Loops which are semidirect products of groups

TL;DR

This paper constructs smooth loops as semidirect products of affine groups, revealing their algebraic structure and associated Akivis algebras, with implications for affine geometry and infinite-dimensional Lie groups.

Contribution

It introduces a new class of loops formed via semidirect products of affine groups, analyzing their smooth structure and Akivis algebras, expanding the understanding of affine loops.

Findings

Constructed smooth loops from affine group semidirect products

Identified Lie groups generated by left translations as affine groups

Determined Akivis algebras of the constructed loops

Abstract

We construct loops which are semidirect products of groups of affinities. As their elements in many cases one may take transversal subspaces of an affine space. In particular we obtain in this manner smooth loops having Lie groups of affine real transformations as the groups generated by left translations, whereas the groups generated by right translations are smooth groups of infinite dimension. We also determine the Akivis algebras of these loops.