关键词: 弱Doi双代数, 弱双交叉积, 弱相关Long双代数, 弱相关重模代数

Abstract: This paper gives a sufficient and
necessary 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 weak
bicrossed 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 $B^{OP}\bowtie_\sigma H$ can be constructed. Espically, for the Sweedler 4-dimensional
Hopf algebra $H_4$, the author gives an example to show that
$(B^{OP}\bowtie_\sigma H_4, \beta)$ is not only a Long bialgebra
but also a non-commutative and non-cocommutative 8-dimensional
Hopf algebra, where $B$ is a sub-Hopf algebra of $H_4$. If $B$ and
$H$ are weak bialgebras, and $\sigma: B\otimes H\rightarrow k$ is
a linear map, then a sufficient and necessary condition
for $(B,\sigma,\leftharpoonup, \Delta_B)$ to be a weak right
relative $(H, B)$-dimodule algebra is given.

Key words: Weak skew paired bialgebra, Weak bicrossed product, Weak relative long bialgebra, Weak relative dimodule algebra