本项目涵盖涉及我们关心的Drinfeld量子化问题和Kaplansky关于有限维Hopf代数分类猜想问题及由此延伸的更高层次的国际热点跨学科研究---张量范畴论中的若干研究问题，包括：（I) 双参多参李超量子群的结构和表示论中的结构实现和构造问题；已知Lusztig小量子群表示论中涉及BGG理论及一些特定模结构的量子微分算子实现问题，关注量子群的范畴化研究；(II) 用AS-广义提升方法于给定维数的Hopf代数的构造和Nichols代数的提升分类及Green环刻画；对有限维点的拟Hopf代数（拟量子群）、拟Nichols代数根系的分类；(III) 聚焦有限秩的点的融合（fusion) 范畴或模基 (modular) 范畴分类问题；刻画某些具体构造的有限维（拟）Hopf代数的表示范畴（作为有限张量范畴）上的模范畴；挖掘Hopf双Galois理论对Hopf 2-上圈及Drinfeld扭的贡献。

This project contains some research open questions we concerned/involved on Drinfeld's quantiozation question, Kaplansky's question on classifying finite-dimensional Hopf algebras, and newly developed and advanced topics on Tensor Categories arising from several interdisciplinary research areas in mathematics and physics like representation theory of finite-dimensional Hopf algebras and 3D topological quantum field theory and 2D conformal field theory etc, which includes: (I) realization and construction questions in establishing two-(multi-)parameter quantum algebras corresponding Lie superalgebras; to study BGG theory of Lusztig small quantum groups and realize some particular modules in terms of quantum differential operators; and continue to pay more attention to the development of catigorification works of quantum groups or categorified quantum groups like those did by Lauda and Khovanov, Mazorchuk, etc. (II) to construct some new finite-dimensional Hopf algebras of given dimenions in terms of Andruskiewitsch-Schneider's generalized lifting method, to classify those Nichols algebras of non-diagonal type and consider their liftings via non-abelian groups, as well as describe Green rings; to concentrate on classifying the pointed finite-dimensional quasi-Hopf algebras or quasi-quantum groups and try to make a classification on quasi-Nichols algebras of diagonal type or their arithematic root systems via mimicking Heckenberger's combinatorical approach; (III) to focus on the classification questions on (pointed) integral fusion categories or modular categories of finite rank; to describe some module categories over some finite tensor categories like representation categories for a couple of interesting finite-dimensional (quasi-) Hopf algebras; to dig a method of constructing Hopf 2-cocycles or dually Drinfeld twists out of Hopf bi-Galois theory since these are crucial for the classification or isomorphism question of some special classes of finite-dimensional Hopf algebras of given dimensions. etc.