On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^*$ of Kashina

TL;DR

This paper classifies finite-dimensional Hopf algebras over a specific 16-dimensional dual semisimple Hopf algebra by analyzing Nichols algebras and their tensor decompositions, advancing the understanding of their structure and growth.

Contribution

It provides a complete classification of certain finite-dimensional Hopf algebras over $H_{b:1}^*$ using Nichols algebra decompositions, a novel approach in this context.

Findings

Classification of all Nichols algebras with tensor product structure

Complete determination of finite-dimensional Hopf algebras over $H_{b:1}^*$

Identification of conditions for finite-dimensional growth

Abstract

Let $H$ be the dual of $16$-dimensional nontrivial semisimple Hopf algebra $H_{b:1}$ in the classification work of Kashina \cite{K00}. We completely determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(N)\cong \bigotimes_{i\in I}\mathcal{B}(N_i)$, where $N=\bigoplus_{i\in I}N_i$, each $N_i$ is a simple object in $_H^H\mathcal{YD}$. Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra $H$ via the relevant Nichols algebras $\mathcal B(N)$.