Braided join comodule algebras of bi-Galois objects

A bi-Galois object A is a bicomodule algebra for Hopf-Galois coactions with trivial invariants. In the spirit of Milnor's construction, we define the join of noncommutative bi-Galois objects (quantum torsors). To ensure that the diagonal coaction on the join algebra of the right-coacting Hopf algebra is an algebra homomorphism, we braid the tensor product A circle times A with the help of the left-coacting Hopf algebra. Our main result is that the diagonal coaction is principal. Then we show that an anti-Drinfeld double is a symmetric bi-Galois object with the Drinfeld-double Hopf algebra coacting on both left and right. In this setting, we consider a finite quantum covering as an example. Finally, we take the noncommutative torus with the natural free action of the classical torus as an example of a symmetric bi-Galois object equipped with a *-structure. It yields a noncommutative deformation of a nontrivial torus bundle. \ua9 2016, University at Albany. All rights reserved