$3$-dimensional loops on non-solvable reductive spaces

TL;DR

This paper classifies 3-dimensional strongly left alternative almost differentiable left A-loops on non-solvable reductive homogeneous spaces, expanding understanding of their structure and classification within differential geometry.

Contribution

It provides a complete classification of 3D strongly left alternative loops on non-solvable reductive spaces, a novel result in the theory of differentiable loops.

Findings

Classified all such loops on non-solvable reductive spaces.

Identified structural properties of these loops.

Extended the theory of differentiable loops in geometric contexts.

Abstract

We treat the almost differentiable left A-loops as images of global differentiable sharply transitive sections $\sigma :G/H \to G$ for a Lie group $G$ such that $G/H$ is a reductive homogeneous manifold. In this paper we classify all $3$-dimensional connected strongly left alternative almost differentiable left A-loops $L$, such that for the corresponding section $\sigma :G/H \to G$ the Lie group $G$ is non-solvable.