The Hyperoctahedral Quantum Group

We present a definition for free quantum groups. The idea is that these must satisfy S n ⊂ G ⊂ U + n , along with a technical representation theory condition. We work out in detail the case of quantum analogues of the hyperoctahedral group Hn. We first consider the hypercube in R , and show that its quantum symmetry group is in fact a q-deformation of On at q = −1. Then we consider the space formed by n segments, and show that its quantum symmetry group is a free analogue of Hn. This latter quantum group, denoted H n , enlarges Wang’s series S + n , O + n , U + n . Introduction The idea of noncommuting coordinates goes back to Heisenberg. Several theories emerged from Heisenberg’s work, most complete being Connes’ noncommutative geometry, where the base space is a Riemannian manifold. See [18]. A natural question is about studying algebras of free coordinates on algebraic groups. Given a group G ⊂ Un, the matrix coordinates uij ∈ C(G) commute with each other, and satisfy certain relations R. One can define then the universal algebra generated by abstract variables uij, subject to the relations R. The spectrum of this algebra is an abstract object, called noncommutative version of G. The noncommutative version is not unique, because it depends on R. We have the following examples: (1) Free quantum semigroups. The algebra generated by variables uij with relations making u = (uij) a unitary matrix was considered by Brown [13]. This algebra has a comultiplication and a counit, but no antipode. In other words, the corresponding noncommutative version Unc n is a quantum semigroup. (2) Quantum groups. A remarkable discovery is that for a compact Lie group G, once commutativity is included into the relations R, one can introduce a complex parameter q, as to get deformed relations Rq. The corresponding noncommutative versions Gq are called quantum groups. See Drinfeld [20]. (3) Compact quantum groups. The quantum group Gq is semisimple for q not a root of unity, and compact under the slightly more restrictive assumption q ∈ R. Woronowicz gave in [33] a simple list of axioms for compact quantum groups, which allows construction of many other noncommutative versions. (4) Free quantum groups. The Brown algebras don’t fit into Woronowicz’s axioms, but a slight modification leads to free quantum groups. The quantum 2000 Mathematics Subject Classification. 20G42 (46L54).