群余环的可分函子及Picard群

As the generalization of corings, group-corings become increasingly the research focus. In the project, we will use the category of comodules over group corings as object of research, and adopt some basic functorial techniques to make some basic scientific research on the induction functor stemming from every morphism of corings. Detailed contents are as follows: discussing the adjoint functor of the induction functor and characterizing its separability; studying equivalences between comodule categories over group corings and establishing the comatrix coring of a quasi-finte comodule over a group coring; introducing the Picard group of a group coring and the relations among the automorphism group、inner automorphism group and the picard group, and characterizing the group corings which have the Aut-Pic property in order to get the meaningful results. Developing the project research not only enrichs the theory of group corings, also unifies the important results in Hopf algebra setting and lays the foundation for further research on group corings.

作为余环的推广，群余环日益成为研究的热点。本项目以群余环的余模范畴为研究对象，采用函子化手段，对基于群余环态射的导出函子进行基础科学研究。具体内容包括：讨论导出函子的伴随性，拟刻画导出函子的可分性；探讨群余环的余模范畴的等价关系，试图建立群余环的拟有限余模的余矩阵余环；拟建立群余环的Picard群及自同构群、内自同构群及Picard群之间的关系，刻画群余环具有Aut-Pic性质的条件，以期得到更有意义的结果。开展本项目研究所得成果不仅丰富群余环理论，而且统一了Hopf代数中重要结果，为进一步研究群余环奠定一定的基础。

余环的概念由Sweedler于1975年最先提出, 它统一了各种Hopf型模，例如，Hopf模、Doi-Hopf模、Yetter-Drinfeld模和entwined模。作为余环的推广，群余环理论得以建立和发展。本项目主要刻画了任意群余环上的余模范畴的忘却函子的可分性，建立了群余环的Morita关系及一般的Galois理论。同时引入和发展了Hopf群余代数的偏作用理论， 包括了一般的Hopf代数偏作用的相应理论。其次，进一步发展了弱Hopf群代数理论，通过弱Hopf群代数，构造了一类新的群交叉范畴；证明了广义量子交换的Smash积的Maschke型定理。最后，构造了一类拟三角和余拟三角量子（超）群结构，为Yang-Baxter方程提供一类解；证明了L-R交叉积的对偶定理，推广了Hopf代数中相应的经典结果。