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q-Fuzzy Spheres and Quantum Differentials on B q [SU 2] and U q (su 2)

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Abstract

We provide a new unified construction of the two-parameter Podleś two-spheres as characterised by a projector e with trace q (e) = 1 + λ. In our formulation the limit in which q → 1 with λ fixed is the fuzzy sphere, while the limit λ → 0 with q fixed is the standard q-deformed sphere. We show further that the non-standard Podleś spheres arise geometrically as ‘constant time slices’ of the unit hyperboloid in q-Minkowski space viewed as the braided group B q [SU 2]. Their localisations are then isomorphic to quotients of U q (su 2) at fixed values of the q-Casimir precisely q-deforming the fuzzy case. We also use transmutation and twisting theory to introduce a \({C_q[G_\mathbb {C}]}\) -covariant differential calculus on general B q [G] and U q (g), with Ω(B q [SU 2]) and Ω(U q (su 2) given in detail. To complete the picture, we show how the covariant calculus on the 3D bicrossproduct spacetime arises from Ω(C q [SU 2]) prior to twisting.

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  1. School of Mathematics, Queen Mary University of London, Mile End Rd., London, E1 4NS, UK

    Shahn Majid

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Majid, S. q-Fuzzy Spheres and Quantum Differentials on B q [SU 2] and U q (su 2). Lett Math Phys 98, 167–191 (2011). https://doi.org/10.1007/s11005-011-0523-8

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  • DOI: https://doi.org/10.1007/s11005-011-0523-8

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