Differential Algebras in Non-Commutative Geometry

TL;DR

This paper explores the structure of differential algebras in non-commutative geometry, specifically in the context of Connes' approach to Yang-Mills theories with spontaneous symmetry breaking, providing a general formula and characterization.

Contribution

It introduces a general formula for differential algebras based on tensor products and characterizes those relevant in models with symmetry breaking.

Findings

Differential algebras are skew tensor products of forms and matrix algebras.

A general formula for tensor product-based differential algebras is derived.

Differential algebras in symmetry breaking models are characterized.

Abstract

We discuss the differential algebras used in Connes' approach to Yang-Mills theories with spontaneous symmetry breaking. These differential algebras generated by algebras of the form functions $\otimes$ matrix are shown to be skew tensorproducts of differential forms with a specific matrix algebra. For that we derive a general formula for differential algebras based on tensor products of algebras. The result is used to characterize differential algebras which appear in models with one symmetry breaking scale.