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

C*-代数是Hilbert空间上算子的范闭自伴随代数。它们具有显著的特性，并提供了一个自然的框架来研究不同领域之间的联系，如功能分析、代数、拓扑、几何和动力系统。也许该领域最重要的单个项目是通过k -理论数据对核C*-代数进行分类。这个项目通常被称为艾略特计划。在过去的40年里，经过数百名研究人员的深入研究，艾略特程序已经达到了一定的成熟程度，现在似乎是时候将该程序扩展到新的方向了。在一些数学分支中，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，是C*-代数量子群。我们提出将Elliott规划推广到合适的C*-代数量子群。我们已经在一些有趣的案例中实施了这一建议。这是一个独特的、原创的提议，我很兴奋。

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 Kä hler manifolds.