Connections on central bimodules

TL;DR

This paper develops a theory of derivation-based connections on bimodules over noncommutative algebras, generalizing linear connections and exploring noncommutative differential forms and geometric structures.

Contribution

It introduces a new framework for connections on bimodules in noncommutative geometry, extending classical concepts to the noncommutative setting.

Findings

Defined derivation-based connections on bimodules

Generalized linear connections to noncommutative algebras

Explored noncommutative differential forms and geometric structures

Abstract

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.