Connections over Twisted Tensor Products of Algebras

Motivated from some results in classical differential geometry, we give a constructive procedure for building up a connection over a (twisted) tensor product of two algebras, starting from connections defined on the factors. The curvature for the product connection is explicitly calculated, and shown to be independent of the choice of the twisting map and the module twisting map used to define the product connection. As a consequence, we obtain that a product of two flat connections is again a flat connection. We show that our constructions also behves well with respesct to bimodule structures, namely being the product of two bimodule connections again a bimodule connection. As an application of our theory, all the product connections on the quantum plane are computed.