Twisting quantum groups at the roots of unity

Let G Be a simply connected compact Lie group and g be its complexified Lie algebra. Building on the work of H. Wenzl, we present a weak tensor structure on the unitary modular categories arising from quantum groups U_q(g) specialised at the root of 1 q, following the work about weak quasi-tensor categories by S. Carpi, S. Ciamprone, C. Pinzari and myself. The theory therein developed allows us to reconstruct these categories as representation categories of discrete unitary coboundary weak Hopf algebras. Furthermore, we consider the twisted categories obtained by modifying the associator using 3-cocycles on the dual of the centre of G and reconstruct them as representation categories of suitable discrete unitary weak Hopf algebras; this is done by adaptation of a result by S. Neshveyev and M. Yamashita in the analogous scenario of the compact quantum group corresponding to U_q(g) for q>1, i.e. for generic values of q