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Representations of fusion categories and their commutants

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Abstract

A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant \({\mathcal {C}}'\) of a fully faithful representation \({\mathcal {C}}\rightarrow \textrm{Bim}(R)\) of a unitary fusion category \({\mathcal {C}}\). Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if \({\mathcal {C}}\) and \({\mathcal {D}}\) are Morita equivalent unitary fusion categories, then their commutant categories \({\mathcal {C}}'\) and \({\mathcal {D}}'\) are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories:

$$\begin{aligned} \Big (\,\,{\mathcal {C}}\,\,\simeq _{\textrm{Morita}}\,\,{\mathcal {D}}\,\,\Big ) \qquad \Longrightarrow \qquad \Big (\,\,{\mathcal {C}}' \,\,\simeq _{\textrm{tensor}}\,\,{\mathcal {D}}'\,\,\Big ). \end{aligned}$$

This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional \(\textrm{C}^*\)-algebras are isomorphic von Neumann algebras, provided the representations are ‘big enough’. We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a \(\textrm{C}^*\)-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and \(\textrm{Bim}(R)\) admit distinguished positive structures, and that fully faithful representations \({\mathcal {C}}\rightarrow \textrm{Bim}(R)\) automatically respect these positive structures. This is the published version of arXiv:2004.08271.

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Notes

  1. In the original definition of bicommutant categories [14], positive structures were not mentioned. We believe that this was a mistake. We fix this by slightly altering the definition.

  2. We omit various unitors and associators to keep the equations compact.

  3. We omit the structure isomorphisms \(\mu \) and i for better readability.

  4. Note that, by Lemma 3.4, such functors are almost always fully faithful.

  5. The equivalence between this definition and the one presented in [42, Definition 2.16] follows along the same lines as the proof of [28, Lemma 8.3].

  6. An endomorphism \(\rho \in {{\,\textrm{End}\,}}(R)\) has finite depth if the \(\textrm{C}^{*}\)-tensor category generated by \(\rho \) is fusion.

  7. Similarly to [18, Lemma 3.5] the collection of pairs \(({\mathcal {C}},X)\) forms a 2-category which is equivalent to a 1-category.

  8. The results in [1] are phrased in the context of von Neumann algebras with finite dimensional centers but extend verbatim to the case of von Neumann algebras with atomic centers.

  9. The letters cp stands for “completely positive”. We warn the reader that positive maps \(a\otimes \bar{a}\rightarrow a\otimes \bar{a}\) (i.e., maps which can be written as \(f^*{\circ } f\)) are typically not cp.

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Acknowledgements

The authors would like to thank Reiji Tomatsu for helping us understand his article [42]. We thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Operator Algebras: Subfactors and their Applications where work on this paper was undertaken. This work was supported by EPSRC grant no EP/K032208/1. André Henriques was supported by the Leverhulme trust and the EPSRC grant “Quantum Mathematics and Computation”. David Penneys was partially supported by NSF DMS grants 1500387/1655912 and 1654159.

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    André Henriques

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    David Penneys

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Henriques, A., Penneys, D. Representations of fusion categories and their commutants. Sel. Math. New Ser. 29, 38 (2023). https://doi.org/10.1007/s00029-023-00841-2

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