Braided autoequivalences and quantum commutative bi-Galois objects

TL;DR

This paper establishes a connection between braided autoequivalences of Yetter-Drinfeld categories over a quasitriangular weak Hopf algebra and quantum commutative bi-Galois objects, revealing their role in the structure of the Brauer group.

Contribution

It proves that braided autoequivalences correspond to quantum commutative bi-Galois objects in the setting of weak Hopf algebras, extending the understanding of the Brauer group.

Findings

Braided autoequivalences are characterized by quantum commutative Galois objects.

In the semisimple case, autoequivalences trivial on $_H\mathscr{M}$ are determined by these objects.

Quantum commutative Galois objects form a group related to the Brauer group of $(H,R)$.

Abstract

Let $(H,R)$ be a quasitriangular weak Hopf algebra over a field $k$. We show that there is a braided monoidal equivalence between the Yetter-Drinfeld module category $^H_H\mathscr{YD}$ over $H$ and the category of comodules over some braided Hopf algebra ${}_RH$ in the category $_H\mathscr{M}$. Based on this equivalence, we prove that every braided bi-Galois object $A$ over the braided Hopf algebra ${}_RH$ defines a braided autoequivalence of the category $^H_H\mathscr{YD}$ if and only if $A$ is quantum commutative. In case $H$ is semisimple over an algebraically closed field, i.e. the fusion case, then every braided autoequivalence of $^H_H\mathscr{YD}$ trivializable on $_H\mathscr{M}$ is determined by such a quantum commutative Galois object. The quantum commutative Galois objects in $_H\mathscr{M}$ form a group measuring the Brauer group of $(H,R)$ as studied in [20] in the Hopf algebra case.