Representations of fusion categories and their commutants
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 $\mathcal{C}\to\operatorname{Bim}(R)$ 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: \[ \Big(\,\,\mathcal{C}\,\,\simeq_{\text{Morita}}\,\,\mathcal{D}\,\,\Big) \qquad\Longrightarrow\qquad \Big(\,\,\mathcal{C}'\,\,\simeq_{\text{tensor}}\,\,\mathcal{D}'\,\,\Big). \] This categorifies the wellknown result according to which the commutants (in some representations) of Morita equivalent finite dimensional $\rm C^*$algebras are isomorphic von Neumann algebras, provided the representations are `big enough'. We also introduce a notion of positivity for biinvolutive tensor categories. For dagger categories, positivity is a property (the property of being a $\rm C^*$category). But for biinvolutive tensor categories, positivity is extra structure. We show that unitary fusion categories and $\operatorname{Bim}(R)$ admit distinguished positive structures, and that fully faithful representations $\mathcal{C}\to\operatorname{Bim}(R)$ automatically respect these positive structures.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.08271
 Bibcode:
 2020arXiv200408271H
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Category Theory;
 Mathematics  Quantum Algebra;
 18D10 (Primary);
 46L10 (Secondary)
 EPrint:
 40 pages, many figures