W*-superrigidity for discrete quantum groups

Abstract: A (countable) group is called W*-superrigid if it is completely remembered by its group von Neumann algebra in the following sense: if another group gives rise to an isomorphic group von Neumann algebra, the groups must be isomorphic. In the past fifteen years, several classes of W*-superrigid groups have been found. However, it turns out that many of these groups are not W*-superrigid in the larger class of compact quantum groups: their group von Neumann algebras admit different quantum group structures. In a recent work with Stefaan Vaes, we found the first examples of compact quantum groups which are 'quantum W*-superrigid'. To obtain quantum W*-superrigidity, we had to combine three different types of results: vanishing of cohomology, rigidity of (quantum) groups relative to a family of (quantum) group automorphisms, and deformation/rigidity theory. I will explain why each of these three parts is essential and how they come together to prove our main result.

operator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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