Invariant noncommutative connections

TL;DR

This paper classifies invariant noncommutative connections within the algebra of endomorphisms of a complex vector bundle, extending classical geometric methods to a noncommutative setting.

Contribution

It introduces a classification framework for invariant noncommutative connections using algebraic and geometric tools, generalizing ordinary connection theory.

Findings

Classification achieved via a reduced algebra and differential calculus

Extension of classical geometric constructions to noncommutative geometry

Framework applicable to algebra of endomorphisms of vector bundles

Abstract

In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitely some geometric constructions usually introduced to classify ordinary invariant connections, and we expand them using algebraic objects coming from the noncommutative setting. The main result is that the classification can be performed using a ``reduced'' algebra, an associated differential calculus and a module over this algebra.