On the geometry of quantum spheres and hyperboloids

We study two classes of quantum spheres and hyperboloids which are $*$-quantum spaces for the quantum orthogonal group O(SOq(3)). We construct line bundles over the quantum homogeneous space of invariant elements for the quantum subgroup $SO(2)$ of SOq(3). These are associated to the quantum principal bundle via corepresentations of $SO(2)$ and are given by finitely-generated projective modules Eₙ of rank $1$ and even degree $-2n$. The corresponding idempotents, representing classes in K-theory, are explicitly worked out. For q real, we diagonalise the Casimir operator of the Hopf algebra U_q1/2(sl₂) dual to O(SOq(3)).