Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 076, 18 pages      arXiv:1307.4850      https://doi.org/10.3842/SIGMA.2014.076
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Quantum Isometry Groups of Noncommutative Manifolds Obtained by Deformation Using Dual Unitary 2-Cocycles

Debashish Goswami and Soumalya Joardar
Indian Statistical Institute, 203, B.T. Road, Kolkata 700108, India

Received January 29, 2014, in final form July 11, 2014; Published online July 17, 2014

Abstract
It is proved that the (volume and orientation-preserving) quantum isometry group of a spectral triple obtained by deformation by some dual unitary 2-cocycle is isomorphic with a similar twist-deformation of the quantum isometry group of the original (undeformed) spectral triple. This result generalizes similar work by Bhowmick and Goswami for Rieffel-deformed spectral triples in [Comm. Math. Phys. 285 (2009), 421-444].

Key words: cocycle twist; quantum isometry group; Rieffel deformation; spectral triple.

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References

  1. Banica T., Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005), 243-280, math.QA/0311402.
  2. Banica T., Goswami D., Quantum isometries and noncommutative spheres, Comm. Math. Phys. 298 (2010), 343-356, arXiv:0905.3814.
  3. Bhowmick J., Goswami D., Quantum group of orientation-preserving Riemannian isometries, J. Funct. Anal. 257 (2009), 2530-2572, arXiv:0806.3687.
  4. Bhowmick J., Goswami D., Quantum isometry groups: examples and computations, Comm. Math. Phys. 285 (2009), 421-444, arXiv:0707.2648.
  5. Bhowmick J., Goswami D., Skalski A., Quantum isometry groups of 0-dimensional manifolds, Trans. Amer. Math. Soc. 363 (2011), 901-921, arXiv:0807.4288.
  6. Bichon J., Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (2003), 665-673, math.QA/9902029.
  7. Bichon J., De Rijdt A., Vaes S., Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (2006), 703-728, math.OA/0502018.
  8. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  9. Das B., Goswami D., Joardar S., Rigidity of action of compact quantum groups on compact, connected manifolds, arXiv:1309.1294.
  10. Donin J., Shnider S., Deformation of certain quadratic algebras and the corresponding quantum semigroups, Israel J. Math. 104 (1998), 285-300.
  11. Goswami D., Twisted entire cyclic cohomology, J-L-O cocycles and equivariant spectral triples, Rev. Math. Phys. 16 (2004), 583-602, math-ph/0204010.
  12. Goswami D., Quantum group of isometries in classical and noncommutative geometry, Comm. Math. Phys. 285 (2009), 141-160, arXiv:0704.0041.
  13. Kustermans J., Locally compact quantum groups in the universal setting, Internat. J. Math. 12 (2001), 289-338, math.OA/9902015.
  14. Kustermans J., Vaes S., Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), 68-92, math.OA/0005219.
  15. Lance E.C., Hilbert $C^*$-modules. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, Vol. 210, Cambridge University Press, Cambridge, 1995.
  16. Maes A., Van Daele A., Notes on compact quantum groups, Nieuw Arch. Wisk. (4) 16 (1998), 73-112, math.FA/9803122.
  17. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  18. Neshveyev S., Tuset L., The Dirac operator on compact quantum groups, J. Reine Angew. Math. 641 (2010), 1-20, math.OA/0703161.
  19. Neshveyev S., Tuset L., Deformation of $C^\ast$-algebras by cocycles on locally compact quantum groups, Adv. Math. 254 (2014), 454-496, arXiv:1301.4897.
  20. Podleś P., Symmetries of quantum spaces. Subgroups and quotient spaces of quantum ${\rm SU}(2)$ and ${\rm SO}(3)$ groups, Comm. Math. Phys. 170 (1995), 1-20, hep-th/9402069.
  21. Rieffel M.A., van Daele A., A bounded operator approach to Tomita-Takesaki theory, Pacific J. Math. 69 (1977), 187-221.
  22. Sołtan P.M., On actions of compact quantum groups, Illinois J. Math. 55 (2011), 953-962, arXiv:1003.5526.
  23. Takesaki M., Theory of operator algebras. I, Springer-Verlag, New York - Heidelberg, 1979.
  24. Vaes S., The unitary implementation of a locally compact quantum group action, J. Funct. Anal. 180 (2001), 426-480, math/0005262.
  25. Van Daele A., Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994), 917-932.
  26. Wang S., Deformations of compact quantum groups via Rieffel's quantization, Comm. Math. Phys. 178 (1996), 747-764.
  27. Wang S., Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195-211, math.OA/9807091.
  28. Woronowicz S.L., Compact quantum groups, in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 845-884.

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