Finite-dimensional Nichols algebras over the Suzuki algebras III: simple Yetter-Drinfeld modules

TL;DR

This paper advances the classification of finite-dimensional Nichols algebras over simple Yetter-Drinfeld modules associated with Suzuki algebras, addressing specific cases and highlighting open problems related to dihedral rack types.

Contribution

It completes the classification for certain Suzuki algebra cases and discusses open problems in calculating Nichols algebra dimensions for dihedral rack types.

Findings

Finished classification for $A_{N, 2n}^{oldsymbol{ u heta}}$ cases.

Identified open problem for $A_{N, 2n+1}^{oldsymbol{ u heta}}$ cases.

Highlighted the set-theoretical nature of Suzuki algebras.

Abstract

In this paper, we continue to investigate finite-dimensional Nichols algebras over simple Yetter-Drinfeld modules of the Suzuki algebras $A_{N\, n}^{\mu\lambda}$. It is finished for the case $A_{N\, 2n}^{\mu\lambda}$. As for the case $A_{N\, 2n+1}^{\mu\lambda}$, it boils down to the long-standing open problem: calculate dimensions of Nichols algebras of dihedral rack type $\Bbb D_{2n+1}$. It is interesting to see that the Suzuki algebras are of set-theoretical type. We pose some question or problems for our future research. In particular, we are curious about how to generalize the correspondence between braidings of rack type and group algebras to braidings and Hopf algebras of set-theoretical type.