辫子交叉范畴在构造3-维流形的量子不变量中起着关键作用。它不仅是Hopf代数理论中的重要研究对象，而且在许多领域内都有重要应用。本项目主要研究辫子交叉范畴及其在3-维流形的量子不变量中的应用，主要研究内容有：（1）Hom-缠绕模范畴和Hom-Long dimodules范畴的结构和性质的刻画；（2）（余）冲积Hopf群余代数和Hom-Hopf代数的表示范畴；（3）弱Hopf群余代数和群余分次乘子Hopf代数上新的ribbon范畴，及其上的Kirby元素集合。这些研究内容一方面丰富了Hopf代数中的辫子交叉范畴理论，另一方面为3-维流形的量子不变量和扭结理论的研究提供了新思路。

Braided crossed categories, which not only are important objects in the theory of Hopf algebra but also have wide applications in many fields, play a key role in the construction of quantum invariants for 3-dimensional manifolds. This project will devote to the braided crossed category theory and the applications on quantum invariants for 3-dimensional manifolds. The main research contents of this project are as follows: (1) The structures and properties of categories of Hom-entwined modules and Hom-Long dimodules; (2) The representation categories of smash (co)products of Hopf group coalgebras and Hom-Hopf algebras; (3) The ribbon categories by weak Hopf group coalgebras and group-cograded multiplier Hopf algebras and the set of Kirby elements on them. On the one hand, the above research enriches the braided crossed category theory in Hopf algebras, and on the other hand, it provides a new idea for studying the quantum invariants for 3-dimensional manifolds and knot theory.