Non-commutative connections of the second kind

TL;DR

This paper introduces hom-connections in non-commutative geometry, based on homomorphisms, and explores their relation to principal bundles, curvature, and chain complexes.

Contribution

It defines hom-connections using homomorphisms, links them to non-commutative principal bundles, and describes their induction and curvature properties.

Findings

Hom-connections naturally arise from strong connections in non-commutative principal bundles.

A curvature for hom-connections is defined and analyzed.

Flat hom-connections induce chain complexes.

Abstract

A connection-like objects, termed {\em hom-connections} are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.