Quantum automorphism groups and SO(3)-deformations

We show that any compact quantum group having the same fusion rules as the ones of SO(3) is the quantum automorphism group of a pair (A,ϕ), where A is a finite dimensional C∗-algebra endowed with a homogeneous faithful state. We also study the representation category of the quantum automorphism group of (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). 1 Introduction and main results The quantum automorphism group of a measured finite dimensional C∗-algebra (A,ϕ) (i.e. a finite-dimensional C∗-algebra A endowed with a faithful state ϕ) has been defined by Wang in [24] as the universal object in the category of compact quantum groups acting on (A,ϕ). The corresponding compact Hopf algebra is denoted by Aaut(A,ϕ)