*Citation:* Müger, Michael. "From subfactors to categories and topology II: The quantum double of tensor categories and subfactors." Journal of Pure and Applied Algebra 180.1-2 (2003): 159-219.

*Web:* https://arxiv.org/abs/math/0111205

*Tags:* Foundational, Mathematical, Subfactors, Modular-tensor-categories, Monoidal-categories

This is the paper which properly introduces the notion of Drinfeld center ("quantum double") to monoidal categories, and establishes the import foundational facts. In particular, it is proved that Z(C) is modular whenever C is spherical, and that the quantum dimension of Z(C) is the square of the quantum dimension of C.

According to Muger, the history of the definition of the Drinfeld center of monoidal categories is as follows: "Given an arbitrary monoidal category (or tensor category) C its center Z(C) is a braided monoidal category which was defined independently by Drinfel'd (unpublished), Majid [36] and Joyal and Street [20]". The papers which Muger is referencing here are

> S. Majid: Representations, duals and quantum doubles of monoidal categories. Rend. Circ. Mat. Palermo Suppl. 26, 197-206 (1991).

and

> A. Joyal & R. Street: Tortile Yang-Baxter operators in tensor categories. J. Pure Appl. Alg. 71, 43-51 (1991).

The quantum double construction itself has a much older history, which is summarized in

> Evans, David, and Yasuyuki Kawahigashi. "Subfactors and mathematical physics." Bulletin of the American Mathematical Society (2023).

Namely, the quantum double construction was first introduced by Ocneanu via asymptotic inclusion in

> Ocneanu, Adrian. "Quantized groups, string algebras and Galois theory for algebras." Operator algebras and applications 2 (1988): 119-172.

and then re-cast through symmetric enveloping algebras in

> Popa, Sorin. "Symmetric enveloping algebras, amenability and AFD properties for subfactors." Mathematical Research Letters 1.4 (1994): 409-425.

it was then understood via the Longo-Rehren subfactor in

> Longo, Roberto, and K-H. Rehren. "Nets of subfactors." Reviews in Mathematical Physics 7.04 (1995): 567-597.

and two works of Izumi,

> Izumi, Masaki. "The Structure of Sectors¶ Associated with Longo-Rehren Inclusions¶ I. General Theory." Communications in Mathematical Physics 213 (2000): 127-179.

> Izumi, Masaki. "The structure of sectors associated with Longo-Rehren inclusions II: examples." Reviews in Mathematical Physics 13.05 (2001): 603-674.

One can also think about the quantum double in totally elementary terms. This is best understood when paired with mixed-state TQFTs, as discussed in my review of the mixed-state TQFT paper:

> Zini, Modjtaba Shokrian, and Zhenghan Wang. "Mixed-state TQFTs." arXiv preprint arXiv:2110.13946 (2021).