On the multiplication groups of three-dimensional topological loops

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.