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(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 U_q1/2(sl₂) U q 1 / 2 ( s l 2 ) which is dual to O(SOq(3)) O ( S O q ( 3 ) ) .