Let $G$ be a finite group and $D(G)$ its quantum double. As in [my previous question](https://mathoverflow.net/questions/347944/classification-of-operatornamerep-dg), a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\theta$ is a conjugacy class of $G$ and $\pi$ an irreducible representation of the centralizer of $\theta$.


By reconstruction theorems (cf. Etingof et al. *Tensor Categories*), the category $\operatorname{Rep}D(G)$ is naturally isomorphic to the categorical center of $\operatorname{Rep}(G)$, whose typical objects are in the form $(X,\gamma)$, where $X$ is an object of $\operatorname{Rep}(G)$ and $\gamma$ a half-braiding.


### Questions


1. Is there a known translation between both descriptions under the natural isomorphism $\operatorname{Rep}D(G) \simeq Z\operatorname{Rep}G$?
2. More generally, replacing $\mathbb{C}[G]$ by any finite dimensional Hopf algebra $H$, a typical representation of $H$ is a Drinfeld-Yetter module, i.e. a $H$-module with suitable comodule structure. In this case, is there a known translation from the D-Y module description to the center side?
3. My impression is that $\operatorname{Rep}D(H)$ is wildly unknown for most finite dimensional Hopf algebras $H$. Is this impression correct? Is there at least a criterion for simplicity?



Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$
1)2) is standard for an arbitrary f.d. Hopf algebra $H$, as you say it's not hard to idenfity $D(H)$-modules with Yetter-Drinfeld modules. Then, given two of those, say $V,W$ you can define a braiding by
 $$V \otimes W \rightarrow H \otimes V \otimes W \rightarrow H \otimes W \otimes V \rightarrow W \otimes V$$
where the first map is the coaction of $V$, the middle map the flip, and the last one the action on $W$.


3) it really depends on what you mean by unknown, there are lots of things that can be said. For example, if $C$ is an arbitrary finite tensor category over $\mathbb C$ say, then $Z(C)$ is semi-simple iff $C$ is, and the global dimension (the sum of squares of dimensions of simples) in $Z(C)$ is the square of the global dimension of $C$.


