# Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

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?

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 $$C$$.

• In 1)2), doesn’t the second map have $V \otimes W$ as codomain, instead of the permuted? – Student Dec 20 '19 at 17:06
• of course, sorry, you need a flip on the middle (and V is a left comodule). I'll edit. – Adrien Dec 20 '19 at 17:09
• of course, sorry, you need a flip on the middle (and V is a left comodule). I'll edit. – Adrien Dec 20 '19 at 17:10
• @Student I think this is correct now. All of this is done carefully in e.g. Kassel's book "quantum groups". – Adrien Dec 20 '19 at 17:13
• you're right, I definitely answered too fast, I meant the so-called global dimension, not the number of simples. I edited. – Adrien Dec 21 '19 at 13:03

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.