Bicrossed products with the Taft algebra

TL;DR

This paper classifies bicrossed products involving the Taft algebra and group Hopf algebras, showing they are essentially smash products and linking their classification to automorphisms of the underlying group.

Contribution

It proves that bicrossed products with the Taft algebra are isomorphic to smash products and provides a complete classification for dihedral groups.

Findings

Bicrossed products are isomorphic to smash products.

Classification reduces to group automorphisms.

Complete description for dihedral groups.

Abstract

Let $G$ be a group which admits a generating set consisting of finite order elements. We prove that any Hopf algebra which factorizes through the Taft algebra and the group Hopf algebra $K[G]$ (equivalently, any bicrossed product between the aforementioned Hopf algebras) is isomorphic to a smash product between the same two Hopf algebras. The classification of these smash products is shown to be strongly linked to the problem of describing the group automorphisms of $G$. As an application, we completely describe by generators and relations and classify all bicrossed products between the Taft algebra and the group Hopf algebra $K[D_{2n}]$, where $D_{2n}$ denotes the dihedral group.