拟Frobenius-Lusztig核

The aim of this project is to study quasi-Frobenius-Lusztig kernels systematically, including their structures and realization ways, their representation categories and applications in the following three fields: field theory & representation theory of vertex operator algebras, the classification of quasi-Hopf algebras and gauge invariant theory. Explicitly, we will give the construction of quasi-Frobenius-Lusztig kernels; give the description of the representation category of the quasi-Frobenius-Lusatig kernel corresponding to sl(2) in detail, including all finite-dimensional indecomposable modules and the corresponding tensor product decomposition formula; give the description of some important properites of the representation categories of general quasi-Frobenius-Lusztig kernels and their Hochschild cohomology algebras; give some effective gauge invariants which can be applied to non-semisimple quasi-Hopf algebras; complete the classification of basic quasi-Hopf algebras according to their representation type; generalize the Andruskiewitsch-Schneider-classification to quasi-Hopf algebras; settle the FGST-conjecture and try to give a general Kazhdan-Lusztig corrspondence.

本项目将系统研究拟Frobenius-Lusztig核，包括她的结构及实现方式、表示范畴以及她在场论及顶点算子代数的表示、拟Hopf代数的分类及gauge不变量理论三方面的应用。具体有：给出拟Frobenius-Lusztig核的完整构造；完成与sl(2)对应的拟Frobenius-Lusztig核的表示范畴的详细刻画，包括给出所有的有限维不可分解模以及对应的张量积分解公式；给出一般拟Frobenius-Lusztig核的表示范畴的某些重点性质以及Hochschild上同调代数的刻画；给出能适用于非半单拟Hopf代数的有效的gauge不变量；完成基本（basic）拟Hopf代数的表示型分类；推广Andruskiewitsch-Schneider-分类至拟Hopf代数；解决FGST-猜想并尽可能建立一般的Kazhdan-Lusztig对应。

本项目旨在研究（拟）Hopf代数的分类及相关表示, 是Hopf代数研究的中心任务之一。通过本项目，我们取得的成果有：完成了拟Frobenius-Lusztig核的构造；完成了GK-维数1的Hopf代数的完整分类；为拟Hopf代数的分类进行了大量的前期准备等。共发表文章12篇，其中SCI论文7篇，发表在含《Adv. Math.》、《Trans. Amer. Math. Soc.》等知名数学杂志上。