On the Geometry of Quantum Spheres and Hyperboloids

We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are *-quantum spaces for the quantum orthogonal group O(SO_q(3)). We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of SO_q(3).The line bundles are associated to the quantum principal bundle via representations of SO(2) and are described dually by finitely-generated projective modules E_n of rank 1 and of degree computed to be an even integer -2n. The corresponding idempotents, that represent classes in the K-theory of the base space, are explicitly worked out and are paired with two suitable Fredhom modules that compute the rank and the degree of the bundles. For q real, we show how to diagonalise the action (on the base space algebra) of the Casimir operator of the Hopf algebra U_{q^{1/2}}(sl_2)} which is dual O(SO_q(3)).