[Here](http://www.ams.org/journals/proc/1996-124-04/S0002-9939-96-03199-1/S0002-9939-96-03199-1.pdf) Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
$$\mathcal{B}\_n=\mathbb{Z}\_n\times\mathbb{Z}\_n=\{(i,j):i,j=0,1,\dots,n-1\},$$
 and define a block matrix (of unitaries) $U\in M\_n(M\_n(\mathbb{C}))$ by:
 $$u\_{ij}=\sum\_{m=1}^n\zeta^{im}E\_{m,m+j},$$
 where $E\_{i,j}$ is the usual matrix unit in $M\_n(\mathbb{C})$ and $m+j$ is understood as $\mod n$. Note the indexing of the blocks is from $0\rightarrow k-1$ rather than $1\rightarrow k$ although the indexing inside the blocks is as standard.


Consider $n^2$ one-dimensional spaces $\mathbb{C} e\_{(i,j)}$ spanned by elements indexed by $\mathcal{B}\_n$, $\{e\_{(i,j)}:i,j\in \mathcal{B}\_n\}$. Together with a copy of $M\_n(\mathbb{C})$, a direct sum of these $n^2+1$ spaces, the $2n^2$ dimensional space
 $$A\_n=\left(\bigoplus\_{(i,j)\in\mathcal{B}\_n}\mathbb{C} e\_{(i,j)}\right)\oplus M\_n(\mathbb{C}),$$
 can be given the structure of the algebra of functions on a finite group which is denoted by $\mathbb{KP}\_n$ (so that $A\_n=F(\mathbb{KP}\_n)$). On the one dimensional elements the coproduct is given by:
 $$\Delta(e\_{(i,j)})=\sum\_{(m,n)\in\mathcal{B}\_n}e\_{(m,n)}\otimes e\_{(i-m,j-n)}+\frac{1}{n}\sum\_{m,n,s,t=1}^n\left(u\_{(i,j)}\right)\_{m,s}\overline{\left(u\_{(i,j)}\right)\_{n,t}}e\_{mn}\otimes e\_{st}.$$
 On elements in the $M\_n(\mathbb{C})$ factor:
 $$\Delta(a)=\sum\_{(i,j)\in\mathcal{B}\_n}e\_{(-i,-j)}\otimes u\_{(i,j)}a u\_{(i,j)}^\*+\sum\_{(i,j)\in\mathcal{B}\_n}\overline{u\_{(i,j)}}au\_{(i,j)}^T\otimes e\_{(i,j)}.$$


The antipode is given by $S(e\_{(i,j)})=e\_{(-i,-j)}$ on the one dimensional factors and the transpose for the $M\_n(\mathbb{C})$ factor. Sekine doesn't give the counit but by noting that $u\_{(0,0)}=I\_{n}$ it can be seen that the projection onto the $e\_{(0,0)}$ one-dimensional factor satisfies the counital property. The Haar state $h\_n\in M\_p(\mathbb{KP}\_n)$ is given by:
 $$h\_n\left(\sum\_{(i,j)\in \mathcal{B}\_n}x\_{(i,j)}e\_{(i,j)}+a\right)=\frac{1}{2n^2}\left(\sum\_{(i,j)\in\mathcal{B}\_n}x\_{(i,j)}+n\cdot \text{Tr}(a)\right).$$



> 
> I am looking for a reference where the (co)representation theory is
>  discussed. Such a reference may not exist --- Google isn't showing
>  anything for me. I am also interested in the states.
> 
> 
> 


In this [question](https://mathoverflow.net/questions/135241/the-irreducible-corepresentations-of-the-eight-dimensional-kac-paljutkin-quantum) the quantum group is self-dual but this isn't the case as far as I know for these quantum groups. The linked paper above describes central projections in 'a' convolution algebra...



The Irreducible Representations of the Sekine Quantum Groups
Thank you to [Zahlendreher](https://mathoverflow.net/users/33854/zahlendreher) and [Sébastien Palcoux](https://mathoverflow.net/users/34538/s%C3%A9bastien-palcoux) for the help.


Zahlendereher led me to the states 


With Sébastien's help I was happy that brute force would reveal the matrix elements of the corepresentations to be related to the central minimal projections in the convolution algebra. I was unable to adapt his exact methods as our definitions were at odds but from what he told me I knew brute force wasn't futile.


I was able to see that for $n$ odd at least (all I need at this time --- I suggest that things are similar for $n$ even), the one-dimensional central minimal projections of [Lemma 4](http://www.ams.org/journals/proc/1996-124-04/S0002-9939-96-03199-1/S0002-9939-96-03199-1.pdf) in the convolution algebra coincide with the matrix elements of the one-dimensional representations. 


Dividing the matrix units in the range of the two-dimensional central minimal projections of [Lemma 5](http://www.ams.org/journals/proc/1996-124-04/S0002-9939-96-03199-1/S0002-9939-96-03199-1.pdf) by two gave the matrix elements of the two-dimensional representations.


