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 ${^H}_A{\mathscr M}_A$, where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist ${\mathcal F}$ of H we then quantize (deform) H to $H^{\mathcal F}$, A to A⋆ and correspondingly the category ${^H}_A{\mathscr M}_A$ to ${^H}^{\mathcal F}_{A_\star}{\mathscr M}_{A_\star}$. 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⋆-bimodule connections. Their curvatures and those on tensor product modules are also determined.