On characteristic classes of vector bundles over quantum spheres

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 Podleś sphere $S_q^2$ and a quantum $4$-sphere $S_q^4$ 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 $S_q^4$ associated to the principal $S{U}_q(2)$-bundle $S_q^7\to S_q^4$ via irreducible corepresentations of $S{U}_q(2)$, and compute their characteristic classes.

31 pages, no figures