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.