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Quantum Isometry Group for Spectral Triples with Real Structu
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

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Given a spectral triple of compact type with a real structure in the sense of [Dąbrowski L., J. Geom. Phys., 56 (2006), 86–107] (which is a modification of Connes' original definition to accommodate examples coming from quantum group theory) and references therein, we prove that there is always a universal object in the category of compact quantum group acting by orientation preserving isometries (in the sense of [Bhowmick J., Goswami D., J. Funct. Anal., 257 (2009), 2530–2572]) and also preserving the real structure of the spectral triple. This gives a natural definition of quantum isometry group in the context of real spectral triples without fixing a choice of “volume form” as in [Bhowmick J., Goswami D., J. Funct. Anal., 257 (2009), 2530–2572].
Keywords: quantum isometry groups, spectral triples, real structures. 
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     author = {Debashish Goswami},
     title = {Quantum {Isometry} {Group} for {Spectral} {Triples} with {Real} {Structu}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2010},
     volume = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a6/}
}
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Debashish Goswami. Quantum Isometry Group for Spectral Triples with Real Structu. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a6/

[1] Banica T., “Quantum automorphism groups of small metric spaces”, Pacific J. Math., 219 (2005), 27–51 ; math.QA/0304025 | DOI | MR | Zbl

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

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

[4] Bhowmick J., “Quantum isometry group of the $n$-tori”, Proc. Amer. Math. Soc., 137 (2009), 3155–3161 ; arXiv:0803.4434 | DOI | MR | Zbl

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

[6] Bhowmick J., Goswami D., Skalski A., “Quantum isometry groups of 0-dimensional manifolds”, Trans. Amer. Math. Soc. (to appear)

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

[8] Bhowmick J., Goswami D., Quantum isometry groups of the Podles sphere, arXiv:0810.0658

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

[10] Da̧browski L., “Geometry of quantum spheres”, J. Geom. Phys., 56 (2006), 86–107 ; math.QA/0501240 | DOI | MR

[11] Da̧browski L., Landi G., Paschke M., Sitarz A., “The spectral geometry of the equatorial Podleś sphere”, C. R. Math. Acad. Sci. Paris, 340 (2005), 819–822 ; math.QA/0408034 | MR

[12] Da̧browski L., D'Andrea F., Landi G., Wagner E., “Dirac operators on all Podleś quantum spheres”, J. Noncommut. Geom., 1 (2007), 213–239 ; math.QA/0606480 | MR | Zbl

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

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

[15] Varilly J. C., An introduction to noncommutative geometry, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2006 | MR | Zbl

[16] Wang S., “Free products of compact quantum groups”, Comm. Math. Phys., 167 (1995), 671–692 | DOI | MR | Zbl

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

[18] Wang S., “Structure and isomorphism classification of compact quantum groups $A_u(Q)$ and $B_u(Q)$”, J. Operator Theory, 48 (2002), 573–583 ; math.OA/9807095 | MR | Zbl

[19] Wang S., “Ergodic actions of universal quantum groups on operator algebras”, Comm. Math. Phys., 203 (1999), 481–498 ; math.OA/9807093 | DOI | MR | Zbl

[20] Woronowicz S. L., “Compact matrix pseudogroups”, Comm. Math. Phys., 111 (1987), 613–665 | DOI | MR | Zbl

[21] Woronowicz S. L., “Compact quantum groups”, Symétries Quantiques (Les Houches, 1995), eds. A. Connes et al., North-Holland, Amsterdam, 1998, 845–884 | MR | Zbl