$3$-dimensional Bol loops as sections in non-solvable Lie groups

TL;DR

This paper classifies 3-dimensional differentiable Bol loops with non-solvable groups, exploring their structure as sections in Lie groups and their connections to metric space geometries.

Contribution

It provides a classification of global differentiable Bol loops with non-solvable groups and describes their geometric relations, highlighting differences from local Bol loop classifications.

Findings

Classification of 3D Bol loops with non-solvable groups

Description of Bol loops as sections in Lie groups

Relations to metric space geometries

Abstract

Our aim in this paper is to classify the $3$-dimensional connected differentiable global Bol loops, which have a non-solvable group as the group topologically generated by their left translations and to describe their relations to metric space geometries. The classification of global differentiable Bol loops significantly differs from the classification of local differentiable Bol loops. We treat the differentiable Bol loops as images of global differentiable sections $\sigma :G/H \to G$ such that for all $r,s \in \sigma (G/H)$ the element $rsr$ lies in $\sigma (G/H)$, where $H$ is the stabilizer of the identity $e$ of $L$ in $G$.