On the absence of continuous symmetries for noncommutative 3-spheres

A large class of noncommutative spherical manifolds was obtained recently from cohomology considerations. A one-parameter family of twisted three-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 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 coactions, our results suggest that this is not the case.