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

SIGMA 16 (2020), 008, 29 pages      arXiv:1811.10913
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

On the Relationship between Classical and Deformed Hopf Fibrations

Tomasz Brzeziński ab, James Gaunt c and Alexander Schenkel c
a)  Department of Mathematics, Swansea University, Swansea University Bay Campus, Fabian Way, Swansea SA1 8EN, UK
b)  Faculty of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland
c)  School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Received September 17, 2019, in final form February 17, 2020; Published online February 23, 2020

The $\theta$-deformed Hopf fibration $\mathbb{S}^3_\theta\to \mathbb{S}^2$ over the commutative $2$-sphere is compared with its classical counterpart. It is shown that there exists a natural isomorphism between the corresponding associated module functors and that the affine spaces of classical and deformed connections are isomorphic. The latter isomorphism is equivariant under an appropriate notion of infinitesimal gauge transformations in these contexts. Gauge transformations and connections on associated modules are studied and are shown to be sensitive to the deformation parameter. A homotopy theoretic explanation for the existence of a close relationship between the classical and deformed Hopf fibrations is proposed.

Key words: noncommutative geometry; principal comodule algebras; noncommutative principal bundles; Hopf fibrations; homotopy equivalence.

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