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
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
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
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
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 $a,a'\in A$,
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
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 $c\in C$,
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 $c\in A$,
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 $a,a'\in A$,
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.
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
Proof For all $a\in A,h\in H$, we have
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:
for all $h\in H$ and $u\in U$.
Proposition 4.2 When $(H,\beta)$ is a Hom-Hopf algebra, $(HYD)$ is equivalent to
for all $h\in H,u\in U$.
Proof $(HYD)\Longrightarrow (HYD)'$. We have
$(HYD)'\Longrightarrow (HYD)$ is proved as follows:
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
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
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
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
and
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
Lemma 4.6 Let $(H,\beta)$ be a Hom-bialgebra, and
With notation as above, define the linear map
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
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
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
and the rest is obvious. These complete the proof.