Non-Associative Geometry of Quantum Tori
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Mots-clés : noncommutative torus; quasi-Hopf algebras; cochain quantization.
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Francesco D'Andrea; Davide Franco. Non-Associative Geometry of Quantum Tori. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a14/

[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, arXiv: hep-th/9208007 | DOI | MR | Zbl

[4] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[5] Connes A., “$C^{\ast} $ algèbres et géométrie différentielle”, C. R. Acad. Sci. Paris Sér. A-B, 290 (1980), A599–A604, arXiv: hep-th/0101093 | MR

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

[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 | DOI | MR | Zbl

[9] Dieng M., Schwarz A., “Differential and complex geometry of two-dimensional noncommutative tori”, Lett. Math. Phys., 61 (2002), 263–270, arXiv: math.QA/0203160 | DOI | MR | Zbl

[10] Drinfeld V. G., “Quasi-Hopf algebras”, Leningrad Math. J., 1 (1989), 1419–1457 | MR

[11] Drinfeld V. G., “On quasitriangular quasi-{H}opf algebras and on a group that is closely connected with ${\rm Gal}(\overlineQ/Q)$”, Leningrad Math. J., 2 (1990), 829–860 | MR

[12] Folland G. B., Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, Princeton, NJ, 1989 | MR | Zbl

[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 | DOI | MR | Zbl

[14] Hajac P. M., “Strong connections on quantum principal bundles”, Comm. Math. Phys., 182 (1996), 579–617, arXiv: hep-th/9406129 | DOI | MR | Zbl

[15] Landi G., An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics Monographs, 51, Springer-Verlag, Berlin, 1997, arXiv: hep-th/9701078 | DOI | MR

[16] Mac Lane S., Categories for the working mathematician, Graduate Texts in Mathematics, 5, 2nd ed., Springer-Verlag, New York, 1998 | MR | Zbl

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

[18] Plazas J., “Arithmetic structures on noncommutative tori with real multiplication”, Int. Math. Res. Not., 2008, rnm147, 41 pp., arXiv: math.QA/0610127 | DOI | MR | Zbl

[19] Polishchuk A., Schwarz A., “Categories of holomorphic vector bundles on noncommutative two-tori”, Comm. Math. Phys., 236 (2003), 135–159, arXiv: math.QA/0211262 | DOI | MR | Zbl

[20] Rieffel M. A., “$C^{\ast} $-algebras associated with irrational rotations”, Pacific J. Math., 93 (1981), 415–429 | DOI | MR | Zbl

[21] Rieffel M. A., “The cancellation theorem for projective modules over irrational rotation $C^{\ast} $-algebras”, Proc. London Math. Soc., 47 (1983), 285–302 | DOI | MR | Zbl

[22] Vlasenko M., “The graded ring of quantum theta functions for noncommutative torus with real multiplication”, Int. Math. Res. Not., 2006 (2006), 15825, 19 pp., arXiv: math.QA/0601405 | DOI | MR | Zbl