Liberations and twists of real and complex spheres

We study the 10 noncommutative spheres obtained by liberating, twisting, and liberating +twisting the real and complex spheres $S_{\mathbb{R}}^{N-1},S_{\mathbb{C}}^{N-1}$. At the axiomatic level, we show that, under very strong axioms, these 10 spheres are the only ones. Our main results concern the computation of the quantum isometry groups of these 10 spheres, taken in an affine real/complex sense. We formulate as well a proposal for an extended formalism, comprising 18 spheres.