Noncommutative Geometry and Topology in Prague

Classical spaces and quantum symmetry

One direction of research in the theory of quantum groups is to use them as generalized symmetries of spaces. For non-commutative spaces, this has lead to the discovery of very important examples but for classical spaces the picture is less clear. While finite spaces are known to have quantum symmetries embodied in Wang’s quantum permutation group, many classical spaces have been proven to have no quantum symmetry. In this talk I will survey some results on that topic and present recent progress obtained in collaboration with F. Taipe and S. Wang. We prove that the quantum permutation group cannot act ergodically on a connected classical compact space, answering a question of Goswami as modified by Huang. The proof uses tools which are very different from the usual ones in quantum rigidity theory and rely on a categorical description of ergodic actions of compact quantum groups.