The multiplication groups of 2-dimensional topological loops

TL;DR

This paper characterizes the multiplication groups of 2-dimensional topological loops, showing they are elementary filiform nilpotent Lie groups of dimension at least 4, and classifies loops with such groups when the dimension is 3.

Contribution

It provides a classification of 2D topological loops with elementary filiform Lie groups as their multiplication groups, including necessary and sufficient conditions for certain dimensions.

Findings

Multiplication groups are elementary filiform nilpotent Lie groups of dimension ≥ 4.

Complete classification of loops with 3-dimensional elementary filiform Lie groups.

Necessary and sufficient conditions for loops to have specified multiplication groups.

Abstract

We prove that if the multiplication group $Mult(L)$ of a connected $2$-dimensional topological loop is a Lie group, then $Mult(L)$ is an elementary filiform nilpotent Lie group of dimension at least $4$. Moreover, we describe loops having elementary filiform Lie groups $\mathbb F$ as the group topologically generated by their left translations and obtain a complete classification for these loops $L$ if $\hbox{dim} \ \mathbb F=3$. In this case necessary and sufficient conditions for $L$ are given that $Mult(L)$ is an elementary filiform Lie group for a given allowed dimension.