李代数量子化、Hopf代数及张量范畴论中的若干问题

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.

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。

本课题聚焦于我们关注的Drinfeld量子化和Kaplansky关于有限维Hopf代数分类猜想及国际热点研究领域---张量范畴论中若干研究问题，课题组发表论文33篇，其中SCI论文31篇，2篇美国数学会国际会议论文集Contemp. Math.收入发表，这包括（I）量子群的结构与表示论中若干研究问题及其应用方面（23篇），其中涉及量子化的结构与表示论的有13篇：[1] 与胡红梅合作解决G_2型量子群的Majid猜想，发表在Israel J. Math.;与胡夏合作将该猜想推广至可对称化量子Kac-Moody代数---由此构造出一些不定型新量子群结构Frontier Math. China发表。[2] 与郜云夏利猛合作，构造了广义相交矩阵李代数的量子代数，发表于Alg. Rep.Theory。[3]与黎允楠冯鸽合作在Pac.J. Math.发表纽结量子不变量工作，得到著名Jones多项式的推广。[4]黎允楠与郭锂合作解决了根树形Hopf代数的辫子化（量子化的前奏）5篇系列工作，及夏利猛与人合作表示论的10篇论文。（II）在有限维Hopf代数分类（6篇SCI论文，与熊荣川合作在阿根廷科学院数学杂志上发表第一篇分类论文；石聿兴也发表了H_8的有限生长分类工作；与郑艺伟-郜云合作在JPAA和JA上发表2篇论文16维半单代数有限生长分类结果）、有限张量范畴结构与分类（4篇SCI论文）这些国际近10多年非常活跃热门而国内近乎空白的研究领域（本项目聚焦的重点研究方面）培养出5名博士，带出了一支强有力的年轻队伍：石聿兴、熊荣川、郑英[中德双博士学位]、于志强、冯鸽（后4位是项目执行期间毕业博士），熊、于分别获得2018和2019年度博士生国家奖学金，其博士论文送国际评审均获国际同行高度优评，冯鸽获校优博论文奖。熊获天元基金访问上海数学中心、郑、于均获得基金委2项青年基金和于还获得省青年基金。

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