Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations
- 24 Downloads
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.
Unable to display preview. Download preview PDF.
- [DC17]K. De Commer, Actions of compact quantum groups, in Toplogical Quantum Groups, Banach Center Publications, Vol. 111, Polish Academy of Sciences Institute of Mathematics, Warsaw, 2017, pp. 33–100.Google Scholar
- [KKSS17]M. Kalantar, P. Kasprzak, A. Skalski and P. Sołtan, Induction for locally compact quantum groups revisited, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, to appear.Google Scholar
- [KS14]P. Kasprzak and P. Sołtan, Quantum groups with projection and extensions of locally compact quantum groups, Journal of Noncommutative Geometry.Google Scholar
- [Ped79]G. K. Pedersen, C*-algebras and their Automorphism Groups, London Mathematical Society Monographs, Vol. 14, Academic Press, Harcourt Brace Jovanovich, London–New York, 1979.Google Scholar
- [Wor98]S. L. Woronowicz, Compact quantum groups, in Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 845–884.Google Scholar