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Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, 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?

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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 $C$.

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  • $\begingroup$ In 1)2), doesn’t the second map have $V \otimes W$ as codomain, instead of the permuted? $\endgroup$ – Student Dec 20 '19 at 17:06
  • $\begingroup$ of course, sorry, you need a flip on the middle (and V is a left comodule). I'll edit. $\endgroup$ – Adrien Dec 20 '19 at 17:09
  • $\begingroup$ of course, sorry, you need a flip on the middle (and V is a left comodule). I'll edit. $\endgroup$ – Adrien Dec 20 '19 at 17:10
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    $\begingroup$ @Student I think this is correct now. All of this is done carefully in e.g. Kassel's book "quantum groups". $\endgroup$ – Adrien Dec 20 '19 at 17:13
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    $\begingroup$ you're right, I definitely answered too fast, I meant the so-called global dimension, not the number of simples. I edited. $\endgroup$ – Adrien Dec 21 '19 at 13:03
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This is my study note that spells out @Adrien 's answer to 1) and 2). As suggested by @Adrien, we will follow Kassel's Quantum Groups, mainly chapter XIII.5. It is a very detailed account.

Explicit equivalence between $Z\operatorname{Rep}(H)\simeq \operatorname{Rep}(D(H))$

Let $H$ be a finite dimensional complex Hopf algebra, we will sketch the (braided) equivalence between two categories.

A typical object of $Z\operatorname{Rep}(H)$ is a pair $(V,c_{-,V})$, where $V$ is an object of $\operatorname{Rep}(H)$, and $c$ is a half-braiding. Using it, we can define a right comodule structure on $V$ by

$$ \Delta_V : V \to V \otimes H : v \mapsto c_{H,V}(1 \otimes v). $$

We write the image to be $\sum_{(v)} v_V \otimes v_H$ for future use. We can check that this gives $V$ a Drinfeld-Yetter structure over $H$ (or so called a crossed $H$-bimodule structure). Details of this can be found in the proof of XIII Lemma 5.2.

IX.5 tells us that a Drinfeld-Yetter structure over $H$ is naturally equivalent as a $D(H)$ left module structure, so we get a left $D(H)$-module. Furthermore, IX.5 spells the $H\otimes H^{op*} = D(H)$-module structure out:

$$ a \alpha v = \sum_{(v)} <\alpha,v_H>av_V. $$

So far, we associate a $D(H)$-module to an object on the right hand side. This map can be shown to be a faithful, strict monoidal functor (page 335).

The half-braiding $c_{-,V}$ is shown to be equal to $\text{transpose}_{-,V} \circ l_R$, where $R$ is the universal $R$-matrix of $D(H)$. This gives the braided structure of the functor (details in page 336), and also gives the description of the inverse functor (details in page 336, 337). Finishing the proof.

A remark which I find important is that under this equivalence, the restriction from $\operatorname{Rep}D(H)$ to $\operatorname{Rep} H$ is the same as the forgetful functor from $Z\operatorname{Rep}H$ to $\operatorname{Rep}H$. This is immediate from the description of the equivalence.

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