Noncommutative coverings of quantum tori
DOI:
https://doi.org/10.7146/math.scand.a-116147Abstract
We investigate a framework for coverings of noncommutative spaces. Furthermore, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.
References
Baum, P. F., De Commer, K., and Hajac, P. M., Free actions of compact quantum groups on unital $C^*$-algebras, Doc. Math. 22 (2017), 825–849.
Canlubo, C. R., Non-commutative coverings spaces, preprint arxiv:1612.08673 [math.QA], 2016.
Connes, A. and Rieffel, M. A., Yang-Mills for noncommutative two-tori, in “Operator algebras and mathematical physics (Iowa City, Iowa, 1985)”, Contemp. Math., vol. 62, Amer. Math. Soc., Providence, RI, 1987, pp. 237–266. https://doi.org/10.1090/conm/062/878383
Cuntz, J., Skandalis, G., and Tsygan, B., Cyclic homology in non-commutative geometry, Encyclopaedia of Mathematical Sciences, vol. 121, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-662-06444-3
Elliott, G. A., The diffeomorphism group of the irrational rotation $C^\ast $-algebra, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 5, 329–334.
Elliott, G. A. and Evans, D. E., The structure of the irrational rotation $C^*$-algebra, Ann. of Math. (2) 138 (1993), no. 3, 477–501. https://doi.org/10.2307/2946553
Elliott, G. A. and Rørdam, M., The automorphism group of the irrational rotation $C^*$-algebra, Comm. Math. Phys. 155 (1993), no. 1, 3–26.
Ellwood, D. A., A new characterisation of principal actions, J. Funct. Anal. 173 (2000), no. 1, 49–60. https://doi.org/10.1006/jfan.2000.3561
Gardella, E., Hajac, P. M., Tobolski, M., and Wu, J., The local-triviality dimension of actions of compact quantum groups, preprint arxiv:1801.00767 [math.OA], 2018.
Gracia-Bond\'ıa, J. M., Várilly, J. C., and Figueroa, H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001. https://doi.org/10.1007/978-1-4612-0005-5
Hajac, P. M., Krähmer, U., Matthes, R., and Zieliński, B., Piecewise principal comodule algebras, J. Noncommut. Geom. 5 (2011), no. 4, 591–614. https://doi.org/10.4171/JNCG/88
Ivankov, P., Quantization of noncompact coverings and its physical applications, J. Phys. Conf. Ser. 965 (2018), art. 012020, 10 pp. https://doi.org/10.1088/1742-6596/965/1/012020
Kodaka, K., Picard groups of irrational rotation $C^*$-algebras, J. London Math. Soc. (2) 56 (1997), no. 1, 179–188. https://doi.org/10.1112/S0024610797005243
Mahanta, S. and van Suijlekom, W. D., Noncommutative tori and the Riemann-Hilbert correspondence, J. Noncommut. Geom. 3 (2009), no. 2, 261–287. https://doi.org/10.4171/JNCG/37
Meyer, P.-A., Quantum probability for probabilists, 2nd ed., Lecture Notes in Mathematics, vol. 1538, Springer-Verlag, Berlin, 1995. https://doi.org/10.1007/BFb0084701
Milne, J. S., Lectures on étale cohomology (v2.10), Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 2008, availble at www.jmilne.org/math/.
von Neumann, J., Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), no. 1, 570–578. https://doi.org/10.1007/BF01457956
Peligrad, C., Compact actions commuting with ergodic actions and applications to crossed products, Trans. Amer. Math. Soc. 331 (1992), no. 2, 825–836. https://doi.org/10.2307/2154144
Petz, D., An invitation to the algebra of canonical commutation relations, Leuven Notes in Mathematical and Theoretical Physics. Series A: Mathematical Physics, vol. 2, Leuven University Press, Leuven, 1990.
Pflaum, M. J., Quantum groups on fibre bundles, Comm. Math. Phys. 166 (1994), no. 2, 279–315.
Phillips, N. C., Every simple higher dimensional noncommutative torus is an AT algebra, preprint arxiv:math/0609783 [math.OA], 2006.
Phillips, N. C., Freeness of actions of finite groups on $C^*$-algebras, in “Operator structures and dynamical systems”, Contemp. Math., vol. 503, Amer. Math. Soc., Providence, RI, 2009, pp. 217–257. https://doi.org/10.1090/conm/503/09902
Rieffel, M. A., Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), no. 4, 531–562.
Rieffel, M. A., Noncommutative tori—a case study of noncommutative differentiable manifolds, in “Geometric and topological invariants of elliptic operators (Brunswick, ME, 1988)”, Contemp. Math., vol. 105, Amer. Math. Soc., Providence, RI, 1990, pp. 191–211. https://doi.org/10.1090/conm/105/1047281
Rieffel, M. A., Proper actions of groups on $C^*$-algebras, in “Mappings of operator algebras: Proceedings of the Japan—U.S. Joint Seminar, University of Pennsylvania, 1988” (Araki, A. and Kadison, R. V., eds.), Progr. Math., vol. 84, Birkhäuser Boston, Boston, MA, 1991, pp. 141–182. https://doi.org/10.1007/978-1-4612-0453-4_6
Schwieger, K. and Wagner, S., Part I, Free actions of compact Abelian groups on $\rm C^*$-algebras, Adv. Math. 317 (2017), 224–266. https://doi.org/10.1016/j.aim.2017.06.036
Schwieger, K. and Wagner, S., Part II, Free actions of compact groups on $\rm C^*$-algebras, J. Noncommut. Geom. 11 (2017), no. 2, 641–668. https://doi.org/10.4171/JNCG/11-2-6
Schwieger, K. and Wagner, S., Part III, Free actions of compact quantum groups on $\rm C^*$-algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), paper No. 062, 19 pp. https://doi.org/10.3842/SIGMA.2017.062
Wagner, S., On noncommutative principal bundles with finite abelian structure group, J. Noncommut. Geom. 8 (2014), no. 4, 987–1022. https://doi.org/10.4171/JNCG/175
Canlubo, C. R., Non-commutative coverings spaces, preprint arxiv:1612.08673 [math.QA], 2016.
Connes, A. and Rieffel, M. A., Yang-Mills for noncommutative two-tori, in “Operator algebras and mathematical physics (Iowa City, Iowa, 1985)”, Contemp. Math., vol. 62, Amer. Math. Soc., Providence, RI, 1987, pp. 237–266. https://doi.org/10.1090/conm/062/878383
Cuntz, J., Skandalis, G., and Tsygan, B., Cyclic homology in non-commutative geometry, Encyclopaedia of Mathematical Sciences, vol. 121, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-662-06444-3
Elliott, G. A., The diffeomorphism group of the irrational rotation $C^\ast $-algebra, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 5, 329–334.
Elliott, G. A. and Evans, D. E., The structure of the irrational rotation $C^*$-algebra, Ann. of Math. (2) 138 (1993), no. 3, 477–501. https://doi.org/10.2307/2946553
Elliott, G. A. and Rørdam, M., The automorphism group of the irrational rotation $C^*$-algebra, Comm. Math. Phys. 155 (1993), no. 1, 3–26.
Ellwood, D. A., A new characterisation of principal actions, J. Funct. Anal. 173 (2000), no. 1, 49–60. https://doi.org/10.1006/jfan.2000.3561
Gardella, E., Hajac, P. M., Tobolski, M., and Wu, J., The local-triviality dimension of actions of compact quantum groups, preprint arxiv:1801.00767 [math.OA], 2018.
Gracia-Bond\'ıa, J. M., Várilly, J. C., and Figueroa, H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001. https://doi.org/10.1007/978-1-4612-0005-5
Hajac, P. M., Krähmer, U., Matthes, R., and Zieliński, B., Piecewise principal comodule algebras, J. Noncommut. Geom. 5 (2011), no. 4, 591–614. https://doi.org/10.4171/JNCG/88
Ivankov, P., Quantization of noncompact coverings and its physical applications, J. Phys. Conf. Ser. 965 (2018), art. 012020, 10 pp. https://doi.org/10.1088/1742-6596/965/1/012020
Kodaka, K., Picard groups of irrational rotation $C^*$-algebras, J. London Math. Soc. (2) 56 (1997), no. 1, 179–188. https://doi.org/10.1112/S0024610797005243
Mahanta, S. and van Suijlekom, W. D., Noncommutative tori and the Riemann-Hilbert correspondence, J. Noncommut. Geom. 3 (2009), no. 2, 261–287. https://doi.org/10.4171/JNCG/37
Meyer, P.-A., Quantum probability for probabilists, 2nd ed., Lecture Notes in Mathematics, vol. 1538, Springer-Verlag, Berlin, 1995. https://doi.org/10.1007/BFb0084701
Milne, J. S., Lectures on étale cohomology (v2.10), Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 2008, availble at www.jmilne.org/math/.
von Neumann, J., Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), no. 1, 570–578. https://doi.org/10.1007/BF01457956
Peligrad, C., Compact actions commuting with ergodic actions and applications to crossed products, Trans. Amer. Math. Soc. 331 (1992), no. 2, 825–836. https://doi.org/10.2307/2154144
Petz, D., An invitation to the algebra of canonical commutation relations, Leuven Notes in Mathematical and Theoretical Physics. Series A: Mathematical Physics, vol. 2, Leuven University Press, Leuven, 1990.
Pflaum, M. J., Quantum groups on fibre bundles, Comm. Math. Phys. 166 (1994), no. 2, 279–315.
Phillips, N. C., Every simple higher dimensional noncommutative torus is an AT algebra, preprint arxiv:math/0609783 [math.OA], 2006.
Phillips, N. C., Freeness of actions of finite groups on $C^*$-algebras, in “Operator structures and dynamical systems”, Contemp. Math., vol. 503, Amer. Math. Soc., Providence, RI, 2009, pp. 217–257. https://doi.org/10.1090/conm/503/09902
Rieffel, M. A., Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), no. 4, 531–562.
Rieffel, M. A., Noncommutative tori—a case study of noncommutative differentiable manifolds, in “Geometric and topological invariants of elliptic operators (Brunswick, ME, 1988)”, Contemp. Math., vol. 105, Amer. Math. Soc., Providence, RI, 1990, pp. 191–211. https://doi.org/10.1090/conm/105/1047281
Rieffel, M. A., Proper actions of groups on $C^*$-algebras, in “Mappings of operator algebras: Proceedings of the Japan—U.S. Joint Seminar, University of Pennsylvania, 1988” (Araki, A. and Kadison, R. V., eds.), Progr. Math., vol. 84, Birkhäuser Boston, Boston, MA, 1991, pp. 141–182. https://doi.org/10.1007/978-1-4612-0453-4_6
Schwieger, K. and Wagner, S., Part I, Free actions of compact Abelian groups on $\rm C^*$-algebras, Adv. Math. 317 (2017), 224–266. https://doi.org/10.1016/j.aim.2017.06.036
Schwieger, K. and Wagner, S., Part II, Free actions of compact groups on $\rm C^*$-algebras, J. Noncommut. Geom. 11 (2017), no. 2, 641–668. https://doi.org/10.4171/JNCG/11-2-6
Schwieger, K. and Wagner, S., Part III, Free actions of compact quantum groups on $\rm C^*$-algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), paper No. 062, 19 pp. https://doi.org/10.3842/SIGMA.2017.062
Wagner, S., On noncommutative principal bundles with finite abelian structure group, J. Noncommut. Geom. 8 (2014), no. 4, 987–1022. https://doi.org/10.4171/JNCG/175
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Published
2020-03-29
How to Cite
Schwieger, K., & Wagner, S. (2020). Noncommutative coverings of quantum tori. MATHEMATICA SCANDINAVICA, 126(1), 99–116. https://doi.org/10.7146/math.scand.a-116147
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