Affine Nichols algebras of diagonal type and modular tensor categories

根系统和晶体学考克斯特群是研究半单李代数的中心工具。在点Hopf代数的结构理论中，Weyl群和它们的根系统也有类似的作用。有限泛Weyl群对应于申请人引入的晶体排列。这些是超平面的排列，它们满足一定的整体完整性公理。在一系列的论文中，Heckenberger和申请人完成了晶体排列的完整分类，直到同构。我们提出将关于对角型和Weyl群的Nichols代数的结果推广到仿射情况。我们期望得到与经典理论相似的结果。特别地，还应该存在一个奇异的傅立叶变换矩阵，将晶体排列的组合与某些模张量范畴联系起来。我们的第二个目标是为每个有限Weyl群类群W找到一个具有对称结构的相关Hopf代数H。我们建议通过研究相应晶体排列的环向变化上的（偏）微分算子束来解决这个问题。

Root systems and crystallographic Coxeter groups are central tools in the study of semisimple Lie algebras. In the structure theory of pointed Hopf algebras a similar role is expected to be played by Weyl groupoids and their root systems. Finite universal Weyl groupoids correspond to the crystallographic arrangements introduced by the applicant. These are arrangements of hyperplanes which satisfy a certain global axiom of integrality. In a series of papers, Heckenberger and the applicant achieved a complete classification of crystallographic arrangements up to isomorphisms. We propose to extend the results on Nichols algebras of diagonal type and Weyl groupoids to the affine case. We expect similar results as in the classical theory. In particular, there should also exist an exotic Fourier transform matrix connecting the combinatorics of crystallographic arrangements to certain modular tensor categories. Our second objective is to find for each finite Weyl groupoid W an associated Hopf algebra H having Was symmetry structure. We propose to approach this question by the study of the sheaf of (skew) differential operators on the toric variety of the corresponding crystallographic arrangement.