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.
Similar content being viewed by others
References
Banica T.: Quantum automorphism groups of small metric spaces. Pac. J. Math. 219(1), 27–51 (2005)
Banica T.: Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224(2), 243–280 (2005)
Bhowmick J., Goswami D.: Quantum isometry groups: examples and computations. Commun. Math. Phys. 285(2), 421–444 (2009) arXiv0707.2648
Bhowmick J., Goswami D.: Quantum group of orientation preserving Riemannian Isometries. J. Funct. Anal. 257, 2530–2572 (2009)
Bichon J.: Quantum automorphism groups of finite graphs. Proc. Am. Math. Soc. 131(3), 665–673 (2003)
Chakraborty P.S., Pal A.: Spectral triples and associated Connes-de Rham complex for the quantum SU(2) and the quantum sphere. Commun. Math. Phys. 240(3), 447–456 (2003)
Connes A.: Noncommutative Geometry. Academic Press, London (1994)
Connes A., Dubois-Violette M.: Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Commun. Math. Phys. 230(3), 539–579 (2002)
Dabrowski, L.: Spinors and theta deformations. math.QA/0808.0440
Davidson, K.R.: C* Algebras by Examples. Hindustan Book Agency, New Delhi
Goswami D.: Quantum Group of isometries in classical and non commutative geometry. Commun. Math. Phys. 285(1), 141–160 (2009) arXiv 0704.0041
Goswami D.: Twisted entire cyclic cohomology, JLO cocycles and equivariant spectral triples. Rev. Math. Phys. 16(5), 583–602 (2004)
Podles P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987)
Sheu A.J.-L.: Quantization of the Poisson SU(2) and its Poisson homogeneous space—the 2-sphere. Commun. Math. Phys. 135, 217–232 (1991)
Wang S.: Free products of compact quantum groups. Commun. Math. Phys. 167(3), 671–692 (1995)
Wang S.: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195, 195–211 (1998)
Wang S.: Structure and isomorphism classification of compact quantum groups A u (Q) and B u (Q). J. Oper. Theory 48, 573–583 (2002)
Wang S.: Ergodic actions of universal quantum groups on operator algebras. Commun. Math. Phys. 203(2), 481–498 (1999)
Woronowicz S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111(4), 613–665 (1987)
Woronowicz, S.L.: Compact quantum groups. In: Connes, A., et al. (eds.) Symétries quantiques (Quantum symmetries) (Les Houches, 1995), pp. 845–884. Elsevier, Amsterdam (1998)
Woronowicz, S.L.: Pseudogroups, pseudospaces and Pontryagin duality. In: Proceedings of the International Conference on Mathematical Physics. Lecture Notes in Physics, Lausane, vol. 116, pp. 407–412 (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-010-0409-1