Let $\mathcal{G}$ be compact quantum group in the sense of S. L. Woronowicz. As is well-known, every compact quantum group contains a dense Hopf algebra, called the **polynomial Hopf algebra** Pol$(\mathcal{G})$.
For example, consider the famous $C(SU_q(2))$, the $q$-deformation of $SU(2)$, with generators $\alpha$ and $\beta$ :

https://en.wikipedia.org/wiki/Compact_quantum_group

The dense Hopf algebra is now the polynomial $*$-algebra generated by $\alpha$ and $\beta$. A well-known fact about Pol($SU_q(2)$) is that it has no zero-divisors, that is, it is a domain. What is a good example of a compact quantum group $\mathcal{G}$ such that Pol$(\mathcal{G})$ **has** zero divisors? On the other hand, is there any abstract characterization of the compact quantum groups such that the polynomial Hopf algebra is a domain?