Resumen de Representations of fusion categories and their commutants

Andre Henriques, David Penneys

• 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:

  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. This is the published version of arXiv:2004.08271.