Projective Modules over Quantum Projective Line

TL;DR

This paper studies the structure and classification of finitely generated projective modules over quantum projective lines using groupoid C*-algebra techniques, identifying quantum principal bundles within this framework.

Contribution

It provides a detailed analysis of the C*-algebra of quantum projective lines, computes its K-groups, and classifies its projective modules, including quantum principal bundles.

Findings

Determined the K-groups of the quantum projective line C*-algebra.

Classified all finitely generated projective modules over the algebra.

Identified quantum principal U(1)-bundles among the modules.

Abstract

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T}\right) $ constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra $C\left( \mathbb{P}^{1}\left( \mathcal{T}\right) \right) $ realized as a concrete groupoid C*-algebra, and find its $K$-groups. Furthermore after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra $C\left( \mathbb{P}^{1}\left( \mathcal{T}\right) \right) $, we identify those quantum principal $U\left( 1\right) $-bundles introduced by Hajac and collaborators among the projections classified.