Quantum supergroups III. Twistors

We establish direct connections at several levels between quantum groups and supergroups associated to bar-consistent anisotropic super Cartan datum by constructing an automorphism (called twistor) in the setting of covering quantum groups. The canonical bases of the halves of quantum√ groups and supergroups are shown to match under the twistor up to powers of −1. We further show that the modified quantum √ group and supergroup are isomorphic over the rational function field adjoined with −1, by constructing a twistor on the modified covering quantum group. An equivalence of categories of weight modules for quantum groups and supergroups follows.