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.