Topological loops having decomposable solvable multiplication group

TL;DR

This paper classifies certain low-dimensional decomposable solvable Lie groups and their associated topological loops, revealing their structure and properties, including central nilpotency of related loops.

Contribution

It explicitly determines the structure of decomposable solvable Lie groups of dimension up to six and analyzes their role as multiplication groups of topological loops.

Findings

Lie groups in the class are explicitly characterized.

Loops with these groups have centers of dimension one or two.

All such loops are centrally nilpotent of class two.

Abstract

In this paper we deal with the class C of decomposable solvable Lie groups having dimension at most six. We determine those Lie groups in C and their subgroups which are the multiplication group Mult(L) and the inner mapping group Inn(L) for three-dimensional connected simply connected topological loops L. These loops L have one- or two-dimensional centre and their group Mult(L) has two- or three-dimensional commutator subgroup. Together with this result we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.