We study two classes of quantum spheres and hyperboloids which are $*$-quantum spaces for the quantum orthogonal group $\mathcal{ O}(SO_q(3))$. We construct line bundles over the quantum homogeneous space of invariant elements for the quantum subgroup $SO(2)$ of $SO_q(3)$. These are associated to the quantum principal bundle via corepresentations of $SO(2)$ and are given by finitely-generated projective modules $\mathcal{ E}_n$ 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 ${ \mathcal{ U}_{ q^{ 1/2}}(sl_2)}$ dual to $\mathcal{ O}(SO_q(3))$.
Categories: math.QA math-ph math.MP