量子群胚作用与余作用的研究

In this project, we put our emphasis on (co)actions for quantum groupoids and braided monoidal (crossed) categories. Firstly, we provide methods to construct braided groups by quasitriangular structures, weak invertible unit 2-cocycles and weak ad-invariant cocycles of quantum groupoids, and consider relations among them further. Secondly, we introduce a kind of new α-Yetter-Drinfeld module, and use it to construct a class of braided crossed categories. As an application, we give the concept of a Long dimodule category for a weak crossed Hopf group coalgebra with quasitriangular and coquasitriangular structures, and then study the intrinsic relationship between these two classes of braided crossed categories. Finally, according to the theory of braided Lie algebras and the weak version of Radford theorem, we study Schur double centralizer theorem in the Yetter-Drinfeld module category over quantum groupoids.

本项目重点围绕量子群胚的（余）作用以及辫子张量（交叉）范畴展开了讨论。首先，研究由量子群胚的拟三角结构、弱可逆的单位2-余循环以及弱伴随不变余循环，构造辫子群的方法，并进一步对其做了深入的研究。其次，引入一种新的α-Yetter-Drinfeld模，由此构造一类辫子交叉范畴。作为应用，给出带有拟三角和余拟三角结构的弱交叉Hopf群余代数的Long重模范畴，进而研究上述两类辫子交叉范畴之间内在的关系。最后，根据辫子李代数的相关理论及Radford定理的弱形式，研究量子群胚Yetter-Drinfeld模范畴中的Schur双中心化定理。

本项目主要围绕量子群胚和弱Hopf群余代数展开讨论，利用其作用和余作用，研究了辫子交叉范畴以及Majid意义下辫子群的构造问题。首先，构造了新的辫子交叉范畴，即弱交叉Hopf群余代数的弱Yetter-Drinfeld模范畴以及弱Long重模范畴，并证明了弱Long重模范畴在某种条件下是弱Yetter-Drinfeld模范畴的辫子T-子范畴的结论。其次，考虑了量子群胚中心化子代数，给出了分别由量子群胚的拟三角结构以及弱可逆的单位2-余循环构造辫子群的方法，从而推广了经典的转化理论以及量子化方法，并得出了同构定理。最后，讨论了正则乘子Hopf代数Yetter-Drinfeld模范畴中自同构代数的问题。