Generalized Kac-Paljutkin algebras

TL;DR

This paper constructs a broad family of semisimple Hopf algebras generalizing the Kac-Paljutkin algebra, exploring their structure, representations, and actions on quantum polynomial algebras.

Contribution

It introduces a new family of semisimple Hopf algebras $H_{n,m}$ extending known examples, with explicit construction and representation theory analysis.

Findings

The algebras $H_{n,m}$ are semisimple Hopf algebras of dimension $n^m m!$.

Explicit irreducible $m$-dimensional representations are constructed.

A nontrivial inner-faithful action on quantum polynomial algebras is demonstrated.

Abstract

In this note, we construct a family of semisimple Hopf algebras $H_{n,m}$ of dimension $n^m m!$ over a field of characteristic zero containing a primitive $n$th root of unity, where $n, m \geq 2$ are integers. The well-known eight-dimensional Kac--Paljutkin algebra arises as the special case $H_{2,2}$, while the Hopf algebras previously constructed by Pansera correspond to the instances $H_{n,2}$. Each algebra $H_{n,m}$ is defined as an extension of the group algebra $\mathbb{K} \Sigma_m$ of the symmetric group by the $m$-fold tensor product $R = \mathbb{K} \mathbb{Z}_n^{\otimes m}$, where $\mathbb{Z}_n$ denotes the cyclic group of order $n$. This extension admits a realization as a crossed product: $H_{n,m} = \mathbb{K} \mathbb{Z}_n^{\otimes m} \#_\gamma \Sigma_m$. In the final section, we construct a family of irreducible $m$-dimensional representations of $H_{n,m}$ that are inner faithful as $R$-modules and exhibit a nontrivial inner-faithful action of a subalgebra of $H_{n,m}$ on a quantum polynomial algebra.