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。

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.