Finite-dimensional Nichols algebras over dual Radford algebras

TL;DR

This paper classifies and constructs new finite-dimensional Nichols algebras over dual Radford algebras, especially for the case n=2, providing explicit presentations and recovering known results.

Contribution

It identifies new finite-dimensional Nichols algebras over dual Radford algebras and explicitly describes their generators and relations for specific cases.

Findings

18 possible cases for n=2

Explicit generators and relations for 5 cases

Recovered known results for specific modules

Abstract

For $n,m\in \mathbb{N}$, let $H_{n,m}$ be the dual of the Radford algebra of dimension $n^{2}m$. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter-Drinfeld modules over $H_{n,m}$. Along the way, we describe the simple objects in ${}^{H_{n,m}}_{H_{n,m}}\mathcal{YD}$ and their projective envelopes. Then, we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case $n=2$. There are 18 possible cases. We present by generators and relations the corresponding Nichols algebras on five of these eighteen cases. As an application, we characterize finite-dimensional Nichols algebras over indecomposable modules for $n=2=m$ and $n=2$, $m=3$, which recovers some results of the second and third author in the former case, and of Xiong in the latter.