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

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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