Abstract
In this note, we construct a family of non-commutative, non-cocommutative semisimple Hopf algebras
Hn comma m
$ H_{n,m} $
H
n
,
m
of dimension
nmm exclamation mark
$ n^m m! $
n
m
m
!
over a field of characteristic zero containing a primitive
n
$ n $
n
th root of unity, where
n comma m greater than or equals 2
$ n, m \geq 2 $
n
,
m
≥
2
are integers. The well-known eight-dimensional Kac–Paljutkin algebra arises as the special case
upper H 2 comma 2
$ H_{2,2} $
H
2
,
2
, while the Hopf algebras previously constructed by Pansera correspond to the instances
Hn comma 2
$ H_{n,2} $
H
n
,
2
. Each algebra
Hn comma m
$ H_{n,m} $
H
n
,
m
is defined as an extension of the group algebra
double struck upper K upper Sigma m
$ \mathbb{K} \Sigma_m $
𝕂
Σ
m
of the symmetric group by the
m
$ m $
m
-fold tensor product
upper R equals double struck upper K double struck upper Z n circled times m
$ R = \mathbb{K} \mathbb{Z}_n^{\otimes m} $
R
=
𝕂
ℤ
n
⊗
m
, where
double struck upper Z n
$ \mathbb{Z}_n $
ℤ
n
denotes the cyclic group of order
n
$ n $
n
. This extension admits a realization as a crossed product:
Hn comma m equals double struck upper K double struck upper Z n circled times m number sign gamma upper Sigma m period
$H_{n,m} = \mathbb{K} \mathbb{Z}_n^{\otimes m} \#_\gamma \Sigma_m.$
H
n
,
m
=
𝕂
ℤ
n
⊗
m
#
γ
Σ
m
.
In the final section, we construct a family of irreducible
m
$ m $
m
-dimensional representations of
Hn comma m
$ H_{n,m} $
H
n
,
m
that are inner faithful as
upper R
$ R $
R
-modules and exhibit a non-trivial inner-faithful action of a subalgebra of
Hn comma m
$ H_{n,m} $
H
n
,
m
on a quantum polynomial algebra.