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

C*-代数是Hilbert空间上算子的范数闭自共轭代数。它们具有显著的性质，并提供了一个自然的框架来研究不同领域之间的联系，如泛函分析、代数、拓扑、几何和动力系统。也许该领域最重要的一个项目是根据K-理论数据对核C*-代数进行分类。这个项目通常被称为埃利奥特计划。经过过去40年数百名研究人员的深入研究，埃利奥特项目已经达到一定的成熟度，现在似乎是研究向新方向扩展该项目的时候了。在数学的某些分支中，C*-代数自然而然地成为拓扑空间的非对易推广。如果我们寻找拓扑群的推广，而不仅仅是拓扑空间，我们可以得到Hopf C*-代数，更具体地说，我们可以得到C*-代数量子群。我们建议将Elliott程序推广到合适的C*-代数量子群。我们已经在一些有趣的案例中执行了这项建议。*将C*-代数的Elliott分类程序的强大工具应用于Hopf C*-代数是非常创新的。这是因为现有的Hopf代数的部分分类结果，例如有限维点Hopf代数的Andruskiewitsch-Schneider分类，都是受Cartan李群分类的启发。另一方面，埃利奥特分类程序有不同的起源，分类的意义与卡尔坦的分类方法完全不同。我们的结果很可能是对Elliott程序的一个新分支的早期贡献之一，并且这些想法很有可能被不止一个国际研究人员社区采用。*实C*-代数的Elliott分类程序似乎相当具有挑战性。四元数算子代数是实C*-代数的一个子类。如果实C*-代数的Elliott分类程序仅限于四元数算子代数，则待分类的对象类变得更小，分类可能变得更容易。因此，我们进一步提出了一个关于四元数算子代数的Elliott程序的初步研究。除了可能建立Elliott程序的一个新分支之外，这些结果还可能应用于四元数Kähler流形上的指数理论。

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.**