张量范畴中的带状结构及其在扭结不变量中的应用

带状结构源于对纽结和链环在连续变形下的量子不变量的研究，其对3维流形不变量、模性范畴的表示与分类、同伦量子场论等方面的研究具有重要意义。本项目旨在研究带状结构及其子结构的刻画以及在纽结不变量上的应用等问题，这些问题包括：混合双模、Yetter-Drinfeld群模的带状结构的构造及其诱导的扭结不变量；无辫子张量范畴中拟双代数理论的建立；枢纽交叉结构与广义Deligne定理的刻画；更一般的半单带状范畴和模性范畴及量子维数的讨论等。本项目的关键研究手段为单子理论和Tannaka对偶理论、同调方法和低维拓扑理论中的投影方法等。目标是得到某些重要张量范畴的系统构造形式、一些重要代数结构的普遍性质和等价条件等。相信本项目所得结论将统一已知的一些经典结果，进一步丰富Hopf代数和张量范畴理论，并将在低维拓扑、表示论和量子场论中有重要应用。

The theory of ribbon structures and their substructures starts from the classical works of the invariants of knots and links, it have been one of the most powerful tools to study the link invariants of 3-manifolds, the classifications of modular categories and qantum gauge theory. This proposal will focus on the new constructions of ribbon structures and their substructures, and the quantum invariants derived from them. We will consider the ribbon structures in the category of mixed bi-modules and the crossed Yetter-Drinfeld modules, and specially discuss the knot invariants; we will discuss the theory of quasi-bialgebras in the non-braided monoidal categories; we will compute the pivotal crossed structures and discuss the generalized Deligne type theorem; we will discuss the construction of the new modular category and the semisimple ribbon category which is more generally, and compute the quantum dimension of them. The key idea of our study is based on the monads theory and Tannaka-Krein duality, low-dimension topology methods and homological methods. The main aim of the project is to obtain some systematic means of monoidal categories, some general informations and equivalence results of important structures. It is hoped that the results obtained can unify some classical results, enrich Hopf algebras and monoidal catogory theory. The study of the project will have important theoretical significance and application value on the low-dimensional topology, the representation theory and the Topological Qantum Field Theory.