I asked this question on [Math.Stack](https://math.stackexchange.com/questions/430687/the-irreducible-corepresentations-of-the-eight-dimensional-kac-paljutkin-quantum) but have not had any answers.


**Question**


What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?


The trivial corepresentation is given by $\Delta\_{|\mathbb{C}1\_A}$ where $1\_A$ is just the unit. 


By my reckoning there should be seven more one dimensional invariant subspaces and hence irreducible corepresentations.


**The eight-dimensional Kac-Paljutkin Quantum Group**


Here we give the defining relations and the main structure of an eight-dimensional quantum group introduced by Kac and Paljutkin. This is actually the smallest finite quantum groups that is not a group algebra. In other words, it is the non-commutative C\*-Hopf algebra of smallest dimension.


Consider the multi-matrix algebra $$A=\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus M\_2(\mathbb{C}),$$
with the usual multiplication and involution. We shall use the basis $e\_1=(1,0,0,0,0)$ (with $e\_2,\,e\_3,\,e\_4$ defined in the same way) and


$$ a\_{11}=0+ 0+ 0+ 0+\left(\begin{array}{cc}1&0\\0&0\end{array}\right),$$


with the other $a\_{ij}$ defined in the same way. The algebra $A$ is an eight-dimensional C\*-algebra. Its unit is of course $1\_A=e\_1+e\_2+e\_3+e\_4+a\_{11}+a\_{22}$. The following defines a comultiplication on $A$,


$ \scriptsize{ \Delta(e\_1)=e\_1\otimes e\_1+e\_2\otimes e\_2+e\_3\otimes e\_3+e\_4\otimes e\_4+\frac{1}{2}a\_{11}\otimes a\_{11}+\frac{1}{2}a\_{12}\otimes a\_{12}+\frac{1}{2}a\_{21}\otimes a\_{21}+\frac{1}{2}a\_{22}\otimes a\_{22}}$
$ \scriptsize{ \Delta(e\_2)=e\_1\otimes e\_2+e\_2\otimes e\_1+e\_3\otimes e\_4+e\_4\otimes e\_3+ \frac{1}{2}a\_{11}\otimes a\_{22}+\frac{1}{2}a\_{22}\otimes a\_{11}+\frac{i}{2}a\_{21}\otimes a\_{12}-\frac{i}{2}a\_{12}\otimes a\_{21}}$
$ \scriptsize{ \Delta(e\_3)=e\_1\otimes e\_3+e\_3\otimes e\_1+e\_2\otimes e\_4+e\_4\otimes e\_2+ \frac{1}{2}a\_{11}\otimes a\_{22}+\frac{1}{2}a\_{22}\otimes a\_{11}-\frac{i}{2}a\_{21}\otimes a\_{12}+\frac{i}{2}a\_{12}\otimes a\_{21}}$
$ \scriptsize{ \Delta(e\_4)=e\_1\otimes e\_4+e\_4\otimes e\_1+e\_2\otimes e\_3+e\_3\otimes e\_2+ \frac{1}{2}a\_{11}\otimes a\_{11}+\frac{1}{2}a\_{22}\otimes a\_{22}-\frac{1}{2}a\_{12}\otimes a\_{12}-\frac{1}{2}a\_{21}\otimes a\_{21}}$
$ \scriptsize{ \Delta(a\_{11})=e\_1\otimes a\_{11}+a\_{11}\otimes e\_1+e\_2\otimes a\_{22}+a\_{22}\otimes e\_2+e\_3\otimes a\_{22}+a\_{22}\otimes e\_3+e\_4\otimes a\_{11}+a\_{11}\otimes e\_4}$
$ \scriptsize{ \Delta(a\_{12})=e\_1\otimes a\_{12}+a\_{12}\otimes e\_1+ie\_2\otimes a\_{21}-ia\_{12}\otimes e\_2-ie\_3\otimes a\_{21}+ia\_{21}\otimes e\_3-e\_4\otimes a\_{12}-a\_{12}\otimes e\_4}$
$ \scriptsize{ \Delta(a\_{21})=e\_1\otimes a\_{21}+a\_{21}\otimes e\_1-ie\_2\otimes a\_{12}+ia\_{12}\otimes e\_2+ ie\_3\otimes a\_{12}-ia\_{12}\otimes e\_3-e\_4\otimes a\_{21}-a\_{21}\otimes e\_4}$
$ \scriptsize{ \Delta(a\_{22})=e\_1\otimes a\_{22}+a\_{22}\otimes e\_1+e\_2\otimes a\_{11}+a\_{11}\otimes e\_2+e\_3\otimes a\_{11}+a\_{11}\otimes e\_3+e\_{4}\otimes a\_{22}+a\_{22}\otimes e\_4}$ 


The counit is given by (looking at this we can see the relationship between the counit and the comultiplication with respect to the unit and multiplication in a commutative, group algebra, case. To encode $ ge=g=eg$ any time there is a term of the form $ x\otimes e\_1$, there must be a term of the form $ e\_1\otimes x$ --- to capture the left and right symmetry $ R\_\varepsilon=(I\_A\otimes\varepsilon)\circ\Delta=(\varepsilon\otimes I\_A)\circ \Delta=L\_\varepsilon$. This also shows that, in this case, $\Delta(x)$ must contain $ x\otimes e\_1$ and $ e\_1\otimes x$ to encode $L\_\varepsilon=I\_A=R\varepsilon$):


$$ \varepsilon\left(x\_1+x\_2+x\_3+x\_4+\left(\begin{array}{cc}c\_{11} &c\_{12}\\ c\_{21}&c\_{22}\end{array}\right)\right)=x\_1.$$


The antipode is the transpose map, i.e.


$$ S(e\_i)=e\_i\text{, and }S(a\_{jk})=a\_{ji}.$$


**Background**


A *corepresentation* $\chi$ of a finite quantum group $(A,\Delta)$ on a complex vector space $V$ is a linear map $\chi:V\rightarrow V\otimes A$ that satisfies


$$(\chi\otimes I\_A)\circ \chi=(I\_V\otimes \Delta)\circ\chi\text{ and }$$
$$(I\_V\otimes \varepsilon)\circ \chi=I\_V$$


where $\varepsilon$ is the *counit* of $(A,\Delta)$.


The dimension of the vector space is called the *dimension* of $\chi$ and is denoted by $d\_\chi$. 


If $W$ is a vector space which has the property that $\chi(W)\subset W\otimes A$ then $W$ is said to be *invariant* and $\chi\_{|W}$ is called a *subrepresentation*.


It can be shown that $\chi$ is equivalent to a direct sum of irreducible unitary corepresentations.


Two corepresentations $\chi\_1$ and $\chi\_2$ are *equivalent* as corepresentations if they admit and invertible *intertwiner*: an invertible linear map $T:V\_1\rightarrow V\_2$ such that


$$\chi\_2\circ T=(T\otimes I\_A)\circ\chi\_1.$$


The space of intertwiners from $\chi\_1$ to $\chi\_2$ is denoted by $\text{Hom}(\chi\_1,\chi\_2)$.


**Further Background**


I had been under the impression that the comultiplication plays the role of the regular representation in the representation theory of finite groups so that all of the irreducible corepresentations could be found by finding subspaces $W$ of $A$ invariant under the comultiplication in the sense that $\Delta(W)\subset W\otimes A$... however I am either having a stupid misunderstanding or else this isn't as easy as I thought.


I am trying to apply the philosophy of a use of the representation theory of finite groups to the corepresentation theory of finite quantum groups and perhaps I have confused myself a little in the process! 



The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
The Kac-Paljutkin Quantum Group $A$ is self-dual, i.e. the dual space $A^\*$ (which is of course again finite quantum group = finite-dimesional $C^\*$-Hopf algebra, with the multiplication being the dual of the coproduct of $A$ and the coproduct the dual of the multiplication of $A$) is isomorphic to $A$.


This must have been known already to Kac and Paljutkin. You can find the isomorphism, e.g., in Franz and Gohm, Random Walks on Finite Quantum Groups, in 
"Quantum Independent Increment Processes II", Lecture Notes in Mathematics Volume 1866, 2006, pp 1-32 .


