Structure and Isomorphism Classification of Compact Quantum Groups A_u(Q) and B_u(Q)

We 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.