Representations of fusion categories and their commutants

Journal article

A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant C′ of a fully faithful representation C→Bim(R) of a unitary fusion category C. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if C and D are Morita equivalent unitary fusion categories, then their commutant categories C′ and D′ are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: (C≃MoritaD)⟹(C′≃tensorD′). This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional 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 C∗-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and Bim(R) admit distinguished positive structures, and that fully faithful representations C→Bim(R) automatically respect these positive structures.

Authors: Henriques, A; Penneys, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Selecta Mathematica (New Series)
• Volume: 29
• Article number: 38
• Publication date: 2023-04-27
• Acceptance date: 2023-02-08
• DOI: 10.1007/s00029-023-00841-2
• EISSN: 1420-9020
• ISSN: 1022-1824
• Language: English