\bar{t}}))\beta (\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes (\alpha ^{-2}(a_{2\bar{T}})\alpha ^{-3}(1_{AT}\alpha (\alpha ^{-2}(b_{2\bar{t}})_{\bar{r}})))\alpha (\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta ^{-2}(h_{2\bar{r}})_{\bar{R}}g_{2}\\&~{\mathop {=}\limits ^{(HA2)(HA1)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b_{1})_{R})_{r}\otimes \beta ^{-2}(1_{H\bar{T}r}\beta ^{-1}(h_{1})_{TR})(\beta ^{-1}(1_{H\bar{t}})\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-2}(a_{2\bar{T}}1_{AT})(\alpha ^{-1}(\alpha ^{-2}(b_{2\bar{t}})_{\bar{r}})\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta ^{-2}(h_{2\bar{r}})_{\bar{R}}g_{2}\\&~{\mathop {=}\limits ^{(R)}}a_{1}\alpha ^{-1}(\alpha ^{-1}(b_{1})_{Rr})\otimes \beta ^{-2}(\beta ^{-1}(\beta (1_{H\bar{T}})_{r})\beta ^{-1}(h_{1})_{TR})(\beta ^{-1}(1_{H\bar{t}})\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-2}(a_{2\bar{T}}1_{AT})(\alpha ^{-1}(\alpha ^{-2}(b_{2\bar{t}})_{\bar{r}})\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta ^{-2}(h_{2\bar{r}})_{\bar{R}}g_{2}\\&~{\mathop {=}\limits ^{(RS2)}}a_{1}\alpha ^{-1}(b_{1R})\otimes \beta ^{-2}((1_{H\bar{T}}\beta ^{-1}(h_{1})_{T})_{R})(\beta ^{-1}(1_{H\bar{t}})\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-2}(\alpha ^{-1}(\alpha (a_{2}))_{\bar{T}}1_{AT})(\alpha ^{-1}(\alpha ^{-2}(b_{2\bar{t}})_{\bar{r}})\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta ^{-2}(h_{2\bar{r}})_{\bar{R}}g_{2}\\&~{\mathop {=}\limits ^{(B3)}}a_{1}\alpha ^{-1}(b_{1R})\otimes \beta ^{-2}(h_{1TR})\beta ^{-1}(1_{H\bar{t}}\beta (\beta ^{-2}(g_{1})_{t}))\\&~~~~~~~~~\otimes \alpha ^{-2}(\alpha (a_{2})_{T})(\alpha ^{-1}(\alpha ^{-2}(b_{2\bar{t}})_{\bar{r}})\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta ^{-2}(h_{2\bar{r}})_{\bar{R}}g_{2}\\&~{\mathop {=}\limits ^{(HA1)}}a_{1}\alpha ^{-1}(b_{1R})\otimes \beta ^{-2}(h_{1TR})(\beta ^{-1}(1_{H\bar{t}})\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-2}(\alpha (a_{2})_{T})(\alpha ^{-1}(\alpha ^{-2}(b_{2\bar{t}})_{\bar{r}})\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta ^{-2}(h_{2\bar{r}})_{\bar{R}}g_{2}\\&~{\mathop {=}\limits ^{(T)}}a_{1}\alpha ^{-1}(b_{1R})\otimes \beta ^{-2}(h_{1TR})(1_{H\bar{t}}\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-2}(\alpha (a_{2})_{T})(\alpha ^{-1}(\alpha ^{-1}(\alpha ^{-1}(b_{2})_{\bar{t}})_{\bar{r}})\alpha ^{-2}(1_{At})_{\bar{R}})\otimes \beta ^{-2}(h_{2\bar{r}})_{\bar{R}}g_{2}\\&~{\mathop {=}\limits ^{(R)}}a_{1}\alpha ^{-1}(b_{1R})\otimes \beta ^{-2}(h_{1TR})(1_{H\bar{t}}\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-2}(\alpha (a_{2})_{T})(\alpha ^{-2}(\alpha ^{-1}(b_{2})_{\bar{t}})_{\bar{r}}\alpha ^{-1}(\alpha ^{-1}(1_{At})_{\bar{R}}))\otimes \beta ^{-1}(\beta ^{-1}(h_{2})_{\bar{r}\bar{R}})g_{2}\\&~{\mathop {=}\limits ^{(R)}}a_{1}\alpha ^{-1}(b_{1})_{R}\otimes \beta ^{-1}(\beta ^{-1}(h_{1})_{TR})(1_{H\bar{t}}\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-1}(a_{2T})(\alpha ^{-2}(\alpha ^{-1}(b_{2})_{\bar{t}})_{\bar{r}}\alpha ^{-1}(\alpha ^{-1}(1_{At})_{\bar{R}}))\otimes \beta ^{-1}(\beta ^{-1}(h_{2})_{\bar{r}\bar{R}})g_{2}\\&~{\mathop {=}\limits ^{(HA1)}}a_{1}\alpha ^{-1}(b_{1})_{R}\otimes \beta ^{-1}(\beta ^{-1}(h_{1})_{TR})(1_{H\bar{t}}\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-1}(a_{2T})\alpha ^{-1}(\alpha (\alpha ^{-2}(\alpha ^{-1}(b_{2})_{\bar{t}})_{\bar{r}})\alpha (\alpha ^{-2}(1_{At}))_{\bar{R}})\otimes \beta ^{-1}(\beta ^{-1}(h_{2})_{\bar{r}\bar{R}})g_{2}\\&~{\mathop {=}\limits ^{(RS3)}}a_{1}\alpha ^{-1}(b_{1})_{R}\otimes \beta ^{-1}(\beta ^{-1}(h_{1})_{TR})(1_{H\bar{t}}\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-1}(a_{2T})(\alpha ^{-2}(\alpha ^{-1}(b_{2})_{\bar{t}})\alpha ^{-2}(1_{At}))_{r})\otimes \beta ^{-1}(h_{2r})g_{2}\\&~{\mathop {=}\limits ^{(HA1)}}a_{1}\alpha ^{-1}(b_{1})_{R}\otimes \beta ^{-1}(\beta ^{-1}(h_{1})_{TR})(1_{H\bar{t}}\beta ^{-2}(g_{1})_{t})\\&~~~~~~~~~\otimes \alpha ^{-1}(a_{2T})\alpha ^{-2}(\alpha ^{-1}(b_{2})_{\bar{t}}1_{At})_{r}\otimes \beta ^{-1}(h_{2r})g_{2}\\&~{\mathop {=}\limits ^{(B3)}}a_{1}\alpha ^{-1}(b_{1})_{R}\otimes \beta ^{-1}(\beta ^{-1}(h_{1})_{TR})\beta ^{-1}(g_{1})_{t}\otimes \alpha ^{-1}(a_{2T})\alpha ^{-2}(b_{2t})_{r}\otimes \beta ^{-1}(h_{2r})g_{2}\\&~~{\mathop {=}\limits ^{}}\varDelta _{A^{\natural _R}_{\diamond _T} H}(a\otimes h)\varDelta _{A^{\natural _R}_{\diamond _T} H}(b\otimes g), \end{aligned}$$

and $\varDelta _{A^{\natural _R}_{\diamond _T} H}(1_A\otimes 1_H)=1_A\otimes 1_H\otimes 1_A\otimes 1_H$ can be proved directly.

$(\Longrightarrow )$ It is easy to prove that the conditions (B1) and (B6) hold. Next we check the conditions (B2)–(B5) are satisfied as follows.

For all $a,b\in A$ and $h,g\in H$, since $\varDelta _{A^{\natural _R}_{\diamond _T} H}((a\otimes h)(b\otimes g))=\varDelta _{A^{\natural _R}_{\diamond _T} H}(a\otimes h)\varDelta _{A^{\natural _R}_{\diamond _T} H}(b\otimes g)$, we have

$$\begin{aligned} (*)&(a\alpha ^{-1}(b)_{R})_{1}\otimes \beta ^{-1}((\beta ^{-1}(h_{R})g)_{1})_{T}\otimes \alpha ^{-1}((a\alpha ^{-1}(b)_{R})_{2T})\otimes (\beta ^{-1}(h_{R})g)_{2}\\&~~~~=a_{1}\alpha ^{-1}(b_{1})_{R}\otimes \beta ^{-1}(\beta ^{-1}(h_{1})_{TR})\beta ^{-1}(g_{1})_{t}\otimes \alpha ^{-1}(a_{2T})\alpha ^{-2}(b_{2t})_{r}\otimes \beta ^{-1}(h_{2r})g_{2} \end{aligned}$$

Apply $id_A\otimes id_H\otimes id_A\otimes \varepsilon _H$ to Eq.($*$) and then set $h=g=1_H$, we get (B2).

(B3) can be obtained by using $\varepsilon _A\otimes id_H\otimes id_A\otimes \varepsilon _H$ to Eq.($*$) and setting $b=1_A, h=1_H$.

Similarly, we apply $\varepsilon _A\otimes id_H\otimes id_A\otimes id_H$ to Eq.($*$) and set $a=b=1_A$, then (B4) holds.

(B5) can be derived by letting $a=1_A$ and $g=1_H$ in Eq.($*$).    $\square$

Remark 13.2

If $\alpha =id_A$ and $\beta =id_H$, then we can get the twisted tensor biproduct bialgebra in [4].

Corollary 13.3

([2]) Let $(C,\alpha ), (H,\beta )$ be two Hom-bialgebras, and $T: C\otimes H \longrightarrow H\otimes C$ a linear map such that the condition (T) holds. Then the T-smash coproduct Hom-coalgebra $(C\diamond _T H, \alpha \otimes \beta )$ endowed with the tensor product Hom-algebra structure becomes a Hom-bialgebra if and only if T is a map of Hom-algebras.

Proof

Let $R(h\otimes c)=\alpha (c)\otimes \beta (h)$ in Theorem 13.6. Then, by (B2)–(B5), we have

$$\begin{aligned}&(C1)\, \, 1_{HT}\otimes (ab)_T=1_{HT}1_{Ht}\otimes a_Tb_t,\\&(C2)\,\, h_T\otimes a_T=1_{hT}\beta ^{-1}(h)_t\otimes \alpha ^{-1}(a)_T1_{At}=\beta ^{-1}(h)_t1_{hT}\otimes 1_{At}\alpha ^{-1}(a)_T,\\&(C3)\, \, (hg)_T\otimes 1_{AT}=h_Tg_t\otimes 1_{AT}1_{At}. \end{aligned}$$

Next we only prove that $(hg)_T\otimes (ab)_T=h_Tg_t\otimes a_Tb_t$ as follows. And the rest are easy.

$$\begin{aligned}&(hg)_T\otimes (ab)_T\\&~~~~~~~{\mathop {=}\limits ^{(C2)}}1_{HT}\beta ^{-1}(hg)_{t}\otimes \alpha ^{-1}(ab)_{T}1_{At}\\&~~~~~~~{\mathop {=}\limits ^{(HA1)}}1_{HT}(\beta ^{-1}(h)\beta ^{-1}(g))_{t}\otimes (\alpha ^{-1}(a)\alpha ^{-1}(b))_{T}1_{At}\\&~~~~~~~{\mathop {=}\limits ^{(C1)(C3)}}(1_{H\bar{\bar{T}}}1_{HT}(\beta ^{-1}(h)_{\bar{T}}\beta ^{-1}(g)_{t})\otimes (\alpha ^{-1}(a)_{\bar{\bar{T}}}\alpha ^{-1}(b)_{T})(1_{A\bar{T}}1_{At})\\&~~~~~~~{\mathop {=}\limits ^{(HA2)}}(1_{H\bar{\bar{T}}}\beta ^{-1}(1_{HT}\beta ^{-1}(h)_{\bar{T}}))\beta (\beta ^{-1}(g)_{t})\otimes (\alpha ^{-1}(a)_{\bar{\bar{T}}}\alpha ^{-1}(\alpha ^{-1}(b)_{T}1_{A\bar{T}}))\alpha (1_{At})\\&~~~~~~~{\mathop {=}\limits ^{(C2)}}(1_{H\bar{\bar{T}}}\beta ^{-1}(\beta ^{-1}(h)_{\bar{T}}1_{HT}))\beta (\beta ^{-1}(g)_{t})\otimes (\alpha ^{-1}(a)_{\bar{\bar{T}}}\alpha ^{-1}(1_{A\bar{T}}\alpha ^{-1}(b)_{T}))\alpha (1_{At})\\&~~~~~~~{\mathop {=}\limits ^{(HA2)}}(1_{H\bar{\bar{T}}}\beta ^{-1}(h)_{\bar{T}})(1_{HT}\beta ^{-1}(g)_{t})\otimes (\alpha ^{-1}(a)_{\bar{\bar{T}}}1_{A\bar{T}})(\alpha ^{-1}(b)_{T}1_{At})\\&~~~~~~~{\mathop {=}\limits ^{(C2)}}h_Tg_t\otimes a_Tb_t, \end{aligned}$$

finishing the proof.    $\square$

Corollary 13.4

([3]) Let $(A,\alpha ), (H,\beta )$ be two Hom-bialgebras, and $R: H\otimes A \longrightarrow A\otimes H$ a linear map such that the condition (R) holds. Then the R-smash product Hom-algebra $(A\natural _R H, \alpha \otimes \beta )$ endowed with the tensor product Hom-coalgebra structure becomes a Hom-bialgebra if and only if R is a map of Hom-coalgebras.

Proof

Let $T(a\otimes h)=\beta (h)\otimes \alpha (a)$ in Theorem 13.6.    $\square$

Corollary 13.5

([2]) 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:

• $(A^{\natural }_{\diamond } H, \mu _{A\natural H}, 1_A\otimes 1_H, \varDelta _{A\diamond H}, \varepsilon _A\otimes \varepsilon _H, \alpha \otimes \beta )$ is a Hom-bialgebra, where $(A\natural H, \alpha \otimes \beta )$ is a smash product Hom-algebra and $(A\diamond H, \alpha \otimes \beta )$ is a smash coproduct Hom-coalgebra.

• The following conditions hold ($\forall ~a,b\in A$ and $h\in H$):        

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

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

(R3) $\varepsilon _A$ is a Hom-algebra map and $\varDelta _A(1_A)=1_A\otimes 1_A,$    

(R4) $\varDelta _A(ab)=a_1(\beta ^2(a_{2-1})\triangleright \alpha ^{-1}(b_1))\otimes \alpha ^{-1}(a_{20})b_2,$    

(R5) $h_1\beta (a_{-1})\otimes (\beta ^3(h_2)\triangleright a_0)=(\beta ^2(h_1)\triangleright a)_{-1}h_2\otimes (\beta ^2(h_1)\triangleright a)_{0}$.

In this case, we call $(A^{\natural }_{\diamond } H, \alpha \otimes \beta )$ Radford biproduct bialgebra.

Proof

Let $R(h\otimes a)=(h_1\triangleright a)\otimes h_2$ and $T(a\otimes h)=a_{-1}h\otimes a_0$ in Theorem 13.6.

   $\square$

Theorem 13.7

Let $(H,\beta ,S_{H})$ be a Hom-Hopf algebra, and $(A,\alpha )$ be a Hom-algebra and a Hom-coalgebra. Let $R: H\otimes A \longrightarrow A\otimes H$ and $T: A\otimes H \longrightarrow H\otimes A$ be two linear maps such that the conditions (R) and (T) hold. Assume that $(A^{\natural _R}_{\diamond _T} H, \alpha \otimes \beta )$ is a twisted tensor biproduct Hom-bialgebra defined as above, and $S_A: A\longrightarrow A$ 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 $(A^{\natural _R}_{\diamond _T} H, \alpha \otimes \beta , S_{A^{\natural _R}_{\diamond _T} H})$ is a Hom-Hopf algebra, where

$$ S_{A^{\natural _R}_{\diamond _T} H}(a\otimes h)=S_A(\alpha ^{-2}(a_T))_R\otimes \beta ^{-1}(S_H(\beta ^{-1}(h)_T)_{R}). $$

Proof

We can compute that $(A^{\natural }_{\diamond } H, \alpha \otimes \beta , S_{A^{\natural }_{\diamond } H})$ is a Hom-Hopf algebra as follows. For all $a\in A$ and $h\in H$, we have

$$\begin{aligned}&(S_{A^{\natural _R}_{\diamond _T} H}*id_{A^{\natural _R}_{\diamond _T} H})(a\otimes h)\\&~~~~~{\mathop {=}\limits ^{}}S_A(\alpha ^{-2}(a_{1t}))_{R}\alpha ^{-2}(a_{2T})_r\otimes \beta ^{-1}(\beta ^{-1}(S_H(\beta ^{-1}(\beta ^{-1}(h_{1})_{T})_{t})_{R})_{r})h_2\\&~~{\mathop {=}\limits ^{(T)}}S_A(\alpha ^{-2}(a_{1t}))_{R}\alpha ^{-3}(\alpha (a_{2})_{T})_r\otimes \beta ^{-1}(\beta ^{-1}(S_H(\beta ^{-2}(h_{1T})_{t})_{R})_{r})h_2\\&~~{\mathop {=}\limits ^{(T)}}S_A(\alpha ^{-4}(\alpha ^2(a_{1})_{t}))_{R}\alpha ^{-3}(\alpha (a_{2})_{T})_r\otimes \beta ^{-1}(\beta ^{-1}(S_H(\beta ^{-2}(h_{1Tt}))_{R})_{r})h_2\\&~~{\mathop {=}\limits ^{(T)}}S_A(\alpha ^{-4}(\alpha (\alpha (a)_{1})_{t}))_{R}\alpha ^{-4}(\alpha (\alpha (a)_{2T}))_r\otimes \beta ^{-1}(\beta ^{-1}(S_H(\beta ^{-2}(h_{1Tt}))_{R})_{r})h_2\\&~~{\mathop {=}\limits ^{(TS3)}}S_A(\alpha ^{-4}(\alpha ^2(a)_{T1}))_{R}\alpha ^{-4}(\alpha ^2(a)_{T2})_r\otimes \beta ^{-1}(\beta ^{-1}(S_H(\beta ^{-1}(h_{1T}))_{R})_{r})h_2\\&~~~~{\mathop {=}\limits ^{}}\alpha (S_A(\alpha ^{-5}(\alpha ^2(a)_{T1}))_{R})\alpha (\alpha ^{-5}(\alpha ^2(a)_{T2}))_r\otimes \beta ^{-1}(S_H(\beta ^{-2}(h_{1T}))_{Rr})h_2\\&~~{\mathop {=}\limits ^{(RS3)}}\alpha ((S_A(\alpha ^{-5}(\alpha ^2(a)_{T1}))\alpha ^{-5}(\alpha ^2(a)_{T2}))_R)\otimes \beta ^{-1}(S_H(\beta ^{-1}(h_{1T}))_{R})h_2\\&~~{\mathop {=}\limits ^{(HA1)}}\alpha (\alpha ^{-5}(S_A(\alpha ^2(a)_{T1})\alpha ^2(a)_{T2})_R)\otimes \beta ^{-1}(S_H(\beta ^{-1}(h_{1T}))_{R})h_2\\&~~{\mathop {=}\limits ^{(HA1)}}(\alpha (1_{AR})\otimes \beta ^{-1}(S_H(\beta ^{-1}(h_{1T}))_{R})h_2)\varepsilon _A(\alpha ^2(a)_{T})\\&~~{\mathop {=}\limits ^{(TS1)}}(\alpha (1_{AR})\otimes \beta ^{-1}(S_H(h_{1})_{R})h_2)\varepsilon _A(a)\\&~{\mathop {=}\limits ^{(RS1)(HA1)}}(1_{A}\otimes S_H(h_{1})h_2)\varepsilon _A(a)\\&~~~~~{\mathop {=}\limits ^{}}(1_{A}\otimes 1_{H})\varepsilon _A(a)\varepsilon _H(h). \end{aligned}$$

Similarly, we have $(id_{A^{\natural _R}_{\diamond _T} H}*S_{A^{\natural _R}_{\diamond _T} H})(a\otimes h)=(1_{A}\otimes 1_{H})\varepsilon _A(a)\varepsilon _H(h)$.

Finally,

$$\begin{aligned} \begin{array}{rcl} (\alpha \otimes \beta )\circ S_{A^{\natural _R}_{\diamond _T} H}(a\otimes h) &{}{\mathop {=}\limits ^{}}&{}\alpha (S_A(\alpha ^{-2}(a_T))_R)\otimes S_H(\beta ^{-1}(h)_T)_{R}\\ &{}{\mathop {=}\limits ^{(R)}}&{}\alpha (S_A(\alpha ^{-2}(a_T)))_R\otimes \beta ^{-1}(\beta (S_H(\beta ^{-1}(h)_T))_{R})\\ &{}{\mathop {=}\limits ^{}}&{}S_A(\alpha ^{-1}(a_T))_R\otimes \beta ^{-1}(S_H(\beta (\beta ^{-1}(h)_T))_{R})\\ &{}{\mathop {=}\limits ^{(T)}}&{}S_A(\alpha ^{-2}(\alpha (a)_T))_R\otimes \beta ^{-1}(S_H(h_T)_{R})\\ &{}{\mathop {=}\limits ^{(T)}}&{}S_{A^{\natural _R}_{\diamond _T} H}\circ (\alpha (a)\otimes \beta (h)), \end{array} \end{aligned}$$

finishing the proof.    $\square$

Corollary 13.6

([2]) Let $(H,\beta ,S_{H})$ be a Hom-Hopf algebra, and $(A,\alpha )$ be a Hom-algebra and a Hom-coalgebra. Assume that $(A^{\natural }_{\diamond } H, \alpha \otimes \beta )$ is a Radford biproduct Hom-bialgebra defined in Corollary 13.5, and $S_A: A\longrightarrow A$ 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 $(A^{\natural }_{\diamond } H, \alpha \otimes \beta , S_{A^{\natural }_{\diamond } H})$ is a Hom-Hopf algebra, where

$$ S_{A^{\natural }_{\diamond } H}(a\otimes h)=(S_H(a_{-1}\beta ^{-1}(h))_{1}\triangleright S_A(\alpha ^{-2}(a_{0})))\otimes \beta ^{-1}(S_H(a_{-1}\beta ^{-1}(h))_{2}). $$

Proof

Let $R(h\otimes a){=}(h_1\triangleright a)\otimes h_2$ and $T(a\otimes h){=}a_{-1}h\otimes a_0$ in Theorem 13.7.    $\square$

Example 13.1

Let $K\mathbf {Z}_2=K\{1,a\}$ be Hopf group algebra (see [8]). Then $(K\mathbf {Z}_2, id_{K\mathbf {Z}_2})$ is a Hom-Hopf algebra.

Let $T_{2,-1}=K\{1, g, x, gx|g^2=1, x^2=0\}$ be Taft’s Hopf algebra (see [4]), its coalgebra structure and antipode are given by

$$ \varDelta (g)=g\otimes g,~\varDelta (x) = x\otimes g+1\otimes x, ~\varDelta (gx)=gx\otimes 1+g\otimes gx; $$

$$ \varepsilon (g) = 1, \varepsilon (x) = 0, \varepsilon (gx)=0; $$

and

$$ S(g)=g,~S(x)=gx,~S(gx)=-x. $$

Define a linear map $\alpha$: $T_{2,-1}\longrightarrow T_{2,-1}$ by

$$ \alpha (1)=1,~\alpha (g)=g,~\alpha (x)=kx,~\alpha (gx)=kgx $$

where $0\ne k\in K$. Then $\alpha$ is an automorphism of Hopf algebras.

So we can get a Hom-Hopf algebra $H_{\alpha }=(T_{2,-1}, \alpha \circ \mu _{T_{2,-1}}, 1_{T_{2,-1}}, \varDelta _{T_{2,-1}}\circ \alpha , \varepsilon _{T_{2,-1}}, \alpha )$ (see [7]).

With notations above. Define two linear maps as follows:

$$\begin{aligned}&R: K\mathbf {Z}_2\otimes H_{\alpha }\longrightarrow H_{\alpha }\otimes K\mathbf {Z}_2\\&~~~~1_{K\mathbf {Z}_2}\otimes 1_{H_{\alpha }}\mapsto 1_{H_{\alpha }}\otimes 1_{K\mathbf {Z}_2}\\&~~~~~~1_{K\mathbf {Z}_2}\otimes g\mapsto g\otimes 1_{K\mathbf {Z}_2}\\&~~~~~~1_{K\mathbf {Z}_2}\otimes x\mapsto kx\otimes 1_{K\mathbf {Z}_2}\\&~~~~~1_{K\mathbf {Z}_2}\otimes gx\mapsto kgx\otimes 1_{K\mathbf {Z}_2}\\&~~~~~~~~~a\otimes 1_{H_{\alpha }}\mapsto 1_{H_{\alpha }}\otimes a\\&~~~~~~~~~a\otimes g\mapsto g\otimes a\\&~~~~~~~~~a\otimes x\mapsto kx\otimes a\\&~~~~~~~~~a\otimes gx\mapsto kgx\otimes a \end{aligned}$$

and

$$\begin{aligned}&T: H_{\alpha }\otimes K\mathbf {Z}_2\longrightarrow K\mathbf {Z}_2\otimes H_{\alpha }\\&~~~~1_{H_{\alpha }}\otimes 1_{K\mathbf {Z}_2}\mapsto 1_{K\mathbf {Z}_2}\otimes 1_{H_{\alpha }}\\&~~~~~~g\otimes 1_{K\mathbf {Z}_2}\mapsto 1_{K\mathbf {Z}_2}\otimes g\\&~~~~~~x\otimes 1_{K\mathbf {Z}_2}\mapsto ka\otimes x\\&~~~~~gx\otimes 1_{K\mathbf {Z}_2}\mapsto ka\otimes gx\\&~~~~1_{H_{\alpha }}\otimes a\mapsto a\otimes 1_{H_{\alpha }}\\&~~~~~~g\otimes a\mapsto a\otimes g\\&~~~~~~x\otimes a\mapsto k1_{K\mathbf {Z}_2}\otimes x\\&~~~~~gx\otimes a\mapsto k1_{K\mathbf {Z}_2}\otimes gx. \end{aligned}$$

By a direct computation, we have $({H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}, \mu _{H_{\alpha }\natural _R {K\mathbf {Z}_2}},1_{H_{\alpha }}\otimes 1_{K\mathbf {Z}_2}, \varDelta _{H_{\alpha }\diamond _T {K\mathbf {Z}_2}}, \varepsilon _{H_{\alpha }}\otimes \varepsilon _{K\mathbf {Z}_2}, \alpha \otimes id_{K\mathbf {Z}_2})$ is a twisted tensor biproduct Hom-bialgebra. Furthermore, $({H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}, \alpha \otimes id_{K\mathbf {Z}_2}, S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}})$ is a Hom-Hopf algebra, where $S_{{H_{\alpha }}^{\natural }_{\diamond } {K\mathbf {Z}_2}}$ is defined by

$$\begin{aligned}&S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}}(1_{H_{\alpha }}\otimes 1_{K\mathbf {Z}_2})=1_{H_{\alpha }}\otimes 1_{K\mathbf {Z}_2};~~~S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}}(1_{H_{\alpha }}\otimes a)=1_{H_{\alpha }}\otimes a\\&~~~~~~~S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}}(g\otimes 1_{K\mathbf {Z}_2})=g\otimes 1_{K\mathbf {Z}_2};~~~S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}}(g\otimes a)=g\otimes a\\&~~~~~~~~~~~~S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}}(x\otimes 1_{K\mathbf {Z}_2})=y\otimes a;~~~S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}}(x\otimes a)=y\otimes 1_{K\mathbf {Z}_2}\\&~~~~~~~~~~S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}}(y\otimes 1_{K\mathbf {Z}_2})=-x\otimes a;~~~S_{{H_{\alpha }}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}}(y\otimes a)=-x\otimes 1_{K\mathbf {Z}_2}. \end{aligned}$$

Example 13.2

Let $K\mathbf {Z}_2=K\{1,a\}$ be Hopf group algebra (see [8]). Then $(K\mathbf {Z}_2, id_{K\mathbf {Z}_2})$ is a Hom-Hopf algebra. Let $A=K\{1, x\}$ be a vector space. Define the multiplication $\mu _A$ by

$$ 1x=x1=lx,~~x^2=0 $$

and the automorphism $\beta : A\longrightarrow A$ by

$$ \beta (1)=1,~~~\beta (x)=lx $$

where $0\ne l\in K$. Then $(A,\beta )$ is a Hom-algebra.

Define the comultiplication $\varDelta _A$ by

$$ \varDelta _A(1)=1\otimes 1,~~\varDelta _A(x)=lx\otimes 1+l1\otimes x,~~\hbox {and}~~\varepsilon _A(1)=1,~~\varepsilon _A(x)=0. $$

Then $(A, \beta )$ is a Hom-coalgebra.

With notations above. Define two linear maps as follows:

$$\begin{aligned}&R: K\mathbf {Z}_2\otimes A\longrightarrow A\otimes K\mathbf {Z}_2\\&~~~~1_{K\mathbf {Z}_2}\otimes 1_{A}\mapsto 1_{A}\otimes 1_{K\mathbf {Z}_2}\\&~~~~~~1_{K\mathbf {Z}_2}\otimes x\mapsto lx\otimes 1_{K\mathbf {Z}_2}\\&~~~~~~~~~a\otimes 1_{A}\mapsto 1_{A}\otimes a\\&~~~~~~~~~a\otimes x\mapsto -lx\otimes a \end{aligned}$$

and

$$\begin{aligned}&T: A\otimes K\mathbf {Z}_2\longrightarrow K\mathbf {Z}_2\otimes A\\&~~~~1_{A}\otimes 1_{K\mathbf {Z}_2}\mapsto 1_{K\mathbf {Z}_2}\otimes 1_{A}\\&~~~~~~x\otimes 1_{K\mathbf {Z}_2}\mapsto la\otimes x\\&~~~~1_{A}\otimes a\mapsto a\otimes 1_{A}\\&~~~~~~x\otimes a\mapsto l1_{K\mathbf {Z}_2}\otimes x. \end{aligned}$$

By a direct computation, we have $({A}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}, \mu _{A\natural _R {K\mathbf {Z}_2}},1_{A}\otimes 1_{K\mathbf {Z}_2}, \varDelta _{A\diamond _T {K\mathbf {Z}_2}}, \varepsilon _{A}\otimes \varepsilon _{K\mathbf {Z}_2}, \alpha \otimes id_{K\mathbf {Z}_2})$ is a twisted tensor biproduct Hom-bialgebra. Furthermore, $({A}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}, \alpha \otimes id_{K\mathbf {Z}_2}$, $S_{{A}^{\natural _R}_{\diamond _T} {K\mathbf {Z}_2}})$ is a Hom-Hopf algebra, where $S_{{A}^{\natural }_{\diamond } {K\mathbf {Z}_2}}$ is defined by

$$\begin{aligned}&S_{{A}^{\natural }_{\diamond } {K\mathbf {Z}_2}}(1_{A}\otimes 1_{K\mathbf {Z}_2})=1_{A}\otimes 1_{K\mathbf {Z}_2};~~~S_{{A}^{\natural }_{\diamond } {K\mathbf {Z}_2}}(1_{A}\otimes a)=1_{A}\otimes a\\&~~~~~~~~S_{{A}^{\natural }_{\diamond } {K\mathbf {Z}_2}}(x\otimes 1_{K\mathbf {Z}_2})=x\otimes a;~~~S_{{A}^{\natural }_{\diamond } {K\mathbf {Z}_2}}(x\otimes a)=-x\otimes 1_{K\mathbf {Z}_2}. \end{aligned}$$