On order units in the augmentation ideal

TL;DR

This paper investigates the structure of order units within the augmentation ideal of real group rings and matrix algebras, revealing an infinite family linked to cohomological operations and including the Laplacian.

Contribution

It identifies an infinite family of order units in powers of the augmentation ideal and explains their origin through cohomological operations from simpler diagonal units.

Findings

Infinite family of order units in augmentation ideal powers

Order units include the Laplacian

Connection to cohomological operations

Abstract

We study order units in the real group ring and the augmentation ideal, as well as in matrix algebras. We identify an infinite family of order units in the powers of the augmentation ideal, that includes the Laplacian, and show that these order units are naturally obtained via cohomological operations from more simpler diagonal order units in matrix algebras.