2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77175We classify the compact quantum groups $A_u(Q)$ (resp. $B_u(Q)$) up to isomorphism when $Q>0$ (resp. when $Q \bar{Q} \in {\mathbb R} I_n$). We show that the general $A_u(Q)$'s and $B_u(Q)$'s for arbitrary $Q$ have explicit decompositions into free products of these special $A_u(Q)$'s and $B_u(Q)$'s.13 pages. To appear in Journal of Operator Theory. Details left out in the original version are filled inOperator AlgebrasQuantum Algebratext