论文标题
融合类别的表示及其通勤者
Representations of fusion categories and their commutants
论文作者
论文摘要
两种义务类别是von Neumann代数的较高分类类似物。我们研究了一个完全忠实表示的换向$ \ MATHCAL {C}'$的两种类别，$ \ MATHCAL {C} \ to \ operatoratorName {bim}（r）的统一融合类别$ \ Mathcal {c} $。利用Izumi，Popa和Tomatsu的结果以及单一（多）融合类别表示的存在和独特性，我们证明，如果$ \ Mathcal {C} $和$ \ Mathcal {d} $是Morita等效的单一融合类别等效地为两种类别。特别是，它们等效地等于张量类别：\ [\ big（\，\，\ Mathcal {c} \，\，\，\ simeq _ {\ text {morita}}} \，\，\，\，\，\ Mathcal \ big（\，\，\ Mathcal {c}'\，\，\ simeq _ {\ text {tensor}}} \，\，\，\ Mathcal {d}'\，\，\，\ big）。 \]这对众所周知的结果进行分类，根据该结果，莫里塔等效的有限尺寸$ \ rm c^*$ - 代数是同构的von neumann代数，只要表征是“足够大”）。 我们还针对双重张量类别引入了积极性的概念。对于匕首类别，积极性是一种属性（是$ \ rm c^*$ - 类别的属性）。但是对于双重张量类别，阳性是额外的结构。我们表明，统一的融合类别和$ \ operatorName {bim}（r）$允许划定的积极结构，并且完全忠实的表示$ \ MATHCAL {C} \ to \ operatotorname {bim}（bim}（r）$自动尊重这些正面结构。
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 well-known 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 bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a $\rm C^*$-category). But for bi-involutive 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.
