Hopf-Galois objects over bicrossed product Hopf algebras and twisting maps

TL;DR

This paper characterizes Hopf-Galois objects over bicrossed product Hopf algebras, showing they derive from the factors and a twisting map, with the associated Hopf algebra also being a bicrossed product.

Contribution

It provides a detailed description of Hopf-Galois objects over bicrossed products and introduces a method to construct them from simpler components.

Findings

Hopf-Galois objects over bicrossed products are obtained from those over the factors and a twisting map.

The unique Hopf algebra making the object bi-Galois is also a bicrossed product.

The structure of these objects is explicitly characterized.

Abstract

We describe Hopf-Galois objects over bicrossed product Hopf algebras. More precisely, we show that any right Hopf-Galois object over a bicrossed product of Hopf algebras is obtained from Hopf-Galois objects over the two factors and a certain twisting map, while the unique Hopf algebra making it into a bi-Galois object is again a bicrossed product.