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 that there is a morphism $K_0\to \mathbb{Z}[t]∕(t^2)$ 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 $S_q^4$ 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 $S_q^4$ associated to the principal ${\mathrm{SU}<!--FUNCTION\; APPLICATION-->}_q(2)$-bundle $S_q^7\to S_q^4$ via irreducible corepresentations of ${\mathrm{SU}<!--FUNCTION\; APPLICATION-->}_q(2)$, and compute their characteristic classes.

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