EAPSI: Classification of Fusion Categories with one Non-Invertible Object

融合范畴是一种数学结构，用于对各种数学系统的不变量进行编码。数学不变量是数学对象的量或其他性质，在该对象的变换下保持不变。例如，给定几何对象的对称性可以表示为融合范畴的数据；这些对称性的某些性质在对象变形时必须保持不变，这反映在融合范畴的不变理论中。这个项目的一个主要目标是统一几种不同的方法来研究来自数学和物理的融合类别。京都大学教授Masaki Izumi是该项目的东道主科学家，从物理学影响的角度来看，他是聚变范畴理论的国际领先研究人员。这一合作将进一步促进两个群体之间的互动，并将产生新的数学问题，这些问题来自已有的物理理论。本项目旨在对具有一个对象在张量积下不可逆的融合范畴进行分类：这些融合范畴被称为近群融合范畴。Izumi发展了一种利用Hilbert空间上的算子代数理论来进行这种分类的技术；具体地说，他将它们的对象实现为Cuntz C*代数上的自同态，并对它们可能的像进行了分类。Izumi和首席研究员(PI)将以这种方式实现近群，并为具有近群融合规则的融合范畴的可能等价类提供分类参数。还将计算几个不同的重要融合类别不变量家族，包括Drinfel‘d中心和Frobenius-Schur指标的模块数据。东亚和太平洋夏季研究所计划下的这个奖项支持一名美国研究生的夏季研究，由NSF和日本科学促进会共同资助。

Fusion categories are mathematical structures which are used to encode invariants of a wide variety of mathematical systems. A mathematical invariant is a quantity or other property of a mathematical object that remains unchanged under transformations of that object. For example, the symmetries of a given geometric object may be expressed as the data of a fusion category; certain properties of these symmetries must remain the same under deformations of the object, and this is reflected in the invariant theory of fusion categories. One primary goal of this project is to unify several different approaches to the study of fusion categories coming from both mathematics and physics. Kyoto University Professor Masaki Izumi, the host scientist for this project, is a leading international researcher in fusion category theory from the physics-influenced point of view. This collaboration will further the interaction between the two communities and will result in new mathematical questions arising from established physical theories.This project aims to classify fusion categories having one object that is non-invertible under the tensor product: these are called the near-group fusion categories. Izumi has developed a technique utilizing the theory of algebras of operators on Hilbert spaces to make this classification; specifically, he realizes their objects as endomorphisms on Cuntz C* algebras and classifies their possible images. Together Izumi and the principal investigator (PI) will realize the near-groups in this way and provide classification parameters for the possible equivalence classes of fusion categories having the near-group fusion rule. Several different important families of fusion category invariants will also be computed, including the modular data for the Drinfel'd centers and the Frobenius-Schur indicators.This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and the Japan Society for the Promotion of Science.