若干辫子fusion范畴的弱群型性质和分类

Fusion范畴在Hopf代数和拟Hopf代数、顶点算子代数、拓扑量子场、子因子等众多研究领域有着重要的应用。本项目主要研究几类给定Frobenius-Perron维数的辫子fusion范畴的(弱)群型性质以及对给定量子维数预模fusion范畴的分类问题。首先，对范畴的Frobenius-Perron维数或者单对象的Frobenius-Perron维数满足某些限定条件的辫子fusion范畴，利用fusion范畴的等变化、去等变化以及阶化扩张等理论，我们考虑这些fusion范畴的弱群型性质和群型性质。作为应用，我们研究一些半单Hopf代数的结构。然后，从预模fusion范畴的算术性质以及代数数论相关理论角度出发，我们拟研究量子维数为p^2的模fusion范畴和量子维数为2p的超的模fusion范畴的分类，其中，p是奇素数。

The theory of fusion categories has important applications in various mathematical subjects, such as Hopf algebras and quasi-Hopf algebras, vertex operator algebras, topological quantum field theory, subfactor, etc. This research program considers the (weakly) group-theoretical property of some braided fusion categories with given Frobenius-Perron dimensions, and the classifications of certain pre-modular fusion categories of given quantum dimensions. For braided fusion categories whose Frobenius-Perron dimensions or Frobenius-perron dimensions of their simple objects satisfying some restrictions, using equivariantization, de-equivariantization, and graded extension theory of fusion categories, we consider their weakly group-theoretical property and group-theoretical property. As an application, we study the structures of some semi-simple Hopf algebras. Starting from the aspects of arithmetic properties of pre-modular fusion categories and related theories of algebraic number theory, we study the classifications of modular fusion categories of quantum dimensions p^2 and super-modular fusion categories of quantum dimensions 2p, where p is a prime.