Some classifications of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:x^2y}$ of Kashina

TL;DR

This paper classifies finite-dimensional Hopf algebras over a specific 16-dimensional semisimple Hopf algebra by analyzing Yetter-Drinfeld modules, Nichols algebras, and their liftings, enriching the understanding of their structure.

Contribution

It provides a complete classification of certain finite-dimensional Hopf algebras over Kashina's $H_{b:x^2y}$ by examining simple modules, Nichols algebras, and their liftings.

Findings

All simple Yetter-Drinfeld modules over $H_{b:x^2y}$ identified

Classification of Nichols algebras satisfying tensor product constraints achieved

Descriptions of liftings of Radford biproducts provided

Abstract

Let $H$ be the $16$-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:x^2y}$ classified by Kashina. We figure out all simple Yetter-Drinfeld $H$-modules, and then determine all finite-dimensional Nichols algebras satisfying the constraint condition $\mathcal{B}(V)\cong \bigotimes_{i\in I}\mathcal{B}(V_i)$, where $V=\bigoplus_{i\in I}V_i$, each $V_i$ is a simple object in $_H^H\mathcal{YD}$. Finally, we describe some liftings of the corresponding Radford biproducts $\mathcal{B}(V)\sharp H$, which provide some classifications of finite dimensional Hopf algebras with $H$ as their coradical.