We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are \(*\)-quantum spaces for the quantum orthogonal group \(\mathcal {O}(SO_q(3))\). We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of \(SO_q(3)\). The line bundles are associated to the quantum principal bundle via representations of SO(2) and are described dually by finitely-generated projective modules \(\mathcal {E}_n\) of rank 1 and of degree computed to be an even integer \(-2n\). 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 \({\mathcal {U}_{q^{1/2}}(sl_2)}\) which is dual to \(\mathcal {O}(SO_q(3))\).

我们研究两类量子球和双曲面，一类由齐次空间组成，它们是量子正交群 \(\mathcal {O}(SO_q(3))\) 的 \(*\)-量子空间。我们在与 \(SO_q(3)\) 的量子子群 SO(2) 相关的量子齐次空间上构造线丛。线束通过 SO(2) 的表示与量子主束相关联，并由有限生成的射影模块 \(\mathcal {E}_n\) 进行双重描述，其秩为 1，度数计算为偶数 \ (-2n\)。相应的幂等性（表示基空间 K 理论中的类）已明确计算出来，并与两个合适的 Fredhom 模块配对，用于计算束的秩和度。对于 q 实数，我们展示如何对 Hopf 代数的卡西米尔算子的作用（在基空间代数上）进行对角化 \({\mathcal {U}_{q^{1/2}}(sl_2)}\)它是 \(\mathcal {O}(SO_q(3))\) 的对偶。