This note calculates the group of monoidal autoequivalences of the category \(\mathcal C\) of representations of the \(q\)-deformation \(G_q\) of a simply connected semisimple compact Lie group \(G\) (identified up to monoidal natural isomorphisms). Part of this was done by J. R. McMullen, who showed that the group of automorphisms of the fusion ring of \(G\) is isomorphic to \(\text{Out}(G)\), i.e., the automorphism group of the based root datum of \(G\) [Math. Z. 185, 539–552 (1984; Zbl 0513.43007)]. The remaining part is determined by the possible tensor structures one can have on the identity functor, and these are described by the cohomology group defined by invariant 2-cocycles on the dual \(\widehat G_q\) of \(G_q\).
The precise statement of the authors’ main result is that for \(q\in\mathbb C^*\) a nontrivial root of unity, \(g\) the complexified Lie algebra of \(G\), \(H^2(\widehat G_q; \mathbb C^*)\) is isomorphic to \(H^2(P/Q; \pi^*)\), where \(P\) and \(Q\) are the weight and root lattices of \(g\), respectively. This implies that the group of autoequivalences of the tensor category of \(U_q(g)\)-modules is the semidirect product of \(H^2(P/Q; \pi)\) and the automorphism group of the based root datum of \(g\). The authors use techniques from their earlier paper [Adv. Math. 227, No. 1, 146–169 (2011; Zbl 1220.46046)].