BRAIDED JOIN COMODULE ALGEBRAS OF GALOIS OBJECTS LUDWIK D  ABROWSKI, TOM HADFIELD, PIOTR M. HAJAC, AND ELMAR WAGNER 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. Contents 1. Introduction and preliminaries 2 1.1. Classical principal bundles from the join construction 3 1.2. Left and right Hopf-Galois coactions 4 1.3. Principal right coactions 6 1.4. Left Durdevic braiding 6 2. Braided principal join comodule algebras 8 2.1. Left braided right comodule algebras 8 2.2. Braided join comodule algebras 9 2.3. Pullback structure and principality 10 3. *-Galois objects 12 3.1. *-structure 12 3.2. Noncommutative-torus algebra as a Galois object 13 4. Finite quantum coverings 14 4.1. (Anti-)Drinfeld doubles 15 4.2. A finite quantum subgroup of SL e 2πi/3 (2) 16 References 19 2010 Mathematics Subject Classification. Primary *****, Secondary **** . 1 arXiv:1407.6840v2 [math.QA] 31 Jul 2014