Given a Hopf algebra $H$, we study modules and bimodules over an algebra $A$ that carry an $H$-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over $A$ of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily Hequivariant ones, and further extends to the tensor algebra generated by a bimodule and its dual. Examples of these noncommutative structures arise in deformation quantization via Drinfeld twists of the commutative differential geometry of a smooth manifold, where the Hopf algebra $H$ is the universal enveloping algebra of vector fields (or a finitely generated Hopf subalgebra).
We extend the Drinfeld twist deformation theory of modules and algebras to morphisms and connections that are not necessarily $H$-equivariant. The theory canonically lifts to the tensor product structure.
"Noncommutative connections on bimodules and Drinfeld twist deformation." Adv. Theor. Math. Phys. 18 (3) 513 - 612, June 2014.