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Drinfeld Center and Representation Theory for Monoidal Categories

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Abstract

Motivated by the relation between the Drinfeld double and central property (T) for quantum groups, given a rigid C*-tensor category \({\mathcal{C}}\) and a unitary half-braiding on an ind-object, we construct a *-representation of the fusion algebra of \({\mathcal{C}}\). This allows us to present an alternative approach to recent results of Popa and Vaes, who defined C*-algebras of monoidal categories and introduced property (T) for them. As an example we analyze categories \({\mathcal{C}}\) of Hilbert bimodules over a II1-factor. We show that in this case the Drinfeld center is monoidally equivalent to a category of Hilbert bimodules over another II1-factor obtained by the Longo–Rehren construction. As an application, we obtain an alternative proof of the result of Popa and Vaes stating that property (T) for the category defined by an extremal finite index subfactor \({N \subset M}\) is equivalent to Popa’s property (T) for the corresponding SE-inclusion of II1-factors. In the last part of the paper we study Müger’s notion of weakly monoidally Morita equivalent categories and analyze the behavior of our constructions under the equivalence of the corresponding Drinfeld centers established by Schauenburg. In particular, we prove that property (T) is invariant under weak monoidal Morita equivalence.

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316, Oslo, Norway

    Sergey Neshveyev

  2. Department of Mathematics, Ochanomizu University, Otsuka 2-1-1, Bunkyo, 112-8610, Tokyo, Japan

    Makoto Yamashita

Authors
  1. Sergey Neshveyev
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  2. Makoto Yamashita
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Correspondence to Makoto Yamashita.

Additional information

Communicated by Y. Kawahigashi

S. Neshveyev: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement No. 307663.

M. Yamashita: Supported by JSPS KAKENHI Grant Number 25800058.

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Neshveyev, S., Yamashita, M. Drinfeld Center and Representation Theory for Monoidal Categories. Commun. Math. Phys. 345, 385–434 (2016). https://doi.org/10.1007/s00220-016-2642-7

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  • DOI: https://doi.org/10.1007/s00220-016-2642-7

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