权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Hopf algebras, modular categories and their invariants

Hopf 代数、模范畴及其不变量

基本信息

批准号：
1501179
负责人：
Siu-Hung Ng
金额：
$ 11.57万
依托单位：
Louisiana State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-08-16 至 2018-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501179&HistoricalAwards=false
关键词：
Hopf algebras modular categories their

项目摘要

This project will study the structure of certain finite-dimensional Hopf algebras, their categories of representations, and spherical fusion categories. The main part of this project concerns some categorical arithmetic invariants such as dimensions, exponents, Frobenius-Schur indicators, and representations of the modular group associated with modular tensor categories. These arithmetic invariants are closely related among each other, and they serve as important tools for studying these algebraic structures. A basic question is the relationship between the prime factors of the dimension of a simple object and those of the quasi-exponent of the underlying category. Any progress here would help in proving a conjecture of Kaplansky on semisimple Hopf algebras that remains open. The project also concerns the classification of Hopf algebras whose dimensions admit a simple prime factorization. Progress on the preceding basic question would also help in classifying finite-dimensional Hopf algebras.The appeal of symmetry has been guidance for understanding the nature of science. Some symmetry can be described in terms of algebraic structures. For instance, the symmetry of platonic solids can be described by some finite groups of spatial rotations. Hopf algebras and tensor categories are generalizations of groups which can also describe the symmetry of some physical systems. They appear naturally in many areas of mathematics and mathematical physics such as conformal field theory, statistical mechanics, and quantum computations. Thus, a deeper understanding of these algebraic structures is of fundamental importance and it will eventually be useful in the other areas of science.

这个项目将研究某些有限维Hopf代数的结构，它们的表示范畴，以及球面融合范畴。这个项目的主要部分涉及一些范畴算术不变量，如维度，指数，Frobenius-Schur指标，以及与模张量范畴相关的模群的表示。这些算术不变量彼此密切相关，是研究这些代数结构的重要工具。一个基本问题是简单对象的维度的素因数与基础范畴的准指数的素因数之间的关系。这里的任何进展都将有助于证明半单Hopf代数上的Kaplansky猜想仍然是开的。该项目还涉及到维度允许简单素因式分解的Hopf代数的分类。在前面的基本问题上的进展也将有助于对有限维Hopf代数进行分类。对称性的吸引力一直是理解科学本质的指导。某些对称性可以用代数结构来描述。例如，柏拉图式固体的对称性可以用一些空间旋转的有限群来描述。Hopf代数和张量范畴是群的推广，也可以描述某些物理系统的对称性。它们自然而然地出现在数学和数学物理的许多领域，如保角场论、统计力学和量子计算。因此，更深入地了解这些代数结构是至关重要的，它最终将在科学的其他领域发挥作用。