弱斜配对双代数和弱相关Long双代数

该文在弱双代数$H$上给出了扭曲积$(H^\sigma,\cdot_\sigma)$成为弱双代数的充分必要条件.设$B, H, \tau]$是一个弱斜配对, 并且$\tau$可逆,则在某个条件下弱双交叉积$B\bowtie_\tau H$是一个弱双代数. 如果$(B,H, \sigma)$是弱相关Long双代数, 并且$\sigma$可逆,则弱双交叉积$B^{OP}\bowtie_\sigma H$可以被构造. 它的乘法是:$(x\otimes h)(y\otimes g)=\Sigma\sigma(y_1, h_1)y_2x\otimes h_2g\sigma^{-1}(y_3, h_3),$ 特别地, 如果$(B, H,\sigma)$是相关Long双代数, 则$(B^{OP \bowtie_\sigma H,\beta)$是Long双代数当且仅当对任意$b, d\in B^{OP}; g, \ell\in H$,$\Sigma\sigma^{-1}(b, g_2\ell)\sigma(d, g_1)=\Sigma\sigma^{-1}(b,\ell g_1)\sigma(d, g_2),$ 其中$B$为$H$的子Hopf代数,$\beta$定义为$\beta(b\bowtie_\sigma h\otimes c\bowtie_\sigma g)=\varepsilon_H(h)\varepsilon_{B^{OP}}(c)\sigma^{-1}(b, g).$ 对于Sweedler 4维Hopf代数$H$, 作者给出一个例子说明:此弱双交叉积$(B^{OP}\bowtie_\sigma H, \beta)$不仅是一个Long双代数,而且是一个非可换和非余可换的8维Hopf代数. 最后, 设$B,H$都是弱双代数, $\sigma: B\otimes H\rightarrow k$是一个线性映射, 作者给出了$(B,\sigma,\leftharpoonup, \Delta_B)$是弱相关右$(H, B)$ -重模代数的充分必要条件.

Weak Skew Paired Bialgebras and Weak Relative Long Bialgebras

This paper gives a sufficient andnecessary condition for given twisted product$(H^\sigma,\cdot_\sigma)$ to be a weak bialgebra. If $B, H,\tau]$ are weak skew paired bialgebras and $\tau$ is invertible,then, under some condition, the weakbicrossed product $B\bowtie_\tau H$ is a weak bialgebra. If $(B,H, \sigma)$ is a weak relative Long bialgebra and $\sigma$invertible, then the weak bicrossed product $B^{OP}\bowtie_\sigmaH$ can be constructed. Espically, for the Sweedler 4-dimensionalHopf algebra $H_4$, the author gives an example to show that$(B^{OP}\bowtie_\sigma H_4, \beta)$ is not only a Long bialgebrabut also a non-commutative and non-cocommutative 8-dimensionalHopf algebra, where $B$ is a sub-Hopf algebra of $H_4$. If $B$ and$H$ are weak bialgebras, and $\sigma: B\otimes H\rightarrow k$ isa linear map, then a sufficient and necessary conditionfor $(B,\sigma,\leftharpoonup, \Delta_B)$ to be a weak rightrelative $(H, B)$-dimodule algebra is given.