Coxeter-Chein Loops

TL;DR

This paper introduces Coxeter-Chein loops, a new class of Moufang loops constructed from Coxeter groups, and provides the first known amalgam presentations for these loops, linking Coxeter and Chein relations.

Contribution

It develops the concept of Chein-Coxeter systems and derives amalgam presentations for Coxeter-Chein loops, a novel contribution to Moufang loop theory.

Findings

Established Chein-Coxeter systems for these loops

Provided amalgam presentations for Coxeter-Chein loops

First known presentation for a Moufang loop

Abstract

In 1974 Orin Chein discovered a new family of Moufang loops which are now called Chein loops. Such a loop can be created from any group $W$ together with $\mathbb{Z}_2$ by a variation on a semi-direct product. We study these loops in the case where $W$ is a Coxeter group and show that it has what we call a Chein-Coxeter system, a small set of generators of order 2, together with a set of relations closely related to the Coxeter relations and Chein relations. As a result we are able to give amalgam presentations for Coxeter-Chein loops. This is to our knowledge the first such presentation for a Moufang loop.