Generalized Kac-Paljutkin algebras
Christian Lomp
TL;DR
This work generalizes the Kac–Paljutkin construction by producing semisimple Hopf algebras $H_{n,m}$ of dimension $n^m m!$ as twisted crossed products $\mathbb{K}\mathbb{Z}_n^{\otimes m}\#_\gamma\Sigma_m$ over fields with a primitive $n$th root of unity. The authors develop a twist-based framework using skew monoid algebras to extend bialgebra structures along the symmetric group action, yielding explicit comultiplications and a $2$-cocycle $\gamma$ that defines the crossed product. They provide concrete representations: irreducible $m$-dimensional modules $V_{a,b}$ that are inner-faithful on the $R$-part under a gcd condition, and construct quantum polynomial module algebras $A_{a,b}$ on which $H_{n,m}$ acts, with invariants linked to cyclic polynomials when $n$ is even. These results give new, nontrivial inner-faithful Hopf actions on noncommutative domains, including reproducing the classic $H_8$ as $H_{2,2}$ and Pansera’s algebras as $H_{n,2}$, and they illuminate how twist data controls representation theory and invariants.
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 $\mathbb{K} Σ_m$ 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: $H_{n,m} = \mathbb{K} \mathbb{Z}_n^{\otimes m} \#_γΣ_m$. 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.