Liberation of orthogonal Lie groups

We show that under suitable assumptions, we have a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: On, Sn, Hn, Bn, , . We investigate the representation theory aspects of the correspondence, with the result that for On, Sn, Hn, Bn, this is compatible with the Bercovici–Pata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.