
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 of dimension over a field of characteristic zero containing a primitive th root of unity, where are integers. The well-known eight-dimensional Kac--Paljutkin algebra arises as the special case , while the Hopf algebras previously constructed by Pansera correspond to the instances . Each algebra is defined as an extension of the group algebra of the symmetric group by the -fold tensor product , where denotes the cyclic group of order . This extension admits a realization as a crossed product: . In the final section, we construct a family of irreducible -dimensional representations of that are inner…
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