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

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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.