Dirac-Connes Operator on Discrete Abelian Groups and Lattices

TL;DR

This paper introduces a Dirac-Connes operator within Connes' noncommutative geometry framework on discrete abelian groups and lattices, satisfying a Junk-free condition and connecting different noncommutative geometries.

Contribution

It presents a new Dirac-Connes operator that bridges two noncommutative geometry frameworks and applies it to d-dimensional lattices.

Findings

Operator satisfies Junk-free condition

Connects different NCG frameworks

Applied to d-dimensional lattices

Abstract

A kind of Dirac-Connes operator defined in the framework of Connes' NCG is introduced on discrete abelian groups; it satisfies a Junk-free condition, and bridges the NCG composed by Dimakis, M\"{u}ller-Hoissen and Sitarz and the NCG of Connes. Then we apply this operator to d-dimensional lattices.