Braided join comodule algebras of Galois objects

TL;DR

This paper introduces a method to construct braided joins of noncommutative Galois objects over Hopf algebras, resulting in principal coactions and new quantum geometric structures.

Contribution

It develops a braided tensor product approach for Galois objects, ensuring principal coactions and providing explicit examples involving noncommutative tori and quantum coverings.

Findings

Constructed braided join algebras with principal coactions

Provided examples using noncommutative tori and quantum doubles

Demonstrated noncommutative deformations of classical bundles

Abstract

We construct the join of noncommutative Galois objects (quantum torsors) over a Hopf algebra H. To ensure that the join algebra enjoys the natural (diagonal) coaction of H, we braid the tensor product of the Galois objects. Then we show that this coaction is principal. Our examples are built from the noncommutative torus with the natural free action of the classical torus, and arbitrary anti-Drinfeld doubles of finite-dimensional Hopf algebras. The former yields a noncommutative deformation of a non-trivial torus bundle, and the latter a finite quantum covering.