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Twisting all the way: from algebras to morphisms and connections

Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules, where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist F of H we then quantize (deform) H to H^F, A to A_\star and correspondingly the category of left H-modules and A-bimodules to the category of left H^F-modules and A_\star-bimodules. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphisms, and their twist quantization. This study leads to the definition of arbitrary (i.e., not necessarily H-equivariant) connections on quasi-commutative A-bimodules, to extend these connections to tensor product modules and to quantize them to A_\star-bimodule connections. Their curvatures and those on tensor product modules are also determined.