Noncommutative coverings of quantum tori

  • Kay Schwieger
  • Stefan Wagner


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.


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.

Cuntz, J., Skandalis, G., and Tsygan, B., Cyclic homology in non-commutative geometry, Encyclopaedia of Mathematical Sciences, vol. 121, Springer-Verlag, Berlin, 2004.

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.

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.

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.

Hajac, P. M., Krähmer, U., Matthes, R., and Zieliński, B., Piecewise principal comodule algebras, J. Noncommut. Geom. 5 (2011), no. 4, 591–614.

Ivankov, P., Quantization of noncompact coverings and its physical applications, J. Phys. Conf. Ser. 965 (2018), art. 012020, 10 pp.

Kodaka, K., Picard groups of irrational rotation $C^*$-algebras, J. London Math. Soc. (2) 56 (1997), no. 1, 179–188.

Mahanta, S. and van Suijlekom, W. D., Noncommutative tori and the Riemann-Hilbert correspondence, J. Noncommut. Geom. 3 (2009), no. 2, 261–287.

Meyer, P.-A., Quantum probability for probabilists, 2nd ed., Lecture Notes in Mathematics, vol. 1538, Springer-Verlag, Berlin, 1995.

Milne, J. S., Lectures on étale cohomology (v2.10), Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 2008, availble at

von Neumann, J., Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), no. 1, 570–578.

Peligrad, C., Compact actions commuting with ergodic actions and applications to crossed products, Trans. Amer. Math. Soc. 331 (1992), no. 2, 825–836.

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.

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.

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.

Schwieger, K. and Wagner, S., Part I, Free actions of compact Abelian groups on $\rm C^*$-algebras, Adv. Math. 317 (2017), 224–266.

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.

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.

Wagner, S., On noncommutative principal bundles with finite abelian structure group, J. Noncommut. Geom. 8 (2014), no. 4, 987–1022.
How to Cite
Schwieger, K., & Wagner, S. (2020). Noncommutative coverings of quantum tori. MATHEMATICA SCANDINAVICA, 126(1), 99-116.