TWISTING ALL THE WAY: FROM ALGEBRAS TO MORPHISMS AND CONNECTIONS

Abstract

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 [Formula: see text], where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist [Formula: see text] of H we then quantize (deform) H to [Formula: see text], A to A⋆ and correspondingly the category [Formula: see text] to [Formula: see text]. 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.