On the Geometry of Quantum Spheres and Hyperboloids

Abstract 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 (SOq (3) ) O (S O q (3) ). We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO (2) of SOq (3) S O 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 Eₙ E n of rank 1 and of degree computed to be an even integer -2n - 2 n. 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 {Uₐ^₁/₂ (sl₂) } U q 1 / 2 (s l 2) which is dual to O (SOq (3) ) O (S O q (3) ).