Free products of compact quantum groups

We construct and study compact quantum groups from free products ofC ^*-algebras. In this connection, we discover two mysterious classes of natural compact quantum groups,A _u (m) andA _o (m). TheA _u (m)'s (respectivelyA _o (m)'s) are non-isomorphic to each other for differentm's, and are not obtainable by the ordinary quantization method. We also clarify some basic concepts in the theory of compact quantum groups.