Twisting the q-deformations of compact semisimple Lie groups

Given a compact semisimple Lie group G of rank r, and a parameter q > 0, we can define new associativity morphisms in Rep(G(q)) using a 3-cocycle Phi, on the dual of the center of G, thus getting a new tensor category Rep(G(q))(Phi). For a class of cocycles Phi we construct compact quantum groups G(q)(tau) with representation categories Rep(G(q))(Phi). The construction depends on the choice of an r-tuple tau of elements in the center of G. In the simplest case of G = SU (2) and tau = -1, our construction produces Woronowicz's quantum group SU-q(2) out of SUq(2). More generally, for G = SU(n), we get quantum group realizations of the Kazhdan-Wenzl categories.