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, S ′ n , B n . 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. Introduction The notion of free quantum group appeared in Wang’s papers [24], [25]. The idea is as follows: let G ⊂ Un be a compact group. The n matrix coordinates uij satisfy certain relations R, and generate the algebra C(G). One can define then the universal algebra A generated by n noncommuting variables uij, satisfying the relations R. For a suitable choice of R we get a Hopf algebra in the sense of Woronowicz [27], and we have the heuristic formula A = C(G), where G is a compact quantum group, called free version of G. (Clearly, if A is not commutative then G is a fictional object and any statement about G has to be interpreted in terms of A to make rigorous sense.) This construction is not axiomatized, in the sense that G depends on the relations R, and it is not known in general what the good choice of R is. For instance any choice with R including the commutativity relations uijukl = ukluij would be definitely a bad one, because in this case we would get G = G. Moreover, any choice with R including certain relations which imply these commutativity relations would be a bad one as well. The study of free quantum groups basically belongs to combinatorics, and can be divided into three main areas, having interactions between them: (1) Quantum permutation groups. This area is concerned with the general study of free quantum groups G, in the case G ⊂ Sn. Most results here were obtained in the last few years, and we refer to [5] for a survey. 2000 Mathematics Subject Classification. 16W30 (46L54).