Finite dimensional Hopf algebras over the Kac-Paljutkin algebra $H_8$.

Yuxing Shi

doi:10.33044/revuma.v60n1a17

Abstract

Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects and direct sums of two simple objects in ${}_{H_8}^{H_8}\mathcal{YD}$, we calculated dimensions for the corresponding Nichols algebras, except four semisimple cases which are generally difficult. Under the assumption that the four undetermined Nichols algebras are all infinite dimensional, we determine all the finite dimensional Nichols algebras over $H_8$. It turns out that the already known finite dimensional Nichols algebras are all diagonal type. In fact, they are Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$. By the way, we calculate Gelfand-Kirillov dimensions for some Nichols algebras. As an application, we obtain five families of new finite dimensional Hopf algebras over $H_8$ according to the lifting method.

Highlights

Let K be an algebraically closed field of characteristic zero
We describe the procedure for the lifting method briefly
The lifting method was extensively used in the classification of finite-dimensional pointed Hopf algebras such as [15], [12], [25], [23], [2], [1], [9], [8] and so on

Summary

Introduction

Let K be an algebraically closed field of characteristic zero. The question of classification of all Hopf algebras over K of a given dimension up to isomorphism was posed by Kaplansky in 1975 [40]. The lifting method was extensively used in the classification of finite-dimensional pointed Hopf algebras such as [15], [12], [25], [23], [2], [1], [9], [8] and so on. We would like to initiate a project for the study of Hopf algebras whose coradicals are low-dimensional neither commutative nor cocommutative semisimple Hopf algebras by running procedures of the lifting method. The problem of classifying finite-dimensional Nichols algebras over non-abelian groups is difficult in general for lack of systematic method; for related works please refer to [12], [24], [29], [32],. Let H be a finite-dimensional Hopf algebra over H8 such that its infinitesimal braiding is. Except for the case (2), the remaining four families of Hopf algebras contain non-trivial lifting relations

Preliminaries

Simple Yetter–Drinfel’d modules of H8

Hopf algebras over H8