Comparison between two differential graded algebras in Noncommutative Geometry

TL;DR

This paper compares two canonical differential graded algebras derived from spectral triples in noncommutative geometry, showing that Connes' dga provides more detailed information than Fröhlich et al.'s in a precise sense.

Contribution

It provides a detailed comparison between Connes' and Fröhlich et al.'s differential graded algebras, highlighting the greater informativeness of Connes' construction.

Findings

Connes' dga is more informative than Fröhlich et al.'s in a precise sense.

Both dgas coincide with the de-Rham dga for classical spectral triples.

The comparison clarifies the roles of these dgas in noncommutative geometry.

Abstract

Starting with a spectral triple one can associate two canonical differential graded algebras (dga) defined by Connes and Fr\"ohlich et al. For the classical spectral triples associated with compact Riemannian spin manifolds both these dgas coincide with the de-Rham dga. Therefore, both are candidates for the noncommutative space of differential forms. Here we compare these two dgas and observe that in a very precise sense Connes' dga is more informative than that of Fr\"ohlich et al.