Expanding the boundaries of the Elliott classification program: Quantum groups and Quaternions

扩展艾略特分类程序的边界：量子群和四元数

• 批准号: RGPIN-2016-05768
• 负责人: Kucerovsky, Dan
• 金额: $ 1.31万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676874
• 关键词: Expanding; boundaries; Elliott; classification; program

C*-algebras are norm-closed self-adjoint algebras of operators on Hilbert space. They have remarkable properties, and provide a natural framework to study connections between disparate areas such as functional analysis, algebra, topology, geometry, and dynamical systems. Perhaps the most important single project in the field is to classify nuclear C*-algebras by K-theoretical data. This project is often called the Elliott program. As a result of intensive study by hundreds of researchers over the last 40 years, the Elliott program has achieved a certain maturity, and the time seems right to investigate branching out the program in new directions. In some branches of mathematics, C*-algebras appear naturally as noncommutative generalizations of topological spaces. If we look for generalizations of topological groups rather than just topological spaces, we can obtain Hopf C*-algebras, more specifically, C*-algebraic quantum groups. We propose to generalize the Elliott program to suitable classes of C*-algebraic quantum groups. We have already carried out this proposal in some interesting cases. This is an unique and original proposal that I am excited about.***Applying the powerful tools of the Elliott classification program for C*-algebras to Hopf C*-algebras is very innovative. This is because the existing partial classification results for Hopf algebras, for example the Andruskiewitsch-Schneider classification of pointed finite-dimensional Hopf algebras, are inspired by Cartan's classification of Lie groups. The Elliott classification program, on the other hand, has different origins, and classifies in a sense that is entirely different than in Cartan's approach to classification. It is likely that our results will be one of the early contributions to a new branch of the Elliott program, and there is a very strong potential that these ideas will be adopted by more than one international community of researchers.***The Elliott classification program for real C*-algebras appears to be quite challenging. Quaternionic operator algebras are a subclass of the real C*-algebras. If the Elliott classification program for real C*-algebras is restricted to quaternionic operator algebras, the class of objects to be classified gets smaller, and it is plausible that classification becomes easier. Thus, we further propose a preliminary study of an Elliott program for quaternionic operator algebras. Beyond possibly founding a new branch of the Elliott program, the results may have applications to index theory over quaternionic Kähler manifolds.**

C* - 代数是希尔伯特空间上运营商的规范封闭的自动辅助代数。它们具有显着的特性，并提供了一个自然框架来研究不同区域之间的联系，例如功能分析，代数，拓扑，几何和动态系统。该领域中最重要的单一项目也许是通过K理论数据对核C*代数进行分类。该项目通常称为Elliott计划。在过去40年中，数百名研究人员进行了深入研究的结果，埃利奥特计划已经达到了一定的成熟度，并且是时候调查朝新方向进行分支计划的时候了。在数学的某些分支中，C* - 代数自然是拓扑空间的非交通概括。如果我们寻找拓扑组的概括，而不仅仅是拓扑空间，则可以获得更具体地说，更具体地说，是c* - 代数量子群。我们建议将Elliott计划推广到合适的C* - 代数量子组。在一些有趣的情况下，我们已经提出了这一建议。这是我很兴奋的独特而原始的建议。这是因为HOPF代数的现有部分分类结果，例如，尖锐的有限维霍普夫代数的Andruskiewitsch-Schneider分类受到cartan lie群体的分类的启发。另一方面，Elliott分类程序具有不同的起源，并且在某种意义上与Cartan的分类方法完全不同。我们的结果很可能是对埃利奥特计划的新分支的早期贡献之一，并且有很强的潜力，即多个国际研究人员社区将采用这些想法。 Quaternionic操作员代数是实际C* - 代数的子类。如果对真正的C*-Algebras的Elliott分类程序仅限于Quaternionic Operator代数，则要分类的对象类会变小，并且合理的分类变得更加容易。这，我们进一步提出了针对Quaternionic操作员代数的Elliott计划的初步研究。除了找到Elliott计划的新分支外，结果可能还针对QuaternionicKähler歧管索引理论的应用。**