Classifying bicrossed products of two Sweedler's Hopf algebras

TL;DR

This paper classifies all Hopf algebras that factorize through two Sweedler's Hopf algebras, identifying three isomorphism classes and describing their automorphism groups, thus advancing the understanding of their structure.

Contribution

It explicitly describes all bicrossed products of two Sweedler's Hopf algebras and classifies their isomorphism classes, including automorphism groups.

Findings

Three isomorphism classes of Hopf algebras factorizing through two Sweedler's Hopf algebras.

Explicit description of all bicrossed products parameterized by the ground field.

Automorphism group of the Drinfel'd double of H_4 is a semidirect product of k^× and Z_2.

Abstract

In this paper we continue the study started recently in \cite{ABMbp} 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 \ot 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: $H_4 \ot H_4$, $\mathcal{H}_{16, \, 0}$ 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 $\Aut_{\rm Hopf}\big(D(H_4)\big)$ is isomorphic to a semidirect product of groups, $k^{\times} \rtimes \mathbb{Z}_2$.