Quantum automorphism groups and SO(3)SO(3)-deformations

We show that any compact quantum group having the same fusion rules as the ones of SO(3)SO(3) is the quantum automorphism group of a pair (A,φ)(A,φ), where A   is a finite dimensional C⁎C⁎-algebra endowed with a homogeneous faithful state. We also study the representation category of the quantum automorphism group of (A,φ)(A,φ) when φ   is not necessarily positive, generalizing some known results, and we discuss the possibility of classifying the cosemisimple (not necessarily compact) Hopf algebras whose corepresentation semi-ring is isomorphic to that of SO(3)SO(3).