Here 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 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...


2 Answers 2


Disclamer: This is just what some digging gave me, maybe some of it can be helpful:

Representation Theory: The paper on nilpotent fusion categories [1] (see example 4.5(2)) claims that representations over the Kac-Paljutkin algebras (or even Sekine's generalizations) give examples of nilpotent semisimple tensor categories. In particular, it is claimed that these categories are examples of Tambara-Yamagami categories (introduced in [2], see e.g. [3] for a definition).

States: About the states, there is a paper by Franz and Skalski [4] (section 6, Lemma 6.4) investigates all quantum subgroups and examples of idempotent states that are not Haar states on subgroups.

[1] Gelaki, Shlomo; Nikshych, Dmitri. Nilpotent fusion categories. Adv. Math. 217 (2008), no. 3, 1053--1071.

[2] Tambara, Daisuke; Yamagami, Shigeru. Tensor categories with fusion rules of self-duality for finite abelian groups. J. Algebra 209 (1998), no. 2, 692--707.

[3] Turaev, Vladimir; Vainerman, Leonid. The Tambara-Yamagami categories and 3-manifold invariants. Enseign. Math. (2) 58 (2012), no. 1-2, 131--146.

[4] Franz, Uwe; Skalski, Adam. On idempotent states on quantum groups. J. Algebra 322 (2009), no. 5, 1774--1802.

  • 1
    $\begingroup$ Thank you for these. Only the Franz & Skalski paper have I seen before. I am going to get my hands on these. Unless a better answer comes in I will award you the accept and the bounty. $\endgroup$ Commented Oct 14, 2015 at 8:11
  • $\begingroup$ ...they are in the Sekine paper! Note that the construction for k=2 gives the group algebra of the Dihedral Group. $\endgroup$ Commented Dec 10, 2015 at 20:54
  • $\begingroup$ ...update they are not in the Sekine paper after all... mathoverflow.net/questions/234332/… $\endgroup$ Commented Mar 23, 2016 at 12:39

Thank you to Zahlendreher and Sébastien 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 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 by two gave the matrix elements of the two-dimensional representations.


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