近年来, Hom -结构(Hom-Lie代数、Hom -代数、Hom -余代数、Hom-Hopf代数、Hom -模、Hom -余模和Hom-Hopf模)得到了广泛的研究.简而言之, Hom -型结构把之前结构中的恒等映射替换为广义的扭曲映射$\alpha$.随着Hom -代数研究的深入, 一些学者在文献[1-5]中又陆续引入了Hom -双代数、Hom-Hopf代数和拟三角Hom-Hopf代数等概念, 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(见文献[8]), 是通常的双代数和Hopf代数(见文献[9])的推广.关于弱Hopf代数最简单的例子是群代数.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.

本文的主要目的是在弱Hopf代数上引入扭曲映射$\alpha$, 定义Hom -弱Hopf代数概念.然后, 在Hom -弱Hopf代数和余模结构的基础上, 建立弱左$H$-Hom -余模双代数并通过它构造Hom-smash余积, 并证明Hom-smash余积是Hom -余代数和Hom -弱Hopf代数, 推广了由Molnar定义的smash余积Hopf代数.

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$h$, $\bigtriangleup(h)=\sum h_1\otimes h_2$.

在这一节, 首先介绍Hom -代数和Hom -余代数的概念, 然后给出Hom -弱双代数和Hom -弱Hopf代数的定义, 并给出一些关于Hom -弱双代数的性质.

定义2.1 ^{[1, 2]} 设$A$是线性空间, 并且有线性映射$\mu:A\otimes A\rightarrow A$, $\alpha:A\rightarrow A$和$\eta:k\rightarrow A$, 如果四元组$(A,\mu,\eta,\alpha)$对任意$a,b,c\in A$满足下列条件

$\begin{eqnarray}\alpha(ab)=\alpha(a)\alpha(b),\,\,\,\,\, \alpha(\eta(1))=\eta(1),\end{eqnarray}$

(2.1)

$\begin{eqnarray}\alpha(a)(bc)=(ab)\alpha(c),\,\,\,\,\,a\eta(1)=\alpha(a)=\eta(1)a,\end{eqnarray}$

(2.2)

则称四元组$(A,\mu,\eta,\alpha)$为一个Hom -代数.同时, 称$\eta(1)$为$A$的弱单位元和记$\eta(1)=1_A$.

定义2.2 ^{[2, 3]} 设$C$是线性空间, 并且有线性映射$\triangle : C\rightarrow C\otimes C$, $\alpha:C\rightarrow C$和$\varepsilon: C\rightarrow k$, 如果四元组$(C,\bigtriangleup,\varepsilon,\alpha)$对任意$c\in C$满足下列条件

$\begin{eqnarray}\triangle(\alpha(c))=\sum\alpha(c_{1})\otimes\alpha(c_{2}),\,\,\,\,\,\varepsilon(\alpha(c))=\varepsilon(c),\end{eqnarray}$

(2.3)

$\begin{eqnarray}\sum\alpha(c_{1})\otimes\triangle(c_{2})=\sum\triangle(c_{1})\otimes\alpha(c_{2}),\,\,\,\,\,\sum c_{1}\varepsilon(c_{2})=\alpha(c)=\sum\varepsilon(c_{1})c_{2},\end{eqnarray}$

(2.4)

则称四元组$(C,\bigtriangleup,\varepsilon,\alpha)$为一个Hom -余代数.

定义2.3 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:

$\begin{eqnarray}\bigtriangleup(xy)=\bigtriangleup(x)\bigtriangleup(y),\end{eqnarray}$

(2.5)

$\begin{eqnarray}\varepsilon((xy)\alpha(z))=\sum\varepsilon(xy_1)\varepsilon(y_2\alpha(z)),\end{eqnarray}$

(2.6)

$\begin{eqnarray}\varepsilon(\alpha(x)(yz))=\sum\varepsilon(\alpha(x)y_2)\varepsilon(y_1z),\end{eqnarray}$

(2.7)

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

(2.8)

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

(2.9)

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2.4 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

$\begin{eqnarray}S\circ\alpha=\alpha\circ S,\end{eqnarray}$

(2.10)

$\begin{eqnarray}\sum x_1S(x_2)=\sum \varepsilon(1_1\alpha^2(x))1_2,\end{eqnarray}$

(2.11)

$\begin{eqnarray}\sum S(x_1)x_2=\sum 1_1\varepsilon(\alpha^2(x)1_2),\end{eqnarray}$

(2.12)

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

(2.13)

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2.5 由式(2.11) 和(2.12), 容易得到

$\begin{eqnarray*}&&\sum (x_{11}S(x_{12}))\alpha^2(x_2)=\alpha^4(x),\\ &&\sum \alpha^2(x_1)(S(x_{21})x_{22})=\alpha^4(x).\end{eqnarray*}$

对于式(2.13) 的合理性证明如下:

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

注2.6 Hom -弱Hopf代数既不满足结合律也不满足余结合律, 且不再具有余乘法运算的保单位性和余单位运算的保乘法性, 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.

对Hom -弱双代数$H$, 定义映射$\sqcap^L, \sqcap^R:H\rightarrow H$如下:

$\sqcap^L(x)=\sum \varepsilon(1_1\alpha^2(x))1_2, \hspace{0.4cm}\sqcap^R(x)=\sum 1_1\varepsilon(\alpha^2(x)1_2).$

用$H^L$表示映射$\sqcap^L$的像集$\sqcap^L(H), H^R$表示映射$\sqcap^R$的像集$\sqcap^R(H)$.

设$(H, \mu, 1, \bigtriangleup, \varepsilon, S, \alpha)$是一个有限维的Hom -弱Hopf代数, $H^{*}$是$H$的线性对偶空间, 则$(H^{*}, \bigtriangleup^{*}, \widehat{1}, \mu^{*}, \widehat{\varepsilon}, S^{*}, \alpha^{*})$是一个Hom -弱Hopf代数, 并对任意的$x, y\in H$和$\phi, \psi\in H^{*}$, 有

$\begin{eqnarray*}&&\langle\mu^{*}(\phi), h\otimes g\rangle=\langle\phi, \mu(h, g)\rangle, \ \ \langle\triangle^{*}(\phi, \psi), h\rangle=\langle\phi\otimes\psi, \bigtriangleup(h)\rangle, \\ &&\langle \widehat{1}, x\rangle=\varepsilon(x), \ \ \widehat{\varepsilon}(\phi)=\langle\phi, 1\rangle, \\ &&S^{*}(\phi)=\phi\circ S, \ \ \alpha^{*}(\phi)=\phi\circ\alpha.\end{eqnarray*}$

利用Hom -弱双代数的定义和上面的对偶关系, 容易得到下面的一些命题, 至于Hom -弱双代数的更多性质, 作者将另文讨论.

命题2.7 设$(H, \alpha)$是一个Hom -弱双代数, 则有如下结论:

$\begin{eqnarray*}&&\sqcap^L\circ\sqcap^L=\alpha^3\circ\sqcap^L=\sqcap^L\circ\alpha^3, \\ &&\sqcap^R\circ\sqcap^R=\alpha\circ\sqcap^R=\sqcap^R\circ\alpha.\end{eqnarray*}$

命题2.8 设$(H, \alpha)$是一个Hom -弱双代数, 则有如下结论:

$\begin{eqnarray*}&&\sum 1_1\otimes\sqcap^L(1_2)=\sum 1_1\otimes\alpha^3(1_2), \\ &&\sum \sqcap^R(1_1)\otimes 1_2=\sum \alpha(1_1)\otimes 1_2.\end{eqnarray*}$

命题2.9 设$(H, \alpha)$是一个Hom -弱双代数, 则有如下结论:

$\begin{eqnarray*}&&\sqcap^L(x\sqcap^L(y))=\sqcap^L(x\alpha^3(y)), \\ &&\sqcap^R(\sqcap^R(x)y)=\sqcap^R(\alpha(x)y).\end{eqnarray*}$

命题2.10 设$(H, \alpha)$是一个Hom -弱双代数, 则有如下结论:

$\begin{eqnarray*}&&\bigtriangleup(\sqcap^L(x))=\sum 1_1\sqcap^L(x)\otimes 1_2, \\ &&\bigtriangleup(\sqcap^R(x))=\sum \alpha(1_1)\otimes \sqcap^R(\alpha(x))1_2. \end{eqnarray*}$

命题2.11 设$(H, \alpha)$是一个Hom -弱双代数, 则有如下结论:

$\begin{eqnarray*}&&\sum x_1\otimes\sqcap^L(x_2)=\sum 1_1x\otimes \alpha^4(1_2), \\ &&\sum\sqcap^R(x_1)\otimes x_2=\sum \alpha^3(1_1)\otimes x1_2. \end{eqnarray*}$

命题2.12 设$(H, \alpha)$是一个Hom -弱双代数, 则有如下结论:

$\begin{eqnarray*}&&x\sqcap^L(y)=\sum \varepsilon(x_1\alpha^4(y))x_2, \\ &&\sqcap^R(x)y=\sum y_1\varepsilon(\alpha^3(x)y_2). \end{eqnarray*}$

关于Hom -模、Hom -余模、Hom -模代数和Hom -余模代数的相关定义可参阅文献[4, 5].下面, 给出弱$H$-Hom -余模双代数的概念.

设$(H, \beta)$是Hom -弱双代数, $(C, \alpha)$是Hom -双代数.如果有线性映射$\rho:C\rightarrow H\otimes C, \, \rho(c)=\sum c_{(-1)}\otimes c_{(0)}$, 使得对任意$c, d\in C$, 有

$\begin{eqnarray}(\bigtriangleup_H\otimes\alpha)\circ\rho=(\beta\otimes\rho)\circ\rho, \, \, \, \, \, (\beta\otimes\alpha)\circ\rho=\rho\circ\alpha,\end{eqnarray}$

(3.1)

$\begin{eqnarray}(\mu_H\otimes I_C\otimes I_C)(I_H\otimes \tau\otimes I_C)(\rho\otimes\rho)\bigtriangleup_C=(\beta^2\otimes\bigtriangleup_C)\rho,\end{eqnarray}$

(3.2)

$\begin{eqnarray}(\sqcap^L\otimes\varepsilon_C)\circ\rho=(I_H\otimes\varepsilon_C)\circ\rho, \, \, \, \, (\varepsilon_H\otimes I_C)\circ\rho=\alpha,\end{eqnarray}$

(3.3)

$\begin{eqnarray}\rho(1_C)=(\sqcap^R\otimes I_C)\circ\rho(1_C), \, \, \, \, \rho(cd)=\rho(c)\rho(d).\end{eqnarray}$

(3.4)

这里$\tau$是扭曲映射, 如果$(C, \alpha)$满足条件(3.1)-(3.3), 则称$(C, \alpha)$是一个弱左$H$-Hom -余模余代数; 如果$(C, \alpha)$满足条件(3.1) 和(3.4), 则称$(C, \alpha)$是一个弱左$H$-Hom -余模代数; 如果$(C, \alpha)$满足条件(3.1)-(3.4), 则称$(C, \alpha)$是一个弱左$H$-Hom -余模双代数.如果$\alpha=I_C$和$\beta=I_H$, 则弱左$H$-Hom -余模双代数是弱Hopf代数上的弱左$H$-余模双代数.如果$(H, \beta)$和$(C, \alpha)$是Hom -双代数, 则弱左$H$-Hom -余模双代数是左$H$-Hom -余模双代数.

设$(H, \beta)$是Hom -弱双代数, $(C, \alpha)$是弱左$H$-Hom -余模余代数, 且$\alpha, \beta$都可逆.为方便, 分别记$1_H$为$1$, $1_C$为$\widehat{1}$.定义线性映射$\chi:C\otimes H\rightarrow C\otimes H$如下:

$\chi(c\otimes h)=\sum\varepsilon_H(c_{(-1)}h_1)\alpha^{-1}(c_{(0)})\otimes\beta^{-1}(h_2).$

则有

$\begin{eqnarray*} \chi^2(c\otimes h)&=&\sum\varepsilon_H(c_{(-1)}h_1)\chi(\alpha^{-1}(c_{(0)})\otimes\beta^{-1}(h_2))\\ &=&\sum\varepsilon_H(c_{(-1)}h_1)\varepsilon_H(\beta^{-1}(c_{(0)(-1)})\beta^{-1}(h_{21}))\alpha^{-2}(c_{(0)(0)})\otimes\beta^{-2}(h_{22})\\ &=&\sum\varepsilon_H(\beta^{-1}(c_{(-1)1}h_{11}))\varepsilon_H(\beta^{-1}(c_{(-1)2}h_{12}))\alpha^{-1}(c_{(0)})\otimes\beta^{-1}(h_2)\\ &=&\sum\varepsilon_H(c_{(-1)}h_1)\alpha^{-1}(c_{(0)})\otimes\beta^{-1}(h_2)\\ &=&\chi(c\otimes h). \end{eqnarray*}$

所以$\chi$是一个投射.因此, 定义$C\times H$, 作为向量空间有$C\times H=(C\otimes H)/\ker\chi$, 令线性映射$\gamma=\alpha\otimes\beta:C\otimes H\rightarrow C\otimes H$, 并定义弱余单位和余乘分别为

$\begin{eqnarray*}\varepsilon_{C\times H}(c\times h)&=&\varepsilon_C\otimes\varepsilon_H(c\times h)=\varepsilon_C(c)\varepsilon_H(h)\widehat{1}\times 1, \\ \bigtriangleup_{C\times H}(c\times h)&=&\sum c_1\times\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1)\otimes\alpha^{-1}(c_{2(0)})\times h_2,\end{eqnarray*}$

则称四元组$(C\times H, \bigtriangleup_{C\times H}, \varepsilon_{C\times H}, \gamma)$是一个Hom-smash余积, 并记为$(C\times H, \gamma)$, 如果它是一个Hom -余代数.

定理3.1 设$(H, \beta)$是Hom -弱双代数, $(C, \alpha)$是弱左$H$-Hom -余模余代数, 且$\alpha, \beta$都可逆, 而$C\times H$的定义如上, 若对任意的$c\in C, h\in H$, 满足

$\sum c_{(0)}\otimes c_{(-1)}h=\sum c_{(0)}\otimes hc_{(-1)},$

(3.5)

则四元组$(C\times H, \bigtriangleup_{C\times H}, \varepsilon_{C\times H}, \gamma)$是一个Hom-smash余积.

证 由于$(C, \alpha)$是弱左$H$-Hom -余模余代数, 所以对任意的$c\times h\in C\times H$, 有

$\begin{eqnarray*} &&(\bigtriangleup_{C\times H}\otimes\gamma)\bigtriangleup_{C\times H}(c\times h)\\ &=&\sum c_{11}\times\beta^{-2}(c_{12(-1)})\beta^{-1}((\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_1)\\ &&\otimes\alpha^{-1}(c_{12(0)})\times(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_2\otimes c_{2(0)}\times\beta(h_2)\\ &\stackrel{(2.4)}{=}&\sum\alpha(c_{1})\times\beta^{-2}(c_{21(-1)})(\beta^{-4}(c_{22(-1)1})\beta^{-1}(h_1))\\ &&\otimes\alpha^{-1}(c_{21(0)})\times\beta^{-3}(c_{22(-1)2})\beta^{-1}(h_{21})\otimes\alpha^{-1}(c_{22(0)})\times h_{22}\end{eqnarray*}$

$\begin{eqnarray*}&\stackrel{(3.1)}{=}&\sum\alpha(c_{1})\times\beta^{-2}(c_{21(-1)})(\beta^{-3}(c_{22(-1)})\beta^{-1}(h_1))\\ &&\otimes\alpha^{-1}(c_{21(0)})\times\beta^{-3}(c_{22(0)(-1)})\beta^{-1}(h_{21})\otimes\alpha^{-2}(c_{22(0)(0)})\times h_{22}\\ &\stackrel{(2.2)}{=}&\sum\alpha(c_{1})\times\beta^{-3}(c_{21(-1)}c_{22(-1)})h_1\\ &&\otimes\alpha^{-1}(c_{21(0)})\times\beta^{-3}(c_{22(0)(-1)})\beta^{-1}(h_{21})\otimes\alpha^{-2}(c_{22(0)(0)})\times h_{22}\\ &\stackrel{(3.2)}{=}&\sum\alpha(c_{1})\times\beta^{-1}(c_{2(-1)})h_1\\ &&\otimes\alpha^{-1}(c_{2(0)1})\times\beta^{-3}(c_{2(0)2(-1)})\beta^{-1}(h_{21})\otimes\alpha^{-2}(c_{2(0)2(0)})\times h_{22}\\ &=&(\gamma\otimes\bigtriangleup_{C\times H})\bigtriangleup_{C\times H}(c\times h). \end{eqnarray*}$

直接计算可得$\bigtriangleup_{C\times H}\gamma=(\gamma\otimes\gamma)\bigtriangleup_{C\times H}, \varepsilon_{C\times H}\gamma=\varepsilon_{C\times H}$.由(3.3)、(3.5) 式和$\sum c_{(0)}\otimes\varepsilon_H(c_{(-1)}h_1)h_2=\alpha(c)\otimes\beta(h)$.易证$(\varepsilon_{C\times H}\otimes I_{C\times H})\bigtriangleup_{C\times H}=\gamma$及$(I_{C\times H}\otimes\varepsilon_{C\times H})\bigtriangleup_{C\times H}=\gamma$成立.这证明$(C\times H, \bigtriangleup_{C\times H}, \varepsilon_{C\times H}, \gamma)$是一个Hom-smash余积.

定理3.2 设$(H, \beta)$和$(C, \alpha)$是Hom -弱双代数, 且$(C, \alpha)$是弱左$H$-Hom -余模双代数, 其中$\alpha, \beta$都可逆, 并满足(3.5) 式, 则Hom-smash余积$(C\times H, \gamma)$是Hom -弱双代数, 其中$(C\times H, \gamma)$的Hom -代数结构是张量积Hom -代数.

若此时$(H, \beta)$和$(C, \alpha)$是Hom -弱Hopf代数, 其弱对极分别为$S_H$和$S_C$, 则Hom-smash余积$(C\times H, \gamma)$也是Hom -弱Hopf代数, 其弱对极为

$S_{C\times H}(c\times h)=\sum S_C(\alpha^{-1}(c_{(0)}))\times S_H(\beta^{-2}(c_{(-1)})\beta^{-1}(h)).$

证 显然, Hom-smash余积$(C\times H, \gamma)$是Hom -代数和Hom -余代数.由题设知, 需要证明Hom-smash余积$(C\times H, \gamma)$满足定义2.3的(2.5)-(2.9) 项.对任意的$c, d\in C, h, g\in H$, 由(2.2) 和(3.5) 式及条件$\rho(cd)=\rho(c)\rho(d)$, 可得下式

$\begin{eqnarray*} &&\bigtriangleup_{C\times H}(cd\times hg)\\ &=&\sum c_{1}d_{1}\times\beta^{-2}(c_{2(-1)}d_{2(-1)})\beta^{-1}(h_1g_1)\otimes\alpha^{-1}(c_{2(0)}d_{2(0)})\times h_2g_2\\ &=&\sum c_{1}d_{1}\times\beta^{-1}(c_{2(-1)})\beta^{-2}(d_{2(-1)}(h_1g_1))\otimes\alpha^{-1}(c_{2(0)}d_{2(0)})\times h_2g_2\\ &=&\sum c_{1}d_{1}\times\beta^{-1}(c_{2(-1)})\beta^{-2}((\beta^{-1}(d_{2(-1)})h_1)\beta(g_1))\otimes\alpha^{-1}(c_{2(0)}d_{2(0)})\times h_2g_2\\ &=&\sum c_{1}d_{1}\times\beta^{-1}(c_{2(-1)})\beta^{-2}((h_1\beta^{-1}(d_{2(-1)}))\beta(g_1))\otimes\alpha^{-1}(c_{2(0)}d_{2(0)})\times h_2g_2\\ &=&\sum c_{1}d_{1}\times\beta^{-1}(c_{2(-1)})\beta^{-2}(\beta(h_1)(\beta^{-1}(d_{2(-1)})g_1))\otimes\alpha^{-1}(c_{2(0)}d_{2(0)})\times h_2g_2\\ &=&\sum c_{1}d_{1}\times(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))(\beta^{-2}(d_{2(-1)})\beta^{-1}(g_1))\otimes\alpha^{-1}(c_{2(0)}d_{2(0)})\times h_2g_2\\ &=&\bigtriangleup_{C\times H}(c\times h)\bigtriangleup_{C\times H}(d\times g). \end{eqnarray*}$

对于余单位的弱乘运算, 事实上, 有

$\begin{eqnarray*} &&\sum\varepsilon_{C\times H}((a\times h)(b\times g)_1)\varepsilon_{C\times H}((b\times g)_2\gamma(c\times k))\\ &=&\sum\varepsilon_{C\times H}(ab_1\times h(\beta^{-2}(b_{2(-1)})\beta^{-1}(g_1)))\varepsilon_{C\times H}(\alpha^{-1}(b_{2(0)})\alpha(c)\times g_2\beta(k))\\ &=&\sum\varepsilon_{C}(ab_1)\varepsilon_H(h(\beta^{-2}(b_{2(-1)})\beta^{-1}(g_1)))\varepsilon_{C}(\alpha^{-1}(b_{2(0)})\alpha(c))\varepsilon_H(g_2\beta(k))\end{eqnarray*}$

$\begin{eqnarray*}&\stackrel{(2.2)}{=}&\sum\varepsilon_{C}(ab_1)\varepsilon_H((\beta^{-1}(h)\beta^{-2}(b_{2(-1)}))g_1)\varepsilon_{C}(\alpha^{-1}(b_{2(0)})\alpha(c))\varepsilon_H(g_2\beta(k))\\ &\stackrel{(3.5)}{=}&\sum\varepsilon_{C}(ab_1)\varepsilon_H((\beta^{-2}(b_{2(-1)})\beta^{-1}(h))g_1)\varepsilon_{C}(\alpha^{-1}(b_{2(0)})\alpha(c))\varepsilon_H(g_2\beta(k))\\ &\stackrel{(2.3)}{=}&\sum\varepsilon_{C}(ab_1)\varepsilon_H((\beta^{-1}(b_{2(-1)})h)\beta(g_1))\varepsilon_{C}(\alpha^{-1}(b_{2(0)})\alpha(c))\varepsilon_H(g_2\beta(k))\\ &\stackrel{(2.6)}{=}&\sum\varepsilon_{C}(ab_1)\varepsilon_H(\beta^{-1}(b_{2(-1)})h_1)\varepsilon_H(h_2\beta(g_1))\varepsilon_{C}(\alpha^{-1}(b_{2(0)})\alpha(c))\varepsilon_H(g_2\beta(k))\\ &\stackrel{ }{=}&\sum\varepsilon_{C}(ab_1)\varepsilon_H(\beta(h)\beta(g_1))\varepsilon_{C}(b_{2}\alpha(c))\varepsilon_H(g_2\beta(k))\\ &\stackrel{(2.6)}{=}&\sum\varepsilon_{C}((ab)\alpha(c))\varepsilon_H((hg)\beta(k))\\ &=&\varepsilon_{C\times H}(((a\times h)(b\times g))\gamma(c\times k)). \end{eqnarray*}$

同理可得

$\varepsilon_{C\times H}(\gamma(a\times h)((b\times g)(c\times k)))=\sum\varepsilon_{C\times H}(\gamma(a\times h)(b\times g)_2)\varepsilon_{C\times H}((b\times g)_1(c\times k)).$

下面计算单位的余乘运算

$\begin{eqnarray*} &&(\bigtriangleup_{C\times H}\otimes\gamma)\bigtriangleup_{C\times H}(\widehat{1}\times 1)\\ &=&\sum\widehat{1}_{11}\times\beta^{-2}(\widehat{1}_{12(-1)})(\beta^{-3}(\widehat{1}_{2(-1)1})\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}(\widehat{1}_{12(0)})\times\beta^{-2}(\widehat{1}_{2(-1)2})\beta^{-1}(1_{12})\otimes \widehat{1}_{2(0)}\times\beta(1_2)\\ &\stackrel{(2.8)}{=}&\sum\widehat{1}_{1}\times\beta^{-2}((\widehat{1}_2\widehat{1}_{1'})_{(-1)})(\beta^{-3}(\widehat{1}_{2'(-1)1})\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}((\widehat{1}_2\widehat{1}_{1'})_{(0)})\times\beta^{-2}(\widehat{1}_{2'(-1)2})\beta^{-1}(1_{12})\otimes \widehat{1}_{2'(0)}\times\beta(1_2)\\ &\stackrel{(3.4)}{=}&\sum\widehat{1}_{1}\times\beta^{-2}(\widehat{1}_{2(-1)}\widehat{1}_{1'(-1)})(\beta^{-3}(\widehat{1}_{2'(-1)1})\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)}\widehat{1}_{1'(0)})\times\beta^{-2}(\widehat{1}_{2'(-1)2})\beta^{-1}(1_{12})\otimes \widehat{1}_{2'(0)}\times\beta(1_2)\\ &\stackrel{(3.1)}{=}&\sum\widehat{1}_{1}\times\beta^{-2}(\widehat{1}_{2(-1)}\widehat{1}_{1'(-1)})(\beta^{-2}(\widehat{1}_{2'(-1)})\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)}\widehat{1}_{1'(0)})\times\beta^{-2}(\widehat{1}_{2'(0)(-1)})\beta^{-1}(1_{12})\otimes\alpha^{-1}(\widehat{1}_{2'(0)(0)})\times\beta(1_2)\\ &\stackrel{(2.2)}{=}&\sum\widehat{1}_{1}\times\beta^{-1}(\widehat{1}_{2(-1)})(\beta^{-3}(\widehat{1}_{1'(-1)}\widehat{1}_{2'(-1)})\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)}\widehat{1}_{1'(0)})\times\beta^{-2}(\widehat{1}_{2'(0)(-1)})\beta^{-1}(1_{12})\otimes\alpha^{-1}(\widehat{1}_{2'(0)(0)})\times\beta(1_2)\\ &\stackrel{(3.2)}{=}&\sum\widehat{1}_{1}\times\beta^{-1}(\widehat{1}_{2(-1)})(\beta^{-1}(\widehat{1}_{(-1)})\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)}\widehat{1}_{(0)1})\times\beta^{-2}(\widehat{1}_{(0)2(-1)})\beta^{-1}(1_{12})\otimes\alpha^{-1}(\widehat{1}_{(0)2(0)})\times\beta(1_2)\\ &\stackrel{(3.4)}{=}&\sum\widehat{1}_{1}\times\beta^{-1}(\widehat{1}_{2(-1)})(\beta^{-1}(\sqcap^R(\widehat{1}_{(-1)}))\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)}\widehat{1}_{(0)1})\times\beta^{-2}(\widehat{1}_{(0)2(-1)})\beta^{-1}(1_{12})\otimes\alpha^{-1}(\widehat{1}_{(0)2(0)})\times\beta(1_2)\\ &\stackrel{(2.6)}{=}&\sum\widehat{1}_{1}\times\beta^{-1}(\widehat{1}_{2(-1)})(\beta^{-1}(1_{1'})\varepsilon_H(\varepsilon_H(\beta(\widehat{1}_{(-1)})1_{1''})1_{2''}1_{2'})\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)}\widehat{1}_{(0)1})\times\beta^{-2}(\widehat{1}_{(0)2(-1)})\beta^{-1}(1_{12})\otimes\alpha^{-1}(\widehat{1}_{(0)2(0)})\times\beta(1_2)\\ &\stackrel{}{=}&\sum\widehat{1}_{1}\times\beta^{-1}(\widehat{1}_{2(-1)})(\beta^{-1}(1_{1'})\varepsilon_H(\beta(1)1_{2'})\beta^{-2}(1_{11}))\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)})\widehat{1}_{1'}\times\beta^{-1}(\widehat{1}_{2'(-1)})\beta^{-1}(1_{12})\otimes\widehat{1}_{2'(0)}\times\beta(1_2)\end{eqnarray*}$

$\begin{eqnarray*}&\stackrel{(2.4)}{=}&\sum\widehat{1}_{1}\times\beta^{-1}(\widehat{1}_{2(-1)})\beta^{-1}(1_{11})\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)})\widehat{1}_{1'}\times\beta^{-1}(\widehat{1}_{2'(-1)})\beta^{-1}(1_{12})\otimes\widehat{1}_{2'(0)}\times\beta(1_2)\\ &\stackrel{(2.8)}{=}&\sum\widehat{1}_{1}\times\beta^{-1}(\widehat{1}_{2(-1)})\beta^{-1}(1_{1})\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)})\widehat{1}_{1'}\times\beta^{-1}(\widehat{1}_{2'(-1)})\beta^{-1}(1_21_{1'})\otimes\widehat{1}_{2'(0)}\times\beta(1_{2'})\\ &\stackrel{(2.2)}{=}&\sum\widehat{1}_{1}\times\beta^{-2}(\widehat{1}_{2(-1)})\beta^{-1}(1_{1})\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)})\widehat{1}_{1'}\times(\beta^{-2}(\widehat{1}_{2'(-1)})\beta^{-1}(1_2))1_{1'}\otimes\widehat{1}_{2'(0)}\times\beta(1_{2'})\\ &\stackrel{(3.5)}{=}&\sum\widehat{1}_{1}\times\beta^{-2}(\widehat{1}_{2(-1)})\beta^{-1}(1_{1})\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)})\widehat{1}_{1'}\times(\beta^{-1}(1_2)\beta^{-2}(\widehat{1}_{2'(-1)}))1_{1'}\otimes\widehat{1}_{2'(0)}\times\beta(1_{2'})\\ &\stackrel{(2.2)}{=}&\sum\widehat{1}_{1}\times\beta^{-2}(\widehat{1}_{2(-1)})\beta^{-1}(1_{1})\\ &&\otimes\alpha^{-1}(\widehat{1}_{2(0)})\widehat{1}_{1'}\times 1_2(\beta^{-2}(\widehat{1}_{2'(-1)})\beta^{-1}(1_{1'}))\otimes\widehat{1}_{2'(0)}\times\beta(1_{2'})\\ &\stackrel{ }{=}&\sum(\widehat{1}\times 1)_1\otimes(\widehat{1}\times 1)_2(\widehat{1}\times 1)_{1'}\otimes\gamma((\widehat{1}\times 1)_{2'}). \end{eqnarray*}$

同理可得

$(\gamma\otimes\bigtriangleup_{C\times H})\bigtriangleup_{C\times H}(\widehat{1}\times 1)=\sum\gamma((\widehat{1}\times 1)_1) \otimes(\widehat{1}\times 1)_{1'}(\widehat{1}\times 1)_{2}\otimes(\widehat{1}\times 1)_{2'}.$

因此Hom-smash余积$(C\otimes H, \gamma)$是Hom -弱双代数.

最后, 设$(H, \beta)$和$(C, \alpha)$是Hom -弱Hopf代数, 由于$S_{C\times H}\circ\gamma=\gamma\circ S_{C\times H}$直接验证可得.现需证明$S_{C\times H}$满足定义2.4中的(2.11)-(2.13) 项.设任意$c\in C$和$h\in H$, 有

$\begin{eqnarray*} &&I\ast S_{C\times H}(c\times h)\\ &=&\sum(c_1\times\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))S_{C\times H}(\alpha^{-1}(c_{2(0)})\times h_2)\\ &=&\sum c_1S_C(\alpha^{-2}(c_{2(0)(0)}))\times(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))S_{H}(\beta^{-3}(c_{2(0)(-1)})\beta^{-1}(h_2))\\ &\stackrel{(2.2)}{=}&\sum c_1S_C(\alpha^{-2}(c_{2(0)(0)}))\times\beta^{-1}(c_{2(-1)})(\beta^{-2}(h_1S_{H}(h_2))S_H(\beta^{-3}(c_{2(0)(-1)})))\\ &\stackrel{(3.5)}{=}&\sum c_1S_C(\alpha^{-2}(c_{2(0)(0)}))\times\beta^{-1}(c_{2(-1)})(S_H(\beta^{-3}(c_{2(0)(-1)}))\sqcap^L(\beta^{-2}(h)))\\ &\stackrel{(2.2)}{=}&\sum c_1S_C(\alpha^{-2}(c_{2(0)(0)}))\times(\beta^{-2}(c_{2(-1)})S_H(\beta^{-3}(c_{2(0)(-1)})))\sqcap^L(\beta^{-1}(h))\\ &\stackrel{(3.1)}{=}&\sum c_1S_C(\alpha^{-1}(c_{2(0)}))\times\beta^{-3}(c_{2(-1)1}S_H(c_{2(-1)2}))\sqcap^L(\beta^{-1}(h))\\ &\stackrel{(3.5)}{=}&\sqcap^L(c)\times\sqcap^L(h)\\ &\stackrel{ }{=}&\sum\varepsilon_C(\widehat{1}_1\alpha^2(c))\widehat{1}_2\times\varepsilon_H(1_1\beta^2(h))1_2\\ &\stackrel{ }{=}&\sum\varepsilon_C(\widehat{1}_1\alpha^2(c))\alpha^{-1}(\widehat{1}_{2(0)})\times \varepsilon_H(\beta^{-1}(\varepsilon_H(\beta^{-1}(\widehat{1}_{2(-1)})1_{11})1_{12})\beta^2(h))1_2\\ &\stackrel{(2.6)}{=}&\sum\varepsilon_C(\widehat{1}_1\alpha^2(c))\alpha^{-1}(\widehat{1}_{2(0)})\times \varepsilon_H((\beta^{-2}(\widehat{1}_{2(-1)})\beta^{-1}(1_{1}))\beta^2(h))1_2\\ &=&\sum\varepsilon_{C\times H}(\widehat{1}_1\alpha^2(c)\times(\beta^{-2}(\widehat{1}_{2(-1)})\beta^{-1}(1_{1}))\beta^2(h))\alpha^{-1}(\widehat{1}_{2(0)})\times 1_2\\ &=&\sum\varepsilon_{C\times H}((\widehat{1}\times 1)_1\gamma^2(c\times h))(\widehat{1}\times 1)_2=\sqcap^L(c\times h). \end{eqnarray*}$

因此$I\ast S_{C\times H}(c\times h)=\sqcap^L(c\times h)$.同理可得$S_{C\times H}\ast I(c\times h)=\sqcap^R(c\times h)$.

$\begin{eqnarray*} &&(S_{C\times H}((c\times h)_{11})(c\times h)_{12})S_{C\times H}(\gamma^2((c\times h)_{2}))\\ &=&\sum(S_{C\times H}(c_{11}\times\beta^{-2}(c_{12(-1)})\beta^{-1}((\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_1))\\ &&(\alpha^{-1}(c_{12(0)})\times(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_2))S_{C\times H}(\alpha(c_{2(0)})\times\beta^2(h_2))\\ &=&\sum(S_{C}(\alpha^{-1}(c_{11(0)}))\alpha^{-1}(c_{12(0)}))S_C(c_{2(0)(0)}) \\ &&\times(S_H(\beta^{-2}(c_{11(-1)})\beta^{-1}(\beta^{-2}(c_{12(-1)})\beta^{-1}((\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_1)))\\ &&(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_2)S_{H}(\beta^{-1}(c_{2(0)(-1)})\beta(h_2))\\ &\stackrel{(2.2)}{=}&\sum(S_{C}(\alpha^{-1}(c_{11(0)}))\alpha^{-1}(c_{12(0)}))S_C(c_{2(0)(0)}) \\ &&\times(S_H(\beta^{-3}(c_{11(-1)}c_{12(-1)})\beta^{-1}((\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_1))\\ &&(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_2)S_{H}(\beta^{-1}(c_{2(0)(-1)})\beta(h_2))\\ &\stackrel{(3.2)}{=}&\sum(S_{C}(\alpha^{-1}(c_{1(0)1}))\alpha^{-1}(c_{1(0)2}))S_C(c_{2(0)(0)}) \\ &&\times(S_H(\beta^{-1}(c_{1(-1)})\beta^{-1}((\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_1))\\ &&(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_2)S_{H}(\beta^{-1}(c_{2(0)(-1)})\beta(h_2))\\ &\stackrel{(3.5)}{=}&\sum(S_{C}(\alpha^{-1}(c_{1(0)1}))\alpha^{-1}(c_{1(0)2}))S_C(c_{2(0)(0)}) \\ &&\times((S_H(\beta^{-1}(c_{1(-1)}))S_H(\beta^{-1}((\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_1)))\\ &&(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_2)S_{H}(\beta^{-1}(c_{2(0)(-1)})\beta(h_2))\\ &\stackrel{(2.2)}{=}&\sum(S_{C}(\alpha^{-1}(c_{1(0)1}))\alpha^{-1}(c_{1(0)2}))S_C(c_{2(0)(0)}) \\ &&\times(S_H(c_{1(-1)})\beta^{-1}(S_H((\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_1)(\beta^{-2}(c_{2(-1)})\beta^{-1}(h_1))_2))\\ &&S_{H}(\beta^{-1}(c_{2(0)(-1)})\beta(h_2))\\ &\stackrel{(3.1)}{=}&\sum(S_{C}(\alpha^{-1}(c_{1(0)1}))\alpha^{-1}(c_{1(0)2}))S_C(\alpha(c_{2(0)})) \\ &&\times(S_H(c_{1(-1)})\beta^{-1}(S_H((\beta^{-3}(c_{2(-1)1})\beta^{-1}(h_1))_1)(\beta^{-3}(c_{2(-1)1})\beta^{-1}(h_1))_2))\\ &&S_{H}(\beta^{-1}(c_{2(-1)2})\beta(h_2))\\ &\stackrel{(2.2)}{=}&\sum(S_{C}(\alpha^{-1}(c_{1(0)1}))\alpha^{-1}(c_{1(0)2}))S_C(\alpha(c_{2(0)}))\times\beta(S_H(c_{1(-1)}))\\ &&\beta^{-1}((S_H((\beta^{-3}(c_{2(-1)1})\beta^{-1}(h_1))_1)(\beta^{-3}(c_{2(-1)1})\beta^{-1}(h_1))_2)\\ &&S_{H}(\beta^{-1}(c_{2(-1)2})\beta(h_2)))\\ &\stackrel{(2.13)}{=}&\sum(S_{C}(\alpha^{-1}(c_{1(0)1}))\alpha^{-1}(c_{1(0)2}))S_C(\alpha(c_{2(0)}))\times\beta(S_H(c_{1(-1)}))\\ &&\beta^{-1}(S_H(\beta(c_{2(-1)})\beta^{3}(h)))\\ &&\stackrel{(3.5)}{=}\sum(S_{C}(\alpha^{-1}(c_{1(0)1}))\alpha^{-1}(c_{1(0)2}))S_C(\alpha(c_{2(0)}))\times S_H(\beta(c_{1(-1)})(c_{2(-1)}\beta^{2}(h)))\\ &\stackrel{(2.2)}{=}&\sum(S_{C}(\alpha^{-1}(c_{1(0)1}))\alpha^{-1}(c_{1(0)2}))S_C(\alpha(c_{2(0)}))\times S_H((c_{1(-1)}c_{2(-1)})\beta^{3}(h))\\ &\stackrel{(3.2)}{=}&\sum(S_{C}(\alpha^{-1}(c_{(0)11}))\alpha^{-1}(c_{(0)12}))S_C(\alpha(c_{(0)2}))\times S_H(\beta^2(c_{(-1)})\beta^{3}(h))\\ &\stackrel{(2.13)}{=}&\sum S_{C}(\alpha^3(c_{(0)}))\times S_H(\beta^2(c_{(-1)})\beta^3(h))\\ &=&S_{C\times H}(\gamma^4(c\times h)). \end{eqnarray*}$

所以Hom-smash余积$(C\times H, \gamma)$是Hom -弱Hopf代数.

注3.3 如果线性映射$\alpha$和$\beta$是恒等映射, 即对任意的$c\in C$和$h\in H$, 有$\gamma(c\otimes h)=c\otimes h$, 则Hom-smash余积$(C\times H, \gamma)$是由文献[8]定义的弱Hopf代数.如果$(C, \alpha)$和$(H, \beta)$是Hom-Hopf代数, 则Hom-smash余积$(C\times H, \gamma)$是Hom-Hopf代数, 并可得文献[7]中的例2.11, 并推广了文献[10]中由Molnar定义的smash余积Hopf代数.