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Weak Skew Paired Bialgebras and Weak Relative Long Bialgebras
Zhang Liangyun
Acta mathematica scientia,Series A. 2006, 26 (4):
601-611.
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.
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