Connections over twisted tensor products of algebras

TL;DR

This paper develops a method to construct connections over twisted tensor products of algebras, showing that flatness is preserved and applying the theory to quantum planes.

Contribution

It introduces a constructive procedure for product connections on twisted tensor products, including explicit curvature calculations and bimodule compatibility.

Findings

Product of two flat connections remains flat.

Constructed connections are compatible with bimodule structures.

Explicitly computed all product connections on the quantum plane.

Abstract

Motivated by 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 behaves well with respect 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.