Twist deformations of module homomorphisms and connections

Let H be a Hopf algebra, A a left H-module algebra and V a left H-module A-bimodule. We study the behavior of the right A-linear endomorphisms of V under twist deformation. We in particular construct a bijective quantization map to the right A_\star-linear endomorphisms of V_\star, with A_\star,V_\star denoting the usual twist deformations of A,V. The quantization map is extended to right A-linear homomorphisms between left H-module A-bimodules and to right connections on V. We then investigate the tensor product of linear maps between left H-modules. Given a quasitriangular Hopf algebra we can define an H-covariant tensor product of linear maps, which restricts for left H-module A-bimodules to a well-defined tensor product of right A-linear homomorphisms on tensor product modules over A. This also requires a quasi-commutativity condition on the algebra and bimodules. Using this tensor product we can construct a new lifting prescription of connections to tensor product modules, generalizing the usual prescription to also include nonequivariant connections.