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

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