Projections over Quantum Homogeneous Odd-dimensional Spheres

We give a complete classification of isomorphism classes of finitely generated projective modules, or equivalently, unitary equivalence classes of projections, over the C*-algebra $C\left({\mathbb{S}}_q^{2n+1}\right)$ of the quantum homogeneous sphere ${\mathbb{S}}_q^{2n+1}$. Then we explicitly identify as concrete elementary projections the quantum line bundles $L_k$ over the quantum complex projective space $\mathbb{C}{P}_q^n$ associated with the quantum Hopf principal $U\left(1\right)$-bundle $$\mathbb{S}$ {q}^{2n+1}\rightarrow$\mathbb{C}$P{q}^{n}$.