q-Fuzzy Spheres and Quantum Differentials on Bq[SU2] and Uq(su2)

We provide a new unified construction of the two-parameter Podle's two-spheres as characterised by a projector e with trace(q)(e) = 1 + lambda. In our formulation the limit in which q -> 1 with lambda fixed is the fuzzy sphere, while the limit lambda -> 0 with q fixed is the standard q-deformed sphere. We show further that the non-standard Podle's spheres arise geometrically as 'constant time slices' of the unit hyperboloid in q-Minkowski space viewed as the braided group B-q[SU2]. 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(C)]-covariant differential calculus on general B-q[G] and U-q(g), with Omega(B-q[SU2]) and Omega(U-q(su(2)) given in detail. To complete the picture, we show how the covariant calculus on the 3D bicrossproduct spacetime arises from Omega(C-q[SU2]) prior to twisting.