Geodesic
Quantum Isometry Groups of Noncommutative Manifolds Obtained by Deformation Using Dual Unitary 2-Cocycles
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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].
Keywords: cocycle twist; quantum isometry group; Rieffel deformation; spectral triple. 
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     title = {Quantum {Isometry} {Groups} of {Noncommutative} {Manifolds} {Obtained} {by~Deformation} {Using} {Dual} {Unitary} {2-Cocycles}},
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Debashish Goswami; Soumalya Joardar. Quantum Isometry Groups of Noncommutative Manifolds Obtained by Deformation Using Dual Unitary 2-Cocycles. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a75/

[1] Banica T., “Quantum automorphism groups of homogeneous graphs”, J. Funct. Anal., 224 (2005), 243–280, arXiv: math.QA/0311402 | DOI | MR | Zbl

[2] Banica T., Goswami D., “Quantum isometries and noncommutative spheres”, Comm. Math. Phys., 298 (2010), 343–356, arXiv: 0905.3814 | DOI | MR | Zbl

[3] Bhowmick J., Goswami D., “Quantum group of orientation-preserving {R}iemannian isometries”, J. Funct. Anal., 257 (2009), 2530–2572, arXiv: 0806.3687 | DOI | MR | Zbl

[4] Bhowmick J., Goswami D., “Quantum isometry groups: examples and computations”, Comm. Math. Phys., 285 (2009), 421–444, arXiv: 0707.2648 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[6] Bichon J., “Quantum automorphism groups of finite graphs”, Proc. Amer. Math. Soc., 131 (2003), 665–673, arXiv: math.QA/9902029 | DOI | MR | Zbl

[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, arXiv: math.OA/0502018 | DOI | MR | Zbl

[8] Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994 | MR | Zbl

[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 | DOI | MR | Zbl

[11] Goswami D., “Twisted entire cyclic cohomology, {J}-{L}-{O} cocycles and equivariant spectral triples”, Rev. Math. Phys., 16 (2004), 583–602, arXiv: math-ph/0204010 | DOI | MR | Zbl

[12] Goswami D., “Quantum group of isometries in classical and noncommutative geometry”, Comm. Math. Phys., 285 (2009), 141–160, arXiv: 0704.0041 | DOI | MR | Zbl

[13] Kustermans J., “Locally compact quantum groups in the universal setting”, Internat. J. Math., 12 (2001), 289–338, arXiv: math.OA/9902015 | DOI | MR | Zbl

[14] Kustermans J., Vaes S., “Locally compact quantum groups in the von {N}eumann algebraic setting”, Math. Scand., 92 (2003), 68–92, arXiv: math.OA/0005219 | MR | Zbl

[15] Lance E. C., Hilbert {$C^*$}-modules. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, 210, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[16] Maes A., Van Daele A., “Notes on compact quantum groups”, Nieuw Arch. Wisk. (4), 16 (1998), 73–112, arXiv: math.FA/9803122 | MR | Zbl

[17] Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[18] Neshveyev S., Tuset L., “The {D}irac operator on compact quantum groups”, J. Reine Angew. Math., 641 (2010), 1–20, arXiv: math.OA/0703161 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[20] Podleś P., “Symmetries of quantum spaces. {S}ubgroups and quotient spaces of quantum {${\rm SU}(2)$} and {${\rm SO}(3)$} groups”, Comm. Math. Phys., 170 (1995), 1–20, arXiv: hep-th/9402069 | DOI | MR | Zbl

[21] Rieffel M. A., van Daele A., “A bounded operator approach to {T}omita–{T}akesaki theory”, Pacific J. Math., 69 (1977), 187–221 | DOI | MR

[22] Sołtan P. M., “On actions of compact quantum groups”, Illinois J. Math., 55 (2011), 953–962, arXiv: 1003.5526 | MR

[23] Takesaki M., Theory of operator algebras, v. I, Springer-Verlag, New York–Heidelberg, 1979 | MR

[24] Vaes S., “The unitary implementation of a locally compact quantum group action”, J. Funct. Anal., 180 (2001), 426–480, arXiv: math/0005262 | DOI | MR | Zbl

[25] Van Daele A., “Multiplier {H}opf algebras”, Trans. Amer. Math. Soc., 342 (1994), 917–932 | DOI | MR | Zbl

[26] Wang S., “Deformations of compact quantum groups via {R}ieffel's quantization”, Comm. Math. Phys., 178 (1996), 747–764 | DOI | MR | Zbl

[27] Wang S., “Quantum symmetry groups of finite spaces”, Comm. Math. Phys., 195 (1998), 195–211, arXiv: math.OA/9807091 | DOI | MR | Zbl

[28] Woronowicz S. L., “Compact quantum groups”, Symétries Quantiques ({L}es {H}ouches, 1995), North-Holland, Amsterdam, 1998, 845–884 | MR | Zbl