I will start with the general before moving to the specific.

Consider for a moment the two (very) soft definitions.

An

abstractionof an object $X$ is a category $\mathcal{C}_0$ such that $X$ is an object in the class $\operatorname{ob}(\mathcal{C}_0)$.A

generalisationof an abstraction $\mathcal{C}_0$ is a category $\mathcal{C}$ such that $\mathcal{C}_0$ is a proper subcategory of $\mathcal{C}$ (so that in this soft definition regime, a generalisation is also an abstraction).

It is a storied theme of mathematics that by abstracting an object $X$ to $\mathcal{C}_0$, that we can prove theorems for a whole class of objects, rather than just for the single object $X$.

Moreover, often when interested just in the object $X$, it can be easier to work in the abstraction $\mathcal{C}_0$, as this sometimes allows us to disregard the irrelevant idiosyncrasies of $X$.

Mathematical history --- with all its humanity --- is littered with examples of theorems proved in an abstraction $\mathcal{C}_0$ before they were known or considered in the specific context of $X$. This is of a subjectively different flavour to just putting together a slicker proof or proving a general result.

Of course, when you move from $X$ to $\mathcal{C}_0$ some theorems are no longer true.

The same is true when we look at a generalisation $\mathcal{C}$ of $\mathcal{C}_0$. However, of course, theorems true in $\mathcal{C}$ will be true for $X$ but moreover $\mathcal{C}_0$.

Moving towards the specific, the Peter-Weyl Theorem in the category of compact groups is also true (with suitable definitions) in the generalisation to compact matrix *quantum* groups.

There are many definitions/categories of quantum groups. In those categories which are (in the sense above) generalisations of categories of classical groups (classical in the sense of "has a set of points $G$" --- I believe all such definitions of quantum groups include at the very least the category of finite groups), have the quantum group theorists ever 'discovered' something that group theorists either were interested in, or would plausibly be interested in?

When a generalisation $\mathcal{C}$ of an abstraction $\mathcal{C}_0$ is developed to help study objects in $\mathcal{C}_0$, you can imagine that this happens, but as quantum groups are, arguably, studied for their non-commutative aspects, rather than as an attempt to understand classical groups better, this may not have happened.

To bookend; my question:

Have quantum group theorists discovered something

newabout groups that is interesting to group theorists?