Some Counterexamples in the Theory of Quantum Isometry Groups

Abstract

By considering spectral triples on \({S^{2}_{\mu, c}\,\, (c >0 )}\) constructed by Chakraborty and Pal (Commun Math Phys 240(3):447–456, 2000), we show that in general the quantum group of volume and orientation preserving isometries (in the sense of Bhowmick and Goswami in J Funct Anal 257:2530–2572, 2009) for a spectral triple of compact type may not have a C*-action, and moreover, it can fail to be a matrix quantum group. It is also proved that the category with objects consisting of those volume and orientation preserving quantum isometries which induce C*-action on the C* algebra underlying the given spectral triple, may not have a universal object.

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Correspondence to Jyotishman Bhowmick.

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J. Bhowmick gratefully acknowledges the support from National Board of Higher Mathematics, India.

D. Goswami was partially supported by a project on ‘Noncommutative Geometry and Quantum Groups’ funded by Indian National Science Academy.

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Bhowmick, J., Goswami, D. Some Counterexamples in the Theory of Quantum Isometry Groups. Lett Math Phys 93, 279–293 (2010). https://doi.org/10.1007/s11005-010-0409-1

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Mathematics Subject Classification (2010)

  • Primary 58B34
  • Secondary 16T05
  • 46L89

Keywords

  • spectral triples
  • compact quantum groups
  • quantum isometry groups
  • quantum spheres