代数量子群胚的分析结构与相关量子不变量的研究

Algebraic quantum groupoids gave an answer to the duality problem of a class of infinite dimensional weak Hopf algebras, and made a further progress in Pontryagin duality theorem. In this project, the analytic structures and relevant quantum invariants for algebraic quantum groupoids will be mainly studied. Firstly, we are going to investigate the analytic structures of algebraic quantum groupoids, consider the relation between algebraic quantum groupoids and measured quantum groupoids, and prove the statement: algebraic quantum groupoids are the algebraic version of measured quantum groupoids. Secondly, we will construct several quantum invariants by the algebraic structures and representation categories of algebraic quantum groupoids, and also we use the generalized Yetter-Drinfeld categories of (group-cograded) algebraic quantum groupoids to construct the braided crossed categories, which are associative to some invariants. Finally, as an application, we will consider a special class of Galois objects for algebraic quantum groupoids, i.e., the coinvariants of Galois coaction are only the scalars, and give their analytic structures and relevant quantum invariants.

代数量子群胚解决了一类无限维弱Hopf代数的对偶问题，进一步发展了Pontryain对偶定理。本项目主要研究代数量子群胚的分析结构与相关的量子不变量。首先，研究代数量子群胚的分析结构，考察代数量子群胚与可测量子群胚之间的关系，并证明结论：代数量子群胚是可测量子群胚的代数版本。其次，利用代数量子群胚的代数结构及其表示范畴构造量子不变量，并且利用（群余分次）代数量子群胚的广义Yetter-Drinfeld范畴构造与不变量相关的辫子交叉范畴。最后，作为应用考察代数量子群胚一类特殊的Galois对象，即Galois余作用的余不变量是标量的情形，给出其分析结构及相关的量子不变量。

本项目从Hopf代数学的角度，以模论和范畴理论为主要工具，研究了乘子Hopf代数与代数量子群胚的相关量子不变量和分析结构。首先推广对角交叉积构造了一类群余分次乘子Hopf代数，并验证了其拟三角结构的存在性，然后讨论了拟三角群余分次乘子Hopf代数的表示范畴，并给出了其范畴刻画；利用乘子Hopf代数的Drinfeld扭曲构造出新的乘子Hopf代数，并考察拟三角结构和积分在扭曲形变下如何变化；利用广义Drinfeld扭曲构造了弱Hopf代数的双协变微分运算，并将结果应用到了face代数；研究了乘子Hopf代数的ribbon结构，并证实了Hennings的不变量的构造对于乘子Hopf代数依然适用；考察了Hopf代数的拟Yetter-Drinfeld模，并给出其两个等价的范畴刻画；研究了一类特殊代数量子群胚的分析结构，给出了对极的Radford公式。项目推广了Hopf代数与量子群的经典结论，解决了同行提出的一些问题，进一步丰富了Hopf代数理论。