Abstract
We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.
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Debashish Goswamir was partially supported by Swarnajayanti Grant and J.C. Bose fellowship given by D.S.T., Government of India. Sutanu Roy was partially supported by the visiting scientist program at the Indian Statistical Institute, Kolkata and Inspire faculty award given by D.S.T., Government of India.
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Goswami, D., Roy, S. Faithful actions of locally compact quantum groups on classical spaces. Lett Math Phys 107, 1375–1390 (2017). https://doi.org/10.1007/s11005-017-0951-1
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DOI: https://doi.org/10.1007/s11005-017-0951-1