On the Absence of Continuous Symmetries for Noncommutative 3-Spheres

TL;DR

This paper investigates whether noncommutative 3-spheres possess continuous symmetries similar to classical spheres, concluding that such symmetries do not generally extend to these quantum geometries.

Contribution

It demonstrates that noncommutative 3-spheres of Connes and Dubois-Violette lack continuous symmetries from deformations of Spin(4) and SO(4).

Findings

Noncommutative 3-spheres are not homogeneous spaces for continuous quantum group deformations.

Symmetry extension from classical to noncommutative spheres is generally not possible.

Results suggest limitations on symmetry properties of noncommutative spherical manifolds.

Abstract

A large class of noncommutative spherical manifolds was obtained recently from cohomology considerations. A one-parameter family of twisted 3-spheres was discovered by Connes and Landi, and later generalized to a three-parameter family by Connes and Dubois-Violette. The spheres of Connes and Landi were shown to be homogeneous spaces for certain compact quantum groups. Here we investigate whether or not this property can be extended to the noncommutative three-spheres of Connes and Dubois-Violette. Upon restricting to quantum groups which are continuous deformations of Spin(4) and SO(4) with standard co-actions, our results suggest that this is not the case.