On characteristic classes of vector bundles over quantum spheres

We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\Z[t]/(t^2)$. 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 that there is a morphism $K_0 \to \Z[t]/(t^2)$ of abelian groups that is compatible with the tensor product of bimodules. Applications include 2-dimensional and 4-dimensional quantum spheres. We discuss in detail the quantum $4$-sphere $S^4_q$ coming from quantum symplectic groups, whose 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.