Israel Journal of Mathematics

, Volume 226, Issue 1, pp 475–503 | Cite as

Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

  • Kenny De CommerEmail author
  • Paweł Kasprzak
  • Adam Skalski
  • Piotr M. Sołtan


To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors are finite-dimensional (i.e., when the action is on a discrete quantum space) the relation has finite orbits. We then apply this to generalize the classical theory of Clifford, concerning the restrictions of representations to normal subgroups, to the framework of quantum subgroups of discrete quantum groups, itself extending the context of closed normal quantum subgroups of compact quantum groups. Finally, a link is made between our equivalence relation in question and another equivalence relation defined by R. Vergnioux.


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Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Kenny De Commer
    • 1
    Email author
  • Paweł Kasprzak
    • 2
  • Adam Skalski
    • 3
  • Piotr M. Sołtan
    • 2
  1. 1.Vakgroep WiskundeVrije Universiteit BrusselBrusselBelgium
  2. 2.Department of Mathematical Methods in Physics, Faculty of PhysicsUniversity of WarsawWarsawPoland
  3. 3.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland

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