# Quantum group representations from (convolution) matrix units?

Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$

There is a convolution product on $A=F(\mathbb{G})$ given by

$$a\star_A b=b_{(2)}\int_{\mathbb{G}}S\left(b_{(1)}\right)a,$$

where

$$\Delta(b)=\sum b_{1,i}\otimes b_{2,i}=:b_{(1)}\otimes b_{(2)}.$$

There is a link between the convolution in $F(\mathbb{G})$ and the convolution product in $A'=F(\mathbb{G})'=\mathbb{CG}$. The convolution product in $\mathbb{CG}$, $\star$ (with no subscript), is the dual of the comultiplication of $F(\mathbb{G})$ and thus for $\varphi,\,\nu\in\mathbb{CG}$ $$\varphi\star\nu=(\varphi\otimes\nu)\Delta.$$

Let $\mathcal{F}:F(\mathbb{G})\rightarrow \mathbb{CG}$ be the map $b\mapsto \mathcal{F}(b)$ where

$$\mathcal{F}(b)a=\int_{\mathbb{G}}ab.$$

What could be called the Van Daele convolution theorem says:

$$\mathcal{F}(a\star_A b)=\mathcal{F}(a)\star \mathcal{F(b)}.$$

I am wondering

1. Is there any connection between projections in the algebra $(A,\star_A)$ and group-like elements in the coalgebra $(A,\Delta)$:

$$p\star_A p=p\qquad\overset{?}{\sim}\qquad\Delta(p)=p\otimes p.$$

1. Also is there any connection between matrix units in the algebra $(A,\star_A)$ and matrix elements in the coalgebra $(A,\Delta)$:

$$e_{ij}\star_Ae_{k\ell}=\delta_{jk}e_{i\ell}\qquad\overset{?}{\sim}\qquad\Delta(e_{ij})=\sum_ke_{ik}\otimes e_{kj}.$$

A question in a similar vein.

Context

I had thought that I had found what I was looking for but in fact I was mistaken.

I thought Lemmas 4, 5, 8 and 9 were the matrix elements of the irreducible representations of the Sekine quantum group.

Calculations using a 'spectral-type formula' --- involving a sum over irreducible representations --- were giving me nonsensical answers. Unable to find any error in the derivation of the 'spectral-type formula', and seeing the formula fail to give the correct answers, it dawned on me to check were those elements of the algebra of functions actually matrix elements of irreducible representations... no they were not. Specifically, for $k$ odd,

$$\varepsilon\left(e^{u,v}_{11}\right)=2\qquad\text{and}\qquad h\left(\left(e_{11}^{u,v}\right)^*e_{11}^{u,v}\right)=2,$$

when the first should be one and the second should be $\displaystyle \frac12$.

Looking at $k$ odd in particular --- Lemmas 4 and 5 --- I have a feeling that the $p_{l,\pm}$ are indeed the matrix elements of the one dimensional representations.

I could try and adjust the functions in Lemma 5 according to:

$$\rho_{ij}^{u,v}=\frac{c_{ij}^{u,v}}{2}e_{kk}^{u,v}$$

with $c_{ij}^{u,v}\in\mathbb{C}$ --- equal to one for $i=j$ and of modulus one otherwise. This would clean up the issues with the counit and the Haar state but this cannot be correct. My 'spectral-type formula' is still failing in a simple case with this adjustment.

Rather than going back to the drawing board and essentially starting again on the last question, I was wondering is there a way to exploit the fact that in the convolution algebra, the $p_{l,\pm}$ are projections and the $e_{ij}^{u,v}$ are matrix units.

• My question has a positive answer for the non-weak case (see prop 8.16 p40 here). This should answers your question for the Kac type. Mar 23 '16 at 18:03
• @SébastienPalcoux thank you... I will have a look at this... I can see already that being a projection alone certainly isn't enough. Even in the classical case $|G|\delta_0$ is a projection in $(F(G),\star_{F(G)})$ but $\Delta(|G|\delta_0)\neq|G|\delta_0\otimes|G|\delta_0$. Mar 23 '16 at 18:12

As explained here, with help from Sébastien I was able to show that, in the context of what I was working on, the one-dimensional minimal central projections satisfied

$$\Delta(p)=p\otimes p,$$

and the matrix units in the range of the two-dimensional central minimal projections satisfied

$$\Delta(E_{i\ell})=\sum_{k=1}^2\frac12 \left(E_{ik}\otimes E_{k\ell}\right).$$

There wasn't a problem with the 'spectral-type formula' when I made this

$$\rho_{ij}^{u,v}=\frac{1}{2}e_{ij}^{u,v}$$

adjustment --- I incorrectly assumed that the random walk driven by $\nu=(\delta^{(0,1)}+\delta^{(1,0)})/2$ on $\mathbb{Z}_n\times\mathbb{Z}_n$ was ergodic --- it is not.