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Keywords:
bicrossed product of Hopf algebras; Sweedler's Hopf algebra; Drinfel'd double
bicrossed product of Hopf algebras; Sweedler's Hopf algebra; Drinfel'd double
Summary:
We continue the study started recently by Agore, Bontea and Militaru in ``Classifying bicrossed products of Hopf algebras'' (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler's Hopf algebras. Equivalently, we classify all bicrossed products $H_4 \bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4, H_4, \triangleright , \triangleleft )$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for any $\lambda \in k$, we describe by generators and relations the associated bicrossed product, $\mathcal {H}_{16, \lambda }$. This is a $16$-dimensional, pointed, unimodular and non-semisimple Hopf algebra. A Hopf algebra $E$ factorizes through $H_4$ and $H_4$ if and only if $ E \cong H_4 \otimes H_4$ or $E \cong {\mathcal H}_{16, \lambda }$. In the last step we classify these quantum groups by proving that there are only three isomorphism classes represented by: $H_4 \otimes H_4$, ${\mathcal H}_{16, 0}$ and ${\mathcal H}_{16, 1} \cong D(H_4)$, the Drinfel'd double of $H_4$. The automorphism group of these objects is also computed: in particular, we prove that ${\rm Aut}_{\rm Hopf}( D(H_4))$ is isomorphic to a semidirect product of groups, $k^{\times } \rtimes \mathbb {Z}_2$.
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