Finite dimensional Nichols algebras over $H_{c: \sigma_{0}}$ of Kashina

TL;DR

This paper classifies all finite-dimensional Nichols algebras over a specific Hopf algebra, identifying their types and dimensions, thereby advancing understanding of their structure in the context of Kashina's algebra.

Contribution

It provides a complete classification of finite-dimensional Nichols algebras over $H_{c: \sigma_{0}}$, including diagonal and non-diagonal types, with explicit dimension results.

Findings

Finite-dimensional Nichols algebras of diagonal type are $A_1$, $A_2$, or quantum planes.

Non-diagonal type Nichols algebras are 8 or 16 dimensional.

All simple Yetter-Drinfel'd modules over $H_{c: \sigma_{0}}$ are classified.

Abstract

Let $H$ be the Hopf algebra $H_{c: \sigma_{0}}$ of Kashina [J. Algebra, 232(2000),pp.617-663]. We give all simple Yetter-Drinfel'd modules $V$ over $H$, then classify all finite-dimensional Nichols algebras of $V$. The finite dimensional Nichols algebras of diagonal type are either $A_{1}, A_{2}$ or quantum planes, and non-diagonal type ones are $8$ or $16$ dimensional.