Thesis title: Twisting quantum groups at the roots of unity.
Let G be a simply connected Lie group and g be its complexified Lie algebra.
Building on the work of Wenzl, we present a weak tensor structure on the unitary modular categories arising from representation categories of quantum groups U_q(g) specialised at the root of 1 q, following a paper by Carpi, Ciamprone, Pinzari and the author. The theory therein developed allows us to reconstruct theses categories as representation categories of a discrete unitary coboundary weak Hopf algebra.
Furthermore, we consider the twisted categories obtained by modifying the associator by means of 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 the work of Neshveyev and Yamashita in the analogous scenario of the compact quantum group corresponding to U_q(g) specialised at q>1.