# Projections over Quantum Homogeneous Odd-dimensional Spheres

## TL;DR

This paper classifies all finitely generated projective modules over quantum odd-dimensional spheres and explicitly identifies quantum line bundles as concrete projections, advancing the understanding of quantum homogeneous spaces.

## Contribution

It provides a complete classification of projective modules over quantum spheres and explicitly constructs quantum line bundles as concrete projections.

## Key findings

- Complete classification of projective modules over quantum spheres
- Explicit identification of quantum line bundles as projections
- Enhanced understanding of quantum homogeneous space structures

## Abstract

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 $\mathbb{S} _{q}^{2n+1}\rightarrow\mathbb{C}P_{q}^{n}$.