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

Yihan Wu, Hengyi Wang, Naihong Hu

TL;DR

This work addresses the classification of finite-dimensional Hopf algebras with coradical $H_{b:x^2y}$ by applying the lifting method to Kashina's 16-dimensional semisimple Hopf algebra. It computes the Drinfeld double $D(H^{cop})$, enumerates simple Yetter-Drinfeld modules, and classifies finite-dimensional Nichols algebras under a tensor-product constraint, before describing explicit liftings $\mathcal{B}(V)\# H$ to obtain finite-dimensional Hopf algebras with coradical $H$. The results yield a comprehensive list of finite-dimensional Hopf algebras over $H$ (Theorem B) obtained as liftings of Nichols algebras, including detailed parameterized families. This advances understanding of coradical-homogeneous Hopf algebras beyond group-like coradicals and provides concrete, computable models relevant to the broader lifting program.

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.

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

TL;DR

This work addresses the classification of finite-dimensional Hopf algebras with coradical by applying the lifting method to Kashina's 16-dimensional semisimple Hopf algebra. It computes the Drinfeld double , enumerates simple Yetter-Drinfeld modules, and classifies finite-dimensional Nichols algebras under a tensor-product constraint, before describing explicit liftings to obtain finite-dimensional Hopf algebras with coradical . The results yield a comprehensive list of finite-dimensional Hopf algebras over (Theorem B) obtained as liftings of Nichols algebras, including detailed parameterized families. This advances understanding of coradical-homogeneous Hopf algebras beyond group-like coradicals and provides concrete, computable models relevant to the broader lifting program.

Abstract

Let be the -dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra classified by Kashina. We figure out all simple Yetter-Drinfeld -modules, and then determine all finite-dimensional Nichols algebras satisfying the constraint condition , where , each is a simple object in . Finally, we describe some liftings of the corresponding Radford biproducts , which provide some classifications of finite dimensional Hopf algebras with as their coradical.
Paper Structure (9 sections, 26 theorems, 139 equations)

This paper contains 9 sections, 26 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Yetter-Drinfeld modules and Nichols algebras
  4. Bosonization and Hopf algebras with a projection
  5. The Drinfeld double
  6. The Hopf algebra $H$ and its Drinfeld double
  7. The category $^H_H\mathcal{YD}$
  8. Nichols Algebras in $^{H}_{H}\mathcal{YD}$
  9. Hopf algebras over $H$

Key Result

Lemma 2.3

$($Gn00, Theorem 2.2 $)$ Let $(V,c)$ be a braided vector space, $V=\bigoplus_{i=1}^{n}V_{i},$ such that each $\mathcal{B}(V_i)$ is finite-dimensional. Then $\dim \mathcal{B}(V)\geq \prod_{i=1}^{n}\dim \mathcal{B}(V_{i})$. Furthermore, the equality holds if and only if $b_{ij}=b_{ji}^{-1}$ for all

Theorems & Definitions (69)

  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 59 more