A note on flat noncommutative connections

TL;DR

This paper proves that flat connections on modules over noncommutative algebras naturally induce bimodule structures, linking flatness with bimodule connections in noncommutative geometry.

Contribution

It establishes that flat covariant derivatives on modules induce bimodule structures, connecting flatness with bimodule connections in the context of noncommutative differential calculus.

Findings

Flat connections induce bimodule structures.

Flat hom-connections induce compatible module actions.

Results apply to modules over noncommutative algebras.

Abstract

It is proven that every flat connection or covariant derivative $\nabla$ on a left $A$-module $M$ (with respect to the universal differential calculus) induces a right $A$-module structure on $M$ so that $\nabla$ is a bimodule connection on $M$ or $M$ is a flat differentiable bimodule. Similarly a flat hom-connection on a right $A$-module $M$ induces a compatible left $A$-action.