On the multiplication groups of three-dimensional topological loops

TL;DR

This paper classifies the nilpotent Lie groups that can serve as multiplication groups for 3-dimensional topological loops, showing that certain direct products of filiform groups are the only possibilities.

Contribution

It provides a complete classification of nilpotent Lie groups acting as multiplication groups for 3D topological loops, excluding non-solvable groups.

Findings

Certain direct products of filiform groups occur as multiplication groups.

Non-solvable Lie groups cannot be minimal acting on 3D manifolds as multiplication groups.

Classification of 3D loops with 4D nilpotent Lie groups as translation groups.

Abstract

We clarify the structure of nilpotent Lie groups which are multiplication groups of $3$-dimensional simply connected topological loops and prove that non-solvable Lie groups acting minimally on $3$-dimensional manifolds cannot be the multiplication group of $3$-dimensional topological loops. Among the nilpotent Lie groups for any filiform groups ${\mathcal F}_{n+2}$ and ${\mathcal F}_{m+2}$ with $n, m > 1$, the direct product ${\mathcal F}_{n+2} \times \mathbb R$ and the direct product ${\mathcal F}_{n+2} \times _Z {\mathcal F}_{m+2}$ with amalgamated center $Z$ occur as the multiplication group of $3$-dimensional topological loops. To obtain this result we classify all $3$-dimensional simply connected topological loops having a $4$-dimensional nilpotent Lie group as the group topologically generated by the left translations.