代数量子群的Yetter-Drinfel'd模范畴与Galois理论的研究

Algebraic quantum groups give a solution to the problem on Pontryagin dualities of a class of infinite dimensional Hopf algebras, and have close relationship with locally compact quantum groups, Hopf algebras and operator algebras. In this project, we mainly study the Yetter-Drinfel''d module categories and Galois theory: firstly, consider the properties of Yetter-Drinfel''d module categories introduced by the project applicant, and use this kind of categories to construct a new class of braided crossed categories, which are closely related to some topological invariants; secondly, investigate the Galois theories for algebraic quantum group and multiplier Hopf algebra, and apply them to some infinite dimensional Hopf algebras; thirdly, define the crossed products of algebraic quantum groups, give the quasitriangular structures on these crossed products, and lift them to locally compact quantum groups with analytic structures; finally, consider the analytic structure of algebraic quantum hypergroup and establish the link between the algebraic quantum hypergroup and compact quantum hypergroup.

代数量子群解决了一类无限维Hopf代数的Pontryagin对偶问题，与局部紧量子群、Hopf代数以及算子代数有紧密的联系。本项目主要研究代数量子群的Yetter-Drinfel''d模范畴和Galois理论：首先，考察项目申请人所引入的Yetter-Drinfel''d模范畴的性质，并利用此类范畴构造与拓扑不变量紧密相关的辫子交叉范畴；其次，研究代数量子群和乘子Hopf代数的Galois理论，并将其应用无限维Hopf代数领域；再次，定义代数量子群的交叉积，给出拟三角结构，并将其提升到带有分析结构的局部紧量子群；最后，考虑代数量子超群的分析结构，建立代数量子超群与紧量子超群之间的联系。

本项目以乘子 Hopf 代数结构理论和同调理论为主要研究工具，研究了乘子Hopf代数与代数量子群中关于 Yetter-Drinfel'd 模范畴和 Galois 理论方面的前沿问题，丰富了乘子Hopf代数理论，推广了Hopf代数理论中的一些经典结果。通过对Yetter-Drinfel'd 模范畴对象性质的研究，给出了Yetter-Drinfel'd 模的另一种范畴刻画，同时将其应用到无限维coFrobenius Hopf代数理论中；利用弱Hopf代数和乘子Hopf代数上的广义Yetter-Drinfel'd 模范畴构造了两类与拓扑不变量紧密相关的辫子交叉范畴；研究了Yetter-Drinfel'd 模范畴中的三类代数结构：结合代数、李代数与Hom-李代数，证明了Yetter-Drinfel'd 模范畴中李代数的Kegel定理。