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

编织张量类别、其结构、对称性和分级扩展

基本信息

批准号：
1801198
负责人：
Dmitri Nikshych
金额：
$ 15万
依托单位：
University of New Hampshire
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-15 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801198&HistoricalAwards=false
关键词：
Braided Tensor Categories Their Structures

项目摘要

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.

该项目涉及张量类别的研究。这些是由可以使用某些自然规则进行加法和乘法的对象组成的数学结构。这些结构的抽象性质使它们成为研究经典和量子领域对称性的非常方便的工具。最近，张量类别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，并用于预测此类相的新型类型的存在及其物理实现。该项目涉及张量范畴理论的代数方面：它们的结构、分类和算术属性。重点是应用程序中使用最广泛的类别。这些类别允许额外的对称约束，称为编织，用于模拟量子粒子对的相互作用。张量类别为研究各种量子对称性（例如量子群、顶点算子代数、琼斯-冯·诺依曼子因子和共形场论）提供了统一的框架。霍普夫代数和量子群理论中的分类方法产生了许多重要的分类结果，并不断带来新的见解。该项目将使用已经开发的机制来解决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类别、其模和对称群的扩展相关的问题。具体研究领域包括：（1）辫状张量范畴的分级扩张理论和最小非简并扩张的分类，（2）辫状张量范畴的皮卡德群及其对分类正交格拉斯曼函数的作用及其应用小量子群的表示类别； (3)融合范畴和半简单Hopf代数的算术性质和结构； (4)非半简单尖辫张量类别及其模类别的分类。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。