Generalized Kac-Paljutkin algebras

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 $n,m\ge 2$ 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 $\mathbb{K}{\mathrm{\Sigma }}_m$ 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: $H_{n,m}=\mathbb{K}{\mathbb{Z}}_n^{\otimes m}{\mathrm{\#}}_{\gamma }{\mathrm{\Sigma }}_m$. 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.