q-FUZZY SPHERES AND QUANTUM DIFFERENTIALS ON Bq[SU2] AND Uq(su2)

We show that the 2-parameter Podles sphere is a q-fuzzy sphere precisely interpolating between the fuzzy sphere as quotient of the angular momentum algebra U(su2) and the standard q-sphere Cq [S] as subalgebra of the quantum group Cq [SU2]. Whereas the classical sphere as CP 1 can be defined as the algebra generated by the matrix entries of a projector e with trace(e) = 1, the fuzzy-sphere is defined in the same way by trace(e) = 1 + λ. We show that the standard q-sphere is similarly defined by traceq(e) = 1 and the Podles 2-spheres by traceq(e) = 1 + λ, thereby giving a unified point of view. As a corollary we expect and show that a localisation of the Podles 2-sphere is isomorphic to a quotient of Uq(su2) by fixing a value of the qCasimir, and that it is exactly a ‘time slice’ of the braided group Bq[SU2] (the unit hyperboloid in q-Minkowski space) by fixing the value of the q-trace of the braided matrix. We use transmutation theory to introduce a Cq[G]-covariant calculus on general Bq[G] and Uq(g), and use Ω(Bq [SU2]) in particular to provide a unified point of view on the 3D calculi on fuzzy and Podles spheres.