Finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}$ of Kashina

TL;DR

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

Contribution

It provides a complete classification of finite-dimensional Hopf algebras over the nontrivial semisimple Hopf algebra $H_{b:1}$, including simple modules, Nichols algebras, and their liftings.

Findings

All simple Yetter-Drinfeld modules over $H_{b:1}$ are classified.

All finite-dimensional Nichols algebras over $H_{b:1}$ are determined.

All liftings of these Nichols algebras over $H_{b:1}$ are described.

Abstract

Let $H$ be the 16-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:1}$ appeared in Kashina's work \cite{K00}. We obtain all simple Yetter-Drinfeld modules over $H$ and then determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(V)\cong \bigotimes_{i\in I}\mathcal{B}(V_i)$, where $V=\bigoplus_{i\in I}V_i$, each $V_i$ is a simple object in $_H^H\mathcal{YD}$. Finally, we describe all liftings of those $\mathcal{B}(V)$ over $H$.