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Nontrivial Deformation of a Trivial Bundle
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: quantum prolongations of principal bundles; piecewise trivializable quantum principal bundles. 
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     author = {Piotr M. Hajac and Bartosz Zieli\'nski},
     title = {Nontrivial {Deformation} of {a~Trivial} {Bundle}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a30/}
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Piotr M. Hajac; Bartosz Zieliński. Nontrivial Deformation of a Trivial Bundle. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a30/

[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”, Quantum Symmetry in Noncommutative Geometry (to appear) , arXiv: math.DG/0701033

[4] Brzeziński T., Hajac P. M., “The {C}hern–{G}alois character”, C. R. Math. Acad. Sci. Paris, 338 (2004), 113–116, arXiv: math.KT/0306436 | DOI | MR | Zbl

[5] Brzeziński T., Zieliński B., “Quantum principal bundles over quantum real projective spaces”, J. Geom. Phys., 62 (2012), 1097–1107, arXiv: 1105.5897 | DOI | MR | Zbl

[6] Günther R., “Crossed products for pointed {H}opf algebras”, Comm. Algebra, 27 (1999), 4389–4410 | DOI | MR

[7] Hajac P. M., Krähmer U., Matthes R., Zieliński B., “Piecewise principal comodule algebras”, J. Noncommut. Geom., 5 (2011), 591–614, arXiv: 0707.1344 | DOI | MR | Zbl

[8] Hajac P. M., Matthes R., Sołtan P. M., Szymański W., Zieliński B., “Hopf–Galois extensions and $C^*$ algebras”, 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, v. I, Interscience Publishers, New York–London, 1963 | MR | Zbl

[11] Podleś P., “Symmetries of quantum spaces. {S}ubgroups and quotient spaces of quantum {${\rm SU}(2)$} and {${\rm SO}(3)$} groups”, Comm. Math. Phys., 170 (1995), 1–20, arXiv: hep-th/9402069 | DOI | MR | Zbl

[12] Schauenburg P., “Galois objects over generalized {D}rinfeld doubles, with an application to {$u_q({\mathfrak{sl}}_2)$}”, J. Algebra, 217 (1999), 584–598 | DOI | MR | Zbl

[13] Sołtan P. M., “On actions of compact quantum groups”, Illinois J. Math., 55 (2011), 953–962, arXiv: 1003.5526 | MR

[14] Woronowicz S. L., “Twisted {${\rm SU}(2)$} group. {A}n example of a non-commutative differential calculus”, Publ. Res. Inst. Math. Sci., 23 (1987), 117–181 | DOI | MR | Zbl