Three-dimensional topological loops with solvable multiplication groups

TL;DR

This paper classifies 3-dimensional topological loops with solvable multiplication groups of dimension up to 5, showing they are centrally nilpotent of class 2 and identifying their inner mapping groups.

Contribution

It provides a classification of solvable non-nilpotent Lie groups as multiplication groups for 3D loops with dimension ≤ 5, expanding understanding of loop structures.

Findings

3D loops with solvable multiplication groups are centrally nilpotent of class 2.

Solvable non-nilpotent multiplication groups are direct products with dimension 5.

Inner mapping groups of these loops are explicitly determined.

Abstract

We prove that each $3$-dimensional connected topological loop $L$ having a solvable Lie group of dimension $\le 5$ as the multiplication group of $L$ is centrally nilpotent of class $2$. Moreover, we classify the solvable non-nilpotent Lie groups $G$ which are multiplication groups for $3$-dimensional simply connected topological loops $L$ and $\hbox{dim} \ G \le 5$. These groups are direct products of proper connected Lie groups and have dimension $5$. We find also the inner mapping groups of $L$.