Affine Nichols algebras of diagonal type and modular tensor categories

对角型和模张量范畴的仿射尼科尔斯代数

• 批准号: 219514727
• 负责人: Professor Dr. Michael Cuntz
• 金额: --
• 依托单位: Institut für Algebra, Zahlentheorie und Diskrete Mathematik (IAZD)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/219514727?language=en
• 关键词: Affine; Nichols; algebras; diagonal; type

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.

根系和结晶学Coxeter群是半单李代数研究的中心工具。在点Hopf代数的结构理论中，Weyl群胚及其根系统也应该扮演类似的角色。有限泛Weyl群团对应于申请人介绍的结晶学排列。这些超平面的排列满足一定的整体整体性公理。在一系列论文中，Heckenberger和申请人实现了对晶体排列的完整分类，直至同构。我们建议将关于对角型Nichols代数和Weyl群胚的结果推广到仿射情形。我们期望得到与经典理论类似的结果。特别是，还应该存在一个奇异的傅里叶变换矩阵，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对每个有限的Weyl群胚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环面上的(斜)微分算子束来解决这个问题。