Symmetry of some noncommutative sphere algebras

Two known $q$-deformed (or `quantum’) $7$-spheres, both denoted ${\mathbb{S}}_q^7$ in the literature, may be distinguished by the presence or absence of symmetry under ${\mathrm{S}\mathrm{U}}_q(2)$. The quaternionic version of ${\mathbb{S}}_q^7$ has been shown by Brain and Landi to support such a symmetry. Here we show that this is not the case for the older ${\mathbb{S}}_q^7$ introduced by Vaksman and Soibelman: and as a consequence, these quantum $7$-spheres are not isomorphic.