Classifying bicrossed products of two Sweedler's Hopf algebras

Costel Gabriel Bontea

paper

Classifying bicrossed products of two Sweedler's Hopf algebras

arXiv:1205.7010

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 $H_{4} ⋈ H_{4}$ . There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_{4}, H_{4}, ▹, ◃)$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$ . Then, for any $λ \in k$ , we describe by generators and relations the associated bicrossed product, \mathcal{H}_{16, \, λ} $. T hi s i s a 16 - d im e n s i o na l, p o in t e d, u nim o d u l a r an d n o n - se mi s im pl eH o p f a l g e b r a . A H o p f a l g e b r a$ E $f a c t or i z es t h r o ug h$ H_4 $an d$ H_4 $i f an d o n l y i f$ E \cong H_4 \ot H_4 $or$ E \cong \mathcal{H}_{16,\, λ} $. I n t h e l a s t s t e pw ec l a ss i f y t h ese q u an t u m g r o u p s b y p r o v in g t ha tt h er e a r eo n l y t h r ee i so m or p hi s m c l a ssesr e p r ese n t e d b y :$ H_4 \ot H_4 $,$ \mathcal{H}_{16, \, 0} $an d$ \mathcal{H}_{16, \, 1} \cong D(H_4) $, t h eD r in f e l^{'} dd o u b l eo f$ H_4 $. T h e a u t o m or p hi s m g r o u p o f t h eseo bj ec t s i s a l soco m p u t e d : in p a r t i c u l a r, w e p r o v e t ha t$ \Aut_{\rm Hopf}\big(D(H_4)\big) $i s i so m or p hi c t o a se mi d i r ec tp r o d u c t o f g r o u p s,$ k^{\times} \rtimes \mathbb{Z}_2$.

11 pages, submitted