SIGMA 10 (2014), 031, 7 pages      arXiv:1310.7664      https://doi.org/10.3842/SIGMA.2014.031
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Nontrivial Deformation of a Trivial Bundle

Piotr M. Hajac ^{a, b} and Bartosz Zieliński ^c
a) Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, 00-956 Warszawa, Poland
^b) Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, 00-682 Warszawa, Poland
^c) Department of Computer Science, Faculty of Physics and Applied Informatics, University of Łódź, Pomorska 149/153 90-236 Łódź, Poland

Received October 29, 2013, in final form March 03, 2014; Published online March 27, 2014

Abstract
The ${\rm SU}(2)$-prolongation of the Hopf fibration $S^3\to S^2$ is a trivializable principal ${\rm SU}(2)$-bundle. We present a noncommutative deformation of this bundle to a quantum principal ${\rm SU}_q(2)$-bundle that is not trivializable. On the other hand, we show that the ${\rm SU}_q(2)$-bundle is piecewise trivializable with respect to the closed covering of $S^2$ by two hemispheres intersecting at the equator.

Key words: quantum prolongations of principal bundles; piecewise trivializable quantum principal bundles.

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References

1. Baum P.F., De Commer K., Hajac P.M., Free actions of compact quantum groups of unital $C^*$-algebras, arXiv:1304.2812.
2. Baum P.F., Hajac P.M., Local proof of algebraic characterization of free actions, arXiv:1402.3024.
3. Baum P.F., Hajac P.M., Matthes R., Szymański W., Noncommutative geometry approach to principal and associated bundles, in Quantum Symmetry in Noncommutative Geometry, to appear, math.DG/0701033.
4. Brzeziński T., Hajac P.M., The Chern-Galois character, C. R. Math. Acad. Sci. Paris 338 (2004), 113-116, math.KT/0306436.
5. Brzeziński T., Zieliński B., Quantum principal bundles over quantum real projective spaces, J. Geom. Phys. 62 (2012), 1097-1107, arXiv:1105.5897.
6. Günther R., Crossed products for pointed Hopf algebras, Comm. Algebra 27 (1999), 4389-4410.
7. Hajac P.M., Krähmer U., Matthes R., Zieliński B., Piecewise principal comodule algebras, J. Noncommut. Geom. 5 (2011), 591-614, arXiv:0707.1344.
8. Hajac P.M., Matthes R., Sołtan P.M., Szymański W., Zieliński B., Hopf-Galois extensions and $C^*$ algebras, in Quantum Symmetry in Noncommutative Geometry, to appear.
9. Hajac P.M., Rudnik J., Zieliński B., Reductions of piecewise trivial comodule algebras, arXiv:1101.0201.
10. Kobayashi S., Nomizu K., Foundations of differential geometry. Vol. I, Interscience Publishers, New York - London, 1963.
11. Podleś P., Symmetries of quantum spaces. Subgroups and quotient spaces of quantum ${\rm SU}(2)$ and ${\rm SO}(3)$ groups, Comm. Math. Phys. 170 (1995), 1-20, hep-th/9402069.
12. Schauenburg P., Galois objects over generalized Drinfeld doubles, with an application to $u_q({\mathfrak{sl}}_2)$, J. Algebra 217 (1999), 584-598.
13. Sołtan P.M., On actions of compact quantum groups, Illinois J. Math. 55 (2011), 953-962, arXiv:1003.5526.
14. Woronowicz S.L., Twisted ${\rm SU}(2)$ group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181.