代数形变理论现在已是代数学的重要分支之一.近年来作为代数另一类形变代数-Hom -代数的引入, 引起了许多代数学者的关注. Hom -代数的概念是由Makhlouf和Silvestrov于2006年在研究拟李代数时引入的(见文献[1]). Hom -代数的引入实际上是推广了结合代数的概念, 把结合代数中的结合性法则作了形变, 将其变成了线性变换$\alpha$结合性条件, 即$\alpha(a)(bc)=(ab)\alpha(c)$.随着Hom -代数研究的深入, 一些学者在文献[2-5]中又陆续引入了Hom -余结合余代数、Hom -双代数和Hom-Hopf代数等, 并给出了一些重要的性质.在文献[6, 7]中, 作者定义了Hom-$\omega$-smash积和Hom-$\omega$-smash余积, 并分别研究了它们的拟三角结构和辫化结构.

弱Hopf代数是由Bohm和Nill等人定义的(见文[8]), 作为Hopf代数(见文[9])的推广, 弱Hopf代数与Hopf代数有着相似的构成, 只是用更弱的条件去代替余乘法运算的保单位性和余单位运算的保乘法性.因此, 弱Hopf代数的结构远比Hopf代数复杂.

综合上述讨论, 在弱Hopf代数上引入Hom -代数的结合性条件成为自然的问题, 这也是写这篇文章的动机.在Hom -弱Hopf代数和模结构的基础上, 建立弱左$H$ -模Hom -代数的结构并通过它构造Hom-smash积, 证明Hom-smash积是Hom -代数, 且给出使之成为Hom -弱Hopf代数的充分条件.

本文的所有工作都在域$k$上进行的.所讨论的张量积和线性映射均指域$k$上的.文中将使用Sweedler关于余代数余乘法的记号, 即对于$H$中的任意元$h$, $\bigtriangleup(h)=\sum h_1\otimes h_2$.关于Hom -代数和Hom -余代数的概念请参阅文献[1-3].

定义2.1 ^[10]如果$(H,\mu,\eta,\alpha)$是一个Hom -代数, $(H,\bigtriangleup,\varepsilon,\alpha)$是一个Hom -余代数, 且代数和余代数结构满足下列相容性

$$\bigtriangleup(xy)=\bigtriangleup(x)\bigtriangleup(y),$$

(2.1)

$$\varepsilon((xy)\alpha(z))=\sum\varepsilon(xy_1)\varepsilon(y_2\alpha(z)),$$

(2.2)

$$\varepsilon(\alpha(x)(yz))=\sum\varepsilon(\alpha(x)y_2)\varepsilon(y_1z),$$

(2.3)

$$(\bigtriangleup\otimes\alpha)\bigtriangleup(1)=\sum 1_1\otimes 1_21_{1'}\otimes\alpha(1_{2'}),$$

(2.4)

$$(\alpha\otimes\bigtriangleup)\bigtriangleup(1)=\sum \alpha(1_1)\otimes 1_{1'}1_2\otimes 1_{2'},$$

(2.5)

则称六元组$(H,\mu,\eta,\bigtriangleup,\varepsilon,\alpha)$为一个Hom -弱双代数, 并简记为$(H,\alpha)$.

定义2.2 ^[10]设$(H,\alpha)$是一个Hom -弱双代数, $S : H\rightarrow H$是一个线性映射, 如果满足

$$S\circ\alpha=\alpha\circ S,$$

(2.6)

$$\sum x_1S(x_2)=\sum \varepsilon(1_1\alpha^2(x))1_2,$$

(2.7)

$$\sum S(x_1)x_2=\sum 1_1\varepsilon(\alpha^2(x)1_2),$$

(2.8)

$$\sum (S(x_{11})x_{12})S(\alpha^2(x_2))=S(\alpha^4(x)),$$

(2.9)

则称$(H,\alpha)$是一个Hom -弱Hopf代数, 并称$S$是Hom -弱Hopf代数$(H,\alpha)$的对极映射.

注2.3 由(2.7) 和(2.8) 式, 容易得到$\sum (x_{11}S(x_{12}))\alpha^2(x_2)=\alpha^4(x)$和$\sum \alpha^2(x_1)(S(x_{21})x_{22})=\alpha^4(x)$.对于(2.9) 式的合理性证明如下

$\begin{eqnarray*} \sum (S(x_{11})x_{12})S(\alpha^2(x_2)) &=&\sum (1_1\varepsilon(\alpha^2(x_1)1_2))S(\alpha^2(x_2))\\ &=&\sum (\alpha(1_1)\varepsilon((\alpha(x_1)1_{1'})\alpha(1_2)))S(\alpha(x_2)1_{2'})\\ &=&\sum (\alpha(1_1)\varepsilon(\alpha^2(x_1)(1_{1'}1_2)))S(\alpha(x_2)1_{2'})\\ &=&\sum (\alpha(1_1)\varepsilon(\alpha^2(x_1)1_{1'2})\varepsilon(1_{1'1}1_2))S(\alpha(x_2)1_{2'})\\ &=&\sum (\alpha(1_1)\varepsilon(\alpha(1_{1'})1_2))S(\varepsilon(\alpha^2(x_1)1_{2'1})\alpha^{-1}(\alpha^2(x_2)1_{2'2}))\\ &=&\sum (\alpha(1_1)\varepsilon(\alpha(1_{1'})1_2))S(\alpha^2(x)1_{2'})\\ &=&\sum (\alpha(1_1)\varepsilon(1_{1'}1_2))S(\alpha^2(x)\alpha^{-1}(1_{2'}))\\ &=&\sum (\alpha(1_1)\varepsilon(1_{21}))S(\alpha^2(x)\alpha^{-1}(1_{22}))\\ &=&\sum \alpha(1_1)S(\alpha^2(x)1_2)\\ &=&\sum (1_1S(1_2))S(\alpha^3(x))\\ &=& S(\alpha^4(x)). \end{eqnarray*}$

注2.4 Hom -弱Hopf代数既不满足结合律也不满足余结合律, 但当扭曲映射$\alpha=Id$时, 它就是弱Hopf代数.但当余单位$\varepsilon$是代数映射时, Hom -弱Hopf代数就是Hom-Hopf代数.相对于(余)结合性, Hom -弱Hopf代数也有Hom-(余)结合性, 即$\mu\circ(\alpha\otimes\mu)=\mu\circ(\mu\otimes\alpha)$和$(\alpha\otimes\bigtriangleup)\circ\bigtriangleup= (\bigtriangleup\otimes\alpha)\circ\bigtriangleup$.因此Hom -弱Hopf代数的非(余)结合性的程度是由扭曲映射$\alpha$偏离恒等映射的距离决定的.关于Hom -弱Hopf代数的相关性质请参阅文献[10].

命题2.5 设$(H,\alpha)$是一个Hom -弱Hopf代数, 且满足条件$\varepsilon\circ S=\varepsilon$, 则有如下结论

$$S\circ \sqcap^L=\sqcap^R\circ S, \,\,S\circ\sqcap^R=\sqcap^L\circ S.$$

(2.10)

证 对任意$x\in H$, 有

$\begin{eqnarray*} S\circ\sqcap^L(x)&=&\sum\varepsilon(1_1\alpha^2(x))S(1_2)\\ &=&\sum\varepsilon(S(1_1\alpha^2(x)))S(1_2)\\ &=&\sum\varepsilon(S(\alpha^2(x))S(1_1))S(1_2)\\ &=&\sum 1_1\varepsilon(\alpha^2(S(x))1_2)\\ &=&\sqcap^R\circ S(x). \end{eqnarray*}$

同理可证$S\circ\sqcap^R=\sqcap^L\circ S$成立.

上面的命题说明$S(H^L)\subset H^R$和$S(H^R)\subset H^L$.由于$\alpha(1)=1$, 因此有$\sum\alpha(1_1)\otimes\alpha(1_2)=\sum 1_1\otimes 1_2$, 所以有$\alpha(H^L)\subset H^L$和$\alpha(H^R)\subset H^R$.

命题2.6 设$(H,\alpha)$是一个Hom -弱Hopf代数, 则有如下结论

$$\sqcap^L(x)\sqcap^R(y)=\alpha(\sqcap^R(y))\alpha^{-1}(\sqcap^L(x)).$$

(2.11)

证 对任意$x,y\in H$, 有

$\begin{eqnarray*} \sqcap^L(x)\sqcap^R(y)&=&\sum\varepsilon(1_1\alpha^2(x))1_21_{1'}\varepsilon(\alpha^2(y)1_{2'})\\ &=&\sum\varepsilon(1_{11}\alpha^2(x))1_{12}\varepsilon(\alpha^2(y)1_{2})\\ &=&\sum\varepsilon(\alpha(1_{1})\alpha^2(x))1_{21}\varepsilon(\alpha^2(y)\alpha^{-1}(1_{22}))\\ &=&\sum\varepsilon(\alpha(1_{1})\alpha^2(x))1_{1'}1_2\varepsilon(\alpha^2(y)\alpha^{-1}(1_{2'}))\\ &=&\sum\alpha(1_{1'})\varepsilon(\alpha^2(y)1_{2'})\varepsilon(1_{1}\alpha^2(x))\alpha^{-1}(1_2)\\ &=&\alpha(\sqcap^R(y))\alpha^{-1}(\sqcap^L(x)). \end{eqnarray*}$

关于Hom -模、Hom -余模、Hom -模代数和Hom -模余代数的相关概念可参阅文献[5].下面, 给出弱左$H$ -模Hom -代数的概念.

定义3.1 设$(H,\alpha)$是Hom -弱Hopf代数, $(A,\beta)$是Hom -代数.如果有一个线性映射$\rho:H\otimes A\rightarrow A,\, \rho(h\otimes a)=h\cdot a$, 使得对任意$h,g\in H$和$a,b\in A$, 有下面条件成立

$$\alpha^2(h)\cdot(ab)=\sum (h_1\cdot a)(h_2\cdot b),\,\,\,\,\, h\cdot 1_A=\sqcap^L(h)\cdot 1_A,$$

(3.1)

$$(hg)\cdot\beta(a)=\alpha(h)\cdot(g\cdot a),\,\,\,\,\,\beta(h\cdot a)=\alpha(h)\cdot\beta(a),\,\,\,\,\,1_H\cdot a=\beta(a),$$

(3.2)

则称Hom -代数$(A,\beta)$是一个弱左$H$ -模Hom -代数.若$\alpha=Id$和$\beta=Id$, 则弱左$H$ -模Hom -代数是文献[11]中的一个弱左$H$ -模代数.若$(H,\alpha)$是Hom-Hopf代数, 则弱左$H$ -模Hom -代数是文献[5]中的一个左$H$ -模Hom -代数.

本节设$(H,\alpha)$是Hom -弱Hopf代数, 其弱对极$S$是双射, $(A,\beta)$是弱左$H$ -模Hom -代数, 设$\gamma=\beta\otimes\alpha$.为方便, 分别记$1_H$为$1$, $1_A$为$\widehat{1}$.

$(A,\beta)$在$(H,\alpha)$上的Hom-smash积$(A\sharp H,\gamma)$是指带有下面乘法运算的向量空间$A\otimes_{H^L} H$, 运算规定如下

$$(a\sharp h)(b\sharp g)=\sum a(\alpha^{-2}(h_1)\cdot\beta^{-1}(b))\sharp\alpha^{-1}(h_2)g,\,\,\,\,\,a,b\in A,\,\,h,g\in H,$$

其中$a\sharp h$表示向量$a\otimes h$在$A\sharp H$中所在的类, 即

$$A\otimes_{H^L} H=\{a\otimes h|a\cdot x\otimes h=\beta(a)\otimes\alpha^{-1}(x)\alpha^{-1}(h) \mbox{或} a\otimes xh=\beta^{-1}(a)\cdot\alpha(x)\otimes\alpha(h),x\in H^L\}.$$

$H$通过乘法构成左$H^L$模, $A$通过下面作用构成右$H$ -模

$$a\cdot x=S^{-1}(\alpha^{-1}(x))\cdot a=a(x\cdot \widehat{1}),\,a\in A,x\in H^L.$$

引理3.2 $(A,\beta)$在$(H,\alpha)$上的Hom-smash积$(A\sharp H,\gamma)$是一个带有单位元$\widehat{1}\sharp 1$的Hom -代数.

证 对任意$a,b,c\in A$和$h,g,k\in H$, 有

$\begin{eqnarray*} \gamma((a\sharp h)(b\sharp g))&=&\sum\beta(a)(\alpha^{-1}(h_1)\cdot b)\sharp h_2\alpha(g) =\gamma(a\sharp h)\gamma(b\sharp g),\\ (a\sharp h)(\widehat{1}\sharp 1)&=&\sum a(\alpha^{-2}(h_1)\cdot \widehat{1})\sharp h_2 =\sum a(\sqcap^L(\alpha^{-2}(h_1))\cdot \widehat{1})\sharp h_2\\ &=&\sum a\cdot\sqcap^L(\alpha^{-2}(h_1))\sharp h_2 =\sum \beta(a)\sharp\alpha^{-1}(\sqcap^L(\alpha^{-2}(h_1)))\alpha^{-1}(h_2)\\ &=&\beta(a)\sharp\alpha(h),\\ (\widehat{1}\sharp 1)(a\sharp h)&=&\sum \alpha^{-1}(1_1)\cdot a\sharp \alpha^{-1}(1_2)h =\sum 1_1\cdot a\sharp 1_2h\\ &=&\sum\beta^{-1}(1_1\cdot a)\cdot\alpha(1_2)\sharp\alpha(h) =\sum S^{-1}(1_2)\cdot(\alpha^{-1}(1_1)\cdot\beta^{-1}(a))\sharp\alpha(h)\\ &=&\sum \alpha^{-1}(S^{-1}(1_2)1_1)\cdot a\sharp\alpha(h) =\sum \alpha^{-1}(S^{-1}(\sqcap^R(1)))\cdot a\sharp\alpha(h)\\ &=&\beta(a)\sharp\alpha(h),\\ \gamma(a\sharp h)((b\sharp g)(c\sharp k)) &=&\sum(\beta(a)\sharp\alpha(h))(b(\alpha^{-2}(g_1)\cdot\beta^{-1}(c))\sharp\alpha^{-1}(g_2)k)\\ &=&\sum\beta(a)(\alpha^{-1}(h_1)\cdot(\beta^{-1}(b)(\alpha^{-3}(g_1)\cdot\beta^{-2}(c))))\sharp h_2(\alpha^{-1}(g_2)k)\\ &=&\sum\beta(a)((\alpha^{-3}(h_{11})\cdot\beta^{-1}(b))(\alpha^{-3}(h_{12})\cdot(\alpha^{-3}(g_1)\cdot\beta^{-2}(c))))\sharp h_2(\alpha^{-1}(g_2)k)\\ &=&\sum\beta(a)((\alpha^{-2}(h_{1})\cdot\beta^{-1}(b))(\alpha^{-3}(\alpha^{-1}(h_{21})g_1)\cdot\beta^{-1}(c)))\sharp\alpha^{-1}(h_{22})(\alpha^{-1}(g_2)k)\\ &=&\sum(a(\alpha^{-2}(h_{1})\cdot\beta^{-1}(b)))(\alpha^{-2}(\alpha^{-1}(h_{21})g_1)\cdot c)\sharp\alpha^{-1}(\alpha^{-1}(h_{22})g_2)\alpha(k)\\ &=&\sum (a(\alpha^{-2}(h_1)\cdot\beta^{-1}(b))\sharp\alpha^{-1}(h_2)g)(\beta(c)\sharp\alpha(k))\\ &=&((a\sharp h)(b\sharp g))\gamma(c\sharp k). \end{eqnarray*}$

这证明Hom-smash积$(A\sharp H,\gamma)$是一个带有单位元$\widehat{1}\sharp 1$的Hom -代数.

注3.3 若$\alpha=Id$和$\beta=Id$, 则Hom-smash积$(A\sharp H,\gamma)$是文献[12]中的smash积; 若$(H,\alpha)$是Hom-Hopf代数, 则Hom-smash积$(A\sharp H,\gamma)$是文献[6]中的Hom-smash积.

引理3.4 设$(A\sharp H,\gamma)$是一个Hom-smash积, 则对任意$a,b\in A$和$h,g\in H$, 下面关系式成立

(1) $(a\sharp 1)(b\sharp 1)=ab\sharp 1$;

(2) $(\widehat{1}\sharp h)(\widehat{1}\sharp g)=\widehat{1}\sharp hg$;

(3) $(a\sharp 1)(\widehat{1}\sharp g)=\beta(a)\sharp\alpha(g)$;

(4) $(\widehat{1}\sharp h)(b\sharp 1)=\sum\alpha^{-1}(h_1)\cdot b\sharp h_2$.

证 对任意$a,b\in A$和$h,g\in H$, 有

$\begin{eqnarray*} (a\sharp 1)(b\sharp 1)&=&\sum a(\alpha^{-2}(1_1)\cdot\beta^{-1}(b))\sharp 1_2 =\sum a(S^{-1}(\alpha^{-1}(S(\alpha^{-1}(1_1))))\cdot\beta^{-1}(b))\sharp 1_2\\ &=&\sum a(\beta^{-1}(b)(S(\alpha^{-1}(1_1))\cdot \widehat{1}))\sharp 1_2 =\sum\beta^{-1}(ab)(S(1_1)\cdot \widehat{1})\sharp 1_2\\ &=&\sum\beta^{-1}(ab)\cdot S(1_1)\sharp 1_2 =\sum ab\sharp \alpha^{-1}(S(1_1))\alpha^{-1}(1_2) =ab\sharp 1,\\ (\widehat{1}\sharp h)(\widehat{1}\sharp g)&=&\sum \widehat{1}(\alpha^{-2}(h_1)\cdot \widehat{1})\sharp\alpha^{-1}(h_2)g =\sum \widehat{1}(\sqcap^L(\alpha^{-2}(h_1))\cdot \widehat{1})\sharp\alpha^{-1}(h_2)g\\ &=&\sum \widehat{1}\cdot\sqcap^L(\alpha^{-2}(h_1))\sharp\alpha^{-1}(h_2)g =\sum \widehat{1}\sharp\alpha^{-1}(\sqcap^L(\alpha^{-2}(h_1))(\alpha^{-1}(h_2)g))\\ &=&\sum \widehat{1}\sharp\alpha^{-1}(\alpha^{-1}(\sqcap^L(\alpha^{-2}(h_1))h_2)\alpha(g)) =\widehat{1}\sharp hg,\\ (a\sharp 1)(\widehat{1}\sharp g)&=&\sum a(\alpha^{-2}(1_1)\cdot \widehat{1})\sharp\alpha^{-1}(1_2)g =\sum a(\sqcap^L(\alpha^{-2}(1_1))\cdot \widehat{1})\sharp\alpha^{-1}(1_2)g\\ &=&\sum a\cdot\sqcap^L(\alpha^{-2}(1_1))\sharp\alpha^{-1}(1_2)g =\sum \beta(a)\sharp\alpha^{-1}(\sqcap^L(\alpha^{-2}(1_1))(\alpha^{-1}(1_2)g))\\ &=&\sum \beta(a)\sharp\alpha^{-1}(\alpha^{-1}(\sqcap^L(\alpha^{-2}(1_1))1_2)\alpha(g)) =\beta(a)\sharp\alpha(g),\\ (\widehat{1}\sharp h)(b\sharp 1)&=&\sum \widehat{1}(\alpha^{-2}(h_1)\cdot\beta^{-1}(b))\sharp\alpha^{-1}(h_2)1\\ &=&\sum\alpha^{-1}(h_1)\cdot b\sharp h_2. \end{eqnarray*}$

定理3.5 设$(H,\alpha)$是Hom -弱Hopf代数, 其弱对极为$S$, $(A,\beta)$是Hom -弱双代数.如果$(A,\beta)$是弱左$H$ -模Hom -代数, 并且对于任意$h\in H,a\in A$, 下面条件成立

$\begin{eqnarray*} &&\varepsilon_A(h\cdot a)=\varepsilon_H(h)\varepsilon_A(a),\,\varepsilon_H\circ S=\varepsilon_H,\\ &&\bigtriangleup_A(h\cdot a)=\sum h_1\cdot a_1\otimes h_2\cdot a_2,\,\sum h_1\otimes h_2\cdot a=\sum h_2\otimes h_1\cdot a,\end{eqnarray*}$

则Hom-smash积$(A\sharp H,\gamma)$是Hom -弱双代数, 其中$(A\sharp H,\gamma)$的余代数结构是张量积Hom -余代数, 其余乘和余单位定义为

$$\bigtriangleup_{A\sharp H}(a\sharp h)=\sum (a_1\sharp h_1)\otimes (a_2\sharp h_2),\varepsilon_{A\sharp H}(a\sharp h)=\varepsilon_A(a)\varepsilon_H(h).$$

若此时$(A,\beta)$又是Hom -弱Hopf代数, 其弱对极为$S_A$, 满足性质$H^L\cdot \widehat{1}\subseteq Z(A)$, 则Hom-smash积$(A\sharp H,\gamma)$也是Hom -弱Hopf代数, 其弱对极为

$$S_{A\sharp H}(a\sharp h)=\sum S(\alpha^{-2}(h_2))\cdot S_A(\beta^{-1}(a))\sharp S(\alpha^{-1}(h_1)).$$

证 显然, Hom-smash积$(A\sharp H,\gamma)$是Hom -代数和Hom -余代数.假设定理条件成立, 证明Hom-smash积$(A\sharp H,\gamma)$满足定义2.1的(2.1)--(2.5) 项.对任意$a,b\in A$和$h,g\in H$, 有

$\begin{eqnarray*} &&\bigtriangleup_{A\sharp H}((a\sharp h)(b\sharp g))\\ &=&\sum\bigtriangleup_{A\sharp H}(a(\alpha^{-2}(h_1)\cdot\beta^{-1}(b))\sharp\alpha^{-1}(h_2)g)\\ &=&\sum a_1(\alpha^{-2}(h_{11})\cdot\beta^{-1}(b_1))\sharp\alpha^{-1}(h_{21})g_1\otimes a_2(\alpha^{-2}(h_{12})\cdot\beta^{-1}(b_2))\sharp\alpha^{-1}(h_{22})g_2\\ &=&\sum a_1(\alpha^{-1}(h_{1})\cdot\beta^{-1}(b_1))\sharp\alpha^{-2}(h_{212})g_1\otimes a_2(\alpha^{-3}(h_{211})\cdot\beta^{-1}(b_2))\sharp\alpha^{-1}(h_{22})g_2\\ &=&\sum a_1(\alpha^{-1}(h_{1})\cdot\beta^{-1}(b_1))\sharp\alpha^{-2}(h_{211})g_1\otimes a_2(\alpha^{-3}(h_{212})\cdot\beta^{-1}(b_2))\sharp\alpha^{-1}(h_{22})g_2\\ &=&\sum a_1(\alpha^{-2}(h_{11})\cdot\beta^{-1}(b_1))\sharp\alpha^{-1}(h_{12})g_1\otimes a_2(\alpha^{-2}(h_{21})\cdot\beta^{-1}(b_2))\sharp\alpha^{-1}(h_{22})g_2\\ &=&\sum (a_1\sharp h_1)(b_1\sharp g_1)\otimes (a_2\sharp h_2)(b_2\sharp g_2)\\ &=&\sum ((a_1\sharp h_1)\otimes (a_2\sharp h_2))((b_1\sharp g_1)\otimes (b_2\sharp g_2))\\ &=&\bigtriangleup_{A\sharp H}(a\sharp h)\bigtriangleup_{A\sharp H}(b\sharp g). \end{eqnarray*}$

运用定理中的条件, 可得如下关于余单位的弱乘运算

$\begin{eqnarray*} &&\varepsilon_{A\sharp H}(((a\sharp h)(b\sharp g))\gamma(c\sharp k))\\ &=&\sum\varepsilon_{A\sharp H}((a(\alpha^{-2}(h_{1})\cdot\beta^{-1}(b)))(\alpha^{-2}(\alpha^{-1}(h_{21})g_1)\cdot c)\sharp\alpha^{-1}(\alpha^{-1}(h_{22})g_2)\alpha(k))\\ &=&\sum\varepsilon_{A}((a(\alpha^{-2}(h_{1})\cdot\beta^{-1}(b)))(\alpha^{-2}(\alpha^{-1}(h_{21})g_1)\cdot c))\varepsilon_H(\alpha^{-1}(\alpha^{-1}(h_{22})g_2)\alpha(k))\\ &=&\sum\varepsilon_{A}(a(\alpha^{-2}(h_{11})\cdot\beta^{-1}(b_1)))\varepsilon_{A}((\alpha^{-2}(h_{12})\cdot\beta^{-1}(b_2))(\alpha^{-2}(\alpha^{-1}(h_{21})g_1)\cdot c))\\ &&\varepsilon_H(\alpha^{-2}(h_{22})\alpha^{-1}(g_{21}))\varepsilon_H(\alpha^{-1}(g_{22})\alpha(k))\\ &=&\sum\varepsilon_{A}(a(\alpha^{-1}(h_{1})\cdot\beta^{-1}(b_1)))\varepsilon_{A}((\alpha^{-3}(h_{211})\cdot\beta^{-1}(b_2))(\alpha^{-3}(h_{212})\cdot(\alpha^{-2}(g_1)\\ &&\cdot\beta^{-1}(c))))\varepsilon_H(\alpha^{-2}(h_{22})\alpha^{-1}(g_{21}))\varepsilon_H(\alpha^{-1}(g_{22})\alpha(k))\\ &=&\sum\varepsilon_{A}(a(\alpha^{-1}(h_{1})\cdot\beta^{-1}(b_1)))\varepsilon_{A}(\alpha^{-1}(h_{21})\cdot(\beta^{-1}(b_2)(\alpha^{-2}(g_1)\cdot\beta^{-1}(c))))\\ &&\varepsilon_H(\alpha^{-2}(h_{22})\alpha^{-1}(g_{21}))\varepsilon_H(\alpha^{-1}(g_{22})\alpha(k))\\ &=&\sum\varepsilon_{A}(a(\alpha^{-1}(h_{1})\cdot\beta^{-1}(b_1)))\varepsilon_H(\alpha^{-1}(h_{21}))\varepsilon_{A}(\beta^{-1}(b_2(\alpha^{-2}(g_{11})\cdot c)))\\ &&\varepsilon_H(\alpha^{-2}(h_{22})\alpha^{-1}(g_{12}))\varepsilon_H(g_{2}\alpha(k))\\ &=&\sum\varepsilon_{A}(a(\alpha^{-1}(h_{1})\cdot\beta^{-1}(b_1)))\varepsilon_{A}(b_2(\alpha^{-2}(g_{12})\cdot c))\\ &&\varepsilon_H(\alpha^{-1}(h_{2})\alpha^{-1}(g_{11}))\varepsilon_H(g_{2}\alpha(k))\\ &=&\sum\varepsilon_{A}(a(\alpha^{-1}(h_{1})\cdot\beta^{-1}(b_1)))\varepsilon_{A}(b_2(\alpha^{-2}(g_{21})\cdot c))\\ &&\varepsilon_H(\alpha^{-1}(h_{2})g_{1})\varepsilon_H(\alpha^{-1}(g_{22})\alpha(k))\\ &=&\sum\varepsilon_{A\sharp H}(a(\alpha^{-1}(h_{1})\cdot\beta^{-1}(b_1))\sharp\alpha^{-1}(h_{2})g_{1})\varepsilon_{A\sharp H}(b_2(\alpha^{-2}(g_{21})\cdot c)\sharp\alpha^{-1}(g_{22})\alpha(k))\\ &=&\sum\varepsilon_{A\sharp H}((a\sharp h)(b_1\sharp g_1))\varepsilon_{A\sharp H}((b_2\sharp g_2)\gamma(c\sharp k)). \end{eqnarray*}$

同理可得

$$\varepsilon_{A\sharp H}(\gamma(a\sharp h)((b\sharp g)(c\sharp k)))=\sum\varepsilon_{A\sharp H}(\gamma(a\sharp h)(b_2\sharp g_2))\varepsilon_{A\sharp H}((b_1\sharp g_1)(c\sharp k)).$$

由于对于任意的$a\in A,x\in H^L$, 有$a\cdot x=S^{-1}(\alpha^{-1}(x))\cdot a=a(x\cdot \widehat{1})$, 因此有$\alpha(x)\cdot\widehat{1}=\beta(x\cdot\widehat{1})=\widehat{1}\cdot x$, 又由于$\varepsilon_H\circ S=\varepsilon_H$成立, 因此由命题2.5可知$S(H^L)\subset H^R$和$S(H^R)\subset H^L$成立, 所以有

$\begin{eqnarray*} &&\sum(\widehat{1}_1\sharp 1_1)\otimes(\widehat{1}_2\sharp 1_2)(\widehat{1}_{1'}\sharp 1_{1'})\otimes\gamma(\widehat{1}_{2'}\sharp 1_{2'})\\ &=&\sum(\widehat{1}_1\sharp 1_1)\otimes(\widehat{1}_2(\alpha^{-2}(1_{21})\cdot\beta^{-1}(\widehat{1}_{1'}))\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2'})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_1\sharp\alpha^{-1}(1_{11}))\otimes(\widehat{1}_2(\alpha^{-2}(1_{12})\cdot\beta^{-1}(\widehat{1}_{1'}))\sharp 1_{2}1_{1'})\otimes(\beta(\widehat{1}_{2'})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_1\sharp\alpha^{-1}(1_{12}))\otimes(\widehat{1}_2(\alpha^{-2}(1_{11})\cdot\beta^{-1}(\widehat{1}_{1'}))\sharp 1_{2}1_{1'})\otimes(\beta(\widehat{1}_{2'})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_1\sharp\alpha^{-1}(1_{21}))\otimes(\widehat{1}_2(\alpha^{-1}(1_{1})\cdot\beta^{-1}(\widehat{1}_{1'}))\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2'})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_1\sharp\alpha^{-1}(1_{21}))\otimes(\widehat{1}_2(\beta^{-1}(\widehat{1}_{1'})\cdot S(1_1))\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2'})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_1\sharp\alpha^{-1}(1_{21}))\otimes(\widehat{1}_2(\beta^{-1}(\widehat{1}_{1'})(S(1_1)\cdot\widehat{1}))\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2'})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_1\sharp\alpha^{-1}(1_{21}))\otimes(\beta^{-1}(\widehat{1}_2\widehat{1}_{1'})(\alpha(S(1_1))\cdot\widehat{1})\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2'})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp\alpha^{-1}(1_{21}))\otimes(\beta^{-1}(\widehat{1}_{12})(\widehat{1}\cdot S(1_1))\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp\alpha^{-1}(1_{21}))\otimes(\beta^{-1}(\widehat{1}_{12})(\alpha^{-1}(1_1)\cdot\widehat{1})\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp\alpha^{-1}(1_{12}))\otimes(\beta^{-1}(\widehat{1}_{12})(\alpha^{-2}(1_{11})\cdot\widehat{1})\sharp 1_{2}1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp\alpha^{-1}(1_{11}))\otimes(\beta^{-1}(\widehat{1}_{12})(\alpha^{-2}(1_{12})\cdot\widehat{1})\sharp 1_{2}1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp 1_{1})\otimes(\beta^{-1}(\widehat{1}_{12})(\alpha^{-2}(1_{21})\cdot\widehat{1})\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp 1_{1})\otimes(\beta^{-1}(\widehat{1}_{12})(\sqcap^L(\alpha^{-2}(1_{21}))\cdot\widehat{1})\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp 1_{1})\otimes(\beta^{-1}(\widehat{1}_{12})\cdot\sqcap^L(\alpha^{-2}(1_{21}))\sharp\alpha^{-1}(1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp 1_{1})\otimes(\widehat{1}_{12}\sharp\alpha^{-1}(\sqcap^L(\alpha^{-2}(1_{21}))(\alpha^{-1}(1_{22})1_{1'})))\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp 1_{1})\otimes(\widehat{1}_{12}\sharp\alpha^{-2}(\sqcap^L(\alpha^{-2}(1_{21}))1_{22})1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp 1_{1})\otimes(\widehat{1}_{12}\sharp 1_{2}1_{1'})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2'}))\\ &=&\sum(\widehat{1}_{11}\sharp 1_{11})\otimes(\widehat{1}_{12}\sharp 1_{12})\otimes(\beta(\widehat{1}_{2})\sharp\alpha(1_{2}))\\ &=&(\bigtriangleup_{A\sharp H}\otimes\gamma)\bigtriangleup_{A\sharp H}(\widehat{1}\sharp 1). \end{eqnarray*}$

同理可得

$$(\gamma\otimes\bigtriangleup_{A\sharp H})\bigtriangleup_{A\sharp H}(\widehat{1}\sharp 1)=\sum\gamma(\widehat{1}_{1}\sharp 1_{1})\otimes(\widehat{1}_{1'}\sharp 1_{1'})(\widehat{1}_2\sharp 1_2)\otimes(\widehat{1}_{2'}\sharp 1_{2'}).$$

因此Hom-smash积$(A\sharp H,\gamma)$是Hom -弱双代数.

最后, 设$(H,\alpha)$和$(A,\beta)$是Hom -弱Hopf代数且$H^L\cdot\widehat{1}\subseteq Z(A)$.由于

$$S_{A\sharp H}\circ\gamma=\gamma\circ S_{A\sharp H},$$

直接验证可得.现需证明$S_{A\sharp H}$满足定义2.2中的(2.6)--(2.8) 项.设任意$a\in A$和$h\in H$, 有

$\begin{eqnarray*} &&(I\ast S_{A\sharp H})(a\sharp h)=\sum (a_1\sharp h_1)S_{A\sharp H}(a_2\sharp h_2)\\ &=&\sum (a_1\sharp h_1)(S(\alpha^{-2}(h_{22}))\cdot S_A(\beta^{-1}(a_2))\sharp S(\alpha^{-1}(h_{21})))\\ &=&\sum a_1(\alpha^{-2}(h_{11})\cdot(S(\alpha^{-3}(h_{22}))\cdot S_A(\beta^{-2}(a_2))))\sharp \alpha^{-1}(h_{12}S(h_{21}))\\ &=&\sum a_1(\alpha^{-3}(h_{11}S(h_{22}))\cdot S_A(\beta^{-1}(a_2)))\sharp \alpha^{-1}(h_{12}S(h_{21}))\\ &=&\sum a_1(\alpha^{-3}((h_{1}S(h_{2}))_1)\cdot S_A(\beta^{-1}(a_2)))\sharp \alpha^{-1}((h_{1}S(h_{2}))_2)\\ &=&\sum (a_1\sharp\alpha^{-1}(h_{1}S(h_{2})))(S_A(a_2)\sharp 1) =\sum (a_1\sharp\alpha^{-1}(\sqcap^L(h)))(S_A(a_2)\sharp 1)\\ &=&\sum (\beta^{-1}(a_1)\cdot\alpha^{-1}(\sqcap^L(h))\sharp 1)(S_A(a_2)\sharp 1) =\sum (\beta^{-1}(a_1)\cdot\alpha^{-1}(\sqcap^L(h)))S_A(a_2)\sharp 1\\ &=&\sum (\beta^{-1}(a_1)(\alpha^{-1}(\sqcap^L(h))\cdot \widehat{1}))S_A(a_2)\sharp 1 =\sum a_1((\alpha^{-1}(\sqcap^L(h))\cdot \widehat{1})\beta^{-1}(S_A(a_2)))\sharp 1\\ &=&\sum a_1(\beta^{-1}(S_A(a_2))(\alpha^{-1}(\sqcap^L(h))\cdot \widehat{1}))\sharp 1 =\sum \beta^{-1}(a_1S_A(a_2))(\sqcap^L(h)\cdot \widehat{1})\sharp 1\\ &=&\sum \beta^{-1}(\sqcap^L(a))\cdot\sqcap^L(h)\sharp 1 =\sqcap^L(a)\sharp\sqcap^L(h). \end{eqnarray*}$

同时有

$\begin{eqnarray*} \sqcap^L(a\sharp h)&=&\sum\varepsilon_{A\sharp H}((\widehat{1}_1\sharp 1_1)\gamma^2(a\sharp h))\widehat{1}_2\sharp 1_2\\ &=&\sum\varepsilon_{A\sharp H}((\widehat{1}_1\sharp 1_1)(\beta^2(a)\sharp\alpha^2(h)))\widehat{1}_2\sharp 1_2\\ &=&\sum\varepsilon_{A\sharp H}(\widehat{1}_1(\alpha^{-2}(1_{11})\cdot\beta(a))\sharp\alpha^{-1}(1_{12})\alpha^2(h))\widehat{1}_2\sharp 1_2\\ &=&\sum\varepsilon_{A\sharp H}(\widehat{1}_1(\alpha^{-1}(1_{1})\cdot\beta(a))\sharp\alpha^{-1}(1_{21})\alpha^2(h))\widehat{1}_2\sharp\alpha^{-1}(1_{22})\\ &=&\sum\varepsilon_{A\sharp H}(\widehat{1}_1(\beta(a)(S(1_{1})\cdot\widehat{1}))\sharp\alpha^{-1}(1_{21})\alpha^2(h))\widehat{1}_2\sharp\alpha^{-1}(1_{22})\\ &=&\sum\varepsilon_{A\sharp H}(\beta^{-1}(\widehat{1}_1\beta^2(a))(\alpha(S(1_{1}))\cdot\widehat{1})\sharp\alpha^{-1}(1_{21})\alpha^2(h))\widehat{1}_2\sharp\alpha^{-1}(1_{22})\\ &=&\sum\varepsilon_{A\sharp H}(\beta^{-1}(\widehat{1}_1\beta^2(a))(\widehat{1}\cdot S(1_1))\sharp\alpha^{-1}(1_{21})\alpha^2(h))\widehat{1}_2\sharp\alpha^{-1}(1_{22})\\ &=&\sum\varepsilon_{A\sharp H}(\beta^{-1}(\widehat{1}_1\beta^2(a))(\alpha^{-1}(1_{1})\cdot\widehat{1})\sharp\alpha^{-1}(1_{21})\alpha^2(h))\widehat{1}_2\sharp\alpha^{-1}(1_{22})\\ &=&\sum\varepsilon_{A\sharp H}(\beta^{-1}(\widehat{1}_1\beta^2(a))(\sqcap^L(\alpha^{-1}(1_{1}))\cdot\widehat{1})\sharp\alpha^{-1}(1_{21})\alpha^2(h))\widehat{1}_2\sharp\alpha^{-1}(1_{22})\\ &=&\sum\varepsilon_{A\sharp H}(\beta^{-1}(\widehat{1}_1\beta^2(a))\cdot\sqcap^L(\alpha^{-1}(1_{1}))\sharp\alpha^{-1}(1_{21})\alpha^2(h))\widehat{1}_2\sharp\alpha^{-1}(1_{22})\\ &=&\sum\varepsilon_{A\sharp H}(\widehat{1}_1\beta^2(a)\sharp\alpha^{-1}(\sqcap^L(\alpha^{-1}(1_{1}))(\alpha^{-1}(1_{21})\alpha^2(h))))\widehat{1}_2\sharp\alpha^{-1}(1_{22})\\ &=&\sum\varepsilon_{A}(\widehat{1}_1\beta^2(a))\widehat{1}_2\sharp\varepsilon_H(\sqcap^L(\alpha^{-1}(1_{1}))(\alpha^{-1}(1_{21})\alpha^2(h)))\alpha^{-1}(1_{22})\\ &=&\sum\sqcap^L(a)\sharp\varepsilon_H(1_{1'}\alpha(1_{1}))\varepsilon_H(1_{2'}(\alpha^{-1}(1_{21})\alpha^2(h)))\alpha^{-1}(1_{22})\\ &=&\sum\sqcap^L(a)\sharp\varepsilon_H(1_{1'}1_{11})\varepsilon_H(\alpha^{-1}(1_{2'}1_{12})\alpha^3(h))1_{2}\\ &=&\sum\sqcap^L(a)\sharp\varepsilon_H(\alpha(1_{1})\alpha^3(h))1_{2}\\ &=&\sqcap^L(a)\sharp\sqcap^L(h). \end{eqnarray*}$

因此$(I\ast S_{A\sharp H})(a\sharp h)=\sqcap^L(a\sharp h)$.同理可得$(S_{A\sharp H}\ast I)(a\sharp h)=\sqcap^R(a\sharp h)$.

对于定义2.2中的(2.9) 式, 利用定义3.1中的(3.1)、(3.2) 式和引理3.3及定理中的条件$H^L\cdot\widehat{1}\subseteq Z(A)$, 有

$\begin{eqnarray*} &&\sum(S_{A\sharp H}(a_{11}\sharp h_{11})(a_{12}\sharp h_{12}))S_{A\sharp H}(\gamma^2(a_{2}\sharp h_{2}))\\ &=&\sum((S_H(\alpha^{-2}(h_{112}))\cdot S_A(\beta^{-1}(a_{11}))\sharp S_H(\alpha^{-1}(h_{111})))(a_{12}\sharp h_{12}))\\ &&(S_H(h_{22})\cdot S_A(\beta(a_{2}))\sharp S_H(\alpha(h_{21})))\\ &=&\sum((S_H(\alpha^{-2}(h_{112}))\cdot S_A(\beta^{-1}(a_{11})))(S_H(\alpha^{-3}(h_{1112}))\cdot\beta^{-1}(a_{12}))\sharp S_H(\alpha^{-2}(h_{1111}))h_{12})\\ &&(S_H(h_{22})\cdot S_A(\beta(a_{2}))\sharp S_H(\alpha(h_{21})))\\ &=&\sum((S_H(\alpha^{-3}(h_{1122}))\cdot S_A(\beta^{-1}(a_{11})))(S_H(\alpha^{-3}(h_{1121}))\cdot\beta^{-1}(a_{12}))\sharp S_H(\alpha^{-1}(h_{111}))h_{12})\\ &&(S_H(h_{22})\cdot S_A(\beta(a_{2}))\sharp S_H(\alpha(h_{21})))\\ &=&\sum(S_H(\alpha^{-1}(h_{112}))\cdot S_A(\beta^{-1}(a_{11}))\beta^{-1}(a_{12})\sharp S_H(\alpha^{-1}(h_{111}))h_{12})\\ &&(S_H(h_{22})\cdot S_A(\beta(a_{2}))\sharp S_H(\alpha(h_{21})))\\ &=&\sum(S_H(\alpha^{-1}(h_{112}))\cdot \sqcap^R(\beta^{-1}(a_1)))((S_H(\alpha^{-3}(h_{1112}))\alpha^{-2}(h_{121}))\cdot\\ &&(S_H(\alpha^{-1}(h_{22}))\cdot S_A(a_{2})))\sharp(S_H(\alpha^{-2}(h_{1111}))\alpha^{-1}(h_{122}))S_H(\alpha(h_{21}))\\ &=&\sum(S_H(\alpha^{-2}(h_{1122}))\cdot \sqcap^R(\beta^{-1}(a_1)))(S_H(\alpha^{-2}(h_{1121}))\cdot(\alpha^{-2}(h_{121})\cdot\\ &&(S_H(\alpha^{-2}(h_{22}))\cdot S_A(\beta^{-1}(a_{2})))))\sharp(S_H(\alpha^{-1}(h_{111}))\alpha^{-1}(h_{122}))S_H(\alpha(h_{21}))\\ &=&\sum S_H(h_{112})\cdot(\sqcap^R(\beta^{-1}(a_1))(\alpha^{-3}(h_{121})S_H(\alpha^{-2}(h_{22}))\cdot S_A(a_{2})))\sharp\\ &&S_H(h_{111})(\alpha^{-1}(h_{122})S_H(h_{21}))\\ &=&\sum (\widehat{1}\sharp S_H(\alpha(h_{11})))(\sqcap^R(\beta^{-1}(a_1))(\alpha^{-2}((\alpha^{-1}(h_{12})S_H(h_{2}))_1)\cdot S_A(a_{2}))\sharp(\alpha^{-1}(h_{12})S_H(h_{2}))_2)\\ &=&\sum (\widehat{1}\sharp S_H(\alpha^2(h_{1})))(\sqcap^R(\beta^{-1}(a_1))(\alpha^{-3}((\sqcap^L(h_{2}))_1)\cdot S_A(a_{2}))\sharp\alpha^{-1}((\sqcap^L(h_{2}))_2))\\ &=&\sum (\widehat{1}\sharp S_H(\alpha^2(h_{1})))((\sqcap^R(\beta^{-1}(a_1))\sharp\alpha^{-1}(\sqcap^L(h_{2})))(S_A(\beta(a_{2}))\sharp 1))\\ &=&\sum (\widehat{1}\sharp S_H(\alpha^2(h_{1})))((\beta^{-1}(\sqcap^R(\beta^{-1}(a_1)))\cdot\alpha^{-1}(\sqcap^L(h_{2}))\sharp 1)(S_A(\beta(a_{2}))\sharp 1))\\ &=&\sum (\widehat{1}\sharp S_H(\alpha^2(h_{1})))((\beta^{-1}(\sqcap^R(\beta^{-1}(a_1)))(\alpha^{-1}(\sqcap^L(h_{2}))\cdot\widehat{1}))S_A(\beta(a_{2}))\sharp 1)\\ &=&\sum (\widehat{1}\sharp S_H(\alpha^2(h_{1})))(\sqcap^R(\beta^{-1}(a_1))((\alpha^{-1}(\sqcap^L(h_{2}))\cdot\widehat{1})S_A(a_{2}))\sharp 1)\\ &=&\sum (\widehat{1}\sharp S_H(\alpha^2(h_{1})))((\beta^{-1}(\sqcap^R(\beta^{-1}(a_1)))S_A(a_{2}))\cdot\sqcap^L(h_{2})\sharp 1)\\ &=&\sum (\widehat{1}\sharp S_H(\alpha^2(h_{1})))(\sqcap^R(\beta^{-1}(a_1))S_A(\beta(a_{2}))\sharp\sqcap^L(h_{2}))\\ &=&\sum (\widehat{1}\sharp S_H(\alpha^2(h_{1})))(S_A(\beta^3(a))\sharp\sqcap^L(h_{2}))\\ &=&\sum S_H(\alpha(h_{12}))\cdot S_A(\beta^3(a))\sharp S_H(\alpha(h_{11}))\sqcap^L(h_{2})\\ &=&\sum S_H(\alpha(h_{21}))\cdot S_A(\beta^3(a))\sharp S_H(\alpha^2(h_{1}))\sqcap^L(\alpha^{-1}(h_{22}))\\ &=&\sum S_H(\alpha(h_{22}))\cdot S_A(\beta^3(a))\sharp S_H(\alpha^2(h_{1}))\sqcap^L(\alpha^{-1}(h_{21}))\\ &=&\sum S_H(\alpha^2(h_{2}))\cdot S_A(\beta^3(a))\sharp S_H(\alpha(h_{11}))\sqcap^L(\alpha^{-1}(h_{12}))\\ &=&\sum S_H(\alpha^2(h_{2}))\cdot S_A(\beta^3(a))\sharp S_H(\alpha^3(h_{1})) =S_{A\sharp H}(\gamma^4(a\sharp h)), \end{eqnarray*}$

所以Hom-smash积$(A\sharp H,\gamma)$是Hom -弱Hopf代数.

注3.6 如果线性映射$\alpha$和$\beta$是恒等映射, 即对任意的$a\in A$和$h\in H$, 有$\gamma(a\otimes h)=a\otimes h$, 则Hom-smash积$(A\sharp H,\gamma)$是由文献[8]定义的弱Hopf代数, 并可得文献[13]中的例1.8或文献[16]中的定理2.2.如果$(H,\alpha)$和$(A,\beta)$是Hom-Hopf代数, 则Hom-smash积$(A\sharp H,\gamma)$是Hom-Hopf代数, 并可得文献[6]中的例2.2.