Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this [Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models](https://pdfs.semanticscholar.org/7d76/3f033462bd49fe945ff9125e050139beb68f.pdf). However, in the paper, the authors claimed the examples obtained by natural inductions are *complete*, but without a proof. My impression is that the representation theory for a Hopf algebra is not completely known yet in general. How would one classify the representations of $D(G)$?


My understanding is still at the level of representation of a finite group $G$. In that case, the classification can be done due to


1. $\mathbb{C}[G]$ is semi-simple
2. $\lvert\operatorname{Irrep}(G)\rvert =\lvert\operatorname{Conj}(G)\rvert$.


Are there similar statements for $D(G)$ (better with proofs)? Pointers towards anything relevant will be appreciated. Thank you!


**EDIT:** While there's only one chosen answer, the others might be valuable for you. Here is a **table of answers** below so far.


1. Konstantinos Kanakoglou pointed out several papers that directly answered my question.
2. I wrote up a note spelling out Konstantinos Kanakoglou's wonderful answer. I am willing to discuss the detail of the proof.
3. zibadawa timmy's higher categorical view point.



Classification of $\operatorname{Rep} D(G)$
There are some classic results on the **classification of the irreducible $D(G)$-modules**:  

If the field is the complex numbers $\mathbb{C}$, it has been shown that a representation of the finite group $G$, induced from an irreducible representation of the centralizer subgroup of an element $g$ of $G$, generates an irreducible rep of $D(G)$ and that furthermore, all the irreducible quantum double modules are obtained in this way. Proofs for these results can be found at:  

[Quantum double finite group algebras and their representations, Bull. Austr. Math. Soc., **48**, 1993, p.275-301](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0004972700015707), by M.D. Gould.  

(See section 6, mainly theorem 6.3).
In there, it is also shown that all such algebras are semisimple and their character theory is developed. 


From a more general viewpoint, representations of $D(G)$ over algebraically closed fields of arbitrary characteristic have been studied at:  

[The representation ring of the quantum double of a finite group, J. of Algebra, **179**, p.305-329, (1996)](https://core.ac.uk/download/pdf/82567222.pdf), by S.J. Witherspoon. In there, some of the previously mentioned results have been generalized: for example an analogue of Maschke's theorem is proved; it is shown that $D(G)$ is semisimple if and only if the characteristic $p$ of the field, does not divide the order of the group $G$.  

Furthermore, the representation ring $R\big(D(G)\big)$ of the quantum double is studied: it is shown to be a commutative algebra, a direct sum decomposition is described and a **classification of the indecomposable $D(G)$-modules** is also achieved (among other results as well). 


