Autoequivalences of the tensor category of \(U_{q}\mathfrak g\)-modules (Q2909343)

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 \textit{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 \(G_q\).NEWLINENEWLINE 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)].