Symmetry of some noncommutative sphere algebras

TL;DR

This paper investigates the symmetry properties of two different $q$-deformed 7-sphere algebras, showing that only the quaternionic version exhibits $ ext{SU}_q(2)$ symmetry, thus proving they are not isomorphic.

Contribution

The paper demonstrates that the older Vaksman-Soibelman quantum 7-sphere lacks the $ ext{SU}_q(2)$ symmetry present in the quaternionic version, establishing their non-isomorphism.

Findings

The quaternionic $ ext{SU}_q(2)$ symmetry exists for the quaternionic quantum 7-sphere.

The older Vaksman-Soibelman quantum 7-sphere does not support this symmetry.

The two quantum 7-spheres are proven to be non-isomorphic.

Abstract

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.