On characteristic classes of vector bundles over quantum spheres

TL;DR

This paper investigates the K-theory of quantum spheres, providing conditions for classical K-theory structures to persist in the quantum setting, and explicitly computing characteristic classes of vector bundles over quantum spheres.

Contribution

It introduces conditions for the K-theory of quantum spaces to mirror classical dual number structures and computes characteristic classes for vector bundles on quantum spheres.

Findings

K-theory of quantum spheres can be linked to classical dual numbers under certain conditions

Explicit formulas for projections of vector bundles on quantum 4-spheres are derived

Characteristic classes of these bundles are computed explicitly

Abstract

We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\mathbb{Z}[t]/(t^2)$. For a compact Hausdorff space we recall necessary and sufficient conditions for this to hold. For a compact quantum space, we give sufficient conditions that guarantee there is a morphism of abelian groups $K_0 \to \mathbb{Z}[t]/(t^2)$ compatible with the tensor product of bimodules. Applications include the standard Podle\'s sphere $S^2_q$ and a quantum $4$-sphere $S^4_q$ coming from quantum symplectic groups. For the latter, the K-theory is generated by the Euler class of the instanton bundle. We give explicit formulas for the projections of vector bundles on $S^4_q$ associated to the principal $SU_q(2)$-bundle $S^7_q \to S^4_q$ via irreducible corepresentations of $SU_q(2)$, and compute their characteristic classes.