Linear Connections in Non-Commutative Geometry

TL;DR

This paper introduces a new method for defining linear connections on non-commutative algebras, extending classical geometric concepts to non-commutative settings using generalized Leibnitz rules and bimodule structures.

Contribution

It presents a novel construction of linear connections in non-commutative geometry, generalizing existing frameworks and including extensions to tensor products and matrix algebras.

Findings

Constructed linear connections on non-commutative algebras.

Extended the concept of torsion and curvature to non-commutative settings.

Illustrated the approach with matrix algebra examples.

Abstract

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $\Omega^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of $\Omega^1$. These constructions are illustrated with the example of the algebra of $ n \times n$ matrices.