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Let $G$ be a finite group. The category of its representations (complex linear, finite dimensional, throughout this whole question) is equivalent to $\mathbb{C}[G]$-modules. V. Drinfeld constructed a quantum double $D(G)$, which is a Hopf algebra, whose category of representations is the categorical center (also called the Drinfeld center) of Rep$(G)$.

The Hopf algebra $D(G)$ can be realized as a crossed module algebra, where the crossed module is given by

$$ 1: G \to G $$

with the conjugation action. By the correspondence between crossed modules and 2-groups [2], one can also realize $D(G)$ as a 2-group, with the set of objects being $G$, and with exactly two morphisms (invertible to each other) between each pair of objects $g$ and $h$. One can show that the Hopf algebra structure of $D(G)$ corresponds to the monoidal structure of the 2-group.

Question

So $D(G)$ is really a 2-group. What I'm interested in is the quantum/Drinfeld double of 2-groups.

  • (1). Under what kind of representations of 2-groups, does the 2-group have representations correspond to the representations of the Hopf algebra $D(G)$?

  • (2). Has one constructed a quantum double of 2-groups, whose category of representations in the sense of (1) give rise to the categorical center of Rep$D(G)$?

Possibly related: The character of the 2-representations of a finite group $G$ is a representation of the Drinfeld double $D(G)$. [Theorem on p.7, [1]].

  • [1] Representation and character theory of finite 2-groups-[Robert Usher]

  • [2] Notes on 2-groupoids, 2-groups and crossed-modules-[Behrang Noohi]-[arXiv:math--0512106]

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  • $\begingroup$ The “double” should just be the free loop space, or equivalently the homotopy quotient of $G$ acting on itself by conjugation as in the 1-group case. $\endgroup$
    – Qiaochu Yuan
    Sep 15, 2020 at 22:59
  • $\begingroup$ That's a bit concerning.. let G be a 2-group. From Hopf algebra point of view the amount of the objects of D(G) should be the square of that of G.. if I let G act on G naively as groupoids, I don't get as many objects. Also, I'd stress the importance of the monoidal structure on G, which is really why 2-groupoids differ from pure groupoids. Without it, the representation falls back to the case of classical groups (see abstract of link.. and one doesn't see the crossed structures $\endgroup$
    – Student
    Sep 16, 2020 at 2:08

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