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

SIGMA 12 (2016), 015, 14 pages      arXiv:1509.05347      https://doi.org/10.3842/SIGMA.2016.015

Non-Associative Geometry of Quantum Tori

Francesco D'Andrea ^ab and Davide Franco ^b
a) Dipartimento di Matematica e Applicazioni, Università di Napoli ''Federico II'', Complesso MSA, Via Cintia, 80126 Napoli, Italy
^b) I.N.F.N. Sezione di Napoli, Complesso MSA, Via Cintia, 80126 Napoli, Italy

Received October 02, 2015, in final form February 04, 2016; Published online February 07, 2016

Abstract
We describe how to obtain the imprimitivity bimodules of the noncommutative torus from a ''principal bundle'' construction, where the total space is a quasi-associative deformation of a 3-dimensional Heisenberg manifold.

Key words: noncommutative torus; quasi-Hopf algebras; cochain quantization.

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References

1. Arici F., D'Andrea F., Landi G., Pimsner algebras and noncommutative circle bundles, arXiv:1506.03109.
2. Arici F., Kaad J., Landi G., Pimsner algebras and Gysin sequences from principal circle actions, J. Noncommut. Geom., to appear, arXiv:1409.5335.
3. Brzeziński T., Majid S., Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 (1993), 591-638, hep-th/9208007.
4. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
5. Connes A., $C^{\ast}$ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A599-A604, hep-th/0101093.
6. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
7. D'Andrea F., Topics in noncommutative geometry, Lecture Notes for the Autumn School ''From Poisson Geometry to Quantum Fields on Noncommutative Spaces'' (Würzburg, 2015), arXiv:1510.07271.
8. D'Andrea F., Fiore G., Franco D., Modules over the noncommutative torus and elliptic curves, Lett. Math. Phys. 104 (2014), 1425-1443, arXiv:1307.6802.
9. Dieng M., Schwarz A., Differential and complex geometry of two-dimensional noncommutative tori, Lett. Math. Phys. 61 (2002), 263-270, math.QA/0203160.
10. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1989), 1419-1457.
11. Drinfeld V.G., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with ${\rm Gal}(\overline{\bf Q}/{\bf Q})$, Leningrad Math. J. 2 (1990), 829-860.
12. Folland G.B., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.
13. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001.
14. Hajac P.M., Strong connections on quantum principal bundles, Comm. Math. Phys. 182 (1996), 579-617, hep-th/9406129.
15. Landi G., An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics Monographs, Vol. 51, Springer-Verlag, Berlin, 1997, hep-th/9701078.
16. Mac Lane S., Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, 2nd ed., Springer-Verlag, New York, 1998.
17. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
18. Plazas J., Arithmetic structures on noncommutative tori with real multiplication, Int. Math. Res. Not. (2008), Art. ID rnm147, 41 pages, math.QA/0610127.
19. Polishchuk A., Schwarz A., Categories of holomorphic vector bundles on noncommutative two-tori, Comm. Math. Phys. 236 (2003), 135-159, math.QA/0211262.
20. Rieffel M.A., $C^{\ast} $-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429.
21. Rieffel M.A., The cancellation theorem for projective modules over irrational rotation $C^{\ast} $-algebras, Proc. London Math. Soc. 47 (1983), 285-302.
22. Vlasenko M., The graded ring of quantum theta functions for noncommutative torus with real multiplication, Int. Math. Res. Not. 2006 (2006), Art. ID 15825, 19 pages, math.QA/0601405.