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 Φ on the dual of the center of G, thus getting a new tensor category Rep(G_q)^Φ. For a class of cocycles Φ we construct compact quantum groups G^τ_q with representation categories Rep(G_q)^Φ. The construction depends on the choice of an r-tuple τ of elements in the center of G. In the simplest case of G = SU(2) and τ = −1, our construction produces Woronowicz's quantum group SU_−q(2) out of SU_q(2). More generally, for G = SU(n), we get quantum group realizations of the Kazhdan–Wenzl categories.

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