Weak Skew Paired Bialgebras and Weak Relative Long Bialgebras
弱斜配对双代数和弱相关Long双代数

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.