A Maximality Result for Orthogonal Quantum Groups

We prove that the quantum group inclusion On ⊂ O ∗ n is “maximal”, where On is the usual orthogonal group and O ∗ n is the half-liberated orthogonal quantum group, in the sense that there is no intermediate compact quantum group On ⊂ G ⊂ O ∗ n . In order to prove this result, we use: (1) the isomorphism of projective versions PO∗ n ≃ PUn, (2) some maximality results for classical groups, obtained by using Lie algebras and some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.