Log-Modular Quantum Groups at Even Roots of Unity and the Quantum Frobenius I

Abstract

We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type $$A_1$$ , and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1, p)-central charge. For example, we explain how one can (conjecturally) employ known linear equivalences between the triplet vertex algebra and quantum $$\mathfrak {sl}_2$$ , in conjunction with a natural $${{\,\mathrm{PSL}\,}}_2$$ -action on quantum $$\mathfrak {sl}_2$$ provided by our de-equivariantization construction, in order to deduce linear equivalences between “extended” quantum groups, the singlet vertex operator algebra, and the (1, p)-Virasoro logarithmic minimal model. We assume some restrictions on the order of our root of unity outside of type $$A_1$$ , which we intend to eliminate in a subsequent paper.