# Vector Bundles over Multipullback Quantum Complex Projective Spaces

## TL;DR

This paper classifies vector bundles over quantum projective spaces and spheres, using groupoid C*-algebras and Rieffel's stable rank results to determine when modules are free, and explicitly identifies quantum line bundles.

## Contribution

It applies groupoid C*-algebra techniques and stable rank theory to classify projective modules over quantum spaces, providing explicit descriptions of quantum line bundles.

## Key findings

- Modules of rank higher than loor(n/2)+3 are free over quantum spheres.
- Identifies a large portion of the positive cone of the K0-group.
- Explicitly represents quantum line bundles as elementary projections.

## Abstract

We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $ and $C\left( \mathbb{S}_{H}^{2n+1}\right) $ of the quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T} \right) $ and the quantum spheres $\mathbb{S}_{H}^{2n+1}$, and the quantum line bundles $L_{k}$ over $\mathbb{P}^{n}\left( \mathcal{T}\right) $, studied by Hajac and collaborators. Motivated by the groupoid approach of Curto, Muhly, and Renault to the study of C*-algebraic structure, we analyze $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, $C\left( \mathbb{S}_{H}^{2n+1}\right) $, and $L_{k}$ in the context of groupoid C*-algebras, and then apply Rieffel's stable rank results to show that all finitely generated projective modules over $C\left( \mathbb{S}_{H} ^{2n+1}\right) $ of rank higher than $\left\lfloor \frac{n}{2}\right\rfloor +3$ are free modules. Furthermore, besides identifying a large portion of the positive cone of the $K_{0}$-group of $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, we also explicitly identify $L_{k}$ with concrete representative elementary projections over $C\left( \mathbb{P} ^{n}\left( \mathcal{T}\right) \right) $.