Affine reductive spaces of small dimension and left A-loops

TL;DR

This paper classifies low-dimensional affine reductive homogeneous manifolds and associated left A-loops for certain Lie groups, providing a comprehensive understanding of their structure and examples.

Contribution

It determines and classifies all low-dimensional affine reductive spaces and left A-loops related to specific Lie groups, extending previous classifications.

Findings

Classified all 4-dimensional affine reductive homogeneous manifolds for certain Lie groups.

Identified all global almost differentiable left A-loops with specified Lie groups as translation groups.

Determined all at most 5-dimensional left A-loops with $PSU_3( ext{C},1)$ as the translation group.

Abstract

In this paper we determine the at least $4$-dimensional affine reductive homogeneous manifolds for an at most $9$-dimensional simple Lie group or an at most $6$-dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable left A-loops. Using this we classify all global almost differentiable left A-loops $L$ having either a $6$-dimensional semi-simple Lie group or the group $SL_3(\mathbb R)$ as the group topologically generated by their left translations. Moreover, we determine all at most $5$-dimensional left A-loops $L$ with $PSU_3(\mathbb C,1)$ as the group topologically generated by their left translations.