Braided Tensor Categories, Their Structures, Symmetries, and Graded Extensions

这个项目是关于张量范畴的研究。这些是由物体组成的数学结构，可以使用某些自然规则进行添加和乘法。这些结构的抽象性质使它们成为研究对称性的一个非常方便的工具，无论是在经典领域还是在量子领域。近年来，张量范畴被提出作为拓扑量子计算的数学模型，这是构建量子计算机最有前途的模型之一。它们还与物质拓扑相的对称性密切相关，并用于预测这种相的新类型的存在及其物理实现。这个项目涉及张量范畴理论的代数方面：它们的结构、分类和算术性质。重点是在应用程序中使用最广泛的类别。这样的分类承认一个额外的对称约束，称为编织，用于模拟量子粒子对的相互作用。张量范畴为研究各种量子对称性提供了一个统一的框架，如量子群、顶点算子代数、琼斯-冯·诺伊曼子因子和共形场论。Hopf代数和量子群理论中的分类方法导致了许多重要的分类结果，并继续带来新的见解。这个项目将使用已经开发的机器来处理有关张量类别的结构和分类的基本问题。它将解决与编织张量范畴、它们的模块和对称群的扩展有关的问题。具体研究领域包括：(1)编织张量范畴的梯度扩展理论和最小非退化扩展的分类；(2)编织张量范畴的Picard群及其对范畴正交格拉斯曼算子的作用及其在小量子群表示范畴中的应用；(3)融合范畴和半简单Hopf代数的算术性质和结构；(4)非半简单点编织张量类及其模类的分类。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns the study of tensor categories. These are mathematical structures consisting of objects that can be added and multiplied using certain natural rules. The abstract nature of these structures makes them a very convenient tool to study symmetries, both in the classical and quantum arenas. Recently, tensor categories were proposed as mathematical models of a topological quantum computation, one of the most promising models for the construction of a quantum computer. They are also closely related to symmetries of topological phases of matter and are used to predict the existence of new types of such phases and their physical realization. This project deals with algebraic aspects of the theory of tensor categories: their structure, classification, and arithmetic properties. The emphasis is on categories that are most widely used in applications. Such categories admit an additional symmetry constraint called braiding that is used to model interaction of pairs of quantum particles. Tensor categories provide a unified framework for studying various quantum symmetries such as quantum groups, vertex operator algebras, Jones-von Neumann subfactors,and conformal field theories. Categorical methods in the theory of Hopf algebras and quantum groups led to a number of important classification results and continue to bring forth new insights. This project will use the machinery that has already been developed to approach fundamental questions concerning the structure and classification of tensor categories. It will address problems related to extensions of braided tensor categories, their modules, and groups of symmetries. The concrete areas of research include the following: (1) the theory of graded extensions of braided tensor categories and classification of minimal non-degenerate extensions, (2) the Picard groups of braided tensor categories and their actions on categorical orthogonal Grassmannians with applications to the representation categories of small quantum groups; (3) arithmetic properties and structure of fusion categories and semisimple Hopf algebras; (4) classification of non-semisimple pointed braided tensor categories and their module categories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.