Symmetry of some noncommutative sphere algebras

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