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

Abstract

In this paper, we continue to investigate finite-dimensional Nichols algebras over simple Yetter-Drinfeld modules of the Suzuki algebras $A_{Nn}^{\mu \lambda }$. It is finished for the case $A_{N2n}^{\mu \lambda }$. As for the case $A_{N2n+1}^{\mu \lambda }$, it boils down to the long-standing open problem: calculate dimensions of Nichols algebras of dihedral rack type ${\mathbb{D}}_{2n+1}$. 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.

26 pages