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伴代数。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。也许这个领域最重要的一个项目是用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。在过去的40年里，经过数百名研究人员的深入研究，埃利奥特计划已经达到了一定的成熟度，现在似乎是时候研究将该计划扩展到新的方向了。在某些数学分支中，C*-代数自然地表现为拓扑空间的非交换推广。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，C*-代数量子群。我们建议推广的艾略特计划，以适当的类的C*-代数量子群。我们已经在一些有趣的案例中实施了这一建议。这是一个独特的和原始的建议，我感到兴奋。 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。这是因为现有的部分分类结果的Hopf代数，例如Andruskiewitsch-Schneider分类的点有限维Hopf代数，是受嘉当分类的李群。另一方面，埃利奥特分类程序有不同的起源，在某种意义上分类是完全不同于嘉当的分类方法。我们的研究结果很可能成为埃利奥特项目新分支的早期贡献之一，而且这些想法很有可能被不止一个国际研究团体所采用。 真实的C*-代数的Elliott分类程序似乎是相当具有挑战性的。四元数算子代数是真实的C*-代数的一个子类。如果将真实的C*-代数的Elliott分类程序限制为四元数算子代数，则待分类的对象的类变得更小，并且分类变得更容易似乎是合理的。因此，我们进一步提出了一个四元数算子代数的Elliott程序的初步研究。除了可能建立一个新的分支的埃利奥特计划，结果可能有应用指数理论的四元数凯勒流形。

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.