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

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Abstract

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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Authors and Affiliations

  1. Vakgroep Wiskunde, Vrije Universiteit Brussel, Brussel, Belgium

    Kenny De Commer

  2. Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland

    Paweł Kasprzak & Piotr M. Sołtan

  3. Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland

    Adam Skalski

Authors
  1. Kenny De Commer
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  2. Paweł Kasprzak
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  3. Adam Skalski
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  4. Piotr M. Sołtan
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Correspondence to Kenny De Commer.

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De Commer, K., Kasprzak, P., Skalski, A. et al. Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations. Isr. J. Math. 226, 475–503 (2018). https://doi.org/10.1007/s11856-018-1709-x

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  • DOI: https://doi.org/10.1007/s11856-018-1709-x

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