Three-dimensional loops as sections in a four-dimensional solvable Lie group

TL;DR

This paper classifies certain three-dimensional loops related to a specific four-dimensional solvable Lie group, showing their structure and limitations within the context of topological loops and Lie groups.

Contribution

It provides a complete classification of three-dimensional topological loops associated with a particular four-dimensional solvable Lie group, highlighting their properties and non-existence as multiplication groups.

Findings

Classified all such three-dimensional topological loops.

Proved that the Lie group G is not the multiplication group of proper loops.

Identified the structure of G with trivial center and two one-dimensional normal subgroups.

Abstract

We classify all three-dimensional connected topological loops such that the group topologically generated by their left translations is the four-dimensional connected Lie group $G$ which has trivial center and precisely two one-dimensional normal subgroups. We show that $G$ is not the multiplication group of connected topological proper loops.