Quantum $\mathrm{SO}(3)$ groups and quantum group actions on $M_2$

Abstract

Answering a question of Shuzhou Wang we give a description of quantum $\mathrm{SO}(3)$ groups of Podleś as universal compact quantum groups acting on the ${\mathrm{C}}^{\ast }$-algebra $M_2$ and preserving the Powers state. We use this result to give a complete classification of all continuous compact quantum group actions on $M_2$.