Autoequivalences of the tensor category of Uq(g)-modules

TL;DR

This paper characterizes the autoequivalence group of the tensor category of Uq(g)-modules for generic q, linking it to cohomology of lattice quotients and root datum automorphisms, with extensions to compact groups at q=1.

Contribution

It establishes a precise description of the autoequivalence group of Uq(g)-modules for generic q, connecting it to lattice cohomology and root datum automorphisms, and extends results to compact groups at q=1.

Findings

Autoequivalence group is a semidirect product involving H^2(P/Q;T).

Cohomology group is isomorphic to H^2(P/Q;T) for generic q.

Results apply to all compact connected groups at q=1.

Abstract

We prove that for q\in\C* not a nontrivial root of unity the cohomology group defined by invariant 2-cocycles in a completion of Uq(g) is isomorphic to H^2(P/Q;\T), where P and Q are the weight and root lattices of g. This implies that the group of autoequivalences of the tensor category of Uq(g)-modules is the semidirect product of H^2(P/Q;\T) and the automorphism group of the based root datum of g. For q=1 we also obtain similar results for all compact connected separable groups.