On the geometry of quantum spheres and hyperboloids

Abstract

We study two classes of quantum spheres and hyperboloids which are $\ast$-quantum spaces for the quantum orthogonal group $\mathcal{O}(S{O}_q(3))$. We construct line bundles over the quantum homogeneous space of invariant elements for the quantum subgroup $SO(2)$ of $S{O}_q(3)$. 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 ${\mathcal{U}}_{q^{1/2}}(s{l}_2)$ dual to $\mathcal{O}(S{O}_q(3))$.

27 pages, no figures