Quantum Supergroups III. Twistors

TL;DR

This paper introduces a twistor automorphism linking quantum groups and supergroups, demonstrating their canonical bases correspondence and establishing an equivalence of their module categories.

Contribution

It constructs a twistor automorphism in covering quantum groups, revealing deep connections and isomorphisms between quantum groups and supergroups.

Findings

Canonical bases match under the twistor up to powers of i

Modified quantum groups and supergroups are isomorphic over a field with i

Categories of weight modules for quantum groups and supergroups are equivalent

Abstract

We establish direct connections at several levels between quantum groups and supergroups associated to bar-consistent 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 a square root of -1. We further show that the modified quantum group and supergroup are isomorphic over the rational function field adjoint with a square root of -1, by constructing a twistor on the modified covering quantum group. An equivalence of categories of weight modules for quantum groups and supergroups follows.