Quantum weighted projective and lens spaces

TL;DR

This paper extends the study of quantum projective spaces to weighted variants, constructing explicit K-homology classes, and demonstrates their non-triviality through quantum principal bundles and Fredholm modules.

Contribution

It introduces quantum weighted projective spaces in any dimension, constructs explicit Fredholm modules, and establishes their non-triviality via quantum principal circle bundles.

Findings

Explicit Fredholm modules are linearly independent in K-homology.

Quantum weighted projective spaces serve as base spaces for quantum lens spaces.

Constructed projective modules are non-trivial and paired with Fredholm modules.

Abstract

We generalize to quantum weighted projective spaces in any dimension previous results of us on K-theory and K-homology of quantum projective spaces `tout court'. For a class of such spaces, we explicitly construct families of Fredholm modules, both bounded and unbounded (that is spectral triples), and prove that they are linearly independent in the K-homology of the corresponding C*-algebra. We also show that the quantum weighted projective spaces are base spaces of quantum principal circle bundles whose total spaces are quantum lens spaces. We construct finitely generated projective modules associated with the principal bundles and pair them with the Fredholm modules, thus proving their non-triviality.