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: O(n), S(n), H(n), B(n), S(n)', B(n)'. We investigate the representation theory aspects of the correspondence, with the result that for O(n), S(n), H(n), B(n), 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. (C) 2009 Elsevier Inc. All rights reserved.