论文标题
多因素双模型的失真和多率类别的表示
Distortion for multifactor bimodules and representations of multifusion categories
论文作者
论文摘要
我们将具有有限维中心的von Neumann代数称为多因素。我们在$ \ rm II_1 $多因素上引入了一个不变的双模型，我们称之为模块化失真，并使用它来制定两个分类结果。 我们首先将有限的深度有限指数连接的高限$ \ rm II_1 $多因素包含物$ a \子集b $就标准不变式（一个统一的平面代数）而言，以及限制到$ b $ by上的独特markov trace的$ a $ a $ a $ a $。后者确定相关双模块的模块化失真。三种至关重要的成分是Popa的独特定理，对于此类包含物，它也是均匀的，标准不变性是完全不变的，是Ocneanu Compactness定理的广义版本，以及莫里塔（Morita）等价的概念。 其次，我们将单一多源类别类别的完全忠实表示形式分为多限$ \ rm II_1 $多因素，从模块化失真中。每个可能的失真都来自表示形式，我们表征了由连接的$ \ rm II_1 $多因素包含物引起的变形的正确子集。
We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $\rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results. We first classify finite depth finite index connected hyperfinite $\rm II_1$ multifactor inclusions $A\subset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popa's uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions. Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $\rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $\rm II_1$ multifactor inclusions.
