What question did this study set out to answer?

The aim is to explore the structure of nilpotent Lie groups that serve as multiplication groups for three-dimensional topological loops.

March 29, 2026Open Access

Three-Dimensional Topological Loops with Nilpotent Multiplication Groups

Key Points

The aim is to explore the structure of nilpotent Lie groups that serve as multiplication groups for three-dimensional topological loops.
Analyzed the properties of indecomposable nilpotent Lie groups and their subgroups.
Investigated the relationship between three-dimensional topological loops and their multiplication groups.
Examined the nature of the Lie algebra associated with these groups.
Identified that no connected topological loop of dimension 3 exists with its Lie algebra as an elementary filiform Lie algebra.
Established that all multiplication groups feature a 1-dimensional center.
Demonstrated that the corresponding loops are centrally nilpotent of class 2.

Abstract

In this paper we describe the structure of indecomposable nilpotent Lie groups which are multiplication groups of three-dimensional simply connected topological loops.In contrast to the 2 -dimensional loops there is no connected topological loop of dimension 3 such that the Lie algebra of its multiplication group is an elementary filiform Lie algebra.We determine the indecomposable nilpotent Lie groups of dimension 6 and their subgroups which are the multiplication groups and the inner mapping groups of the investigated loops.We prove that all multiplication groups have 1 -dimensional centre and the corresponding loops are centrally nilpotent of class 2 .

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