On two-coloured noncrossing quantum groups

We classify compact quantum groups associated to noncrossing partitions coloured with two elements $x$ and $y$ which are their own inverses. Together with the work of P. Tarrago and M. Weber, this completes the classification of all noncrossing quantum groups on two colours. We also give some general structure results concerning noncrossing quantum groups and suggest some more general classification statements.