Hopf algebras, modular categories and their invariants

本课题将研究某些有限维Hopf代数的结构、它们的表示范畴和球面融合范畴。这个项目的主要部分涉及一些范畴算术不变量，如维数、指数、Frobenius-Schur指标，以及与模张量范畴相关的模群的表示。这些算术不变量之间关系密切，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数的素数因子与基本范畴的拟指数的素数因子之间的关系。这里的任何进展都将有助于证明Kaplansky关于半简单Hopf代数的猜想，这个猜想仍然是开放的。该项目还涉及的Hopf代数的分类，其维度承认一个简单的质因数分解。上述基本问题的进展也有助于对有限维Hopf代数进行分类。对称的吸引力一直是理解科学本质的指南。一些对称性可以用代数结构来描述。例如，柏拉图立体的对称性可以用空间旋转的有限群来描述。Hopf代数和张量范畴是群的推广，它们也可以描述某些物理系统的对称性。它们自然地出现在数学和数学物理的许多领域，如共形场论、统计力学和量子计算。因此，对这些代数结构的深入理解是至关重要的，它最终将在其他科学领域发挥作用。

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.