Majid代数的Yetter-Drinfeld模范畴及形变理论研究

Braided monoidal category is a generalization of symmetric catetory, and it is of great significance to the explanation and construction of low dimension topology. Majid's algebra is the linear duality of quasi-Hopf algebra. Because of its special link with quiver, the research on Majid's algebra has a positive impact on that of quasi-Hopf algebra. This project will focus on the theory of the category of Yetter-Drinfeld modules and conduct the following researches: (1) Construct its category of Yetter-Drinfeld modules, give the braided monoidal structure and construct Radford biproduct. Construct the category of two-sided Hopf modules of Majid's algebra, define Yetter-Drinfeld module of another type via the category equivalence between two-sided Hopf modules category and YD category, and study the relationship between the two kinds of YD categories. (2)Construct the Drinfeld codouble of Majid's algebra and find its ribbon element and coquasitriangular structure. (3) Present the relationship between the YD category and transmutation theory of Majid's algebra. The above reaearches will on one hand enrich the theory of braided monoidal category, and on the other hand play an important role in the classification of Majid's algebras or quasi-Hopf algebras.

辫子张量范畴源于对称范畴的推广，其对低维拓扑中的扭结不变量的解释与构造具有重要意义。 Majid代数是拟Hopf代数的线性对偶，由于它与箭图构造的特殊联系，因此对拟Hopf代数的研究有积极的促进作用。本项目将围绕Majid代数的Yetter-Drinfeld模范畴及相关理论进行以下研究：(1) 构造Majid代数的YD模范畴，给出其辫子张量结构，并借助范畴方法构造Radford 双积。建立Majid代数的双边Hopf 模范畴，并根据其与YD范畴之间的等价构造另一形式的YD模，讨论两种形式的YD范畴之间的等价。(2)构造Majid代数的Drinfeld余偶， 并给出ribbon结构和余拟三角结构。(3)研究Majid代数的YD范畴与形变理论之间的关系。这些研究内容一方面丰富了Hopf代数中的辫子张量范畴理论，另一方面在Majid代数或拟Hopf代数的分类问题中有重要应用。

辫子张量范畴是对称范畴的自然推广，对低维拓扑中的扭结不变量的解释与构造具有重大意义。因V. Drinfeld的杰出工作，辫子张量范畴与Hopf代数之间的深刻联系被揭示出来。本项目旨在研究辫子张量范畴的构造和性质，尤其是余拟Hopf代数上的Yetter-Drinfeld范畴及其性质，主要研究内容有：(1) 给出Majid 代数(或余拟Hopf代数)上的Yetter-Drinfeld范畴中标准同构的具体刻画，并研究了Majid 代数的Drinfeld余偶理论，(2) 给出拟Turaev群余代数上Yetter-Drinfeld范畴的构造及性质，(3) 进行了几类Hopf代数双交叉积构造，从而完成了可通过某种双交叉积分解的Hopf代数的分类，(4) 进行了Hom-Hopf代数的交叉积构造并研究了相关性质。这些研究内容一方面丰富了辫子张量范畴理论，另一方面将在低维拓扑、表示论和量子场论中有重要应用。

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