7
$\begingroup$

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?

$\endgroup$

2 Answers 2

4
$\begingroup$

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 $C$.

$\endgroup$
10
  • $\begingroup$ In 1)2), doesn’t the second map have $V \otimes W$ as codomain, instead of the permuted? $\endgroup$
    – Student
    Dec 20, 2019 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, 2019 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, 2019 at 17:10
  • 2
    $\begingroup$ @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, 2019 at 17:13
  • 1
    $\begingroup$ 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, 2019 at 13:03
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.